← #17189020
Subgroups — #17189020
Order 4352.
520 conjugacy classes of subgroups.
The subgroup lattice has 330 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #14539320 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #14537652 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #14530950 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #14530946 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #14530934 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #14530924 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #14530570 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #14530568 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #14530542 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #14530504 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | C2×C17×#358 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #11763440 | — | 2 | 1 | no | — |
| 13 | #11763287 | — | 2 | 1 | no | — |
| 14 | C2×#10587483 | C22 | 1 | 1 | yes | characteristic |
| 15 | #11763217 | — | 2 | 2 | no | — |
| 16 | #11763213 | C22 | 1 | 2 | yes | — |
| 17 | #11763210 | C22 | 1 | 2 | yes | — |
| 18 | #11762916 | — | 2 | 2 | no | — |
| 19 | Q16×Dic17 | C22 | 1 | 2 | yes | — |
| 20 | Q16×Dic17 | C22 | 1 | 2 | yes | — |
| 21 | #11762849 | — | 2 | 1 | no | — |
| 22 | #11762847 | C22 | 1 | 1 | yes | characteristic |
| 23 | #11762847 | C22 | 1 | 1 | yes | characteristic |
| 24 | #11762818 | — | 2 | 1 | no | — |
| 25 | #11762764 | C22 | 1 | 1 | yes | characteristic |
| 26 | #11762637 | C22 | 1 | 1 | yes | characteristic |
| 27 | #11762637 | C22 | 1 | 1 | yes | characteristic |
| 28 | #11762636 | C22 | 1 | 2 | yes | — |
| 29 | #11762635 | C22 | 1 | 1 | yes | characteristic |
| 30 | #11762635 | C22 | 1 | 1 | yes | characteristic |
| 31 | #11762634 | C22 | 1 | 2 | yes | — |
| 32 | #11762627 | C22 | 1 | 2 | yes | — |
| 33 | #11762627 | C22 | 1 | 2 | yes | — |
| 34 | #11762613 | C22 | 1 | 2 | yes | — |
| 35 | #11762613 | C22 | 1 | 2 | yes | — |
| 36 | C22×Q16×C17 | C22 | 1 | 1 | yes | characteristic |
| 37 | C22×C16×C17 | C22 | 1 | 1 | yes | characteristic |
| 38 | C2×C17×#107 | C22 | 1 | 1 | yes | characteristic |
| 39 | C17×#358 | C22 | 1 | 2 | yes | — |
| 40 | C17×#358 | C22 | 1 | 2 | yes | — |
| 41 | C22×Q8×C17 | — | 2 | 1 | no | — |
| 42 | C2×Q16×C17 | — | 2 | 1 | no | — |
| 43 | C2×Q16×C17 | — | 2 | 2 | no | — |
| 44 | C2×Q16×C17 | Q8 | 1 | 4 | yes | complemented |
| 45 | C2×Q16×C17 | C23 | 1 | 2 | yes | — |
| 46 | C22×C8×C17 | C23 | 1 | 1 | yes | characteristic |
| 47 | C2×C4⋊C4×C17 | — | 2 | 1 | no | — |
| 48 | Q8×Dic17 | — | 2 | 2 | no | — |
| 49 | Q8×Dic17 | — | 2 | 2 | no | — |
| 50 | Q8×Dic17 | — | 4 | 1 | no | — |
| 51 | Q8×Dic17 | — | 4 | 1 | no | — |
| 52 | #10587612 | — | 2 | 1 | no | — |
| 53 | #10587612 | — | 2 | 1 | no | — |
| 54 | #10587612 | — | 4 | 2 | no | — |
| 55 | #10587612 | D4 | 1 | 1 | yes | characteristic |
| 56 | #10587610 | — | 4 | 2 | no | — |
| 57 | C2×#63223 | D4 | 1 | 1 | yes | characteristic |
| 58 | #10587576 | — | 2 | 2 | no | — |
| 59 | #10587576 | — | 4 | 1 | no | — |
| 60 | #10587574 | — | 2 | 2 | no | — |
| 61 | #10587574 | — | 4 | 1 | no | — |
| 62 | C2×C16×C17 | — | 2 | 2 | no | — |
| 63 | C2×C16×C17 | C23 | 1 | 1 | yes | characteristic |
| 64 | C2×C16×C17 | C23 | 1 | 1 | yes | characteristic |
| 65 | C17×#107 | — | 2 | 1 | no | — |
| 66 | C17×#107 | C23 | 1 | 2 | yes | — |
| 67 | #10587506 | — | 2 | 2 | no | — |
| 68 | #10587506 | — | 2 | 2 | no | — |
| 69 | #10587489 | C23 | 1 | 1 | yes | characteristic |
| 70 | #10587489 | C23 | 1 | 1 | yes | characteristic |
| 71 | #10587489 | C23 | 1 | 1 | yes | characteristic |
| 72 | #10587489 | C23 | 1 | 1 | yes | characteristic |
| 73 | #10587488 | — | 2 | 1 | no | — |
| 74 | #10587488 | — | 2 | 1 | no | — |
| 75 | #10587486 | D4 | 1 | 2 | yes | — |
| 76 | C8×Dic17 | C23 | 1 | 2 | yes | — |
| 77 | #10587483 | — | 2 | 2 | no | — |
| 78 | C2×#63214 | C23 | 1 | 2 | yes | — |
| 79 | #10587479 | — | 2 | 2 | no | — |
| 80 | #10587478 | — | 2 | 2 | no | — |
| 81 | C2×Q8×C17 | — | 2 | 4 | no | — |
| 82 | C2×Q8×C17 | — | 2 | 2 | no | — |
| 83 | C2×Q8×C17 | — | 4 | 1 | no | — |
| 84 | C2×Q8×C17 | — | 4 | 2 | no | — |
| 85 | C22×C4×C17 | — | 4 | 1 | no | — |
| 86 | C22×C4×C17 | — | 4 | 1 | no | — |
| 87 | C22×C4×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C2×Dic34 | — | 4 | 2 | no | — |
| 89 | Q16×C17 | — | 2 | 4 | no | — |
| 90 | Q16×C17 | — | 2 | 2 | no | — |
| 91 | Q16×C17 | C2×Q8 | 1 | 4 | yes | — |
| 92 | C2×C8×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C2×C8×C17 | C2×Q8 | 1 | 2 | yes | — |
| 94 | C2×C8×C17 | #41 | 1 | 2 | yes | — |
| 95 | C2×C8×C17 | C24 | 1 | 1 | yes | characteristic |
| 96 | C4⋊C4×C17 | — | 2 | 2 | no | — |
| 97 | C4⋊C4×C17 | — | 4 | 1 | no | — |
| 98 | C22⋊Dic17 | — | 4 | 1 | no | — |
| 99 | C22⋊Dic17 | — | 4 | 1 | no | — |
| 100 | C22⋊Dic17 | D8 | 1 | 2 | yes | — |
| 101 | C4⋊Dic17 | — | 2 | 1 | no | — |
| 102 | C4⋊Dic17 | — | 2 | 1 | no | — |
| 103 | C4⋊Dic17 | — | 2 | 1 | no | — |
| 104 | C4⋊Dic17 | — | 2 | 1 | no | — |
| 105 | C4⋊Dic17 | — | 4 | 2 | no | — |
| 106 | C4⋊Dic17 | — | 4 | 2 | no | — |
| 107 | C4⋊Dic17 | — | 4 | 2 | no | — |
| 108 | C4⋊Dic17 | D8 | 1 | 1 | yes | characteristic |
| 109 | C4⋊Dic17 | D8 | 1 | 1 | yes | characteristic |
| 110 | #63225 | — | 4 | 2 | no | — |
| 111 | #63225 | — | 4 | 2 | no | — |
| 112 | C4×Dic17 | — | 4 | 1 | no | — |
| 113 | C4×Dic17 | — | 4 | 1 | no | — |
| 114 | C4×Dic17 | — | 4 | 2 | no | — |
| 115 | C4×Dic17 | — | 4 | 2 | no | — |
| 116 | C4×Dic17 | C2×D4 | 1 | 2 | yes | — |
| 117 | #63223 | — | 2 | 2 | no | — |
| 118 | C2×#3622 | C2×D4 | 1 | 2 | yes | — |
| 119 | C16×C17 | — | 2 | 1 | no | — |
| 120 | C16×C17 | — | 2 | 1 | no | — |
| 121 | #63214 | — | 2 | 2 | no | — |
| 122 | #33632 | — | 17 | 1 | no | maximal, Sylow |
| 123 | C23×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 124 | Q8×C17 | — | 2 | 4 | no | — |
| 125 | Q8×C17 | — | 4 | 2 | no | — |
| 126 | Q8×C17 | — | 4 | 4 | no | — |
| 127 | C2×C4×C17 | — | 4 | 2 | no | — |
| 128 | C2×C4×C17 | — | 4 | 4 | no | — |
| 129 | C2×C4×C17 | — | 4 | 2 | no | — |
| 130 | C2×C4×C17 | — | 16 | 1 | no | — |
| 131 | C2×C4×C17 | C22⋊Q8 | 1 | 2 | yes | — |
| 132 | C2×C4×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 133 | C2×C4×C17 | #137 | 1 | 2 | yes | — |
| 134 | C2×C4×C17 | C22×D4 | 1 | 1 | yes | characteristic |
| 135 | C2×Dic17 | — | 4 | 1 | no | — |
| 136 | C2×Dic17 | — | 4 | 1 | no | — |
| 137 | C2×Dic17 | — | 4 | 1 | no | — |
| 138 | C2×Dic17 | — | 4 | 1 | no | — |
| 139 | C2×Dic17 | C2×D8 | 1 | 2 | yes | — |
| 140 | C2×Dic17 | C2×D8 | 1 | 2 | yes | — |
| 141 | Dic34 | — | 16 | 2 | no | — |
| 142 | C8×C17 | C22⋊Q8 | 1 | 2 | yes | — |
| 143 | C8×C17 | C22×Q8 | 1 | 1 | yes | characteristic |
| 144 | C8×C17 | C2×#41 | 1 | 1 | yes | characteristic |
| 145 | #3622 | — | 2 | 2 | no | — |
| 146 | #3164 | — | 17 | 1 | no | — |
| 147 | #2950 | — | 17 | 1 | no | — |
| 148 | #2229 | — | 17 | 2 | no | — |
| 149 | #2229 | — | 17 | 2 | no | — |
| 150 | #2227 | — | 17 | 2 | no | — |
| 151 | #2225 | — | 17 | 2 | no | — |
| 152 | #2159 | — | 17 | 1 | no | — |
| 153 | #2158 | — | 17 | 1 | no | — |
| 154 | #2151 | — | 17 | 1 | no | — |
| 155 | #2141 | — | 17 | 1 | no | — |
| 156 | C2×#358 | D17 | 1 | 1 | yes | characteristic |
| 157 | C22×C17 | — | 2 | 1 | no | — |
| 158 | C22×C17 | — | 2 | 1 | no | — |
| 159 | C22×C17 | #474 | 1 | 2 | yes | — |
| 160 | C22×C17 | C22×D8 | 1 | 1 | yes | characteristic |
| 161 | C4×C17 | — | 4 | 4 | no | — |
| 162 | C4×C17 | — | 8 | 2 | no | — |
| 163 | C4×C17 | #474 | 1 | 2 | yes | — |
| 164 | C4×C17 | C2×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 165 | C4×C17 | C2×#137 | 1 | 1 | yes | characteristic |
| 166 | Dic17 | — | 2 | 2 | no | — |
| 167 | Dic17 | — | 2 | 2 | no | — |
| 168 | Dic17 | — | 8 | 1 | no | — |
| 169 | Dic17 | — | 8 | 1 | no | — |
| 170 | Dic17 | — | 8 | 1 | no | — |
| 171 | Dic17 | — | 8 | 1 | no | — |
| 172 | C22×Q16 | C2×D17 | 1 | 1 | yes | characteristic |
| 173 | #544 | — | 34 | 1 | no | — |
| 174 | #519 | — | 34 | 1 | no | — |
| 175 | C2×M32 | — | 17 | 1 | no | — |
| 176 | C22×C16 | C2×D17 | 1 | 1 | yes | characteristic |
| 177 | #501 | — | 34 | 2 | no | — |
| 178 | #500 | — | 17 | 2 | no | — |
| 179 | #499 | — | 17 | 2 | no | — |
| 180 | #441 | — | 34 | 2 | no | — |
| 181 | C4×Q16 | — | 17 | 2 | no | — |
| 182 | C4×Q16 | — | 17 | 2 | no | — |
| 183 | #428 | — | 34 | 1 | no | — |
| 184 | #427 | — | 17 | 1 | no | — |
| 185 | #427 | — | 17 | 1 | no | — |
| 186 | C2×#107 | C2×D17 | 1 | 1 | yes | characteristic |
| 187 | #419 | — | 34 | 1 | no | — |
| 188 | #405 | — | 17 | 1 | no | — |
| 189 | #367 | — | 17 | 1 | no | — |
| 190 | #367 | — | 17 | 1 | no | — |
| 191 | #367 | — | 17 | 2 | no | — |
| 192 | #366 | — | 17 | 1 | no | — |
| 193 | #366 | — | 17 | 1 | no | — |
| 194 | #366 | — | 17 | 2 | no | — |
| 195 | C4⋊C16 | — | 17 | 2 | no | — |
| 196 | C4⋊C16 | — | 17 | 2 | no | — |
| 197 | #358 | — | 17 | 2 | no | — |
| 198 | #358 | — | 17 | 2 | no | — |
| 199 | #358 | C2×D17 | 1 | 2 | yes | — |
| 200 | #358 | C2×D17 | 1 | 2 | yes | — |
| 201 | C2×C17 | — | 2 | 2 | no | — |
| 202 | C2×C17 | C2×#474 | 1 | 1 | yes | characteristic |
| 203 | C2×C17 | #3416 | 1 | 1 | yes | characteristic |
| 204 | C2×C17 | #3417 | 1 | 1 | yes | characteristic |
| 205 | C22×Q8 | — | 2 | 1 | no | — |
| 206 | C2×Q16 | — | 2 | 1 | no | — |
| 207 | C2×Q16 | — | 2 | 2 | no | — |
| 208 | C2×Q16 | Dic34 | 1 | 4 | yes | complemented |
| 209 | C2×Q16 | C22×D17 | 1 | 2 | yes | — |
| 210 | C2×M16 | — | 17 | 1 | no | — |
| 211 | C22×C8 | C22×D17 | 1 | 1 | yes | characteristic |
| 212 | C4⋊Q8 | — | 34 | 2 | no | — |
| 213 | C4⋊Q8 | — | 68 | 1 | no | — |
| 214 | #125 | — | 34 | 2 | no | — |
| 215 | #125 | — | 68 | 1 | no | — |
| 216 | C22⋊Q8 | — | 68 | 2 | no | — |
| 217 | C4×Q8 | — | 34 | 2 | no | — |
| 218 | C4×Q8 | — | 34 | 2 | no | — |
| 219 | C4×Q8 | — | 68 | 1 | no | — |
| 220 | C4×Q8 | — | 68 | 1 | no | — |
| 221 | #117 | — | 17 | 1 | no | — |
| 222 | #117 | — | 34 | 1 | no | — |
| 223 | #117 | — | 34 | 1 | no | — |
| 224 | #117 | — | 68 | 2 | no | — |
| 225 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 226 | M32 | — | 34 | 2 | no | — |
| 227 | C2×C16 | — | 2 | 2 | no | — |
| 228 | C2×C16 | — | 17 | 2 | no | — |
| 229 | C2×C16 | C22×D17 | 1 | 1 | yes | characteristic |
| 230 | C2×C16 | C22×D17 | 1 | 1 | yes | characteristic |
| 231 | #107 | — | 2 | 1 | no | — |
| 232 | #107 | — | 17 | 1 | no | — |
| 233 | #107 | — | 17 | 1 | no | — |
| 234 | #107 | — | 17 | 1 | no | — |
| 235 | #107 | — | 17 | 1 | no | — |
| 236 | #107 | — | 34 | 2 | no | — |
| 237 | #107 | C22×D17 | 1 | 2 | yes | — |
| 238 | #106 | — | 34 | 2 | no | — |
| 239 | #106 | — | 34 | 1 | no | — |
| 240 | #106 | — | 34 | 1 | no | — |
| 241 | #103 | — | 34 | 2 | no | — |
| 242 | #103 | — | 34 | 2 | no | — |
| 243 | #97 | — | 17 | 2 | no | — |
| 244 | C4×C8 | — | 17 | 2 | no | — |
| 245 | C17 | #33632 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 246 | C2×Q8 | — | 2 | 2 | no | — |
| 247 | C2×Q8 | — | 2 | 4 | no | — |
| 248 | C2×Q8 | — | 4 | 1 | no | — |
| 249 | C2×Q8 | — | 4 | 2 | no | — |
| 250 | C2×Q8 | — | 68 | 2 | no | — |
| 251 | C22×C4 | — | 4 | 1 | no | — |
| 252 | C22×C4 | — | 4 | 1 | no | — |
| 253 | C22×C4 | D4×D17 | 1 | 1 | yes | characteristic |
| 254 | Q16 | — | 2 | 2 | no | — |
| 255 | Q16 | — | 2 | 4 | no | — |
| 256 | Q16 | C2×Dic34 | 1 | 4 | yes | — |
| 257 | M16 | — | 34 | 2 | no | — |
| 258 | C2×C8 | — | 17 | 2 | no | — |
| 259 | C2×C8 | C2×Dic34 | 1 | 2 | yes | — |
| 260 | C2×C8 | D4×D17 | 1 | 1 | yes | characteristic |
| 261 | C2×C8 | #63254 | 1 | 2 | yes | — |
| 262 | C2×C8 | C23×D17 | 1 | 1 | yes | characteristic |
| 263 | C4⋊C4 | — | 2 | 2 | no | — |
| 264 | C4⋊C4 | — | 4 | 1 | no | — |
| 265 | C4⋊C4 | — | 17 | 1 | no | — |
| 266 | C4⋊C4 | — | 17 | 1 | no | — |
| 267 | C4⋊C4 | — | 34 | 1 | no | — |
| 268 | C4⋊C4 | — | 34 | 1 | no | — |
| 269 | C4⋊C4 | — | 34 | 1 | no | — |
| 270 | C4⋊C4 | — | 34 | 1 | no | — |
| 271 | C4⋊C4 | — | 68 | 2 | no | — |
| 272 | C4⋊C4 | — | 68 | 2 | no | — |
| 273 | C4⋊C4 | — | 68 | 2 | no | — |
| 274 | C4⋊C4 | — | 68 | 2 | no | — |
| 275 | C4⋊C4 | — | 68 | 2 | no | — |
| 276 | C22⋊C4 | — | 17 | 2 | no | — |
| 277 | C22⋊C4 | — | 68 | 1 | no | — |
| 278 | C22⋊C4 | — | 68 | 1 | no | — |
| 279 | C42 | — | 17 | 2 | no | — |
| 280 | C42 | — | 68 | 2 | no | — |
| 281 | C42 | — | 68 | 2 | no | — |
| 282 | C42 | — | 68 | 1 | no | — |
| 283 | C42 | — | 68 | 1 | no | — |
| 284 | C16 | — | 2 | 1 | no | — |
| 285 | C16 | — | 2 | 1 | no | — |
| 286 | C16 | — | 34 | 2 | no | — |
| 287 | C23 | D8×D17 | 1 | 1 | yes | characteristic |
| 288 | Q8 | — | 2 | 4 | no | — |
| 289 | Q8 | — | 4 | 4 | no | — |
| 290 | Q8 | — | 4 | 2 | no | — |
| 291 | Q8 | — | 272 | 2 | no | — |
| 292 | C2×C4 | — | 4 | 2 | no | — |
| 293 | C2×C4 | — | 4 | 4 | no | — |
| 294 | C2×C4 | — | 4 | 2 | no | — |
| 295 | C2×C4 | — | 16 | 1 | no | — |
| 296 | C2×C4 | — | 17 | 2 | no | — |
| 297 | C2×C4 | — | 17 | 2 | no | — |
| 298 | C2×C4 | — | 68 | 1 | no | — |
| 299 | C2×C4 | — | 68 | 1 | no | — |
| 300 | C2×C4 | — | 68 | 1 | no | — |
| 301 | C2×C4 | — | 68 | 1 | no | — |
| 302 | C2×C4 | #10587564 | 1 | 2 | yes | — |
| 303 | C2×C4 | D8×D17 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | #10587601 | 1 | 2 | yes | — |
| 305 | C2×C4 | C2×D4×D17 | 1 | 1 | yes | characteristic |
| 306 | C8 | — | 34 | 2 | no | — |
| 307 | C8 | #10587564 | 1 | 2 | yes | — |
| 308 | C8 | C22×Dic34 | 1 | 1 | yes | characteristic |
| 309 | C8 | C2×#63254 | 1 | 1 | yes | characteristic |
| 310 | C22 | — | 2 | 1 | no | — |
| 311 | C22 | — | 2 | 1 | no | — |
| 312 | C22 | #11763087 | 1 | 2 | yes | — |
| 313 | C22 | C2×D8×D17 | 1 | 1 | yes | characteristic |
| 314 | C4 | — | 4 | 4 | no | — |
| 315 | C4 | — | 8 | 2 | no | — |
| 316 | C4 | — | 34 | 2 | no | — |
| 317 | C4 | — | 34 | 2 | no | — |
| 318 | C4 | — | 136 | 1 | no | — |
| 319 | C4 | — | 136 | 1 | no | — |
| 320 | C4 | — | 136 | 1 | no | — |
| 321 | C4 | — | 136 | 1 | no | — |
| 322 | C4 | #11763087 | 1 | 2 | yes | — |
| 323 | C4 | C2×#10587564 | 1 | 1 | yes | characteristic |
| 324 | C4 | C2×#10587601 | 1 | 1 | yes | characteristic |
| 325 | C2 | — | 2 | 2 | no | — |
| 326 | C2 | C2×#11763087 | 1 | 1 | yes | characteristic |
| 327 | C2 | #14541926 | 1 | 1 | yes | characteristic |
| 328 | C2 | #14541933 | 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