Atlas of subgroup lattices of finite almost simple groups

An atlas of subgroup lattices of finite almost simple groups

Thomas Connor and Dimitri Leemans


This atlas contains all subgroup lattices of almost simple groups G such that S≤G≤Aut(S) and S is a simple group of order less than 1 million appearing in the Atlas of Finite Groups by Conway et al. Some simple groups and almost simple groups or order larger than 1 million have also been included, but not in a systematic way.
You are free to use those lattices and distribute them. We only require that you cite the following reference when you do so:

Connor, T. and Leemans, D. An atlas of subgroup lattices of finite almost simple groups. arXiv:1306.4820. 6 pages. 2013.

By clicking on a link in the tables below, you will get a PDF file containing the subgroup lattice of the corresponding group. For a description on the format in which the subgroup lattices are given, please read the howto. You can find an implementation in Magma code of the algorithm that we used to compute the subgroup lattice here. You can contact the authors at tconnor 'at' ulb.ac.be or d.leemans 'at' auckland.ac.nz if you have any remark or suggestion.

The groups are divided in the following families :

  • Sporadic groups and their automorphism groups
  • Alternating groups and their automorphism groups
  • Symplectic groups and their automorphism groups
  • Unitary groups and their automorphism groups
  • Exceptional groups of Lie type and their automorphism groups (untwisted)
  • Exceptional groups of Lie type and their automorphism groups (twisted)

    Sporadic groups and their automorphism groups
    G Aut(G) Order of G Number of conjugacy classes of subgroups
    M11 M11 7920 39
    M12 M12:2 95040 147
    M12:2 M12:2 190080 213
    M22 M22:2 443510 156
    M22:2 M22:2 887040 490
    M23 M23 10200960 204
    M24 M24 244823040 1529
    J1 J1 175560 40
    J2 J2:2 604800 146
    J2:2 J2:2 1209600 373
    J3 J3:2 50232960 137
    J3:2 J3:2 100465920 354
    HS HS:2 44352000 589
    HS:2 HS:2 88704000 2057
    McL McL:2 898128000 373
    McL:2 McL:2 1796256000 733
    He He:2 4030387200 1698
    He:2 He:2 8060744400 1930
    Ru Ru 145926144000 6035
    Suz Suz:2 448345497600 6381
    Suz:2 Suz:2 896690995200 19105
    O'N O'N:2 460815505920 581
    O'N:2 O'N:2 921631011840 734
    Co3 Co3 495766656000 2483
    Fi22 Fi22:2 64561751654400 111004

    Alternating groups and their automorphism groups
    G Aut(G) Order of G Number of conjugacy classes of subgroups
    Alt(5) = PSL(2,4) = PSL(2,5) Sym(5) 60 9
    Sym(5) Sym(5) 120 19
    Alt(6) = PSL(2,9) = Sp(4,2)' = M10' PΓL(2,9) 360 22
    PGL(2,9) PΓL(2,9) 720 26
    Sym(6) = PΣL(2,9) = Sp(4,2) PΓL(2,9) 720 56
    M10 PΓL(2,9) 720 25
    PΓL(2,9) PΓL(2,9) 1440 65
    Alt(7) Sym(7) 2520 40
    Sym(7) Sym(7) 5040 96
    Alt(8) = PSL(4,2) Sym(8) 20160 137
    Sym(8) Sym(8) 40320 296
    Alt(9) Sym(9) 181440 223
    Sym(9) Sym(9) 362880 554
    Alt(10) Sym(10) 1814400 430
    Sym(10) Sym(10) 3628800 1593
    Alt(11) Sym(11) 19958400 788
    Sym(11) Sym(11) 39916800 3094
    Alt(12) Sym(12) 239500800 2537
    Sym(12) Sym(12) 479001600 10723
    Alt(13) Sym(13) 3113510400 4552
    Sym(13) Sym(13) 6227020800 20832
    Alt(14) Sym(14) 43589145600 12136

    Symplectic groups and their automorphism groups
    G Aut(G) Order of G Number of conjugacy classes of subgroups
    Sp(4,2)' = Alt(6) = PSL(2,9) = M10' PΓL(2,9) 360 22
    PGL(2,9) PΓL(2,9) 720 26
    Sp(4,2) = Sym(6) = PΣL(2,9) PΓL(2,9) 720 56
    M10 PΓL(2,9) 720 25
    PΓL(2,9) PΓL(2,9) 1440 65
    Sp(4,3) = PSU(4,2) = PGU(4,2) PΓU(4,2) 25920 116
    PΓU(4,2) = PΣU(4,2) PΓU(4,2) 51840 350
    Sp(4,4) Sp(4,4):4 979200 496
    Sp(4,4):2 Sp(4,4):4 1958400 1001
    Sp(4,4):4 Sp(4,4):4 3916800 620
    Sp(4,5) Sp(4,5):2 4680000 307
    Sp(4,5):2 Sp(4,5):2 9360000 881
    Sp(4,7) Sp(4,7):2 138297600 603
    Sp(4,7):2 Sp(4,7):2 276595200 1367
    Sp(6,2) Sp(6,2) 1451520 1369

    Linear groups and their automorphism groups
    G Aut(G) Order of G Number of conjugacy classes of subgroups
    Alt(5) = PSL(2,4) = PSL(2,5) Sym(5) 60 9
    Sym(5) Sym(5) 120 19
    PSL(3,2) = PSL(2,7) PΓL(2,7) 168 15
    PGL(2,7) = PΓL(2,7) PΓL(2,7) 336 23
    Alt(6) = PSL(2,9) = Sp(4,2)' = M10' PΓL(2,9) 360 22
    PGL(2,9) PΓL(2,9) 720 26
    Sym(6) = PΣL(2,9) = Sp(4,2) PΓL(2,9) 720 56
    M10 PΓL(2,9) 720 25
    PΓL(2,9) PΓL(2,9) 1440 65
    PSL(2,8) = PGL(2,8) PΓL(2,8) 504 12
    PΓL(2,8) = PΣL(2,8) PΓL(2,8) 1512 25
    PSL(2,11) = PΣL(2,11) PΓL(2,11) 660 16
    PGL(2,11) = PΓL(2,11) PΓL(2,11) 1320 29
    PSL(2,13) = PΣL(2,13) PΓL(2,13) 1092 16
    PGL(2,13) = PΓL(2,13) PΓL(2,13) 2184 30
    PSL(2,17) = PΣL(2,17) PΓL(2,17) 2448 22
    PGL(2,17) = PΓL(2,17) PΓL(2,17) 4896 32
    PSL(2,19) = PΣL(2,19) PΓL(2,19) 3420 19
    PGL(2,19) = PΓL(2,19) PΓL(2,19) 6840 36
    PSL(2,16) = PGL(2,16) PΓL(2,16) = PΣL(2,16) 4080 21
    PSL(2,16):2 PΓL(2,16) = PΣL(2,16) 8160 47
    PΓL(2,16) = PΣL(2,16) PΓL(2,16) = PΣL(2,16) 16320 69
    PSL(3,3) = PGL(3,3) = PΣL(3,3) = PΓL(3,3) PSL(3,3):2 5616 51
    PSL(3,3):2 PSL(3,3):2 11232 78
    PSL(2,23) = PΣL(2,23) PΓL(2,23) 6072 23
    PGL(2,23) = PΓL(2,23) PΓL(2,23) 12144 34
    PSL(2,25) PΓL(2,25) 7800 37
    PGL(2,25) PΓL(2,25) 15600 43
    PΣL(2,25) PΓL(2,25) 15600 103
    PSL(2,25).2 PΓL(2,25) 15600 39
    PΓL(2,25) PΓL(2,25) 31200 115
    PSL(2,27) PΓL(2,27) 9828 16
    PGL(2,27) PΓL(2,27) 19656 32
    PΣL(2,27) PΓL(2,27) 29484 38
    PΓL(2,27) PΓL(2,27) 58968 67
    PSL(2,29) = PΣL(2,29) PΓL(2,29) 12180 22
    PGL(2,29) = PΓL(2,29) PΓL(2,29) 24360 41
    PSL(2,31) = PΣL(2,31) PΓL(2,31) 14880 29
    PGL(2,31) = PΓL(2,31) PΓL(2,31) 29760 44
    PSL(3,4) PSL(3,4):D12 20160 95
    PSL(3,4):21 PSL(3,4):D12 40320 151
    PSL(3,4).3 = PGL(3,4) PSL(3,4):D12 60480 100
    PSL(3,4):S3 PSL(3,4):D12 120960 142
    PSL(3,4):S3 = PΓL(3,4) PSL(3,4):D12 120960 226
    PSL(3,4):6 PSL(3,4):D12 120960 143
    PSL(3,4).D12 PSL(3,4):D12 241920 409
    PSL(3,4):23 PSL(3,4):22 40320 102
    PSL(3,4).22 = PΣL(3,4) PSL(3,4):22 40320 152
    PSL(3,4):22 PSL(3,4):22 80640 288
    PSL(2,32) = PGL(2,32) PΓL(2,32) = PΣL(2,32) 32736 24
    PΓL(2,32) = PΣL(2,32) PΓL(2,32) 163680 30
    PSL(3,5) = PΣL(3,5) = PGL(3,5) = PΓL(3,5) PSL(3,5):2 372000 140
    PSL(3,5):2 PSL(3,5):2 744000 208
    PSL(2,49) PΓL(2,49) 58800 51
    PSL(2,64) PΓL(2,64) = PΣL(2,64) 262080 76
    PSL(2,128) PΓL(2,128) = PΣL(2,128) 2097024 242
    PΓL(2,128) = PΣL(2,128) PΓL(2,128) = PΣL(2,128) 14679168 68
    PSL(2,256) PΓL(2,256) = PΣL(2,256) 16776960 1678
    PΓL(2,256) = PΣL(2,256) PΓL(2,256) = PΣL(2,256) 134215680 523

    Unitary groups and their automorphism groups
    G Aut(G) Order of G Number of conjugacy classes of subgroups
    PSU(3,3) = PGU(3,3) = G2(2)' PΓU(3,3) 6048 36
    PΓU(3,3) = PΣU(3,3) PΓU(3,3) 12096 100
    PSU(4,2) = PGU(4,2) = PSp(4,3) PΓU(4,2) 25920 116
    PΓU(4,2) = PΣU(4,2) PΓU(4,2) 51840 350
    PSU(3,4) = PGU(3,4) PΓU(3,4) 62400 34
    PSU(3,4):2 PΓU(3,4) 124800 80
    PΣU(3,4) = PΓU(3,4) PΓU(3,4) 249600 120
    PSU(3,5) PΓU(3,5) 126000 80
    PGU(3,5) PΓU(3,5) 378000 93
    PΓU(3,5) PΓU(3,5) 756000 244
    PΣU(3,5) PΣU(3,5) 252000 148

    Exceptional groups of Lie type and their automorphism groups (untwisted)
    G Aut(G) Order of G Number of conjugacy classes of subgroups
    G2(2)'=PSU(3,3)=PGU(3,3) PΓU(3,3) 6048 36
    G2(2)=PΣU(3,3)=PΓU(3,3) PΓU(3,3) 12096 100
    G2(3) G2(3):2 4245696 433
    G2(3):2 G2(3):2 8491392 399

    Exceptional groups of Lie type and their automorphism groups (twisted)
    G Aut(G) Order of G Number of conjugacy classes of subgroups
    Sz(8) Sz(8):3 29120 22
    Sz(8):3 Sz(8):3 87360 39
    Sz(32) Sz(32):5 32537600 132
    Sz(32):5 Sz(32):5 162688000 67
    2F4(2)' 2F4(2) 17971200 434
    2F4(2) 2F4(2) 35942400 849
    R(27) R(27):3 10073444472 411


    Last update : August 4, 2013

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