← #28310190
Subgroups — #28310190
Order 13568.
478 conjugacy classes of subgroups.
The subgroup lattice has 194 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20290916 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16811510 | C2 | 1 | 2 | yes | maximal |
| 3 | C2×C16×#6157 | C2 | 1 | 2 | yes | maximal |
| 4 | #20290371 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C4×C16×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C2×D53⋊C16 | C22 | 1 | 2 | yes | — |
| 7 | #16811582 | C4 | 1 | 1 | yes | characteristic |
| 8 | C2×#13133271 | C4 | 1 | 2 | yes | — |
| 9 | C2×C8×#6157 | C22 | 1 | 2 | yes | — |
| 10 | #16811559 | C4 | 1 | 1 | yes | characteristic |
| 11 | #16811512 | C4 | 1 | 2 | yes | complemented |
| 12 | #16811511 | C4 | 1 | 2 | yes | complemented |
| 13 | #16811510 | C4 | 1 | 8 | yes | complemented |
| 14 | C16×#6157 | C4 | 1 | 8 | yes | complemented |
| 15 | C2×C16×D53 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×C16×D53 | C22 | 1 | 1 | yes | characteristic |
| 17 | C4×C8×D53 | C22 | 1 | 1 | yes | characteristic |
| 18 | C4×#11719908 | C4 | 1 | 1 | yes | characteristic, complemented |
| 19 | C16×Dic53 | C4 | 1 | 1 | yes | characteristic, complemented |
| 20 | C16×Dic53 | C4 | 1 | 1 | yes | characteristic, complemented |
| 21 | C4×C16×C53 | C4 | 1 | 1 | yes | characteristic, complemented |
| 22 | C2×D53⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 23 | C2×C4×#6157 | C2×C4 | 1 | 2 | yes | — |
| 24 | D53⋊C16 | C2×C4 | 1 | 4 | yes | — |
| 25 | D53⋊C16 | C2×C4 | 1 | 4 | yes | — |
| 26 | C2×#11719910 | C2×C4 | 1 | 4 | yes | — |
| 27 | #13133272 | C8 | 1 | 4 | yes | — |
| 28 | #13133271 | C8 | 1 | 8 | yes | — |
| 29 | C8×#6157 | C2×C4 | 1 | 4 | yes | — |
| 30 | C8×#6157 | C2×C4 | 1 | 4 | yes | — |
| 31 | (C2×#6157)⋊C4 | C8 | 1 | 4 | yes | — |
| 32 | C2×C8×D53 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C2×C8×D53 | C23 | 1 | 1 | yes | characteristic |
| 34 | C42×D53 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C2×#11719908 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C2×#11719908 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C16×D53 | C2×C4 | 1 | 4 | yes | — |
| 38 | C16×D53 | C2×C4 | 1 | 4 | yes | — |
| 39 | C4×#88322 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C2×C16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C2×C16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | C22×#6157 | C2×C8 | 1 | 4 | yes | — |
| 46 | C2×C4×D53 | C2×C8 | 1 | 2 | yes | — |
| 47 | C2×C4×D53 | C22×C4 | 1 | 1 | yes | characteristic |
| 48 | C2×#88324 | C2×C8 | 1 | 4 | yes | — |
| 49 | C4×#6157 | C42 | 1 | 4 | yes | — |
| 50 | C4×#6157 | C2×C8 | 1 | 4 | yes | — |
| 51 | D53⋊C8 | C42 | 1 | 4 | yes | — |
| 52 | D53⋊C8 | C2×C8 | 1 | 4 | yes | — |
| 53 | C2×C8×C53 | C42 | 1 | 1 | yes | characteristic |
| 54 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 55 | C42×C53 | C42 | 1 | 1 | yes | characteristic |
| 56 | C4×Dic53 | C42 | 1 | 1 | yes | characteristic |
| 57 | C4×Dic53 | C2×C8 | 1 | 2 | yes | — |
| 58 | C2×#88322 | C42 | 1 | 1 | yes | characteristic |
| 59 | C2×#88322 | C22×C4 | 1 | 1 | yes | characteristic |
| 60 | C8×D53 | C2×C8 | 1 | 4 | yes | — |
| 61 | C8×D53 | C22×C4 | 1 | 2 | yes | — |
| 62 | C8×D53 | C22×C4 | 1 | 2 | yes | — |
| 63 | #11719910 | C42 | 1 | 8 | yes | — |
| 64 | C16×C53 | C42 | 1 | 2 | yes | — |
| 65 | C16×C53 | C42 | 1 | 2 | yes | — |
| 66 | #11719908 | C42 | 1 | 2 | yes | — |
| 67 | #11719908 | C42 | 1 | 2 | yes | — |
| 68 | C22×D53 | C22×C8 | 1 | 1 | yes | characteristic |
| 69 | C2×#6157 | — | 2 | 8 | no | — |
| 70 | C2×#6157 | C4×C8 | 1 | 8 | yes | — |
| 71 | C2×C4×C53 | C4×C8 | 1 | 2 | yes | — |
| 72 | C2×C4×C53 | C2×C42 | 1 | 1 | yes | characteristic |
| 73 | C2×Dic53 | C4×C8 | 1 | 2 | yes | — |
| 74 | C2×Dic53 | C22×C8 | 1 | 1 | yes | characteristic |
| 75 | C4×D53 | — | 2 | 4 | no | — |
| 76 | C4×D53 | C2×C42 | 1 | 2 | yes | — |
| 77 | C4×D53 | C22×C8 | 1 | 2 | yes | — |
| 78 | #88324 | C4×C8 | 1 | 8 | yes | — |
| 79 | C8×C53 | C4×C8 | 1 | 2 | yes | — |
| 80 | C8×C53 | C2×C42 | 1 | 1 | yes | characteristic |
| 81 | C8×C53 | C2×C42 | 1 | 1 | yes | characteristic |
| 82 | #88322 | C4×C8 | 1 | 2 | yes | — |
| 83 | #88322 | C2×C42 | 1 | 2 | yes | — |
| 84 | #12539 | — | 53 | 1 | no | maximal, Sylow |
| 85 | C22×C53 | C2×C4×C8 | 1 | 1 | yes | characteristic |
| 86 | C2×D53 | C2×C4×C8 | 1 | 2 | yes | — |
| 87 | C2×D53 | #504 | 1 | 4 | yes | — |
| 88 | #6157 | — | 2 | 8 | no | — |
| 89 | C4×C53 | — | 2 | 2 | no | — |
| 90 | C4×C53 | C43 | 1 | 1 | yes | characteristic |
| 91 | C4×C53 | C2×C4×C8 | 1 | 1 | yes | characteristic |
| 92 | Dic53 | — | 2 | 2 | no | — |
| 93 | Dic53 | C2×C4×C8 | 1 | 2 | yes | — |
| 94 | C2×#346 | — | 53 | 2 | no | — |
| 95 | C2×#346 | — | 53 | 1 | no | — |
| 96 | C2×C4×C16 | — | 53 | 2 | no | — |
| 97 | C2×C4×C16 | — | 53 | 1 | no | — |
| 98 | #1726 | — | 53 | 1 | no | — |
| 99 | C2×C53 | C42×C8 | 1 | 1 | yes | characteristic |
| 100 | C2×C53 | #2108 | 1 | 2 | yes | — |
| 101 | D53 | #2108 | 1 | 4 | yes | — |
| 102 | C22×C16 | — | 53 | 2 | no | — |
| 103 | C22×C16 | — | 53 | 1 | no | — |
| 104 | C22×C16 | — | 53 | 1 | no | — |
| 105 | C2×#97 | — | 53 | 2 | no | — |
| 106 | C2×#97 | — | 53 | 1 | no | — |
| 107 | C2×C4×C8 | — | 53 | 2 | no | — |
| 108 | C2×C4×C8 | — | 53 | 1 | no | — |
| 109 | #376 | — | 53 | 1 | no | — |
| 110 | #346 | — | 53 | 8 | no | — |
| 111 | #346 | — | 53 | 2 | no | — |
| 112 | #346 | — | 53 | 2 | no | — |
| 113 | C4×C16 | — | 53 | 8 | no | — |
| 114 | C4×C16 | — | 53 | 1 | no | — |
| 115 | C4×C16 | — | 53 | 1 | no | — |
| 116 | C4×C16 | — | 53 | 1 | no | — |
| 117 | C4×C16 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 118 | C53 | #12539 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 119 | C22×C8 | — | 53 | 2 | no | — |
| 120 | C22×C8 | — | 53 | 1 | no | — |
| 121 | C22×C8 | — | 53 | 1 | no | — |
| 122 | C2×C42 | — | 53 | 2 | no | — |
| 123 | C2×C42 | — | 53 | 1 | no | — |
| 124 | C2×C16 | — | 53 | 4 | no | — |
| 125 | C2×C16 | — | 53 | 4 | no | — |
| 126 | C2×C16 | — | 53 | 4 | no | — |
| 127 | C2×C16 | — | 53 | 1 | no | — |
| 128 | C2×C16 | — | 53 | 1 | no | — |
| 129 | C2×C16 | — | 53 | 4 | no | — |
| 130 | C2×C16 | — | 53 | 4 | no | — |
| 131 | C2×C16 | C2×#6157 | 1 | 1 | yes | characteristic |
| 132 | C2×C16 | C2×#6157 | 1 | 1 | yes | characteristic |
| 133 | #97 | — | 53 | 8 | no | — |
| 134 | #97 | — | 53 | 4 | no | — |
| 135 | C4×C8 | — | 53 | 4 | no | — |
| 136 | C4×C8 | — | 53 | 4 | no | — |
| 137 | C4×C8 | — | 53 | 1 | no | — |
| 138 | C4×C8 | — | 53 | 1 | no | — |
| 139 | C4×C8 | — | 53 | 1 | no | — |
| 140 | C4×C8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 141 | #95 | — | 53 | 4 | no | — |
| 142 | C22×C4 | — | 53 | 1 | no | — |
| 143 | C22×C4 | — | 53 | 4 | no | — |
| 144 | C22×C4 | — | 53 | 2 | no | — |
| 145 | C2×C8 | — | 53 | 4 | no | — |
| 146 | C2×C8 | — | 53 | 4 | no | — |
| 147 | C2×C8 | — | 53 | 4 | no | — |
| 148 | C2×C8 | — | 53 | 1 | no | — |
| 149 | C2×C8 | — | 53 | 1 | no | — |
| 150 | C2×C8 | — | 53 | 4 | no | — |
| 151 | C2×C8 | — | 53 | 2 | no | — |
| 152 | C2×C8 | — | 53 | 2 | no | — |
| 153 | C2×C8 | C4×#6157 | 1 | 1 | yes | characteristic |
| 154 | C2×C8 | C22×#6157 | 1 | 1 | yes | characteristic |
| 155 | C42 | — | 53 | 4 | no | — |
| 156 | C42 | — | 53 | 4 | no | — |
| 157 | C42 | — | 53 | 2 | no | — |
| 158 | C42 | — | 53 | 1 | no | — |
| 159 | C42 | C4×#6157 | 1 | 1 | yes | characteristic |
| 160 | C16 | — | 53 | 8 | no | — |
| 161 | C16 | — | 53 | 2 | no | — |
| 162 | C16 | — | 53 | 2 | no | — |
| 163 | C16 | C4×#6157 | 1 | 2 | yes | — |
| 164 | C16 | C4×#6157 | 1 | 2 | yes | — |
| 165 | C23 | — | 53 | 1 | no | — |
| 166 | C2×C4 | — | 53 | 1 | no | — |
| 167 | C2×C4 | — | 53 | 2 | no | — |
| 168 | C2×C4 | — | 53 | 2 | no | — |
| 169 | C2×C4 | — | 53 | 8 | no | — |
| 170 | C2×C4 | — | 53 | 2 | no | — |
| 171 | C2×C4 | — | 106 | 4 | no | — |
| 172 | C2×C4 | — | 106 | 8 | no | — |
| 173 | C2×C4 | C8×#6157 | 1 | 2 | yes | — |
| 174 | C2×C4 | C2×C4×#6157 | 1 | 1 | yes | characteristic |
| 175 | C8 | — | 53 | 8 | no | — |
| 176 | C8 | — | 53 | 2 | no | — |
| 177 | C8 | — | 53 | 2 | no | — |
| 178 | C8 | C8×#6157 | 1 | 2 | yes | — |
| 179 | C8 | C2×C4×#6157 | 1 | 1 | yes | characteristic |
| 180 | C8 | C2×C4×#6157 | 1 | 1 | yes | characteristic |
| 181 | C22 | — | 53 | 2 | no | — |
| 182 | C22 | — | 53 | 4 | no | — |
| 183 | C22 | C2×C8×#6157 | 1 | 1 | yes | characteristic |
| 184 | C4 | — | 2 | 2 | no | — |
| 185 | C4 | — | 53 | 2 | no | — |
| 186 | C4 | — | 106 | 8 | no | — |
| 187 | C4 | — | 106 | 2 | no | — |
| 188 | C4 | C42×#6157 | 1 | 1 | yes | characteristic |
| 189 | C4 | C2×C8×#6157 | 1 | 1 | yes | characteristic |
| 190 | C2 | — | 53 | 4 | no | — |
| 191 | C2 | C4×C8×#6157 | 1 | 1 | yes | characteristic |
| 192 | C2 | #20290923 | 1 | 2 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta