← #31896153
Subgroups — #31896153
Order 18176.
298 conjugacy classes of subgroups.
The subgroup lattice has 260 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22589115 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22587098 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#17985751 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | M16×C71⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #22586237 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #22586235 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C71×#1950 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #17986052 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#14556063 | — | 2 | 1 | no | — |
| 10 | #17986023 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C4⋊(C71⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C4×C71⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 13 | #17985763 | — | 2 | 1 | no | — |
| 14 | #17985751 | — | 2 | 1 | no | — |
| 15 | #17985751 | — | 2 | 1 | no | — |
| 16 | #17985751 | C4 | 1 | 2 | yes | — |
| 17 | #17985751 | C4 | 1 | 2 | yes | — |
| 18 | #17985720 | — | 2 | 1 | no | — |
| 19 | C8×C71⋊C8 | — | 2 | 1 | no | — |
| 20 | C2×#108×C71 | — | 2 | 1 | no | — |
| 21 | #423×C71 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C4⋊C8×C71 | C22 | 1 | 1 | yes | characteristic |
| 23 | C4×M16×C71 | C22 | 1 | 1 | yes | characteristic |
| 24 | #341×C71 | — | 2 | 1 | no | — |
| 25 | C2×#11765849 | — | 2 | 1 | no | — |
| 26 | C22×C71⋊C8 | — | 2 | 1 | no | — |
| 27 | C22×C71⋊C8 | Q8 | 1 | 1 | yes | characteristic |
| 28 | C22×C71⋊C8 | Q8 | 1 | 1 | yes | characteristic |
| 29 | #14556063 | — | 4 | 2 | no | — |
| 30 | C4⋊(C71⋊C8) | — | 2 | 1 | no | — |
| 31 | C4⋊(C71⋊C8) | — | 2 | 1 | no | — |
| 32 | C4⋊(C71⋊C8) | — | 2 | 1 | no | — |
| 33 | C4⋊(C71⋊C8) | — | 2 | 1 | no | — |
| 34 | C4⋊(C71⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 35 | C22⋊(C71⋊C8) | Q8 | 1 | 2 | yes | — |
| 36 | C4×C71⋊C8 | — | 2 | 1 | no | — |
| 37 | C4×C71⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C4×C71⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C4×C71⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 40 | C4×C71⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 41 | C2×M16×C71 | — | 2 | 1 | no | — |
| 42 | C2×M16×C71 | D4 | 1 | 1 | yes | characteristic |
| 43 | C2×M16×C71 | D4 | 1 | 1 | yes | characteristic |
| 44 | C22×C8×C71 | — | 2 | 1 | no | — |
| 45 | C2×C42×C71 | C23 | 1 | 1 | yes | characteristic |
| 46 | #108×C71 | — | 4 | 2 | no | — |
| 47 | C4⋊C8×C71 | — | 2 | 1 | no | — |
| 48 | C4⋊C8×C71 | — | 2 | 1 | no | — |
| 49 | C4⋊C8×C71 | — | 2 | 1 | no | — |
| 50 | C4⋊C8×C71 | C2×C4 | 1 | 2 | yes | — |
| 51 | #97×C71 | — | 2 | 1 | no | — |
| 52 | C4×C8×C71 | — | 2 | 1 | no | — |
| 53 | #11765849 | — | 4 | 1 | no | — |
| 54 | #11765849 | — | 4 | 1 | no | — |
| 55 | C2×C71⋊C8 | — | 2 | 2 | no | — |
| 56 | C2×C71⋊C8 | — | 2 | 2 | no | — |
| 57 | C2×C71⋊C8 | — | 2 | 2 | no | — |
| 58 | C2×C71⋊C8 | — | 4 | 1 | no | — |
| 59 | C2×C71⋊C8 | — | 4 | 1 | no | — |
| 60 | C2×C71⋊C8 | — | 4 | 1 | no | — |
| 61 | C2×C71⋊C8 | — | 4 | 1 | no | — |
| 62 | C2×C71⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 63 | C2×C71⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 64 | C2×C71⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 65 | C2×C71⋊C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 66 | C22×C4×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | C22×C4×C71 | C2×Q8 | 1 | 1 | yes | characteristic |
| 68 | C22×C4×C71 | C2×Q8 | 1 | 1 | yes | characteristic |
| 69 | M16×C71 | — | 2 | 1 | no | — |
| 70 | M16×C71 | — | 2 | 1 | no | — |
| 71 | M16×C71 | — | 4 | 1 | no | — |
| 72 | M16×C71 | — | 4 | 1 | no | — |
| 73 | M16×C71 | D8 | 1 | 2 | yes | — |
| 74 | M16×C71 | Q16 | 1 | 2 | yes | — |
| 75 | C2×C8×C71 | — | 2 | 1 | no | — |
| 76 | C2×C8×C71 | — | 2 | 1 | no | — |
| 77 | C2×C8×C71 | — | 2 | 2 | no | — |
| 78 | C2×C8×C71 | — | 4 | 1 | no | — |
| 79 | C2×C8×C71 | — | 4 | 1 | no | — |
| 80 | C2×C8×C71 | — | 4 | 1 | no | — |
| 81 | C42×C71 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C42×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C42×C71 | #41 | 1 | 1 | yes | characteristic |
| 84 | C42×C71 | #41 | 1 | 1 | yes | characteristic |
| 85 | C23×C71 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 86 | C2×C4×C71 | — | 2 | 1 | no | — |
| 87 | C2×C4×C71 | — | 2 | 1 | no | — |
| 88 | C2×C4×C71 | — | 4 | 1 | no | — |
| 89 | C2×C4×C71 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C71 | C4×D4 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C71 | C4×Q8 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×C71 | #121 | 1 | 1 | yes | characteristic |
| 93 | C2×C4×C71 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 94 | C2×C4×C71 | #125 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C71 | C2×D8 | 1 | 1 | yes | characteristic |
| 96 | C2×C4×C71 | C2×Q16 | 1 | 1 | yes | characteristic |
| 97 | C2×C4×C71 | #135 | 1 | 1 | yes | characteristic |
| 98 | C2×C4×C71 | #135 | 1 | 1 | yes | characteristic |
| 99 | C8×C71 | — | 2 | 2 | no | — |
| 100 | C8×C71 | — | 2 | 2 | no | — |
| 101 | C8×C71 | — | 4 | 2 | no | — |
| 102 | C8×C71 | — | 8 | 1 | no | — |
| 103 | C71⋊C8 | — | 2 | 1 | no | — |
| 104 | C71⋊C8 | — | 2 | 1 | no | — |
| 105 | C71⋊C8 | — | 2 | 1 | no | — |
| 106 | C71⋊C8 | — | 4 | 2 | no | — |
| 107 | C71⋊C8 | — | 4 | 2 | no | — |
| 108 | C71⋊C8 | — | 8 | 1 | no | — |
| 109 | C71⋊C8 | #108 | 1 | 2 | yes | complemented |
| 110 | C22×C71 | — | 2 | 1 | no | — |
| 111 | C22×C71 | — | 2 | 1 | no | — |
| 112 | C22×C71 | #389 | 1 | 1 | yes | characteristic |
| 113 | C22×C71 | #499 | 1 | 1 | yes | characteristic |
| 114 | C22×C71 | #500 | 1 | 1 | yes | characteristic |
| 115 | C4×C71 | — | 2 | 1 | no | — |
| 116 | C4×C71 | — | 2 | 1 | no | — |
| 117 | C4×C71 | — | 2 | 1 | no | — |
| 118 | C4×C71 | C2×#108 | 1 | 1 | yes | characteristic |
| 119 | C4×C71 | #430 | 1 | 1 | yes | characteristic |
| 120 | C4×C71 | C4×D8 | 1 | 1 | yes | characteristic |
| 121 | C4×C71 | C4×Q16 | 1 | 1 | yes | characteristic |
| 122 | C4×C71 | #466 | 1 | 1 | yes | characteristic |
| 123 | C4×C71 | #467 | 1 | 1 | yes | characteristic |
| 124 | #9624 | — | 71 | 1 | no | maximal, Sylow |
| 125 | C2×C71 | — | 2 | 1 | no | — |
| 126 | C2×C71 | — | 2 | 1 | no | — |
| 127 | C2×C71 | #1942 | 1 | 1 | yes | characteristic |
| 128 | C2×C71 | #1950 | 1 | 1 | yes | characteristic |
| 129 | C2×C71 | #1953 | 1 | 1 | yes | characteristic |
| 130 | #1950 | — | 71 | 1 | no | — |
| 131 | #1950 | D71 | 1 | 1 | yes | characteristic |
| 132 | #1567 | — | 71 | 1 | no | — |
| 133 | C2×#335 | — | 71 | 1 | no | — |
| 134 | C8×M16 | — | 71 | 1 | no | — |
| 135 | #1301 | — | 71 | 1 | no | — |
| 136 | #1301 | — | 71 | 1 | no | — |
| 137 | C71 | #9624 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 138 | #431 | — | 71 | 1 | no | — |
| 139 | C2×#108 | — | 2 | 1 | no | — |
| 140 | C2×#108 | — | 142 | 1 | no | — |
| 141 | #423 | — | 71 | 1 | no | — |
| 142 | #423 | C2×D71 | 1 | 1 | yes | characteristic |
| 143 | C2×C4⋊C8 | — | 71 | 1 | no | — |
| 144 | C2×C4⋊C8 | C2×D71 | 1 | 1 | yes | characteristic |
| 145 | C4×M16 | C2×D71 | 1 | 1 | yes | characteristic |
| 146 | C2×C4×C8 | — | 71 | 1 | no | — |
| 147 | #341 | — | 2 | 1 | no | — |
| 148 | #341 | — | 142 | 1 | no | — |
| 149 | #335 | — | 71 | 2 | no | — |
| 150 | #335 | — | 71 | 2 | no | — |
| 151 | #335 | — | 142 | 1 | no | — |
| 152 | #335 | — | 142 | 1 | no | — |
| 153 | #322 | — | 142 | 1 | no | — |
| 154 | C82 | — | 142 | 1 | no | — |
| 155 | C2×M16 | — | 2 | 1 | no | — |
| 156 | C2×M16 | — | 142 | 1 | no | — |
| 157 | C2×M16 | C22⋊D71 | 1 | 1 | yes | characteristic |
| 158 | C2×M16 | C22⋊D71 | 1 | 1 | yes | characteristic |
| 159 | C22×C8 | — | 2 | 1 | no | — |
| 160 | C22×C8 | — | 71 | 1 | no | — |
| 161 | C22×C8 | — | 71 | 1 | no | — |
| 162 | C22×C8 | — | 142 | 1 | no | — |
| 163 | C2×C42 | C22×D71 | 1 | 1 | yes | characteristic |
| 164 | #108 | — | 4 | 2 | no | — |
| 165 | #108 | — | 284 | 2 | no | — |
| 166 | C4⋊C8 | — | 2 | 1 | no | — |
| 167 | C4⋊C8 | — | 2 | 1 | no | — |
| 168 | C4⋊C8 | — | 2 | 1 | no | — |
| 169 | C4⋊C8 | — | 71 | 2 | no | — |
| 170 | C4⋊C8 | — | 142 | 1 | no | — |
| 171 | C4⋊C8 | — | 142 | 1 | no | — |
| 172 | C4⋊C8 | — | 142 | 1 | no | — |
| 173 | C4⋊C8 | — | 142 | 1 | no | — |
| 174 | C4⋊C8 | C4×D71 | 1 | 2 | yes | — |
| 175 | C22⋊C8 | — | 71 | 2 | no | — |
| 176 | #97 | — | 2 | 1 | no | — |
| 177 | C4×C8 | — | 2 | 1 | no | — |
| 178 | C4×C8 | — | 71 | 1 | no | — |
| 179 | C4×C8 | — | 71 | 1 | no | — |
| 180 | C4×C8 | — | 71 | 1 | no | — |
| 181 | C4×C8 | — | 71 | 1 | no | — |
| 182 | C4×C8 | — | 142 | 1 | no | — |
| 183 | C22×C4 | C2×C22⋊D71 | 1 | 1 | yes | characteristic |
| 184 | C22×C4 | Q8×D71 | 1 | 1 | yes | characteristic |
| 185 | C22×C4 | Q8×D71 | 1 | 1 | yes | characteristic |
| 186 | M16 | — | 2 | 1 | no | — |
| 187 | M16 | — | 2 | 1 | no | — |
| 188 | M16 | — | 4 | 1 | no | — |
| 189 | M16 | — | 4 | 1 | no | — |
| 190 | M16 | — | 284 | 1 | no | — |
| 191 | M16 | — | 284 | 1 | no | — |
| 192 | M16 | #11765850 | 1 | 2 | yes | — |
| 193 | M16 | #11765855 | 1 | 2 | yes | — |
| 194 | C2×C8 | — | 2 | 1 | no | — |
| 195 | C2×C8 | — | 2 | 1 | no | — |
| 196 | C2×C8 | — | 2 | 2 | no | — |
| 197 | C2×C8 | — | 4 | 1 | no | — |
| 198 | C2×C8 | — | 4 | 1 | no | — |
| 199 | C2×C8 | — | 4 | 1 | no | — |
| 200 | C2×C8 | — | 284 | 1 | no | — |
| 201 | C2×C8 | — | 284 | 1 | no | — |
| 202 | C2×C8 | — | 284 | 1 | no | — |
| 203 | C2×C8 | — | 284 | 1 | no | — |
| 204 | C2×C8 | — | 71 | 1 | no | — |
| 205 | C2×C8 | — | 71 | 1 | no | — |
| 206 | C2×C8 | — | 71 | 1 | no | — |
| 207 | C2×C8 | — | 71 | 1 | no | — |
| 208 | C2×C8 | — | 142 | 2 | no | — |
| 209 | C2×C8 | — | 142 | 2 | no | — |
| 210 | C2×C8 | — | 142 | 2 | no | — |
| 211 | C42 | C2×C4×D71 | 1 | 1 | yes | characteristic |
| 212 | C42 | D4×D71 | 1 | 1 | yes | characteristic |
| 213 | C42 | #11765864 | 1 | 1 | yes | characteristic |
| 214 | C42 | #11765865 | 1 | 1 | yes | characteristic |
| 215 | C23 | #14556126 | 1 | 1 | yes | characteristic |
| 216 | C2×C4 | — | 2 | 1 | no | — |
| 217 | C2×C4 | — | 2 | 1 | no | — |
| 218 | C2×C4 | — | 4 | 1 | no | — |
| 219 | C2×C4 | C4⋊C4×D71 | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | C4×C22⋊D71 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #14556093 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #14556100 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | #14556106 | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | #14556118 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | C2×#11765850 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | C2×#11765855 | 1 | 1 | yes | characteristic |
| 227 | C2×C4 | #14556148 | 1 | 1 | yes | characteristic |
| 228 | C2×C4 | #14556148 | 1 | 1 | yes | characteristic |
| 229 | C8 | — | 2 | 2 | no | — |
| 230 | C8 | — | 2 | 2 | no | — |
| 231 | C8 | — | 4 | 2 | no | — |
| 232 | C8 | — | 8 | 1 | no | — |
| 233 | C8 | — | 284 | 2 | no | — |
| 234 | C8 | — | 284 | 2 | no | — |
| 235 | C8 | — | 568 | 1 | no | — |
| 236 | C8 | — | 71 | 2 | no | — |
| 237 | C8 | — | 142 | 1 | no | — |
| 238 | C8 | — | 142 | 1 | no | — |
| 239 | C8 | — | 142 | 1 | no | — |
| 240 | C22 | — | 2 | 1 | no | — |
| 241 | C22 | — | 2 | 1 | no | — |
| 242 | C22 | #17985894 | 1 | 1 | yes | characteristic |
| 243 | C22 | #17986398 | 1 | 1 | yes | characteristic |
| 244 | C22 | #17986402 | 1 | 1 | yes | characteristic |
| 245 | C4 | — | 2 | 1 | no | — |
| 246 | C4 | — | 2 | 1 | no | — |
| 247 | C4 | — | 2 | 1 | no | — |
| 248 | C4 | #108×D71 | 1 | 1 | yes | characteristic |
| 249 | C4 | #17986046 | 1 | 1 | yes | characteristic |
| 250 | C4 | C4×#11765850 | 1 | 1 | yes | characteristic |
| 251 | C4 | C4×#11765855 | 1 | 1 | yes | characteristic |
| 252 | C4 | #17986233 | 1 | 1 | yes | characteristic |
| 253 | C4 | #17986238 | 1 | 1 | yes | characteristic |
| 254 | C2 | — | 2 | 1 | no | — |
| 255 | C2 | — | 2 | 1 | no | — |
| 256 | C2 | #22589070 | 1 | 1 | yes | characteristic |
| 257 | C2 | #22589112 | 1 | 1 | yes | characteristic |
| 258 | C2 | #22589127 | 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