← #17002332
Subgroups — #17002332
Order 4352.
1016 conjugacy classes of subgroups.
The subgroup lattice has 346 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #14542790 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #14537804 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #14527394 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #14527389 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #14527375 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #14527370 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C17×#2875 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #11763538 | C4 | 1 | 1 | yes | characteristic |
| 9 | #11763503 | — | 2 | 3 | no | — |
| 10 | #11763481 | C4 | 1 | 1 | yes | characteristic |
| 11 | #11763480 | C4 | 1 | 3 | yes | — |
| 12 | #11763478 | — | 2 | 1 | no | — |
| 13 | #11763464 | C4 | 1 | 3 | yes | — |
| 14 | #11763410 | — | 2 | 3 | no | — |
| 15 | #11763359 | C22 | 1 | 3 | yes | — |
| 16 | #11763343 | C22 | 1 | 3 | yes | — |
| 17 | #11763291 | — | 2 | 1 | no | — |
| 18 | Q8×#3622 | — | 2 | 1 | no | — |
| 19 | #11762887 | — | 2 | 3 | no | — |
| 20 | #11762866 | C22 | 1 | 3 | yes | — |
| 21 | C4×#63223 | C22 | 1 | 3 | yes | — |
| 22 | #11762550 | C22 | 1 | 1 | yes | characteristic |
| 23 | #11762550 | C22 | 1 | 3 | yes | — |
| 24 | #11762548 | C22 | 1 | 3 | yes | — |
| 25 | #11762548 | C22 | 1 | 3 | yes | — |
| 26 | #11762548 | C22 | 1 | 6 | yes | — |
| 27 | C2×C17×#131 | — | 2 | 1 | no | — |
| 28 | C4×#41×C17 | C22 | 1 | 1 | yes | characteristic |
| 29 | C17×#405 | — | 2 | 3 | no | — |
| 30 | C4×M16×C17 | C22 | 1 | 3 | yes | — |
| 31 | C2×C17×#97 | C22 | 1 | 3 | yes | — |
| 32 | C2×#41×C17 | D4 | 1 | 1 | yes | characteristic |
| 33 | C2×#63256 | — | 2 | 1 | no | — |
| 34 | C2×#63252 | C2×C4 | 1 | 6 | yes | — |
| 35 | C17×#131 | — | 2 | 8 | no | — |
| 36 | C2×M16×C17 | — | 2 | 3 | no | — |
| 37 | C22×C8×C17 | — | 2 | 3 | no | — |
| 38 | C4×Q8×C17 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C4×Q8×C17 | D4 | 1 | 1 | yes | characteristic |
| 40 | C4×D4×C17 | C2×C4 | 1 | 3 | yes | — |
| 41 | C4×D4×C17 | D4 | 1 | 3 | yes | — |
| 42 | C17×#117 | D4 | 1 | 3 | yes | — |
| 43 | C2×C42×C17 | C23 | 1 | 3 | yes | — |
| 44 | Q8×Dic17 | — | 2 | 1 | no | — |
| 45 | #10587626 | — | 2 | 3 | no | — |
| 46 | #10587624 | — | 2 | 3 | no | — |
| 47 | #10587616 | — | 2 | 3 | no | — |
| 48 | #10587616 | — | 2 | 3 | no | — |
| 49 | #10587616 | C2×C4 | 1 | 3 | yes | — |
| 50 | #10587615 | — | 2 | 3 | no | — |
| 51 | C4×C22⋊D17 | — | 2 | 3 | no | — |
| 52 | #10587610 | C2×C4 | 1 | 3 | yes | — |
| 53 | C2×#63223 | — | 2 | 3 | no | — |
| 54 | #10587581 | — | 2 | 3 | no | — |
| 55 | #10587579 | — | 2 | 3 | no | — |
| 56 | C4⋊C4×D17 | — | 2 | 3 | no | — |
| 57 | #10587576 | — | 2 | 3 | no | — |
| 58 | #10587570 | — | 2 | 6 | no | — |
| 59 | #10587569 | — | 2 | 6 | no | — |
| 60 | #10587562 | — | 2 | 3 | no | — |
| 61 | #10587561 | C23 | 1 | 3 | yes | — |
| 62 | #10587560 | C2×C4 | 1 | 3 | yes | — |
| 63 | C4×D68 | — | 2 | 3 | no | — |
| 64 | C4×D68 | — | 2 | 3 | no | — |
| 65 | C4×D68 | C2×C4 | 1 | 3 | yes | — |
| 66 | #10587558 | — | 2 | 3 | no | — |
| 67 | #10587556 | C23 | 1 | 1 | yes | characteristic |
| 68 | #10587555 | C2×C4 | 1 | 3 | yes | — |
| 69 | C4×Dic34 | C2×C4 | 1 | 3 | yes | — |
| 70 | C17×#97 | — | 2 | 3 | no | — |
| 71 | C17×#97 | C23 | 1 | 1 | yes | characteristic |
| 72 | C17×#97 | C23 | 1 | 3 | yes | — |
| 73 | C4×C8×C17 | — | 2 | 3 | no | — |
| 74 | C22⋊#3622 | — | 2 | 3 | no | — |
| 75 | C4⋊#3622 | — | 2 | 3 | no | — |
| 76 | #10587474 | C23 | 1 | 1 | yes | characteristic |
| 77 | #10587474 | C23 | 1 | 3 | yes | — |
| 78 | C4×#3622 | — | 2 | 3 | no | — |
| 79 | #41×C17 | — | 2 | 4 | no | — |
| 80 | C2×Q8×C17 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 81 | C2×D4×C17 | C22⋊C4 | 1 | 3 | yes | — |
| 82 | C22×C4×C17 | C22×C4 | 1 | 3 | yes | — |
| 83 | C22×C4×C17 | C2×D4 | 1 | 3 | yes | — |
| 84 | C22×C4×C17 | C2×D4 | 1 | 3 | yes | — |
| 85 | C2×C22⋊D17 | — | 2 | 6 | no | — |
| 86 | C2×C22⋊D17 | — | 2 | 3 | no | — |
| 87 | #63256 | — | 4 | 4 | no | — |
| 88 | #63252 | — | 8 | 6 | no | — |
| 89 | C2×D68 | — | 2 | 3 | no | — |
| 90 | C2×D68 | — | 2 | 3 | no | — |
| 91 | C2×D68 | C22×C4 | 1 | 3 | yes | — |
| 92 | C2×C4×D17 | — | 2 | 3 | no | — |
| 93 | C2×C4×D17 | — | 2 | 6 | no | — |
| 94 | C2×Dic34 | C22×C4 | 1 | 3 | yes | — |
| 95 | M16×C17 | — | 2 | 12 | no | — |
| 96 | C2×C8×C17 | — | 2 | 12 | no | — |
| 97 | C2×C8×C17 | — | 2 | 1 | no | — |
| 98 | C2×C8×C17 | — | 2 | 3 | no | — |
| 99 | C4⋊C4×C17 | C22⋊C4 | 1 | 3 | yes | — |
| 100 | C4⋊C4×C17 | C22⋊C4 | 1 | 3 | yes | — |
| 101 | C22⋊C4×C17 | C22⋊C4 | 1 | 6 | yes | — |
| 102 | C42×C17 | C22×C4 | 1 | 3 | yes | — |
| 103 | C42×C17 | C2×D4 | 1 | 3 | yes | — |
| 104 | C42×C17 | C2×D4 | 1 | 3 | yes | — |
| 105 | C42×C17 | C24 | 1 | 1 | yes | characteristic |
| 106 | C22⋊Dic17 | — | 2 | 3 | no | — |
| 107 | #63227 | — | 2 | 3 | no | — |
| 108 | #63227 | — | 2 | 3 | no | — |
| 109 | #63227 | — | 2 | 3 | no | — |
| 110 | #63227 | — | 2 | 3 | no | — |
| 111 | C4⋊Dic17 | — | 2 | 3 | no | — |
| 112 | C4⋊Dic17 | — | 2 | 3 | no | — |
| 113 | C4⋊Dic17 | C22×C4 | 1 | 1 | yes | characteristic |
| 114 | C4⋊Dic17 | C22×C4 | 1 | 1 | yes | characteristic |
| 115 | #63225 | — | 2 | 3 | no | — |
| 116 | #63225 | — | 2 | 3 | no | — |
| 117 | #63225 | — | 2 | 1 | no | — |
| 118 | #63225 | — | 2 | 1 | no | — |
| 119 | C4×Dic17 | — | 2 | 3 | no | — |
| 120 | #63223 | — | 4 | 6 | no | — |
| 121 | C2×#3622 | — | 2 | 1 | no | — |
| 122 | C2×#3622 | — | 2 | 3 | no | — |
| 123 | #17655 | — | 17 | 1 | no | maximal, Sylow |
| 124 | C23×C17 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 125 | C22×D17 | — | 2 | 3 | no | — |
| 126 | Q8×C17 | #104 | 1 | 4 | yes | — |
| 127 | D4×C17 | — | 2 | 6 | no | — |
| 128 | C2×C4×C17 | — | 2 | 6 | no | — |
| 129 | C2×C4×C17 | — | 4 | 3 | no | — |
| 130 | C2×C4×C17 | — | 4 | 3 | no | — |
| 131 | C2×C4×C17 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 132 | C2×C4×C17 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 133 | C2×C4×C17 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 134 | C2×C4×C17 | C23×C4 | 1 | 1 | yes | characteristic |
| 135 | C2×C4×C17 | C22×D4 | 1 | 1 | yes | characteristic |
| 136 | C2×C4×C17 | C22×D4 | 1 | 1 | yes | characteristic |
| 137 | C22⋊D17 | — | 8 | 6 | no | — |
| 138 | C22⋊D17 | — | 8 | 3 | no | — |
| 139 | C2×Dic17 | — | 2 | 1 | no | — |
| 140 | C2×Dic17 | — | 2 | 3 | no | — |
| 141 | C2×Dic17 | — | 2 | 1 | no | — |
| 142 | D68 | — | 4 | 3 | no | — |
| 143 | D68 | — | 4 | 6 | no | — |
| 144 | D68 | — | 8 | 3 | no | — |
| 145 | C4×D17 | — | 4 | 6 | no | — |
| 146 | C4×D17 | — | 8 | 6 | no | — |
| 147 | Dic34 | — | 4 | 3 | no | — |
| 148 | C8×C17 | — | 2 | 2 | no | — |
| 149 | C8×C17 | — | 2 | 6 | no | — |
| 150 | #3622 | — | 2 | 2 | no | — |
| 151 | #3622 | — | 4 | 3 | no | — |
| 152 | #3517 | — | 17 | 1 | no | — |
| 153 | #2971 | — | 17 | 1 | no | — |
| 154 | #2875 | D17 | 1 | 1 | yes | characteristic, complemented |
| 155 | #1530 | — | 17 | 3 | no | — |
| 156 | #1529 | — | 17 | 3 | no | — |
| 157 | #1525 | — | 17 | 3 | no | — |
| 158 | #1524 | — | 17 | 3 | no | — |
| 159 | C22×C17 | — | 2 | 3 | no | — |
| 160 | C22×C17 | — | 2 | 3 | no | — |
| 161 | C22×C17 | — | 2 | 3 | no | — |
| 162 | C22×C17 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 163 | C2×D17 | — | 4 | 3 | no | — |
| 164 | C2×D17 | — | 4 | 3 | no | — |
| 165 | C2×D17 | — | 4 | 3 | no | — |
| 166 | C4×C17 | — | 2 | 1 | no | — |
| 167 | C4×C17 | — | 2 | 3 | no | — |
| 168 | C4×C17 | — | 2 | 1 | no | — |
| 169 | C4×C17 | — | 2 | 1 | no | — |
| 170 | C4×C17 | — | 2 | 3 | no | — |
| 171 | C4×C17 | C2×#104 | 1 | 6 | yes | — |
| 172 | Dic17 | — | 4 | 3 | no | — |
| 173 | Dic17 | — | 4 | 1 | no | — |
| 174 | Dic17 | — | 4 | 1 | no | — |
| 175 | C2×#131 | — | 2 | 1 | no | — |
| 176 | #557 | — | 17 | 1 | no | — |
| 177 | #553 | — | 34 | 3 | no | — |
| 178 | #550 | — | 17 | 1 | no | — |
| 179 | #550 | — | 17 | 3 | no | — |
| 180 | #550 | — | 34 | 1 | no | — |
| 181 | #547 | — | 17 | 3 | no | — |
| 182 | #540 | — | 34 | 3 | no | — |
| 183 | #532 | — | 17 | 3 | no | — |
| 184 | #529 | — | 17 | 3 | no | — |
| 185 | #520 | — | 34 | 1 | no | — |
| 186 | C4×#41 | C2×D17 | 1 | 1 | yes | characteristic |
| 187 | C8×Q8 | — | 34 | 1 | no | — |
| 188 | #436 | — | 34 | 3 | no | — |
| 189 | #432 | — | 17 | 3 | no | — |
| 190 | #405 | — | 2 | 3 | no | — |
| 191 | C4×M16 | — | 17 | 3 | no | — |
| 192 | C4×M16 | C2×D17 | 1 | 3 | yes | — |
| 193 | C2×#97 | C2×D17 | 1 | 3 | yes | — |
| 194 | #330 | — | 17 | 1 | no | — |
| 195 | #330 | — | 17 | 3 | no | — |
| 196 | #329 | — | 17 | 3 | no | — |
| 197 | #329 | — | 17 | 3 | no | — |
| 198 | #329 | — | 17 | 6 | no | — |
| 199 | C2×C17 | — | 4 | 3 | no | — |
| 200 | C2×C17 | #2888 | 1 | 1 | yes | characteristic |
| 201 | C2×C17 | C22×#104 | 1 | 1 | yes | characteristic |
| 202 | C2×C17 | #2901 | 1 | 1 | yes | characteristic |
| 203 | D17 | — | 8 | 3 | no | — |
| 204 | C2×#41 | — | 17 | 6 | no | — |
| 205 | C2×#41 | — | 34 | 1 | no | — |
| 206 | C2×#41 | C22⋊D17 | 1 | 1 | yes | characteristic |
| 207 | #131 | — | 2 | 8 | no | — |
| 208 | C2×M16 | — | 2 | 3 | no | — |
| 209 | C2×M16 | — | 34 | 3 | no | — |
| 210 | C22×C8 | — | 2 | 3 | no | — |
| 211 | C4⋊Q8 | — | 17 | 3 | no | — |
| 212 | #127 | — | 17 | 3 | no | — |
| 213 | #127 | — | 34 | 3 | no | — |
| 214 | #126 | — | 34 | 3 | no | — |
| 215 | #125 | — | 17 | 1 | no | — |
| 216 | #125 | — | 34 | 3 | no | — |
| 217 | #124 | — | 17 | 3 | no | — |
| 218 | #123 | — | 34 | 3 | no | — |
| 219 | #123 | — | 34 | 6 | no | — |
| 220 | C22⋊Q8 | — | 17 | 3 | no | — |
| 221 | #121 | — | 17 | 3 | no | — |
| 222 | #121 | — | 34 | 3 | no | — |
| 223 | #121 | — | 34 | 3 | no | — |
| 224 | #121 | — | 34 | 3 | no | — |
| 225 | #121 | — | 34 | 3 | no | — |
| 226 | #121 | — | 34 | 6 | no | — |
| 227 | C4×Q8 | — | 17 | 3 | no | — |
| 228 | C4×Q8 | — | 34 | 1 | no | — |
| 229 | C4×Q8 | C4×D17 | 1 | 1 | yes | characteristic |
| 230 | C4×Q8 | D68 | 1 | 1 | yes | characteristic |
| 231 | C4×D4 | — | 17 | 3 | no | — |
| 232 | C4×D4 | — | 34 | 3 | no | — |
| 233 | C4×D4 | — | 34 | 3 | no | — |
| 234 | C4×D4 | — | 34 | 3 | no | — |
| 235 | C4×D4 | — | 34 | 3 | no | — |
| 236 | C4×D4 | C4×D17 | 1 | 3 | yes | — |
| 237 | C4×D4 | D68 | 1 | 3 | yes | — |
| 238 | #117 | — | 34 | 3 | no | — |
| 239 | #117 | C22⋊D17 | 1 | 3 | yes | — |
| 240 | C2×C4⋊C4 | — | 34 | 3 | no | — |
| 241 | C2×C42 | C22×D17 | 1 | 3 | yes | — |
| 242 | C4⋊C8 | — | 34 | 3 | no | — |
| 243 | C22⋊C8 | — | 34 | 3 | no | — |
| 244 | #97 | — | 2 | 3 | no | — |
| 245 | #97 | — | 17 | 1 | no | — |
| 246 | #97 | — | 17 | 3 | no | — |
| 247 | #97 | C22×D17 | 1 | 1 | yes | characteristic |
| 248 | #97 | C22×D17 | 1 | 3 | yes | — |
| 249 | C4×C8 | — | 2 | 3 | no | — |
| 250 | C4×C8 | — | 34 | 3 | no | — |
| 251 | C17 | #17655 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 252 | #41 | — | 2 | 4 | no | — |
| 253 | #41 | — | 68 | 4 | no | — |
| 254 | #41 | — | 136 | 6 | no | — |
| 255 | C2×Q8 | — | 17 | 3 | no | — |
| 256 | C2×Q8 | #63227 | 1 | 1 | yes | characteristic |
| 257 | C2×D4 | — | 17 | 3 | no | — |
| 258 | C2×D4 | — | 34 | 3 | no | — |
| 259 | C2×D4 | — | 34 | 6 | no | — |
| 260 | C2×D4 | — | 34 | 3 | no | — |
| 261 | C2×D4 | — | 34 | 3 | no | — |
| 262 | C2×D4 | #63227 | 1 | 3 | yes | — |
| 263 | C22×C4 | — | 34 | 3 | no | — |
| 264 | C22×C4 | — | 34 | 6 | no | — |
| 265 | C22×C4 | C2×C4×D17 | 1 | 3 | yes | — |
| 266 | C22×C4 | C2×D68 | 1 | 3 | yes | — |
| 267 | C22×C4 | C2×C22⋊D17 | 1 | 3 | yes | — |
| 268 | M16 | — | 2 | 12 | no | — |
| 269 | M16 | — | 68 | 6 | no | — |
| 270 | C2×C8 | — | 2 | 12 | no | — |
| 271 | C2×C8 | — | 2 | 3 | no | — |
| 272 | C2×C8 | — | 2 | 1 | no | — |
| 273 | C2×C8 | — | 34 | 1 | no | — |
| 274 | C2×C8 | — | 34 | 3 | no | — |
| 275 | C4⋊C4 | — | 17 | 1 | no | — |
| 276 | C4⋊C4 | — | 17 | 1 | no | — |
| 277 | C4⋊C4 | — | 34 | 1 | no | — |
| 278 | C4⋊C4 | — | 34 | 1 | no | — |
| 279 | C4⋊C4 | — | 34 | 3 | no | — |
| 280 | C4⋊C4 | — | 34 | 3 | no | — |
| 281 | C4⋊C4 | — | 34 | 3 | no | — |
| 282 | C4⋊C4 | — | 34 | 3 | no | — |
| 283 | C4⋊C4 | #63227 | 1 | 3 | yes | — |
| 284 | C4⋊C4 | #63227 | 1 | 3 | yes | — |
| 285 | C22⋊C4 | — | 34 | 3 | no | — |
| 286 | C22⋊C4 | — | 34 | 3 | no | — |
| 287 | C22⋊C4 | — | 34 | 3 | no | — |
| 288 | C22⋊C4 | — | 34 | 3 | no | — |
| 289 | C22⋊C4 | — | 34 | 3 | no | — |
| 290 | C22⋊C4 | #63227 | 1 | 6 | yes | — |
| 291 | C42 | — | 34 | 3 | no | — |
| 292 | C42 | C2×C4×D17 | 1 | 3 | yes | — |
| 293 | C42 | C2×D68 | 1 | 3 | yes | — |
| 294 | C42 | C2×C22⋊D17 | 1 | 3 | yes | — |
| 295 | C42 | C23×D17 | 1 | 1 | yes | characteristic |
| 296 | C23 | — | 34 | 3 | no | — |
| 297 | C23 | C2×#63227 | 1 | 3 | yes | — |
| 298 | Q8 | — | 68 | 3 | no | — |
| 299 | Q8 | #10587476 | 1 | 4 | yes | — |
| 300 | D4 | — | 2 | 6 | no | — |
| 301 | D4 | — | 68 | 3 | no | — |
| 302 | D4 | — | 68 | 6 | no | — |
| 303 | D4 | — | 136 | 3 | no | — |
| 304 | D4 | — | 136 | 6 | no | — |
| 305 | D4 | — | 136 | 3 | no | — |
| 306 | C2×C4 | — | 2 | 6 | no | — |
| 307 | C2×C4 | — | 4 | 3 | no | — |
| 308 | C2×C4 | — | 4 | 3 | no | — |
| 309 | C2×C4 | — | 34 | 1 | no | — |
| 310 | C2×C4 | — | 34 | 3 | no | — |
| 311 | C2×C4 | — | 34 | 1 | no | — |
| 312 | C2×C4 | — | 68 | 6 | no | — |
| 313 | C2×C4 | — | 136 | 6 | no | — |
| 314 | C2×C4 | C2×#63227 | 1 | 3 | yes | — |
| 315 | C2×C4 | C2×#63227 | 1 | 3 | yes | — |
| 316 | C2×C4 | C2×#63227 | 1 | 3 | yes | — |
| 317 | C2×C4 | C22×C4×D17 | 1 | 1 | yes | characteristic |
| 318 | C2×C4 | C22×D68 | 1 | 1 | yes | characteristic |
| 319 | C2×C4 | C22×C22⋊D17 | 1 | 1 | yes | characteristic |
| 320 | C8 | — | 2 | 2 | no | — |
| 321 | C8 | — | 2 | 6 | no | — |
| 322 | C8 | — | 34 | 2 | no | — |
| 323 | C8 | — | 68 | 3 | no | — |
| 324 | C22 | — | 2 | 3 | no | — |
| 325 | C22 | — | 2 | 3 | no | — |
| 326 | C22 | — | 2 | 3 | no | — |
| 327 | C22 | — | 68 | 3 | no | — |
| 328 | C22 | — | 68 | 3 | no | — |
| 329 | C22 | — | 68 | 3 | no | — |
| 330 | C22 | C22×#63227 | 1 | 1 | yes | characteristic |
| 331 | C4 | — | 2 | 1 | no | — |
| 332 | C4 | — | 2 | 3 | no | — |
| 333 | C4 | — | 2 | 1 | no | — |
| 334 | C4 | — | 2 | 1 | no | — |
| 335 | C4 | — | 2 | 3 | no | — |
| 336 | C4 | — | 68 | 3 | no | — |
| 337 | C4 | — | 68 | 1 | no | — |
| 338 | C4 | — | 68 | 1 | no | — |
| 339 | C4 | C2×#10587476 | 1 | 6 | yes | — |
| 340 | C2 | — | 4 | 3 | no | — |
| 341 | C2 | — | 136 | 3 | no | — |
| 342 | C2 | #14537262 | 1 | 1 | yes | characteristic |
| 343 | C2 | C22×#10587476 | 1 | 1 | yes | characteristic |
| 344 | C2 | #14537329 | 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