← #17192070
Subgroups — #17192070
Order 4352.
474 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 | Q8×Dic68 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #14531020 | C2 | 1 | 2 | yes | maximal |
| 3 | #14531011 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #14530931 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #14530926 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #14530843 | C2 | 1 | 2 | yes | maximal |
| 7 | #14530831 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #14530693 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #14530653 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C4×#10587498 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | C17×#3393 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | Q8×Dic34 | — | 2 | 1 | no | — |
| 13 | C2×#10587500 | C22 | 1 | 2 | yes | — |
| 14 | C2×#10587498 | C22 | 1 | 1 | yes | characteristic |
| 15 | #11763214 | C22 | 1 | 2 | yes | — |
| 16 | #11763214 | C22 | 1 | 1 | yes | characteristic |
| 17 | #11763105 | — | 2 | 1 | no | — |
| 18 | #11763105 | — | 2 | 2 | no | — |
| 19 | C4×Dic68 | C22 | 1 | 1 | yes | characteristic |
| 20 | C4×Dic68 | C22 | 1 | 2 | yes | — |
| 21 | #11762636 | C22 | 1 | 1 | yes | characteristic |
| 22 | #11762636 | C22 | 1 | 2 | yes | — |
| 23 | C4⋊#63214 | C22 | 1 | 1 | yes | characteristic |
| 24 | C4⋊#63214 | C22 | 1 | 2 | yes | — |
| 25 | #11762613 | C22 | 1 | 2 | yes | — |
| 26 | #11762613 | C22 | 1 | 2 | yes | — |
| 27 | #11762613 | C22 | 1 | 2 | yes | — |
| 28 | #11762612 | C22 | 1 | 1 | yes | characteristic |
| 29 | #11762610 | C22 | 1 | 2 | yes | — |
| 30 | #11762610 | C22 | 1 | 1 | yes | characteristic |
| 31 | #11762585 | C22 | 1 | 2 | yes | — |
| 32 | C4×#63214 | C22 | 1 | 1 | yes | characteristic |
| 33 | C17×#556 | — | 2 | 1 | no | — |
| 34 | C17×#495 | C22 | 1 | 2 | yes | — |
| 35 | C17×#494 | C22 | 1 | 1 | yes | characteristic |
| 36 | C17×#464 | — | 2 | 2 | no | — |
| 37 | C17×#461 | — | 2 | 1 | no | — |
| 38 | C8×Q8×C17 | C22 | 1 | 1 | yes | characteristic |
| 39 | C4×Q16×C17 | C22 | 1 | 2 | yes | — |
| 40 | C4×D8×C17 | C22 | 1 | 1 | yes | characteristic |
| 41 | C2×Q16×C17 | — | 2 | 1 | no | — |
| 42 | C2×Q16×C17 | C23 | 1 | 2 | yes | — |
| 43 | C2×SD16×C17 | — | 2 | 2 | no | — |
| 44 | C2×D8×C17 | C23 | 1 | 1 | yes | characteristic |
| 45 | C17×C4⋊Q8 | — | 2 | 1 | no | — |
| 46 | C17×#126 | — | 4 | 2 | no | — |
| 47 | C17×#124 | — | 2 | 2 | no | — |
| 48 | C17×C22⋊Q8 | — | 4 | 1 | no | — |
| 49 | C4×Q8×C17 | — | 2 | 2 | no | — |
| 50 | C4×Q8×C17 | D4 | 1 | 1 | yes | characteristic |
| 51 | C4×D4×C17 | — | 2 | 1 | no | — |
| 52 | C17×#117 | — | 4 | 1 | no | — |
| 53 | Q8×Dic17 | — | 4 | 1 | no | — |
| 54 | C2×Dic68 | C23 | 1 | 1 | yes | characteristic |
| 55 | #10587574 | — | 4 | 2 | no | — |
| 56 | #10587574 | — | 4 | 1 | no | — |
| 57 | #10587555 | — | 2 | 2 | no | — |
| 58 | #10587555 | — | 2 | 1 | no | — |
| 59 | C4×Dic34 | — | 2 | 2 | no | — |
| 60 | C4×Dic34 | — | 2 | 1 | no | — |
| 61 | C17×#107 | C23 | 1 | 1 | yes | characteristic |
| 62 | C17×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 63 | C17×C4⋊C8 | D4 | 1 | 2 | yes | — |
| 64 | C17×#103 | — | 2 | 2 | no | — |
| 65 | C17×#102 | — | 2 | 1 | no | — |
| 66 | C4×C8×C17 | C23 | 1 | 1 | yes | characteristic |
| 67 | C4×C8×C17 | C23 | 1 | 2 | yes | — |
| 68 | #10587500 | — | 2 | 4 | no | — |
| 69 | #10587498 | — | 2 | 2 | no | — |
| 70 | #10587489 | — | 2 | 2 | no | — |
| 71 | #10587489 | — | 2 | 1 | no | — |
| 72 | #10587489 | C23 | 1 | 1 | yes | characteristic |
| 73 | #10587489 | C23 | 1 | 2 | yes | — |
| 74 | #10587487 | — | 2 | 2 | no | — |
| 75 | #10587487 | — | 2 | 1 | no | — |
| 76 | C2×#63214 | C23 | 1 | 1 | yes | characteristic |
| 77 | C2×#63214 | C23 | 1 | 2 | yes | — |
| 78 | C2×#63214 | C23 | 1 | 1 | yes | characteristic |
| 79 | C2×Q8×C17 | — | 2 | 2 | no | — |
| 80 | C2×Q8×C17 | D8 | 1 | 1 | yes | characteristic |
| 81 | C2×D4×C17 | — | 2 | 1 | no | — |
| 82 | C22×C4×C17 | — | 4 | 1 | no | — |
| 83 | C2×Dic34 | — | 2 | 1 | no | — |
| 84 | C2×Dic34 | — | 4 | 2 | no | — |
| 85 | C2×Dic34 | — | 4 | 1 | no | — |
| 86 | Q16×C17 | — | 2 | 2 | no | — |
| 87 | Q16×C17 | — | 2 | 2 | no | — |
| 88 | Q16×C17 | — | 8 | 1 | no | — |
| 89 | SD16×C17 | — | 8 | 2 | no | — |
| 90 | D8×C17 | — | 2 | 1 | no | — |
| 91 | D8×C17 | — | 2 | 1 | no | — |
| 92 | C2×C8×C17 | C2×D4 | 1 | 2 | yes | — |
| 93 | C2×C8×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×C8×C17 | C24 | 1 | 1 | yes | characteristic |
| 95 | C4⋊C4×C17 | — | 2 | 1 | no | — |
| 96 | C4⋊C4×C17 | — | 4 | 2 | no | — |
| 97 | C4⋊C4×C17 | — | 4 | 1 | no | — |
| 98 | C4⋊C4×C17 | — | 4 | 1 | no | — |
| 99 | C4⋊C4×C17 | D8 | 1 | 2 | yes | — |
| 100 | C4⋊C4×C17 | D8 | 1 | 1 | yes | characteristic |
| 101 | C22⋊C4×C17 | — | 4 | 2 | no | — |
| 102 | C22⋊C4×C17 | — | 4 | 2 | no | — |
| 103 | C22⋊C4×C17 | — | 4 | 1 | no | — |
| 104 | C42×C17 | — | 4 | 2 | no | — |
| 105 | C42×C17 | C2×D4 | 1 | 2 | yes | — |
| 106 | C42×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 107 | C4⋊Dic17 | — | 2 | 2 | no | — |
| 108 | C4⋊Dic17 | — | 2 | 1 | no | — |
| 109 | C4⋊Dic17 | — | 4 | 2 | no | — |
| 110 | C4⋊Dic17 | — | 4 | 1 | no | — |
| 111 | #63225 | — | 4 | 2 | no | — |
| 112 | #63225 | — | 4 | 1 | no | — |
| 113 | C4×Dic17 | — | 4 | 2 | no | — |
| 114 | C4×Dic17 | — | 4 | 1 | no | — |
| 115 | Dic68 | — | 2 | 1 | no | — |
| 116 | Dic68 | #41 | 1 | 2 | yes | — |
| 117 | #63214 | — | 2 | 1 | no | — |
| 118 | #63214 | — | 2 | 2 | no | — |
| 119 | #63214 | — | 2 | 1 | no | — |
| 120 | #33936 | — | 17 | 1 | no | maximal, Sylow |
| 121 | C23×C17 | — | 4 | 1 | no | — |
| 122 | Q8×C17 | — | 2 | 2 | no | — |
| 123 | Q8×C17 | — | 4 | 2 | no | — |
| 124 | Q8×C17 | — | 4 | 2 | no | — |
| 125 | D4×C17 | — | 4 | 1 | no | — |
| 126 | D4×C17 | — | 4 | 1 | no | — |
| 127 | C2×C4×C17 | — | 4 | 1 | no | — |
| 128 | C2×C4×C17 | — | 4 | 2 | no | — |
| 129 | C2×C4×C17 | — | 8 | 2 | no | — |
| 130 | C2×C4×C17 | C2×D8 | 1 | 2 | yes | — |
| 131 | C2×C4×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C17 | C2×D8 | 1 | 2 | yes | — |
| 133 | C2×C4×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 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 | 2 | no | — |
| 138 | Dic34 | — | 2 | 2 | no | — |
| 139 | Dic34 | — | 4 | 1 | no | — |
| 140 | Dic34 | — | 16 | 2 | no | — |
| 141 | Dic34 | — | 16 | 1 | no | — |
| 142 | C8×C17 | — | 2 | 2 | no | — |
| 143 | C8×C17 | — | 2 | 1 | no | — |
| 144 | C8×C17 | C2×#41 | 1 | 1 | yes | characteristic |
| 145 | C8×C17 | 2−1+4 | 1 | 1 | yes | characteristic |
| 146 | #3393 | D17 | 1 | 1 | yes | characteristic |
| 147 | Q8×Q16 | — | 17 | 1 | no | — |
| 148 | #2243 | — | 17 | 2 | no | — |
| 149 | #2241 | — | 17 | 1 | no | — |
| 150 | #2226 | — | 17 | 1 | no | — |
| 151 | #2225 | — | 17 | 2 | no | — |
| 152 | #2211 | — | 17 | 2 | no | — |
| 153 | #2209 | — | 17 | 1 | no | — |
| 154 | #2183 | — | 17 | 1 | no | — |
| 155 | #2176 | — | 17 | 2 | no | — |
| 156 | C4×SD32 | — | 17 | 1 | no | — |
| 157 | C22×C17 | — | 8 | 1 | no | — |
| 158 | C22×C17 | — | 8 | 1 | no | — |
| 159 | C22×C17 | — | 8 | 1 | no | — |
| 160 | C22×C17 | C22×D8 | 1 | 1 | yes | characteristic |
| 161 | C4×C17 | — | 2 | 2 | no | — |
| 162 | C4×C17 | — | 2 | 2 | no | — |
| 163 | C4×C17 | — | 2 | 1 | no | — |
| 164 | C4×C17 | — | 8 | 2 | no | — |
| 165 | C4×C17 | — | 8 | 1 | no | — |
| 166 | C4×C17 | #510 | 1 | 2 | yes | — |
| 167 | C4×C17 | #548 | 1 | 1 | yes | characteristic |
| 168 | C4×C17 | #577 | 1 | 1 | yes | characteristic |
| 169 | Dic17 | — | 4 | 2 | no | — |
| 170 | Dic17 | — | 8 | 1 | no | — |
| 171 | Dic17 | — | 8 | 2 | no | — |
| 172 | Q82 | — | 34 | 1 | no | — |
| 173 | #556 | — | 2 | 1 | no | — |
| 174 | C2×Q32 | — | 17 | 2 | no | — |
| 175 | C2×SD32 | — | 17 | 1 | no | — |
| 176 | #500 | — | 17 | 2 | no | — |
| 177 | #500 | — | 17 | 1 | no | — |
| 178 | #495 | C2×D17 | 1 | 2 | yes | — |
| 179 | #494 | C2×D17 | 1 | 1 | yes | characteristic |
| 180 | #477 | — | 34 | 1 | no | — |
| 181 | #477 | — | 34 | 2 | no | — |
| 182 | #464 | — | 2 | 2 | no | — |
| 183 | #461 | — | 2 | 1 | no | — |
| 184 | C8×Q8 | C2×D17 | 1 | 1 | yes | characteristic |
| 185 | C4×Q16 | — | 17 | 1 | no | — |
| 186 | C4×Q16 | — | 17 | 2 | no | — |
| 187 | C4×Q16 | C2×D17 | 1 | 2 | yes | — |
| 188 | C4×D8 | C2×D17 | 1 | 1 | yes | characteristic |
| 189 | #367 | — | 17 | 1 | no | — |
| 190 | #367 | — | 17 | 2 | no | — |
| 191 | C4⋊C16 | — | 17 | 1 | no | — |
| 192 | C4⋊C16 | — | 17 | 2 | no | — |
| 193 | #358 | — | 17 | 2 | no | — |
| 194 | #358 | — | 17 | 2 | no | — |
| 195 | #358 | — | 17 | 2 | no | — |
| 196 | #358 | — | 17 | 1 | no | — |
| 197 | #357 | — | 17 | 1 | no | — |
| 198 | #357 | — | 17 | 2 | no | — |
| 199 | #346 | — | 17 | 2 | no | — |
| 200 | C4×C16 | — | 17 | 1 | no | — |
| 201 | C2×C17 | — | 16 | 1 | no | — |
| 202 | C2×C17 | #3358 | 1 | 1 | yes | characteristic |
| 203 | C2×C17 | C2×#510 | 1 | 1 | yes | characteristic |
| 204 | C2×C17 | #3416 | 1 | 1 | yes | characteristic |
| 205 | C2×Q16 | — | 2 | 1 | no | — |
| 206 | C2×Q16 | — | 17 | 1 | no | — |
| 207 | C2×Q16 | C22×D17 | 1 | 2 | yes | — |
| 208 | C2×SD16 | — | 2 | 2 | no | — |
| 209 | C2×D8 | C22×D17 | 1 | 1 | yes | characteristic |
| 210 | C4⋊Q8 | — | 2 | 1 | no | — |
| 211 | C4⋊Q8 | — | 34 | 1 | no | — |
| 212 | C4⋊Q8 | — | 34 | 2 | no | — |
| 213 | C4⋊Q8 | — | 68 | 1 | no | — |
| 214 | C4⋊Q8 | — | 68 | 2 | no | — |
| 215 | #126 | — | 4 | 2 | no | — |
| 216 | #124 | — | 2 | 2 | no | — |
| 217 | C22⋊Q8 | — | 4 | 1 | no | — |
| 218 | C4×Q8 | — | 2 | 2 | no | — |
| 219 | C4×Q8 | — | 34 | 1 | no | — |
| 220 | C4×Q8 | — | 34 | 2 | no | — |
| 221 | C4×Q8 | — | 68 | 1 | no | — |
| 222 | C4×Q8 | C22⋊D17 | 1 | 1 | yes | characteristic |
| 223 | C4×D4 | — | 2 | 1 | no | — |
| 224 | #117 | — | 4 | 1 | no | — |
| 225 | Q32 | — | 34 | 4 | no | — |
| 226 | SD32 | — | 34 | 2 | no | — |
| 227 | C2×C16 | — | 17 | 1 | no | — |
| 228 | C2×C16 | — | 17 | 1 | no | — |
| 229 | C2×C16 | — | 17 | 2 | no | — |
| 230 | #107 | — | 17 | 1 | no | — |
| 231 | #107 | — | 17 | 2 | no | — |
| 232 | #107 | — | 34 | 1 | no | — |
| 233 | #107 | — | 34 | 2 | no | — |
| 234 | #107 | C22×D17 | 1 | 1 | yes | characteristic |
| 235 | C4⋊C8 | C22⋊D17 | 1 | 1 | yes | characteristic |
| 236 | C4⋊C8 | C22⋊D17 | 1 | 2 | yes | — |
| 237 | #103 | — | 2 | 2 | no | — |
| 238 | #103 | — | 34 | 1 | no | — |
| 239 | #103 | — | 34 | 2 | no | — |
| 240 | #102 | — | 2 | 1 | no | — |
| 241 | C4×C8 | C22×D17 | 1 | 1 | yes | characteristic |
| 242 | C4×C8 | C22×D17 | 1 | 2 | yes | — |
| 243 | C17 | #33936 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 244 | C2×Q8 | — | 2 | 2 | no | — |
| 245 | C2×Q8 | — | 34 | 1 | no | — |
| 246 | C2×Q8 | — | 68 | 2 | no | — |
| 247 | C2×Q8 | — | 68 | 1 | no | — |
| 248 | C2×Q8 | #63228 | 1 | 1 | yes | characteristic |
| 249 | C2×D4 | — | 2 | 1 | no | — |
| 250 | C22×C4 | — | 4 | 1 | no | — |
| 251 | Q16 | — | 2 | 2 | no | — |
| 252 | Q16 | — | 2 | 2 | no | — |
| 253 | Q16 | — | 8 | 1 | no | — |
| 254 | Q16 | — | 17 | 2 | no | — |
| 255 | Q16 | — | 34 | 1 | no | — |
| 256 | SD16 | — | 8 | 2 | no | — |
| 257 | D8 | — | 2 | 1 | no | — |
| 258 | D8 | — | 2 | 1 | no | — |
| 259 | C2×C8 | C2×C22⋊D17 | 1 | 1 | yes | characteristic |
| 260 | C2×C8 | C2×C22⋊D17 | 1 | 2 | yes | — |
| 261 | C2×C8 | C23×D17 | 1 | 1 | yes | characteristic |
| 262 | C4⋊C4 | — | 2 | 1 | no | — |
| 263 | C4⋊C4 | — | 4 | 1 | no | — |
| 264 | C4⋊C4 | — | 4 | 2 | no | — |
| 265 | C4⋊C4 | — | 4 | 1 | no | — |
| 266 | C4⋊C4 | — | 34 | 1 | no | — |
| 267 | C4⋊C4 | — | 34 | 2 | no | — |
| 268 | C4⋊C4 | — | 68 | 2 | no | — |
| 269 | C4⋊C4 | — | 68 | 1 | no | — |
| 270 | C4⋊C4 | — | 68 | 2 | no | — |
| 271 | C4⋊C4 | — | 68 | 1 | no | — |
| 272 | C4⋊C4 | #63228 | 1 | 1 | yes | characteristic |
| 273 | C4⋊C4 | #63228 | 1 | 2 | yes | — |
| 274 | C22⋊C4 | — | 4 | 2 | no | — |
| 275 | C22⋊C4 | — | 4 | 1 | no | — |
| 276 | C22⋊C4 | — | 4 | 2 | no | — |
| 277 | C42 | — | 4 | 2 | no | — |
| 278 | C42 | — | 68 | 2 | no | — |
| 279 | C42 | — | 68 | 1 | no | — |
| 280 | C42 | C2×C22⋊D17 | 1 | 1 | yes | characteristic |
| 281 | C42 | C2×C22⋊D17 | 1 | 2 | yes | — |
| 282 | C16 | — | 34 | 1 | no | — |
| 283 | C16 | — | 34 | 2 | no | — |
| 284 | C16 | — | 34 | 1 | no | — |
| 285 | C23 | — | 4 | 1 | no | — |
| 286 | Q8 | — | 2 | 2 | no | — |
| 287 | Q8 | — | 4 | 2 | no | — |
| 288 | Q8 | — | 4 | 2 | no | — |
| 289 | Q8 | — | 272 | 2 | no | — |
| 290 | Q8 | — | 272 | 1 | no | — |
| 291 | Q8 | — | 34 | 2 | no | — |
| 292 | Q8 | — | 68 | 1 | no | — |
| 293 | D4 | — | 4 | 1 | no | — |
| 294 | D4 | — | 4 | 1 | no | — |
| 295 | C2×C4 | — | 4 | 1 | no | — |
| 296 | C2×C4 | — | 4 | 2 | no | — |
| 297 | C2×C4 | — | 8 | 2 | no | — |
| 298 | C2×C4 | — | 68 | 1 | no | — |
| 299 | C2×C4 | — | 68 | 2 | no | — |
| 300 | C2×C4 | — | 68 | 1 | no | — |
| 301 | C2×C4 | C2×#63228 | 1 | 2 | yes | — |
| 302 | C2×C4 | C2×#63228 | 1 | 2 | yes | — |
| 303 | C2×C4 | C2×#63228 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | C2×#63228 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | C22×C22⋊D17 | 1 | 1 | yes | characteristic |
| 306 | C8 | — | 2 | 2 | no | — |
| 307 | C8 | — | 2 | 1 | no | — |
| 308 | C8 | #41×D17 | 1 | 1 | yes | characteristic |
| 309 | C8 | #10587697 | 1 | 1 | yes | characteristic |
| 310 | C22 | — | 8 | 1 | no | — |
| 311 | C22 | — | 8 | 1 | no | — |
| 312 | C22 | — | 8 | 1 | no | — |
| 313 | C22 | C22×#63228 | 1 | 1 | yes | characteristic |
| 314 | C4 | — | 2 | 2 | no | — |
| 315 | C4 | — | 2 | 2 | no | — |
| 316 | C4 | — | 2 | 1 | no | — |
| 317 | C4 | — | 8 | 1 | no | — |
| 318 | C4 | — | 8 | 2 | no | — |
| 319 | C4 | — | 68 | 2 | no | — |
| 320 | C4 | — | 136 | 2 | no | — |
| 321 | C4 | — | 136 | 1 | no | — |
| 322 | C4 | #11763249 | 1 | 2 | yes | — |
| 323 | C4 | #11763466 | 1 | 1 | yes | characteristic |
| 324 | C4 | #11763620 | 1 | 1 | yes | characteristic |
| 325 | C2 | — | 16 | 1 | no | — |
| 326 | C2 | #14541415 | 1 | 1 | yes | characteristic |
| 327 | C2 | C2×#11763249 | 1 | 1 | yes | characteristic |
| 328 | C2 | #14541928 | 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