← #28506792
Subgroups — #28506792
Order 15104.
286 conjugacy classes of subgroups.
The subgroup lattice has 252 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16837231 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×Dic59⋊C16 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #21424266 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #108×Dic59 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #21422190 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #21422189 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C2×C59×#368 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C22×C59⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#14273663 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#14273663 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#14273662 | — | 2 | 1 | no | — |
| 12 | #16837425 | — | 2 | 1 | no | — |
| 13 | M16×Dic59 | — | 2 | 1 | no | — |
| 14 | C2×C8×Dic59 | C22 | 1 | 1 | yes | characteristic |
| 15 | #16837231 | — | 2 | 2 | no | — |
| 16 | Dic59⋊C16 | C4 | 1 | 2 | yes | — |
| 17 | Dic59⋊C16 | C4 | 1 | 2 | yes | — |
| 18 | #16837166 | — | 2 | 1 | no | — |
| 19 | C22×C16×C59 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#108×C59 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#108×C59 | C22 | 1 | 1 | yes | characteristic |
| 22 | C59×#368 | — | 2 | 2 | no | — |
| 23 | C2×#11745664 | — | 2 | 1 | no | — |
| 24 | C2×#11745664 | — | 2 | 1 | no | — |
| 25 | C22×C59⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 26 | C2×C4×Dic59 | D4 | 1 | 1 | yes | characteristic |
| 27 | C2×C59⋊C16 | — | 2 | 1 | no | — |
| 28 | C2×C59⋊C16 | — | 2 | 1 | no | — |
| 29 | C2×C59⋊C16 | C2×C4 | 1 | 2 | yes | — |
| 30 | #14273663 | — | 2 | 1 | no | — |
| 31 | #14273663 | — | 2 | 1 | no | — |
| 32 | #14273663 | — | 2 | 1 | no | — |
| 33 | #14273663 | Q8 | 1 | 2 | yes | — |
| 34 | #14273662 | — | 4 | 2 | no | — |
| 35 | Dic59⋊C8 | — | 4 | 1 | no | — |
| 36 | Dic59⋊C8 | — | 4 | 1 | no | — |
| 37 | #14273637 | — | 4 | 1 | no | — |
| 38 | C8×Dic59 | — | 4 | 1 | no | — |
| 39 | C8×Dic59 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C8×Dic59 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C8×Dic59 | Q8 | 1 | 1 | yes | characteristic |
| 42 | C8×Dic59 | Q8 | 1 | 1 | yes | characteristic |
| 43 | C2×M16×C59 | — | 2 | 1 | no | — |
| 44 | C2×M16×C59 | — | 2 | 1 | no | — |
| 45 | C22×C8×C59 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×C16×C59 | — | 2 | 1 | no | — |
| 47 | C2×C16×C59 | — | 2 | 1 | no | — |
| 48 | C2×C16×C59 | C2×C4 | 1 | 2 | yes | — |
| 49 | #108×C59 | — | 2 | 1 | no | — |
| 50 | #108×C59 | — | 2 | 1 | no | — |
| 51 | #108×C59 | — | 2 | 1 | no | — |
| 52 | #108×C59 | D4 | 1 | 2 | yes | — |
| 53 | C22×C4×C59 | C2×D4 | 1 | 1 | yes | characteristic |
| 54 | C22×Dic59 | Q16 | 1 | 1 | yes | characteristic |
| 55 | C22×Dic59 | Q16 | 1 | 1 | yes | characteristic |
| 56 | M16×C59 | — | 2 | 2 | no | — |
| 57 | M16×C59 | — | 4 | 1 | no | — |
| 58 | M16×C59 | — | 4 | 1 | no | — |
| 59 | M16×C59 | — | 4 | 1 | no | — |
| 60 | C2×C8×C59 | — | 4 | 1 | no | — |
| 61 | C2×C8×C59 | — | 4 | 1 | no | — |
| 62 | C2×C8×C59 | C22×C4 | 1 | 1 | yes | characteristic |
| 63 | C2×C8×C59 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C2×C8×C59 | C2×Q8 | 1 | 1 | yes | characteristic |
| 65 | C2×C8×C59 | C2×Q8 | 1 | 1 | yes | characteristic |
| 66 | C2×C8×C59 | #41 | 1 | 1 | yes | characteristic |
| 67 | C2×C8×C59 | #41 | 1 | 1 | yes | characteristic |
| 68 | C4×Dic59 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 69 | C4×Dic59 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 70 | C4×Dic59 | D8 | 1 | 1 | yes | characteristic |
| 71 | C4×Dic59 | D8 | 1 | 1 | yes | characteristic |
| 72 | #11745664 | — | 2 | 2 | no | — |
| 73 | #11745664 | — | 4 | 1 | no | — |
| 74 | #11745664 | — | 4 | 1 | no | — |
| 75 | #11745664 | — | 4 | 1 | no | — |
| 76 | C2×C59⋊C8 | — | 2 | 1 | no | — |
| 77 | C2×C59⋊C8 | — | 2 | 1 | no | — |
| 78 | C2×C59⋊C8 | — | 4 | 1 | no | — |
| 79 | C2×C59⋊C8 | — | 4 | 1 | no | — |
| 80 | C2×C59⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 81 | C2×C59⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 82 | C16×C59 | — | 2 | 2 | no | — |
| 83 | C59⋊C16 | — | 2 | 2 | no | — |
| 84 | C23×C59 | C2×Q16 | 1 | 1 | yes | characteristic |
| 85 | C2×C4×C59 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 86 | C2×C4×C59 | #121 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C59 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C59 | C2×D8 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C59 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C59 | #137 | 1 | 1 | yes | characteristic |
| 91 | C2×Dic59 | — | 2 | 2 | no | — |
| 92 | C2×Dic59 | — | 4 | 1 | no | — |
| 93 | C2×Dic59 | #107 | 1 | 1 | yes | characteristic |
| 94 | C2×Dic59 | #107 | 1 | 1 | yes | characteristic |
| 95 | C2×Dic59 | #107 | 1 | 1 | yes | characteristic |
| 96 | C2×Dic59 | #107 | 1 | 1 | yes | characteristic |
| 97 | C8×C59 | — | 4 | 2 | no | — |
| 98 | C8×C59 | — | 8 | 1 | no | — |
| 99 | C8×C59 | C4×D4 | 1 | 1 | yes | characteristic |
| 100 | C8×C59 | C4×Q8 | 1 | 1 | yes | characteristic |
| 101 | C8×C59 | #125 | 1 | 1 | yes | characteristic |
| 102 | C8×C59 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 103 | C59⋊C8 | — | 2 | 1 | no | — |
| 104 | C59⋊C8 | — | 2 | 1 | no | — |
| 105 | C59⋊C8 | — | 4 | 2 | no | — |
| 106 | C59⋊C8 | — | 8 | 1 | no | — |
| 107 | #13179 | — | 59 | 1 | no | maximal, Sylow |
| 108 | C22×C59 | — | 2 | 1 | no | — |
| 109 | C22×C59 | — | 2 | 1 | no | — |
| 110 | C22×C59 | C2×#107 | 1 | 1 | yes | characteristic |
| 111 | C22×C59 | #462 | 1 | 1 | yes | characteristic |
| 112 | C22×C59 | #477 | 1 | 1 | yes | characteristic |
| 113 | C4×C59 | #389 | 1 | 1 | yes | characteristic |
| 114 | C4×C59 | #428 | 1 | 1 | yes | characteristic |
| 115 | C4×C59 | #459 | 1 | 1 | yes | characteristic |
| 116 | C4×C59 | #474 | 1 | 1 | yes | characteristic |
| 117 | Dic59 | — | 2 | 1 | no | — |
| 118 | Dic59 | — | 2 | 1 | no | — |
| 119 | Dic59 | — | 2 | 1 | no | — |
| 120 | Dic59 | #368 | 1 | 2 | yes | complemented |
| 121 | C2×#368 | — | 59 | 1 | no | — |
| 122 | C2×#368 | D59 | 1 | 1 | yes | characteristic |
| 123 | C2×C4⋊C16 | — | 59 | 1 | no | — |
| 124 | #1848 | — | 59 | 1 | no | — |
| 125 | C4×#108 | — | 59 | 1 | no | — |
| 126 | #1381 | — | 59 | 1 | no | — |
| 127 | #1381 | — | 59 | 1 | no | — |
| 128 | C2×C59 | — | 2 | 1 | no | — |
| 129 | C2×C59 | — | 2 | 1 | no | — |
| 130 | C2×C59 | #1947 | 1 | 1 | yes | characteristic |
| 131 | C2×C59 | C2×#368 | 1 | 1 | yes | characteristic |
| 132 | C2×C59 | #2161 | 1 | 1 | yes | characteristic |
| 133 | C22×C16 | — | 59 | 1 | no | — |
| 134 | C22×C16 | C2×D59 | 1 | 1 | yes | characteristic |
| 135 | C2×#108 | — | 59 | 1 | no | — |
| 136 | C2×#108 | — | 59 | 1 | no | — |
| 137 | C2×#108 | — | 118 | 1 | no | — |
| 138 | C2×#108 | C2×D59 | 1 | 1 | yes | characteristic |
| 139 | C2×#108 | C2×D59 | 1 | 1 | yes | characteristic |
| 140 | #423 | — | 118 | 1 | no | — |
| 141 | C4×M16 | — | 118 | 1 | no | — |
| 142 | C2×C4×C8 | — | 59 | 1 | no | — |
| 143 | #368 | — | 2 | 2 | no | — |
| 144 | #368 | — | 118 | 2 | no | — |
| 145 | C4⋊C16 | — | 59 | 2 | no | — |
| 146 | C4⋊C16 | — | 59 | 2 | no | — |
| 147 | #341 | — | 118 | 1 | no | — |
| 148 | C59 | #13179 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 149 | C2×M16 | — | 2 | 1 | no | — |
| 150 | C2×M16 | — | 2 | 1 | no | — |
| 151 | C2×M16 | — | 118 | 1 | no | — |
| 152 | C2×M16 | — | 118 | 1 | no | — |
| 153 | C22×C8 | — | 59 | 1 | no | — |
| 154 | C22×C8 | C22×D59 | 1 | 1 | yes | characteristic |
| 155 | C2×C42 | — | 59 | 1 | no | — |
| 156 | C2×C16 | — | 2 | 1 | no | — |
| 157 | C2×C16 | — | 2 | 1 | no | — |
| 158 | C2×C16 | — | 59 | 2 | no | — |
| 159 | C2×C16 | — | 118 | 1 | no | — |
| 160 | C2×C16 | — | 118 | 1 | no | — |
| 161 | C2×C16 | C4×D59 | 1 | 2 | yes | — |
| 162 | #108 | — | 2 | 1 | no | — |
| 163 | #108 | — | 2 | 1 | no | — |
| 164 | #108 | — | 2 | 1 | no | — |
| 165 | #108 | — | 59 | 2 | no | — |
| 166 | #108 | — | 118 | 1 | no | — |
| 167 | #108 | — | 118 | 1 | no | — |
| 168 | #108 | — | 118 | 1 | no | — |
| 169 | #108 | — | 236 | 2 | no | — |
| 170 | #108 | C22⋊D59 | 1 | 2 | yes | — |
| 171 | C4⋊C8 | — | 236 | 1 | no | — |
| 172 | C4⋊C8 | — | 236 | 1 | no | — |
| 173 | #97 | — | 236 | 1 | no | — |
| 174 | C4×C8 | — | 59 | 1 | no | — |
| 175 | C4×C8 | — | 59 | 1 | no | — |
| 176 | C4×C8 | — | 59 | 1 | no | — |
| 177 | C4×C8 | — | 59 | 1 | no | — |
| 178 | C4×C8 | — | 236 | 1 | no | — |
| 179 | C22×C4 | — | 59 | 1 | no | — |
| 180 | C22×C4 | — | 59 | 1 | no | — |
| 181 | C22×C4 | D4×D59 | 1 | 1 | yes | characteristic |
| 182 | M16 | — | 2 | 2 | no | — |
| 183 | M16 | — | 4 | 1 | no | — |
| 184 | M16 | — | 4 | 1 | no | — |
| 185 | M16 | — | 4 | 1 | no | — |
| 186 | M16 | — | 118 | 2 | no | — |
| 187 | M16 | — | 236 | 1 | no | — |
| 188 | M16 | — | 236 | 1 | no | — |
| 189 | M16 | — | 236 | 1 | no | — |
| 190 | C2×C8 | — | 4 | 1 | no | — |
| 191 | C2×C8 | — | 4 | 1 | no | — |
| 192 | C2×C8 | — | 59 | 1 | no | — |
| 193 | C2×C8 | — | 59 | 1 | no | — |
| 194 | C2×C8 | — | 118 | 1 | no | — |
| 195 | C2×C8 | — | 118 | 1 | no | — |
| 196 | C2×C8 | — | 236 | 1 | no | — |
| 197 | C2×C8 | — | 236 | 1 | no | — |
| 198 | C2×C8 | C2×C4×D59 | 1 | 1 | yes | characteristic |
| 199 | C2×C8 | #11745685 | 1 | 1 | yes | characteristic |
| 200 | C2×C8 | #11745687 | 1 | 1 | yes | characteristic |
| 201 | C2×C8 | Q8×D59 | 1 | 1 | yes | characteristic |
| 202 | C2×C8 | Q8×D59 | 1 | 1 | yes | characteristic |
| 203 | C2×C8 | C2×C22⋊D59 | 1 | 1 | yes | characteristic |
| 204 | C42 | — | 59 | 1 | no | — |
| 205 | C42 | — | 59 | 1 | no | — |
| 206 | C42 | — | 59 | 1 | no | — |
| 207 | C42 | — | 59 | 1 | no | — |
| 208 | C16 | — | 2 | 2 | no | — |
| 209 | C16 | — | 118 | 2 | no | — |
| 210 | C23 | Q16×D59 | 1 | 1 | yes | characteristic |
| 211 | C2×C4 | — | 59 | 1 | no | — |
| 212 | C2×C4 | — | 59 | 1 | no | — |
| 213 | C2×C4 | — | 59 | 1 | no | — |
| 214 | C2×C4 | — | 59 | 1 | no | — |
| 215 | C2×C4 | — | 118 | 2 | no | — |
| 216 | C2×C4 | — | 236 | 1 | no | — |
| 217 | C2×C4 | C4⋊C4×D59 | 1 | 1 | yes | characteristic |
| 218 | C2×C4 | #14273699 | 1 | 1 | yes | characteristic |
| 219 | C2×C4 | #14273705 | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | D8×D59 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #14273751 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #14273759 | 1 | 1 | yes | characteristic |
| 223 | C8 | — | 4 | 2 | no | — |
| 224 | C8 | — | 8 | 1 | no | — |
| 225 | C8 | — | 118 | 1 | no | — |
| 226 | C8 | — | 118 | 1 | no | — |
| 227 | C8 | — | 472 | 1 | no | — |
| 228 | C8 | — | 236 | 2 | no | — |
| 229 | C8 | C4×C22⋊D59 | 1 | 1 | yes | characteristic |
| 230 | C8 | #14273692 | 1 | 1 | yes | characteristic |
| 231 | C8 | #14273717 | 1 | 1 | yes | characteristic |
| 232 | C8 | #14273725 | 1 | 1 | yes | characteristic |
| 233 | C22 | — | 2 | 1 | no | — |
| 234 | C22 | — | 2 | 1 | no | — |
| 235 | C22 | #107×D59 | 1 | 1 | yes | characteristic |
| 236 | C22 | #16837610 | 1 | 1 | yes | characteristic |
| 237 | C22 | #16837696 | 1 | 1 | yes | characteristic |
| 238 | C4 | — | 59 | 2 | no | — |
| 239 | C4 | — | 118 | 1 | no | — |
| 240 | C4 | — | 118 | 1 | no | — |
| 241 | C4 | — | 118 | 1 | no | — |
| 242 | C4 | #16837297 | 1 | 1 | yes | characteristic |
| 243 | C4 | #16837443 | 1 | 1 | yes | characteristic |
| 244 | C4 | #16837595 | 1 | 1 | yes | characteristic |
| 245 | C4 | #16837681 | 1 | 1 | yes | characteristic |
| 246 | C2 | — | 2 | 1 | no | — |
| 247 | C2 | — | 2 | 1 | no | — |
| 248 | C2 | #21424851 | 1 | 1 | yes | characteristic |
| 249 | C2 | #368×D59 | 1 | 1 | yes | characteristic |
| 250 | C2 | #21425889 | 1 | 1 | yes | characteristic |
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