Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
30862 conjugacy classes of subgroups; only the 816 normal subgroups are shown.
This group has 30862 conjugacy classes of subgroups — too many to draw. Only its 816 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#16811564 | C2 | 1 | 14 | yes | maximal, complemented |
| 2 | C26×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C22×C22⋊#6157 | C22 | 1 | 28 | yes | complemented |
| 4 | #16811564 | — | 2 | 56 | no | — |
| 5 | C25×D53 | — | 2 | 56 | no | — |
| 6 | C25×D53 | C4 | 1 | 7 | yes | complemented |
| 7 | C25×D53 | C22 | 1 | 7 | yes | — |
| 8 | C26×C53 | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | C23×#6157 | C23 | 1 | 8 | yes | complemented |
| 10 | C2×C22⋊#6157 | — | 2 | 168 | no | — |
| 11 | C24×D53 | — | 2 | 896 | no | — |
| 12 | C24×D53 | — | 2 | 28 | no | — |
| 13 | C24×D53 | — | 2 | 168 | no | — |
| 14 | C24×D53 | — | 2 | 168 | no | — |
| 15 | C24×D53 | C2×C4 | 1 | 21 | yes | — |
| 16 | C24×D53 | D4 | 1 | 56 | yes | — |
| 17 | C24×D53 | C23 | 1 | 7 | yes | — |
| 18 | C25×C53 | — | 2 | 28 | no | — |
| 19 | C25×C53 | C2×C4 | 1 | 7 | yes | — |
| 20 | C24×C53 | — | 2 | 224 | no | — |
| 21 | C24×C53 | — | 2 | 84 | no | — |
| 22 | C24×C53 | C22⋊C4 | 1 | 28 | yes | complemented |
| 23 | C24×C53 | C22×C4 | 1 | 7 | yes | — |
| 24 | C23×D53 | — | 2 | 2048 | no | — |
| 25 | C23×D53 | — | 2 | 672 | no | — |
| 26 | C23×D53 | — | 2 | 896 | no | — |
| 27 | C23×D53 | — | 2 | 896 | no | — |
| 28 | C23×D53 | — | 2 | 84 | no | — |
| 29 | C23×D53 | — | 2 | 168 | no | — |
| 30 | C23×D53 | — | 2 | 672 | no | — |
| 31 | C23×D53 | — | 2 | 56 | no | — |
| 32 | C23×D53 | C22⋊C4 | 1 | 84 | yes | complemented |
| 33 | C23×D53 | C22×C4 | 1 | 7 | yes | — |
| 34 | C23×D53 | C2×D4 | 1 | 84 | yes | — |
| 35 | C23×D53 | C24 | 1 | 1 | yes | characteristic |
| 36 | C22×#6157 | — | 2 | 56 | no | — |
| 37 | C22⋊#6157 | — | 4 | 112 | no | — |
| 38 | C23×C53 | — | 2 | 256 | no | — |
| 39 | C23×C53 | — | 2 | 224 | no | — |
| 40 | C23×C53 | — | 2 | 28 | no | — |
| 41 | C23×C53 | — | 2 | 168 | no | — |
| 42 | C23×C53 | C2×C22⋊C4 | 1 | 42 | yes | — |
| 43 | C23×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 44 | C22×D53 | — | 2 | 1792 | no | — |
| 45 | C22×D53 | — | 2 | 672 | no | — |
| 46 | C22×D53 | — | 2 | 1792 | no | — |
| 47 | C22×D53 | — | 2 | 28 | no | — |
| 48 | C22×D53 | — | 2 | 168 | no | — |
| 49 | C22×D53 | — | 2 | 336 | no | — |
| 50 | C22×D53 | — | 2 | 336 | no | — |
| 51 | C22×D53 | C2×C22⋊C4 | 1 | 42 | yes | — |
| 52 | C22×D53 | #120 | 1 | 112 | yes | — |
| 53 | C22×D53 | C22×D4 | 1 | 14 | yes | — |
| 54 | C2×#6157 | — | 4 | 56 | no | — |
| 55 | #14679 | — | 53 | 1 | no | maximal, Sylow |
| 56 | C22×C53 | — | 2 | 224 | no | — |
| 57 | C22×C53 | — | 2 | 84 | no | — |
| 58 | C22×C53 | #379 | 1 | 28 | yes | complemented |
| 59 | C22×C53 | C22×C22⋊C4 | 1 | 7 | yes | — |
| 60 | C2×D53 | — | 2 | 672 | no | — |
| 61 | C2×D53 | — | 2 | 84 | no | — |
| 62 | C2×D53 | — | 2 | 224 | no | — |
| 63 | C2×D53 | #379 | 1 | 28 | yes | complemented |
| 64 | C2×D53 | C2×#120 | 1 | 28 | yes | — |
| 65 | #6157 | — | 8 | 8 | no | — |
| 66 | C27 | — | 53 | 1 | no | — |
| 67 | C2×#379 | — | 53 | 14 | no | — |
| 68 | C2×C53 | — | 2 | 28 | no | — |
| 69 | C2×C53 | C2×#379 | 1 | 7 | yes | — |
| 70 | D53 | — | 2 | 28 | no | — |
| 71 | D53 | #2846 | 1 | 8 | yes | — |
| 72 | C26 | — | 53 | 7 | no | — |
| 73 | C26 | — | 53 | 7 | no | — |
| 74 | C26 | — | 106 | 56 | no | — |
| 75 | C26 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 76 | C22×C22⋊C4 | — | 53 | 28 | no | — |
| 77 | #379 | — | 106 | 56 | no | — |
| 78 | C53 | #14679 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 79 | C25 | — | 2 | 28 | no | — |
| 80 | C25 | — | 53 | 7 | no | — |
| 81 | C25 | — | 53 | 56 | no | — |
| 82 | C25 | — | 53 | 21 | no | — |
| 83 | C25 | — | 106 | 168 | no | — |
| 84 | C25 | — | 106 | 168 | no | — |
| 85 | C25 | — | 106 | 896 | no | — |
| 86 | C25 | — | 106 | 28 | no | — |
| 87 | C25 | C2×#6157 | 1 | 7 | yes | — |
| 88 | C23×C4 | — | 53 | 8 | no | — |
| 89 | C2×C22⋊C4 | — | 106 | 168 | no | — |
| 90 | C24 | — | 2 | 224 | no | — |
| 91 | C24 | — | 2 | 84 | no | — |
| 92 | C24 | — | 53 | 1 | no | — |
| 93 | C24 | — | 53 | 84 | no | — |
| 94 | C24 | — | 53 | 7 | no | — |
| 95 | C24 | — | 53 | 84 | no | — |
| 96 | C24 | — | 106 | 672 | no | — |
| 97 | C24 | — | 106 | 56 | no | — |
| 98 | C24 | — | 106 | 168 | no | — |
| 99 | C24 | — | 106 | 896 | no | — |
| 100 | C24 | — | 106 | 896 | no | — |
| 101 | C24 | — | 106 | 2048 | no | — |
| 102 | C24 | — | 106 | 672 | no | — |
| 103 | C24 | — | 106 | 84 | no | — |
| 104 | C24 | C22⋊#6157 | 1 | 28 | yes | complemented |
| 105 | C24 | C22×#6157 | 1 | 7 | yes | — |
| 106 | C22×C4 | — | 106 | 56 | no | — |
| 107 | C22⋊C4 | — | 212 | 112 | no | — |
| 108 | C23 | — | 2 | 256 | no | — |
| 109 | C23 | — | 2 | 224 | no | — |
| 110 | C23 | — | 2 | 168 | no | — |
| 111 | C23 | — | 2 | 28 | no | — |
| 112 | C23 | — | 53 | 14 | no | — |
| 113 | C23 | — | 53 | 112 | no | — |
| 114 | C23 | — | 53 | 42 | no | — |
| 115 | C23 | — | 106 | 336 | no | — |
| 116 | C23 | — | 106 | 336 | no | — |
| 117 | C23 | — | 106 | 1792 | no | — |
| 118 | C23 | — | 106 | 1792 | no | — |
| 119 | C23 | — | 106 | 672 | no | — |
| 120 | C23 | — | 106 | 168 | no | — |
| 121 | C23 | — | 106 | 28 | no | — |
| 122 | C23 | C2×C22⋊#6157 | 1 | 42 | yes | — |
| 123 | C23 | C23×#6157 | 1 | 1 | yes | characteristic |
| 124 | C2×C4 | — | 212 | 56 | no | — |
| 125 | C22 | — | 2 | 224 | no | — |
| 126 | C22 | — | 2 | 84 | no | — |
| 127 | C22 | — | 53 | 28 | no | — |
| 128 | C22 | — | 53 | 28 | no | — |
| 129 | C22 | — | 106 | 224 | no | — |
| 130 | C22 | — | 106 | 672 | no | — |
| 131 | C22 | — | 106 | 84 | no | — |
| 132 | C22 | #16811564 | 1 | 28 | yes | complemented |
| 133 | C22 | C22×C22⋊#6157 | 1 | 7 | yes | — |
| 134 | C4 | — | 424 | 8 | no | — |
| 135 | C2 | — | 2 | 28 | no | — |
| 136 | C2 | — | 53 | 8 | no | — |
| 137 | C2 | — | 106 | 28 | no | — |
| 138 | C2 | C2×#16811564 | 1 | 7 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×#6157 |
| commutator, socle | C23×C53 | C23×C4 |
| Fitting | C26×C53 | C4 |
| prime radical | C26×D53 | C2 |
| cosocle | C23×D53 | C24 |
| hypercenter | C26 | #6157 |
| hypocenter | C53 | #14679 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C26 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16811564 | C2 | semidirect |
| 3. C22×C22⋊#6157 | C22 | semidirect |
| 6. C25×D53 | C4 | semidirect |
| 8. C26×C53 | C4 | semidirect |
| 9. C23×#6157 | C23 | semidirect |
| 22. C24×C53 | C22⋊C4 | semidirect |
| 32. C23×D53 | C22⋊C4 | semidirect |
| 58. C22×C53 | #379 | semidirect |
| 63. C2×D53 | #379 | semidirect |
| 75. C26 | #6157 | semidirect |
| 78. C53 | #14679 | semidirect |
| 104. C24 | C22⋊#6157 | semidirect |
| 132. C22 | #16811564 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16811564 | 2 |
| C26×D53 | 2 |
| #14679 | 53 |