Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
196 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 25 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 | #20273829 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20272756 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C53×#1410 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #16811258 | C4 | 1 | 1 | yes | characteristic |
| 5 | #16810910 | — | 2 | 1 | no | — |
| 6 | #16810444 | — | 2 | 1 | no | — |
| 7 | #16810367 | C4 | 1 | 1 | yes | characteristic |
| 8 | #16810352 | — | 2 | 1 | no | — |
| 9 | C53×#492 | C22 | 1 | 1 | yes | characteristic |
| 10 | C53×#356 | — | 2 | 1 | no | — |
| 11 | C2×D4×D53 | — | 4 | 1 | no | — |
| 12 | #13133188 | — | 4 | 1 | no | — |
| 13 | #13133185 | — | 4 | 1 | no | — |
| 14 | C4×D212 | — | 2 | 1 | no | — |
| 15 | #13133157 | — | 4 | 1 | no | — |
| 16 | #13133148 | — | 4 | 1 | no | — |
| 17 | C4⋊#88322 | — | 2 | 1 | no | — |
| 18 | #13133137 | — | 4 | 1 | no | — |
| 19 | C2×SD16×C53 | — | 4 | 1 | no | — |
| 20 | C4⋊Q8×C53 | D4 | 1 | 1 | yes | characteristic |
| 21 | #127×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | #101×C53 | — | 4 | 1 | no | — |
| 23 | C4×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 24 | C23×D53 | — | 4 | 1 | no | — |
| 25 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 26 | C2×D4×C53 | — | 2 | 1 | no | — |
| 27 | C2×D4×C53 | — | 4 | 1 | no | — |
| 28 | C2×C22⋊D53 | — | 4 | 2 | no | — |
| 29 | D4×D53 | — | 8 | 2 | no | — |
| 30 | D4×D53 | — | 8 | 2 | no | — |
| 31 | C2×D212 | — | 2 | 1 | no | — |
| 32 | C2×C4×D53 | — | 4 | 1 | no | — |
| 33 | SD16×C53 | — | 8 | 1 | no | — |
| 34 | SD16×C53 | — | 8 | 1 | no | — |
| 35 | M16×C53 | — | 8 | 1 | no | — |
| 36 | C2×C8×C53 | — | 2 | 1 | no | — |
| 37 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 38 | C42×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 39 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 40 | #11719921 | — | 4 | 1 | no | — |
| 41 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 42 | C2×#88322 | — | 4 | 1 | no | — |
| 43 | #11719908 | — | 8 | 1 | no | — |
| 44 | C23×C53 | — | 4 | 1 | no | — |
| 45 | C22×D53 | — | 4 | 1 | no | — |
| 46 | C22×D53 | — | 8 | 4 | no | — |
| 47 | C22×D53 | — | 16 | 1 | no | — |
| 48 | Q8×C53 | — | 8 | 1 | no | — |
| 49 | D4×C53 | — | 8 | 1 | no | — |
| 50 | D4×C53 | — | 8 | 1 | no | — |
| 51 | D4×C53 | — | 8 | 1 | no | — |
| 52 | C2×C4×C53 | — | 2 | 1 | no | — |
| 53 | C2×C4×C53 | — | 4 | 1 | no | — |
| 54 | C2×C4×C53 | #99 | 1 | 1 | yes | characteristic |
| 55 | C22⋊D53 | — | 8 | 4 | no | — |
| 56 | C2×Dic53 | — | 4 | 1 | no | — |
| 57 | D212 | — | 2 | 2 | no | — |
| 58 | D212 | — | 4 | 1 | no | — |
| 59 | C4×D53 | — | 8 | 1 | no | — |
| 60 | C4×D53 | — | 8 | 1 | no | — |
| 61 | C8×C53 | — | 2 | 1 | no | — |
| 62 | C8×C53 | — | 2 | 1 | no | — |
| 63 | C8×C53 | — | 16 | 1 | no | — |
| 64 | #88322 | — | 8 | 1 | no | — |
| 65 | #7517 | — | 53 | 1 | no | maximal, Sylow |
| 66 | C22×C53 | — | 8 | 2 | no | — |
| 67 | C22×C53 | — | 8 | 1 | no | — |
| 68 | C22×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 69 | C2×D53 | — | 4 | 2 | no | — |
| 70 | C2×D53 | — | 8 | 1 | no | — |
| 71 | C2×D53 | — | 8 | 1 | no | — |
| 72 | C2×D53 | — | 16 | 2 | no | — |
| 73 | C2×D53 | — | 16 | 2 | no | — |
| 74 | C4×C53 | — | 2 | 1 | no | — |
| 75 | C4×C53 | — | 2 | 1 | no | — |
| 76 | C4×C53 | — | 2 | 1 | no | — |
| 77 | C4×C53 | — | 8 | 1 | no | — |
| 78 | Dic53 | — | 8 | 1 | no | — |
| 79 | #1629 | — | 53 | 1 | no | — |
| 80 | #1410 | D53 | 1 | 1 | yes | characteristic, complemented |
| 81 | #1359 | — | 53 | 1 | no | — |
| 82 | C2×C53 | — | 2 | 1 | no | — |
| 83 | C2×C53 | — | 16 | 1 | no | — |
| 84 | C2×C53 | #1402 | 1 | 1 | yes | characteristic |
| 85 | D53 | — | 8 | 2 | no | — |
| 86 | D42 | — | 53 | 1 | no | — |
| 87 | #492 | C2×D53 | 1 | 1 | yes | characteristic |
| 88 | #476 | — | 106 | 1 | no | — |
| 89 | #364 | — | 106 | 1 | no | — |
| 90 | #356 | — | 2 | 1 | no | — |
| 91 | #331 | — | 53 | 1 | no | — |
| 92 | D4⋊C8 | — | 106 | 1 | no | — |
| 93 | C53 | #7517 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 94 | C22×D4 | — | 212 | 1 | no | — |
| 95 | C2×SD16 | — | 4 | 1 | no | — |
| 96 | C4⋊Q8 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 97 | #127 | C4×D53 | 1 | 1 | yes | characteristic |
| 98 | #121 | — | 212 | 1 | no | — |
| 99 | #120 | — | 212 | 1 | no | — |
| 100 | C4×D4 | — | 106 | 1 | no | — |
| 101 | M32 | — | 212 | 1 | no | — |
| 102 | #106 | — | 212 | 1 | no | — |
| 103 | C4⋊C8 | — | 106 | 1 | no | — |
| 104 | #102 | — | 212 | 1 | no | — |
| 105 | #101 | — | 4 | 1 | no | — |
| 106 | C4×C8 | D212 | 1 | 1 | yes | characteristic |
| 107 | C24 | — | 212 | 1 | no | — |
| 108 | C2×Q8 | — | 2 | 1 | no | — |
| 109 | C2×D4 | — | 2 | 1 | no | — |
| 110 | C2×D4 | — | 4 | 1 | no | — |
| 111 | C2×D4 | — | 106 | 1 | no | — |
| 112 | C2×D4 | — | 424 | 2 | no | — |
| 113 | C2×D4 | — | 424 | 2 | no | — |
| 114 | C2×D4 | — | 212 | 2 | no | — |
| 115 | C22×C4 | — | 212 | 1 | no | — |
| 116 | SD16 | — | 8 | 1 | no | — |
| 117 | SD16 | — | 8 | 1 | no | — |
| 118 | M16 | — | 8 | 1 | no | — |
| 119 | C2×C8 | — | 2 | 1 | no | — |
| 120 | C2×C8 | — | 212 | 1 | no | — |
| 121 | C4⋊C4 | — | 4 | 1 | no | — |
| 122 | C4⋊C4 | — | 106 | 1 | no | — |
| 123 | C22⋊C4 | — | 212 | 1 | no | — |
| 124 | C22⋊C4 | — | 212 | 1 | no | — |
| 125 | C42 | #11719921 | 1 | 1 | yes | characteristic |
| 126 | C16 | — | 424 | 1 | no | — |
| 127 | C23 | — | 4 | 1 | no | — |
| 128 | C23 | — | 848 | 1 | no | — |
| 129 | C23 | — | 424 | 4 | no | — |
| 130 | C23 | — | 212 | 1 | no | — |
| 131 | Q8 | — | 8 | 1 | no | — |
| 132 | D4 | — | 8 | 1 | no | — |
| 133 | D4 | — | 8 | 1 | no | — |
| 134 | D4 | — | 8 | 1 | no | — |
| 135 | D4 | — | 106 | 2 | no | — |
| 136 | D4 | — | 424 | 4 | no | — |
| 137 | D4 | — | 212 | 1 | no | — |
| 138 | C2×C4 | — | 2 | 1 | no | — |
| 139 | C2×C4 | — | 4 | 1 | no | — |
| 140 | C2×C4 | — | 424 | 1 | no | — |
| 141 | C2×C4 | — | 424 | 1 | no | — |
| 142 | C2×C4 | — | 212 | 1 | no | — |
| 143 | C2×C4 | #13133131 | 1 | 1 | yes | characteristic |
| 144 | C8 | — | 2 | 1 | no | — |
| 145 | C8 | — | 2 | 1 | no | — |
| 146 | C8 | — | 16 | 1 | no | — |
| 147 | C8 | — | 424 | 1 | no | — |
| 148 | C22 | — | 8 | 2 | no | — |
| 149 | C22 | — | 8 | 1 | no | — |
| 150 | C22 | — | 848 | 2 | no | — |
| 151 | C22 | — | 848 | 2 | no | — |
| 152 | C22 | — | 424 | 1 | no | — |
| 153 | C22 | — | 424 | 1 | no | — |
| 154 | C22 | — | 212 | 2 | no | — |
| 155 | C22 | #16810409 | 1 | 1 | yes | characteristic |
| 156 | C4 | — | 2 | 1 | no | — |
| 157 | C4 | — | 2 | 1 | no | — |
| 158 | C4 | — | 2 | 1 | no | — |
| 159 | C4 | — | 8 | 1 | no | — |
| 160 | C4 | — | 424 | 1 | no | — |
| 161 | C2 | — | 2 | 1 | no | — |
| 162 | C2 | — | 16 | 1 | no | — |
| 163 | C2 | — | 424 | 2 | no | — |
| 164 | C2 | #20272859 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20272756 | C2 | semidirect |
| 3. C53×#1410 | C2 | semidirect |
| 80. #1410 | D53 | semidirect |
| 93. C53 | #7517 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20273829 | 2 |
| #20272756 | 2 |
| C53×#1410 | 2 |
| #7517 | 53 |