← #27302149
Subgroups — #27302149
Order 13568.
274 conjugacy classes of subgroups.
The subgroup lattice has 240 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16810452 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16810448 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20276312 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20275500 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #20272825 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #20272823 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C53×#1952 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#13133157 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16810666 | — | 2 | 1 | no | — |
| 12 | C2×#13133151 | C22 | 1 | 1 | yes | characteristic |
| 13 | #16810452 | C4 | 1 | 4 | yes | complemented |
| 14 | #16810448 | — | 2 | 1 | no | — |
| 15 | #16810448 | — | 2 | 1 | no | — |
| 16 | #16810399 | — | 2 | 1 | no | — |
| 17 | C4×#11719908 | — | 2 | 1 | no | — |
| 18 | #16810391 | — | 2 | 1 | no | — |
| 19 | C2×#108×C53 | C22 | 1 | 1 | yes | characteristic |
| 20 | C53×#428 | — | 2 | 1 | no | — |
| 21 | C2×#107×C53 | C22 | 1 | 1 | yes | characteristic |
| 22 | C53×#405 | C22 | 1 | 1 | yes | characteristic |
| 23 | C53×#343 | — | 2 | 1 | no | — |
| 24 | C2×#11719917 | — | 2 | 1 | no | — |
| 25 | C2×C4⋊Dic53 | — | 2 | 1 | no | — |
| 26 | #13133157 | — | 2 | 1 | no | — |
| 27 | #13133157 | Q8 | 1 | 2 | yes | — |
| 28 | C2×#11719908 | — | 2 | 1 | no | — |
| 29 | C2×#11719908 | C2×C4 | 1 | 2 | yes | — |
| 30 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×#11719908 | Q8 | 1 | 2 | yes | — |
| 33 | #13133153 | — | 2 | 1 | no | — |
| 34 | #13133153 | C2×C4 | 1 | 2 | yes | — |
| 35 | #13133151 | — | 2 | 1 | no | — |
| 36 | #13133151 | — | 2 | 1 | no | — |
| 37 | #13133150 | — | 4 | 1 | no | — |
| 38 | #13133148 | — | 4 | 1 | no | — |
| 39 | C2×M16×C53 | — | 2 | 1 | no | — |
| 40 | C2×M16×C53 | D4 | 1 | 1 | yes | characteristic |
| 41 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 42 | #117×C53 | D4 | 1 | 1 | yes | characteristic |
| 43 | C2×C4⋊C4×C53 | — | 2 | 1 | no | — |
| 44 | #108×C53 | — | 2 | 1 | no | — |
| 45 | #108×C53 | C2×C4 | 1 | 2 | yes | — |
| 46 | #107×C53 | — | 2 | 1 | no | — |
| 47 | #107×C53 | — | 2 | 1 | no | — |
| 48 | #107×C53 | — | 4 | 1 | no | — |
| 49 | #106×C53 | — | 4 | 1 | no | — |
| 50 | #97×C53 | — | 2 | 1 | no | — |
| 51 | C4×C8×C53 | — | 2 | 1 | no | — |
| 52 | C22×C4×C53 | — | 4 | 1 | no | — |
| 53 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 54 | C22×Dic53 | — | 4 | 1 | no | — |
| 55 | M16×C53 | — | 2 | 1 | no | — |
| 56 | M16×C53 | — | 2 | 2 | no | — |
| 57 | M16×C53 | — | 4 | 1 | no | — |
| 58 | M16×C53 | Q16 | 1 | 2 | yes | — |
| 59 | C2×C8×C53 | — | 2 | 1 | no | — |
| 60 | C2×C8×C53 | — | 4 | 1 | no | — |
| 61 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 62 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 64 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 65 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 66 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 67 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 68 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 69 | C4⋊C4×C53 | D8 | 1 | 1 | yes | characteristic |
| 70 | C4⋊C4×C53 | D8 | 1 | 1 | yes | characteristic |
| 71 | C22⋊C4×C53 | — | 2 | 1 | no | — |
| 72 | C42×C53 | — | 2 | 1 | no | — |
| 73 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 74 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 75 | #11719917 | — | 2 | 2 | no | — |
| 76 | #11719917 | — | 4 | 1 | no | — |
| 77 | C2×#88322 | — | 4 | 1 | no | — |
| 78 | #11719908 | — | 2 | 2 | no | — |
| 79 | #11719908 | C4⋊C4 | 1 | 2 | yes | — |
| 80 | #11719908 | C4⋊C4 | 1 | 2 | yes | — |
| 81 | C23×C53 | #135 | 1 | 1 | yes | characteristic |
| 82 | C2×C4×C53 | — | 2 | 1 | no | — |
| 83 | C2×C4×C53 | — | 2 | 1 | no | — |
| 84 | C2×C4×C53 | — | 8 | 1 | no | — |
| 85 | C2×C4×C53 | — | 16 | 1 | no | — |
| 86 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C53 | #121 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C53 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C53 | C2×Q16 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C53 | #135 | 1 | 1 | yes | characteristic |
| 92 | C2×Dic53 | — | 8 | 1 | no | — |
| 93 | C2×Dic53 | — | 16 | 1 | no | — |
| 94 | C8×C53 | — | 2 | 2 | no | — |
| 95 | C8×C53 | — | 4 | 2 | no | — |
| 96 | C8×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 97 | C8×C53 | C4×Q8 | 1 | 1 | yes | characteristic |
| 98 | C8×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 99 | C8×C53 | #125 | 1 | 1 | yes | characteristic |
| 100 | #88322 | — | 4 | 2 | no | — |
| 101 | #13171 | — | 53 | 1 | no | maximal, Sylow |
| 102 | C22×C53 | — | 2 | 1 | no | — |
| 103 | C22×C53 | — | 2 | 1 | no | — |
| 104 | C22×C53 | C4×Q16 | 1 | 1 | yes | characteristic |
| 105 | C22×C53 | #466 | 1 | 1 | yes | characteristic |
| 106 | C22×C53 | #499 | 1 | 1 | yes | characteristic |
| 107 | C4×C53 | — | 4 | 1 | no | — |
| 108 | C4×C53 | — | 4 | 1 | no | — |
| 109 | C4×C53 | — | 16 | 1 | no | — |
| 110 | C4×C53 | #389 | 1 | 1 | yes | characteristic |
| 111 | C4×C53 | C4×D8 | 1 | 1 | yes | characteristic |
| 112 | C4×C53 | #467 | 1 | 1 | yes | characteristic |
| 113 | C4×C53 | #500 | 1 | 1 | yes | characteristic |
| 114 | Dic53 | — | 16 | 1 | no | — |
| 115 | C2×#368 | — | 53 | 1 | no | — |
| 116 | C2×#366 | — | 53 | 1 | no | — |
| 117 | #2108 | — | 53 | 1 | no | — |
| 118 | #1952 | — | 53 | 1 | no | — |
| 119 | #1952 | D53 | 1 | 1 | yes | characteristic |
| 120 | #1387 | — | 53 | 1 | no | — |
| 121 | #1387 | — | 53 | 1 | no | — |
| 122 | C2×C53 | — | 2 | 1 | no | — |
| 123 | C2×C53 | — | 2 | 1 | no | — |
| 124 | C2×C53 | #1942 | 1 | 1 | yes | characteristic |
| 125 | C2×C53 | #2221 | 1 | 1 | yes | characteristic |
| 126 | C2×C53 | #2222 | 1 | 1 | yes | characteristic |
| 127 | C2×M32 | — | 53 | 1 | no | — |
| 128 | C22×C16 | — | 53 | 1 | no | — |
| 129 | C2×#108 | — | 53 | 1 | no | — |
| 130 | C2×#108 | C2×D53 | 1 | 1 | yes | characteristic |
| 131 | #428 | — | 2 | 1 | no | — |
| 132 | #428 | — | 106 | 1 | no | — |
| 133 | C2×#107 | — | 53 | 1 | no | — |
| 134 | C2×#107 | C2×D53 | 1 | 1 | yes | characteristic |
| 135 | #405 | C2×D53 | 1 | 1 | yes | characteristic |
| 136 | #368 | — | 53 | 4 | no | — |
| 137 | #366 | — | 106 | 1 | no | — |
| 138 | #366 | — | 106 | 1 | no | — |
| 139 | #346 | — | 106 | 1 | no | — |
| 140 | C4×C16 | — | 106 | 1 | no | — |
| 141 | #343 | — | 2 | 1 | no | — |
| 142 | #343 | — | 106 | 1 | no | — |
| 143 | C53 | #13171 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 144 | C2×M16 | — | 2 | 1 | no | — |
| 145 | C2×M16 | — | 106 | 1 | no | — |
| 146 | C2×M16 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 147 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 148 | #117 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 149 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 150 | C2×C4⋊C4 | — | 106 | 1 | no | — |
| 151 | M32 | — | 53 | 2 | no | — |
| 152 | M32 | — | 106 | 1 | no | — |
| 153 | C2×C16 | — | 53 | 2 | no | — |
| 154 | C2×C16 | — | 53 | 1 | no | — |
| 155 | C2×C16 | — | 53 | 1 | no | — |
| 156 | C2×C16 | — | 53 | 2 | no | — |
| 157 | C2×C16 | — | 106 | 1 | no | — |
| 158 | #108 | — | 2 | 1 | no | — |
| 159 | #108 | — | 53 | 2 | no | — |
| 160 | #108 | — | 106 | 1 | no | — |
| 161 | #108 | C4×D53 | 1 | 2 | yes | — |
| 162 | #107 | — | 2 | 1 | no | — |
| 163 | #107 | — | 2 | 1 | no | — |
| 164 | #107 | — | 4 | 1 | no | — |
| 165 | #107 | — | 106 | 1 | no | — |
| 166 | #107 | — | 106 | 1 | no | — |
| 167 | #107 | — | 212 | 1 | no | — |
| 168 | #106 | — | 4 | 1 | no | — |
| 169 | #106 | — | 212 | 1 | no | — |
| 170 | #97 | — | 2 | 1 | no | — |
| 171 | C4×C8 | — | 2 | 1 | no | — |
| 172 | C22×C4 | — | 4 | 1 | no | — |
| 173 | C22×C4 | — | 212 | 1 | no | — |
| 174 | C22×C4 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 175 | M16 | — | 2 | 1 | no | — |
| 176 | M16 | — | 2 | 2 | no | — |
| 177 | M16 | — | 4 | 1 | no | — |
| 178 | M16 | — | 106 | 2 | no | — |
| 179 | M16 | — | 212 | 1 | no | — |
| 180 | M16 | #11719925 | 1 | 2 | yes | — |
| 181 | C2×C8 | — | 2 | 1 | no | — |
| 182 | C2×C8 | — | 4 | 1 | no | — |
| 183 | C2×C8 | — | 212 | 1 | no | — |
| 184 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 185 | C2×C8 | #11719945 | 1 | 1 | yes | characteristic |
| 186 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 187 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 188 | C2×C8 | Q8×D53 | 1 | 1 | yes | characteristic |
| 189 | C2×C8 | Q8×D53 | 1 | 1 | yes | characteristic |
| 190 | C4⋊C4 | — | 4 | 1 | no | — |
| 191 | C4⋊C4 | — | 4 | 1 | no | — |
| 192 | C4⋊C4 | — | 212 | 1 | no | — |
| 193 | C4⋊C4 | — | 212 | 1 | no | — |
| 194 | C4⋊C4 | #11719922 | 1 | 1 | yes | characteristic |
| 195 | C4⋊C4 | #11719922 | 1 | 1 | yes | characteristic |
| 196 | C22⋊C4 | — | 2 | 1 | no | — |
| 197 | C42 | — | 2 | 1 | no | — |
| 198 | C16 | — | 53 | 2 | no | — |
| 199 | C16 | — | 53 | 2 | no | — |
| 200 | C16 | — | 106 | 2 | no | — |
| 201 | C23 | #13133237 | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | — | 2 | 1 | no | — |
| 203 | C2×C4 | — | 2 | 1 | no | — |
| 204 | C2×C4 | — | 8 | 1 | no | — |
| 205 | C2×C4 | — | 16 | 1 | no | — |
| 206 | C2×C4 | — | 848 | 1 | no | — |
| 207 | C2×C4 | — | 424 | 1 | no | — |
| 208 | C2×C4 | C4×C22⋊D53 | 1 | 1 | yes | characteristic |
| 209 | C2×C4 | #13133189 | 1 | 1 | yes | characteristic |
| 210 | C2×C4 | #13133215 | 1 | 1 | yes | characteristic |
| 211 | C2×C4 | C2×#11719922 | 1 | 1 | yes | characteristic |
| 212 | C2×C4 | C2×#11719925 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | #13133237 | 1 | 1 | yes | characteristic |
| 214 | C8 | — | 2 | 2 | no | — |
| 215 | C8 | — | 4 | 2 | no | — |
| 216 | C8 | — | 212 | 2 | no | — |
| 217 | C8 | C4⋊C4×D53 | 1 | 1 | yes | characteristic |
| 218 | C8 | #13133182 | 1 | 1 | yes | characteristic |
| 219 | C8 | #13133195 | 1 | 1 | yes | characteristic |
| 220 | C8 | #13133207 | 1 | 1 | yes | characteristic |
| 221 | C22 | — | 2 | 1 | no | — |
| 222 | C22 | — | 2 | 1 | no | — |
| 223 | C22 | C4×#11719925 | 1 | 1 | yes | characteristic |
| 224 | C22 | #16810857 | 1 | 1 | yes | characteristic |
| 225 | C22 | #16811022 | 1 | 1 | yes | characteristic |
| 226 | C4 | — | 4 | 1 | no | — |
| 227 | C4 | — | 4 | 1 | no | — |
| 228 | C4 | — | 16 | 1 | no | — |
| 229 | C4 | — | 848 | 1 | no | — |
| 230 | C4 | #16810518 | 1 | 1 | yes | characteristic |
| 231 | C4 | C4×#11719922 | 1 | 1 | yes | characteristic |
| 232 | C4 | #16810862 | 1 | 1 | yes | characteristic |
| 233 | C4 | #16811026 | 1 | 1 | yes | characteristic |
| 234 | C2 | — | 2 | 1 | no | — |
| 235 | C2 | — | 2 | 1 | no | — |
| 236 | C2 | #20275445 | 1 | 1 | yes | characteristic |
| 237 | C2 | #20276833 | 1 | 1 | yes | characteristic |
| 238 | C2 | #20276838 | 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