Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
448 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 83 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 | #20275949 | C2 | 1 | 3 | yes | maximal |
| 2 | #20272538 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C53×#2810 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#13133214 | C22 | 1 | 1 | yes | characteristic |
| 5 | C2×#13133137 | — | 2 | 3 | no | — |
| 6 | C2×#13133137 | — | 2 | 3 | no | — |
| 7 | #16810378 | C22 | 1 | 3 | yes | — |
| 8 | C53×#400 | — | 2 | 3 | no | — |
| 9 | C53×#389 | C22 | 1 | 3 | yes | — |
| 10 | C53×#387 | — | 2 | 3 | no | — |
| 11 | C22×D212 | — | 2 | 3 | no | — |
| 12 | C22×#88322 | — | 2 | 3 | no | — |
| 13 | #13133214 | — | 4 | 2 | no | — |
| 14 | #13133137 | — | 4 | 6 | no | — |
| 15 | #13133137 | — | 4 | 6 | no | — |
| 16 | C2×C4⋊C4×C53 | — | 2 | 3 | no | — |
| 17 | C2×C4⋊C4×C53 | D4 | 1 | 3 | yes | — |
| 18 | C2×C4⋊C4×C53 | D4 | 1 | 3 | yes | — |
| 19 | C2×C42×C53 | — | 2 | 1 | no | — |
| 20 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 21 | #95×C53 | — | 2 | 3 | no | — |
| 22 | #95×C53 | — | 2 | 3 | no | — |
| 23 | #95×C53 | — | 2 | 1 | no | — |
| 24 | #95×C53 | — | 2 | 3 | no | — |
| 25 | C23×D53 | — | 4 | 1 | no | — |
| 26 | C22×C4×C53 | — | 2 | 3 | no | — |
| 27 | C22×C4×C53 | — | 2 | 3 | no | — |
| 28 | C22×C4×C53 | C2×D4 | 1 | 3 | yes | — |
| 29 | C2×D212 | — | 2 | 6 | no | — |
| 30 | C2×D212 | — | 4 | 3 | no | — |
| 31 | C2×D212 | — | 4 | 3 | no | — |
| 32 | C2×D212 | — | 4 | 3 | no | — |
| 33 | C2×D212 | — | 4 | 3 | no | — |
| 34 | C2×D212 | — | 4 | 3 | no | — |
| 35 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 36 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 37 | C4⋊C4×C53 | — | 4 | 3 | no | — |
| 38 | C4⋊C4×C53 | — | 4 | 3 | no | — |
| 39 | C4⋊C4×C53 | — | 4 | 3 | no | — |
| 40 | C42×C53 | — | 4 | 1 | no | — |
| 41 | C42×C53 | — | 8 | 1 | no | — |
| 42 | C2×#88322 | — | 4 | 3 | no | — |
| 43 | C2×#88322 | — | 4 | 3 | no | — |
| 44 | C2×#88322 | — | 4 | 3 | no | — |
| 45 | C23×C53 | #120 | 1 | 1 | yes | characteristic |
| 46 | C22×D53 | — | 4 | 2 | no | — |
| 47 | C22×D53 | — | 8 | 3 | no | — |
| 48 | C22×D53 | — | 8 | 3 | no | — |
| 49 | C2×C4×C53 | — | 2 | 3 | no | — |
| 50 | C2×C4×C53 | — | 2 | 3 | no | — |
| 51 | C2×C4×C53 | — | 4 | 3 | no | — |
| 52 | C2×C4×C53 | — | 4 | 3 | no | — |
| 53 | C2×C4×C53 | — | 4 | 3 | no | — |
| 54 | C2×C4×C53 | — | 4 | 3 | no | — |
| 55 | C2×C4×C53 | — | 4 | 3 | no | — |
| 56 | C2×C4×C53 | — | 4 | 3 | no | — |
| 57 | C2×C4×C53 | #135 | 1 | 3 | yes | — |
| 58 | C2×C4×C53 | AGL(1,ℤ/8) | 1 | 3 | yes | — |
| 59 | D212 | — | 4 | 6 | no | — |
| 60 | D212 | — | 4 | 6 | no | — |
| 61 | D212 | — | 8 | 3 | no | — |
| 62 | D212 | — | 8 | 3 | no | — |
| 63 | #88322 | — | 8 | 3 | no | — |
| 64 | #7744 | — | 53 | 1 | no | maximal, Sylow |
| 65 | C22×C53 | #449 | 1 | 3 | yes | — |
| 66 | C22×C53 | #454 | 1 | 3 | yes | — |
| 67 | C22×C53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 68 | C2×D53 | — | 8 | 6 | no | — |
| 69 | C2×D53 | — | 16 | 3 | no | — |
| 70 | C2×D53 | — | 16 | 1 | no | — |
| 71 | C4×C53 | — | 2 | 3 | no | — |
| 72 | C4×C53 | — | 2 | 3 | no | — |
| 73 | C4×C53 | — | 8 | 3 | no | — |
| 74 | C4×C53 | — | 8 | 3 | no | — |
| 75 | #2810 | D53 | 1 | 1 | yes | characteristic, complemented |
| 76 | #2041 | — | 53 | 3 | no | — |
| 77 | #1271 | — | 53 | 3 | no | — |
| 78 | C2×C53 | #1597 | 1 | 3 | yes | — |
| 79 | C2×C53 | #1628 | 1 | 3 | yes | — |
| 80 | C2×C53 | #1658 | 1 | 1 | yes | characteristic |
| 81 | D53 | — | 16 | 2 | no | — |
| 82 | C2×#127 | — | 53 | 1 | no | — |
| 83 | C2×#102 | — | 106 | 3 | no | — |
| 84 | C2×#102 | — | 106 | 3 | no | — |
| 85 | #400 | — | 2 | 3 | no | — |
| 86 | #389 | C2×D53 | 1 | 3 | yes | — |
| 87 | #387 | — | 2 | 3 | no | — |
| 88 | #336 | — | 53 | 3 | no | — |
| 89 | C53 | #7744 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 90 | C22×D4 | — | 106 | 3 | no | — |
| 91 | C22×C8 | — | 106 | 3 | no | — |
| 92 | #127 | — | 212 | 2 | no | — |
| 93 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 94 | C2×C4⋊C4 | C22⋊D53 | 1 | 3 | yes | — |
| 95 | C2×C4⋊C4 | C22⋊D53 | 1 | 3 | yes | — |
| 96 | C2×C42 | — | 2 | 1 | no | — |
| 97 | C2×C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 98 | #102 | — | 212 | 6 | no | — |
| 99 | #102 | — | 212 | 6 | no | — |
| 100 | #95 | — | 2 | 3 | no | — |
| 101 | #95 | — | 2 | 3 | no | — |
| 102 | #95 | — | 2 | 1 | no | — |
| 103 | #95 | — | 2 | 3 | no | — |
| 104 | C24 | — | 212 | 1 | no | — |
| 105 | C2×D4 | — | 106 | 6 | no | — |
| 106 | C2×D4 | — | 212 | 3 | no | — |
| 107 | C2×D4 | — | 212 | 3 | no | — |
| 108 | C2×D4 | — | 212 | 3 | no | — |
| 109 | C2×D4 | — | 212 | 3 | no | — |
| 110 | C2×D4 | — | 212 | 3 | no | — |
| 111 | C22×C4 | — | 2 | 3 | no | — |
| 112 | C22×C4 | — | 2 | 3 | no | — |
| 113 | C22×C4 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 114 | C2×C8 | — | 212 | 3 | no | — |
| 115 | C2×C8 | — | 212 | 3 | no | — |
| 116 | C2×C8 | — | 212 | 3 | no | — |
| 117 | C4⋊C4 | — | 2 | 3 | no | — |
| 118 | C4⋊C4 | — | 2 | 3 | no | — |
| 119 | C4⋊C4 | — | 4 | 3 | no | — |
| 120 | C4⋊C4 | — | 4 | 3 | no | — |
| 121 | C4⋊C4 | — | 4 | 3 | no | — |
| 122 | C42 | — | 4 | 1 | no | — |
| 123 | C42 | — | 8 | 1 | no | — |
| 124 | C23 | — | 424 | 3 | no | — |
| 125 | C23 | — | 424 | 3 | no | — |
| 126 | C23 | — | 212 | 2 | no | — |
| 127 | C23 | #13133186 | 1 | 1 | yes | characteristic |
| 128 | D4 | — | 424 | 3 | no | — |
| 129 | D4 | — | 424 | 3 | no | — |
| 130 | D4 | — | 212 | 6 | no | — |
| 131 | D4 | — | 212 | 6 | no | — |
| 132 | C2×C4 | — | 2 | 3 | no | — |
| 133 | C2×C4 | — | 2 | 3 | no | — |
| 134 | C2×C4 | — | 4 | 3 | no | — |
| 135 | C2×C4 | — | 4 | 3 | no | — |
| 136 | C2×C4 | — | 4 | 3 | no | — |
| 137 | C2×C4 | — | 4 | 3 | no | — |
| 138 | C2×C4 | — | 4 | 3 | no | — |
| 139 | C2×C4 | — | 4 | 3 | no | — |
| 140 | C2×C4 | #13133237 | 1 | 3 | yes | — |
| 141 | C2×C4 | #13133242 | 1 | 3 | yes | — |
| 142 | C8 | — | 424 | 3 | no | — |
| 143 | C22 | — | 848 | 1 | no | — |
| 144 | C22 | — | 848 | 3 | no | — |
| 145 | C22 | — | 424 | 6 | no | — |
| 146 | C22 | #16810768 | 1 | 3 | yes | — |
| 147 | C22 | #16810796 | 1 | 3 | yes | — |
| 148 | C22 | #16810810 | 1 | 1 | yes | characteristic |
| 149 | C4 | — | 2 | 3 | no | — |
| 150 | C4 | — | 2 | 3 | no | — |
| 151 | C4 | — | 8 | 3 | no | — |
| 152 | C4 | — | 8 | 3 | no | — |
| 153 | C2 | — | 848 | 2 | no | — |
| 154 | C2 | #20273644 | 1 | 3 | yes | — |
| 155 | C2 | #20273825 | 1 | 3 | yes | — |
| 156 | C2 | #20274017 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20272538 | C2 | semidirect |
| 3. C53×#2810 | C2 | semidirect |
| 75. #2810 | D53 | semidirect |
| 89. C53 | #7744 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20275949 | 2 |
| #20272538 | 2 |
| C53×#2810 | 2 |
| #7744 | 53 |