← #30772025
Subgroups — #30772025
Order 17152.
358 conjugacy classes of subgroups.
The subgroup lattice has 254 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16863812 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16863226 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #22560260 | C2 | 1 | 2 | yes | maximal |
| 4 | #22558325 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C67×#2158 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#14524049 | C22 | 1 | 1 | yes | characteristic |
| 7 | #16863812 | — | 2 | 1 | no | — |
| 8 | #16863812 | — | 2 | 1 | no | — |
| 9 | #16863812 | C4 | 1 | 2 | yes | complemented |
| 10 | #16863812 | C4 | 1 | 2 | yes | complemented |
| 11 | C2×#14523983 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#14523982 | — | 2 | 1 | no | — |
| 13 | C2×C16×D67 | — | 2 | 1 | no | — |
| 14 | C2×#14523979 | C22 | 1 | 2 | yes | — |
| 15 | #16863414 | — | 2 | 2 | no | — |
| 16 | #16863226 | — | 2 | 1 | no | — |
| 17 | #16863226 | — | 2 | 1 | no | — |
| 18 | #16863166 | — | 2 | 2 | no | — |
| 19 | C22×C16×C67 | C22 | 1 | 1 | yes | characteristic |
| 20 | #427×C67 | C22 | 1 | 2 | yes | — |
| 21 | #367×C67 | — | 2 | 1 | no | — |
| 22 | #366×C67 | — | 2 | 1 | no | — |
| 23 | C2×#11761628 | D4 | 1 | 1 | yes | characteristic |
| 24 | #14524049 | — | 2 | 1 | no | — |
| 25 | #14524049 | — | 2 | 1 | no | — |
| 26 | #14524049 | C2×C4 | 1 | 2 | yes | — |
| 27 | #14524049 | Q8 | 1 | 2 | yes | — |
| 28 | C2×#11761612 | — | 2 | 2 | no | — |
| 29 | C2×#11761612 | D4 | 1 | 1 | yes | characteristic |
| 30 | C2×#11761611 | — | 2 | 1 | no | — |
| 31 | C2×C8×D67 | — | 2 | 1 | no | — |
| 32 | #14523983 | — | 2 | 1 | no | — |
| 33 | #14523983 | C2×C4 | 1 | 2 | yes | — |
| 34 | #14523982 | — | 2 | 2 | no | — |
| 35 | #14523982 | — | 2 | 2 | no | — |
| 36 | C2×C67⋊C16 | — | 2 | 1 | no | — |
| 37 | C16×D67 | — | 2 | 2 | no | — |
| 38 | C16×D67 | — | 2 | 2 | no | — |
| 39 | #14523979 | — | 2 | 2 | no | — |
| 40 | #14523979 | — | 2 | 2 | no | — |
| 41 | #14523971 | — | 4 | 4 | no | — |
| 42 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 43 | #117×C67 | — | 2 | 2 | no | — |
| 44 | C2×C16×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | C2×C16×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | C2×C16×C67 | D4 | 1 | 1 | yes | characteristic |
| 47 | C2×C16×C67 | D4 | 1 | 1 | yes | characteristic |
| 48 | C2×C16×C67 | D4 | 1 | 1 | yes | characteristic |
| 49 | C2×C16×C67 | D4 | 1 | 1 | yes | characteristic |
| 50 | #107×C67 | — | 2 | 2 | no | — |
| 51 | #106×C67 | — | 2 | 2 | no | — |
| 52 | #11761628 | — | 2 | 1 | no | — |
| 53 | #11761628 | — | 2 | 1 | no | — |
| 54 | #11761628 | C4⋊C4 | 1 | 2 | yes | — |
| 55 | #11761628 | D8 | 1 | 2 | yes | — |
| 56 | C2×Dic134 | Q16 | 1 | 1 | yes | characteristic |
| 57 | C2×D268 | Q16 | 1 | 1 | yes | characteristic |
| 58 | C2×C22⋊D67 | — | 2 | 1 | no | — |
| 59 | C2×C4×D67 | — | 2 | 1 | no | — |
| 60 | #11761612 | — | 2 | 1 | no | — |
| 61 | #11761612 | — | 4 | 2 | no | — |
| 62 | #11761612 | — | 4 | 2 | no | — |
| 63 | #11761612 | C4⋊C4 | 1 | 2 | yes | — |
| 64 | #11761611 | — | 2 | 2 | no | — |
| 65 | #11761611 | — | 2 | 2 | no | — |
| 66 | C2×C67⋊C8 | — | 2 | 1 | no | — |
| 67 | C2×C67⋊C8 | — | 4 | 2 | no | — |
| 68 | C8×D67 | — | 2 | 2 | no | — |
| 69 | C8×D67 | — | 2 | 2 | no | — |
| 70 | C67⋊C16 | — | 2 | 2 | no | — |
| 71 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C2×C8×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 73 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 74 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C2×C8×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 76 | C2×C8×C67 | #41 | 1 | 1 | yes | characteristic |
| 77 | C2×C8×C67 | #41 | 1 | 1 | yes | characteristic |
| 78 | C4⋊C4×C67 | — | 4 | 2 | no | — |
| 79 | C22⋊C4×C67 | — | 4 | 2 | no | — |
| 80 | C42×C67 | — | 4 | 2 | no | — |
| 81 | C16×C67 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 82 | C16×C67 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 83 | C16×C67 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 84 | C16×C67 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 85 | C23×C67 | #135 | 1 | 1 | yes | characteristic |
| 86 | C22×D67 | — | 2 | 1 | no | — |
| 87 | C2×C4×C67 | — | 8 | 2 | no | — |
| 88 | C2×C4×C67 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C67 | #121 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C67 | #121 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×C67 | C2×Q16 | 1 | 1 | yes | characteristic |
| 93 | C2×C4×C67 | #135 | 1 | 1 | yes | characteristic |
| 94 | C22⋊D67 | — | 2 | 2 | no | — |
| 95 | C22⋊D67 | — | 2 | 2 | no | — |
| 96 | C2×Dic67 | — | 2 | 1 | no | — |
| 97 | D268 | — | 2 | 1 | no | — |
| 98 | D268 | #107 | 1 | 2 | yes | — |
| 99 | C4×D67 | — | 2 | 2 | no | — |
| 100 | C4×D67 | — | 2 | 2 | no | — |
| 101 | Dic134 | — | 2 | 1 | no | — |
| 102 | Dic134 | #107 | 1 | 2 | yes | — |
| 103 | C8×C67 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 104 | C8×C67 | #117 | 1 | 1 | yes | characteristic |
| 105 | C8×C67 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 106 | C8×C67 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 107 | C67⋊C8 | — | 2 | 2 | no | — |
| 108 | C67⋊C8 | — | 8 | 2 | no | — |
| 109 | C22×C67 | — | 4 | 1 | no | — |
| 110 | C22×C67 | #427 | 1 | 1 | yes | characteristic |
| 111 | C22×C67 | #466 | 1 | 1 | yes | characteristic |
| 112 | C22×C67 | #467 | 1 | 1 | yes | characteristic |
| 113 | C2×D67 | — | 2 | 2 | no | — |
| 114 | C2×D67 | — | 4 | 2 | no | — |
| 115 | C4×C67 | — | 16 | 2 | no | — |
| 116 | C4×C67 | #380 | 1 | 1 | yes | characteristic |
| 117 | C4×C67 | C2×#107 | 1 | 1 | yes | characteristic |
| 118 | C4×C67 | #466 | 1 | 1 | yes | characteristic |
| 119 | C4×C67 | #467 | 1 | 1 | yes | characteristic |
| 120 | Dic67 | — | 2 | 2 | no | — |
| 121 | #12841 | — | 67 | 1 | no | maximal, Sylow |
| 122 | C2×C67 | — | 2 | 1 | no | — |
| 123 | C2×C67 | — | 2 | 1 | no | — |
| 124 | C2×C67 | — | 2 | 1 | no | — |
| 125 | C2×C67 | #1808 | 1 | 1 | yes | characteristic |
| 126 | D67 | — | 4 | 2 | no | — |
| 127 | C2×#504 | — | 67 | 1 | no | — |
| 128 | C2×#368 | — | 67 | 1 | no | — |
| 129 | #2158 | D67 | 1 | 1 | yes | characteristic, complemented |
| 130 | #1814 | — | 67 | 2 | no | — |
| 131 | #1389 | — | 67 | 2 | no | — |
| 132 | C67 | #12841 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 133 | C2×#131 | — | 67 | 1 | no | — |
| 134 | #504 | — | 67 | 2 | no | — |
| 135 | #504 | — | 67 | 2 | no | — |
| 136 | #504 | — | 134 | 1 | no | — |
| 137 | #504 | — | 134 | 1 | no | — |
| 138 | C2×M32 | — | 67 | 1 | no | — |
| 139 | C2×M32 | — | 134 | 1 | no | — |
| 140 | C22×C16 | — | 134 | 1 | no | — |
| 141 | C22×C16 | C2×D67 | 1 | 1 | yes | characteristic |
| 142 | C2×#108 | — | 67 | 2 | no | — |
| 143 | #427 | C2×D67 | 1 | 2 | yes | — |
| 144 | #421 | — | 134 | 2 | no | — |
| 145 | #368 | — | 134 | 1 | no | — |
| 146 | #368 | — | 134 | 1 | no | — |
| 147 | #367 | — | 2 | 1 | no | — |
| 148 | #366 | — | 2 | 1 | no | — |
| 149 | #344 | — | 134 | 2 | no | — |
| 150 | C2×#41 | — | 67 | 1 | no | — |
| 151 | #131 | — | 67 | 2 | no | — |
| 152 | #131 | — | 67 | 2 | no | — |
| 153 | #131 | — | 134 | 1 | no | — |
| 154 | #131 | — | 134 | 1 | no | — |
| 155 | C2×M16 | — | 67 | 1 | no | — |
| 156 | C2×M16 | — | 134 | 2 | no | — |
| 157 | C2×M16 | — | 134 | 1 | no | — |
| 158 | C22×C8 | — | 134 | 1 | no | — |
| 159 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 160 | #117 | — | 2 | 2 | no | — |
| 161 | M32 | — | 67 | 2 | no | — |
| 162 | M32 | — | 134 | 1 | no | — |
| 163 | M32 | — | 134 | 2 | no | — |
| 164 | M32 | — | 134 | 2 | no | — |
| 165 | C2×C16 | — | 134 | 1 | no | — |
| 166 | C2×C16 | — | 134 | 2 | no | — |
| 167 | C2×C16 | — | 134 | 2 | no | — |
| 168 | C2×C16 | C4×D67 | 1 | 1 | yes | characteristic |
| 169 | C2×C16 | C4×D67 | 1 | 1 | yes | characteristic |
| 170 | C2×C16 | D268 | 1 | 1 | yes | characteristic |
| 171 | C2×C16 | D268 | 1 | 1 | yes | characteristic |
| 172 | C2×C16 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 173 | C2×C16 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 174 | #108 | — | 134 | 2 | no | — |
| 175 | #108 | — | 134 | 2 | no | — |
| 176 | #107 | — | 2 | 2 | no | — |
| 177 | #106 | — | 2 | 2 | no | — |
| 178 | #104 | — | 268 | 4 | no | — |
| 179 | #41 | — | 67 | 2 | no | — |
| 180 | #41 | — | 67 | 2 | no | — |
| 181 | #41 | — | 134 | 1 | no | — |
| 182 | #41 | — | 134 | 1 | no | — |
| 183 | C2×Q8 | — | 67 | 1 | no | — |
| 184 | C2×D4 | — | 67 | 1 | no | — |
| 185 | C2×D4 | — | 134 | 1 | no | — |
| 186 | C22×C4 | — | 134 | 1 | no | — |
| 187 | C22×C4 | D4×D67 | 1 | 1 | yes | characteristic |
| 188 | M16 | — | 268 | 2 | no | — |
| 189 | M16 | — | 268 | 2 | no | — |
| 190 | M16 | — | 67 | 2 | no | — |
| 191 | M16 | — | 134 | 1 | no | — |
| 192 | M16 | — | 134 | 2 | no | — |
| 193 | M16 | — | 134 | 2 | no | — |
| 194 | C2×C8 | — | 268 | 2 | no | — |
| 195 | C2×C8 | — | 134 | 1 | no | — |
| 196 | C2×C8 | — | 134 | 2 | no | — |
| 197 | C2×C8 | — | 134 | 2 | no | — |
| 198 | C2×C8 | C2×C4×D67 | 1 | 1 | yes | characteristic |
| 199 | C2×C8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 200 | C2×C8 | C2×D268 | 1 | 1 | yes | characteristic |
| 201 | C2×C8 | Q8×D67 | 1 | 1 | yes | characteristic |
| 202 | C2×C8 | #11761627 | 1 | 1 | yes | characteristic |
| 203 | C2×C8 | #11761629 | 1 | 1 | yes | characteristic |
| 204 | C4⋊C4 | — | 4 | 2 | no | — |
| 205 | C22⋊C4 | — | 4 | 2 | no | — |
| 206 | C42 | — | 4 | 2 | no | — |
| 207 | C16 | — | 134 | 2 | no | — |
| 208 | C16 | (C2×D67)⋊C4 | 1 | 1 | yes | characteristic |
| 209 | C16 | (C2×D67)⋊C4 | 1 | 1 | yes | characteristic |
| 210 | C16 | (C2×D67)⋊C4 | 1 | 1 | yes | characteristic |
| 211 | C16 | (C2×D67)⋊C4 | 1 | 1 | yes | characteristic |
| 212 | C23 | — | 134 | 1 | no | — |
| 213 | C23 | #14524062 | 1 | 1 | yes | characteristic |
| 214 | Q8 | — | 67 | 2 | no | — |
| 215 | Q8 | — | 134 | 1 | no | — |
| 216 | D4 | — | 67 | 2 | no | — |
| 217 | D4 | — | 134 | 2 | no | — |
| 218 | D4 | — | 134 | 2 | no | — |
| 219 | D4 | — | 134 | 1 | no | — |
| 220 | C2×C4 | — | 8 | 2 | no | — |
| 221 | C2×C4 | — | 134 | 1 | no | — |
| 222 | C2×C4 | — | 134 | 2 | no | — |
| 223 | C2×C4 | — | 134 | 2 | no | — |
| 224 | C2×C4 | C4⋊C4×D67 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | #14524014 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | #14524018 | 1 | 1 | yes | characteristic |
| 227 | C2×C4 | D8×D67 | 1 | 1 | yes | characteristic |
| 228 | C2×C4 | Q16×D67 | 1 | 1 | yes | characteristic |
| 229 | C2×C4 | #14524066 | 1 | 1 | yes | characteristic |
| 230 | C8 | — | 536 | 2 | no | — |
| 231 | C8 | — | 134 | 2 | no | — |
| 232 | C8 | C2×(C2×D67)⋊C4 | 1 | 1 | yes | characteristic |
| 233 | C8 | #14524001 | 1 | 1 | yes | characteristic |
| 234 | C8 | #14524019 | 1 | 1 | yes | characteristic |
| 235 | C8 | #14524023 | 1 | 1 | yes | characteristic |
| 236 | C22 | — | 4 | 1 | no | — |
| 237 | C22 | — | 268 | 2 | no | — |
| 238 | C22 | — | 134 | 2 | no | — |
| 239 | C22 | #16863433 | 1 | 1 | yes | characteristic |
| 240 | C22 | #16863629 | 1 | 1 | yes | characteristic |
| 241 | C22 | #16863638 | 1 | 1 | yes | characteristic |
| 242 | C4 | — | 16 | 2 | no | — |
| 243 | C4 | — | 134 | 2 | no | — |
| 244 | C4 | #16863253 | 1 | 1 | yes | characteristic |
| 245 | C4 | #107×D67 | 1 | 1 | yes | characteristic |
| 246 | C4 | #16863633 | 1 | 1 | yes | characteristic |
| 247 | C4 | #16863634 | 1 | 1 | yes | characteristic |
| 248 | C2 | — | 2 | 1 | no | — |
| 249 | C2 | — | 2 | 1 | no | — |
| 250 | C2 | — | 2 | 1 | no | — |
| 251 | C2 | — | 268 | 2 | no | — |
| 252 | C2 | #22560235 | 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