← #16950403
Subgroups — #16950403
Order 4352.
410 conjugacy classes of subgroups.
The subgroup lattice has 322 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×Q16×Dic17 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #14530539 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#11762625 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #14528954 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #14526894 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #14526893 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C2×C17×#362 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×Q8×Dic17 | — | 2 | 1 | no | — |
| 9 | C2×#10587483 | C22 | 1 | 1 | yes | characteristic |
| 10 | Q16×Dic17 | — | 2 | 1 | no | — |
| 11 | Q16×Dic17 | — | 2 | 1 | no | — |
| 12 | Q16×Dic17 | C4 | 1 | 2 | yes | — |
| 13 | Q16×Dic17 | C4 | 1 | 2 | yes | — |
| 14 | C2×#10587490 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#10587493 | — | 2 | 1 | no | — |
| 16 | C2×#10587489 | C22 | 1 | 1 | yes | characteristic |
| 17 | #11762832 | — | 2 | 1 | no | — |
| 18 | C2×#10587506 | — | 2 | 1 | no | — |
| 19 | C2×C8×Dic17 | C22 | 1 | 1 | yes | characteristic |
| 20 | #11762627 | — | 2 | 1 | no | — |
| 21 | #11762627 | — | 2 | 1 | no | — |
| 22 | #11762625 | — | 2 | 2 | no | — |
| 23 | C22×Q16×C17 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×C17×M32 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×C17×#108 | C22 | 1 | 1 | yes | characteristic |
| 26 | C17×#362 | — | 2 | 2 | no | — |
| 27 | C22×Q8×C17 | — | 2 | 1 | no | — |
| 28 | C2×Q16×C17 | — | 2 | 1 | no | — |
| 29 | C2×Q16×C17 | — | 2 | 1 | no | — |
| 30 | C2×Q16×C17 | — | 2 | 1 | no | — |
| 31 | C2×Q16×C17 | C2×C4 | 1 | 2 | yes | — |
| 32 | C2×Q16×C17 | D4 | 1 | 2 | yes | — |
| 33 | C2×Q16×C17 | Q8 | 1 | 2 | yes | — |
| 34 | C2×M16×C17 | — | 2 | 1 | no | — |
| 35 | C22×C8×C17 | C23 | 1 | 1 | yes | characteristic |
| 36 | Q8×Dic17 | — | 2 | 2 | no | — |
| 37 | Q8×Dic17 | — | 2 | 2 | no | — |
| 38 | Q8×Dic17 | — | 4 | 1 | no | — |
| 39 | Q8×Dic17 | — | 4 | 1 | no | — |
| 40 | C2×C4⋊Dic17 | — | 2 | 1 | no | — |
| 41 | C2×C4⋊Dic17 | — | 4 | 1 | no | — |
| 42 | C2×C4×Dic17 | — | 4 | 1 | no | — |
| 43 | C2×C4×Dic17 | D4 | 1 | 1 | yes | characteristic |
| 44 | C2×#63223 | — | 2 | 1 | no | — |
| 45 | C22×#3622 | D4 | 1 | 1 | yes | characteristic |
| 46 | C17×M32 | — | 2 | 1 | no | — |
| 47 | C17×M32 | — | 2 | 1 | no | — |
| 48 | C2×C16×C17 | — | 2 | 1 | no | — |
| 49 | C17×#108 | — | 2 | 1 | no | — |
| 50 | C17×#108 | — | 2 | 1 | no | — |
| 51 | #10587506 | — | 2 | 2 | no | — |
| 52 | #10587506 | — | 2 | 2 | no | — |
| 53 | #10587493 | — | 4 | 2 | no | — |
| 54 | #10587490 | — | 2 | 1 | no | — |
| 55 | #10587490 | — | 2 | 1 | no | — |
| 56 | #10587489 | — | 2 | 1 | no | — |
| 57 | #10587489 | C2×C4 | 1 | 2 | yes | — |
| 58 | #10587485 | — | 4 | 1 | no | — |
| 59 | #10587485 | — | 4 | 1 | no | — |
| 60 | C8×Dic17 | C2×C4 | 1 | 1 | yes | characteristic |
| 61 | C8×Dic17 | C2×C4 | 1 | 1 | yes | characteristic |
| 62 | C8×Dic17 | D4 | 1 | 1 | yes | characteristic |
| 63 | C8×Dic17 | D4 | 1 | 1 | yes | characteristic |
| 64 | #10587483 | — | 2 | 1 | no | — |
| 65 | #10587483 | — | 2 | 1 | no | — |
| 66 | C2×#63214 | — | 2 | 1 | no | — |
| 67 | C2×Q8×C17 | — | 2 | 2 | no | — |
| 68 | C2×Q8×C17 | — | 2 | 2 | no | — |
| 69 | C2×Q8×C17 | — | 2 | 2 | no | — |
| 70 | C2×Q8×C17 | — | 4 | 1 | no | — |
| 71 | C2×Q8×C17 | — | 4 | 1 | no | — |
| 72 | C2×Q8×C17 | — | 4 | 1 | no | — |
| 73 | C22×C4×C17 | — | 4 | 1 | no | — |
| 74 | C22×C4×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C22×Dic17 | — | 4 | 1 | no | — |
| 76 | C22×Dic17 | SD16 | 1 | 1 | yes | characteristic |
| 77 | C22×Dic17 | SD16 | 1 | 1 | yes | characteristic |
| 78 | Q16×C17 | — | 2 | 2 | no | — |
| 79 | Q16×C17 | — | 2 | 2 | no | — |
| 80 | Q16×C17 | — | 2 | 2 | no | — |
| 81 | Q16×C17 | C4⋊C4 | 1 | 4 | yes | — |
| 82 | M16×C17 | — | 4 | 1 | no | — |
| 83 | M16×C17 | — | 4 | 1 | no | — |
| 84 | C2×C8×C17 | — | 4 | 1 | no | — |
| 85 | C2×C8×C17 | C22×C4 | 1 | 1 | yes | characteristic |
| 86 | C2×C8×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C2×C8×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C2×C8×C17 | C2×Q8 | 1 | 1 | yes | characteristic |
| 89 | C2×C8×C17 | #41 | 1 | 1 | yes | characteristic |
| 90 | C2×C8×C17 | #41 | 1 | 1 | yes | characteristic |
| 91 | C4⋊Dic17 | — | 2 | 2 | no | — |
| 92 | C4⋊Dic17 | — | 4 | 1 | no | — |
| 93 | C4⋊Dic17 | — | 4 | 2 | no | — |
| 94 | C4⋊Dic17 | — | 4 | 2 | no | — |
| 95 | C4×Dic17 | — | 4 | 2 | no | — |
| 96 | C4×Dic17 | — | 4 | 2 | no | — |
| 97 | C4×Dic17 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 98 | C4×Dic17 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 99 | C4×Dic17 | D8 | 1 | 1 | yes | characteristic |
| 100 | C4×Dic17 | D8 | 1 | 1 | yes | characteristic |
| 101 | #63223 | — | 4 | 1 | no | — |
| 102 | #63223 | — | 4 | 1 | no | — |
| 103 | C2×#3622 | — | 2 | 1 | no | — |
| 104 | C2×#3622 | — | 2 | 1 | no | — |
| 105 | C2×#3622 | — | 4 | 1 | no | — |
| 106 | C2×#3622 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 107 | C2×#3622 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 108 | C16×C17 | — | 4 | 1 | no | — |
| 109 | #63214 | — | 4 | 1 | no | — |
| 110 | #12819 | — | 17 | 1 | no | maximal, Sylow |
| 111 | C23×C17 | C2×SD16 | 1 | 1 | yes | characteristic |
| 112 | Q8×C17 | — | 2 | 4 | no | — |
| 113 | Q8×C17 | — | 4 | 2 | no | — |
| 114 | Q8×C17 | — | 4 | 2 | no | — |
| 115 | Q8×C17 | — | 4 | 2 | no | — |
| 116 | C2×C4×C17 | — | 4 | 2 | no | — |
| 117 | C2×C4×C17 | — | 4 | 2 | no | — |
| 118 | C2×C4×C17 | — | 4 | 2 | no | — |
| 119 | C2×C4×C17 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C17 | #121 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C17 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C17 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C17 | #137 | 1 | 1 | yes | characteristic |
| 125 | C2×Dic17 | — | 2 | 2 | no | — |
| 126 | C2×Dic17 | — | 4 | 1 | no | — |
| 127 | C2×Dic17 | — | 4 | 2 | no | — |
| 128 | C2×Dic17 | — | 8 | 2 | no | — |
| 129 | C2×Dic17 | #102 | 1 | 1 | yes | characteristic |
| 130 | C2×Dic17 | #102 | 1 | 1 | yes | characteristic |
| 131 | C2×Dic17 | #102 | 1 | 1 | yes | characteristic |
| 132 | C2×Dic17 | #102 | 1 | 1 | yes | characteristic |
| 133 | C8×C17 | — | 8 | 1 | no | — |
| 134 | C8×C17 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 135 | C8×C17 | #117 | 1 | 1 | yes | characteristic |
| 136 | C8×C17 | #121 | 1 | 1 | yes | characteristic |
| 137 | C8×C17 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 138 | #3622 | — | 2 | 1 | no | — |
| 139 | #3622 | — | 2 | 1 | no | — |
| 140 | #3622 | — | 8 | 1 | no | — |
| 141 | C2×C4×Q16 | — | 17 | 1 | no | — |
| 142 | #2150 | — | 17 | 1 | no | — |
| 143 | C2×#362 | — | 17 | 1 | no | — |
| 144 | C2×#362 | D17 | 1 | 1 | yes | characteristic |
| 145 | #1848 | — | 17 | 1 | no | — |
| 146 | #1387 | — | 17 | 1 | no | — |
| 147 | #1387 | — | 17 | 1 | no | — |
| 148 | C22×C17 | — | 2 | 1 | no | — |
| 149 | C22×C17 | — | 2 | 1 | no | — |
| 150 | C22×C17 | C2×#102 | 1 | 1 | yes | characteristic |
| 151 | C22×C17 | #461 | 1 | 1 | yes | characteristic |
| 152 | C22×C17 | #476 | 1 | 1 | yes | characteristic |
| 153 | C4×C17 | — | 4 | 4 | no | — |
| 154 | C4×C17 | #380 | 1 | 1 | yes | characteristic |
| 155 | C4×C17 | #417 | 1 | 1 | yes | characteristic |
| 156 | C4×C17 | #459 | 1 | 1 | yes | characteristic |
| 157 | C4×C17 | #474 | 1 | 1 | yes | characteristic |
| 158 | Dic17 | — | 2 | 1 | no | — |
| 159 | Dic17 | — | 2 | 1 | no | — |
| 160 | Dic17 | — | 2 | 1 | no | — |
| 161 | Dic17 | — | 8 | 2 | no | — |
| 162 | Dic17 | #362 | 1 | 2 | yes | complemented |
| 163 | C22×Q16 | C2×D17 | 1 | 1 | yes | characteristic |
| 164 | C2×C4×Q8 | — | 34 | 1 | no | — |
| 165 | C2×M32 | — | 17 | 1 | no | — |
| 166 | C2×M32 | C2×D17 | 1 | 1 | yes | characteristic |
| 167 | C4×Q16 | — | 17 | 2 | no | — |
| 168 | C4×Q16 | — | 17 | 2 | no | — |
| 169 | C4×Q16 | — | 34 | 1 | no | — |
| 170 | C4×Q16 | — | 34 | 1 | no | — |
| 171 | C2×#108 | — | 17 | 1 | no | — |
| 172 | C2×#108 | — | 34 | 1 | no | — |
| 173 | C2×#108 | C2×D17 | 1 | 1 | yes | characteristic |
| 174 | C2×#107 | — | 17 | 1 | no | — |
| 175 | #423 | — | 34 | 1 | no | — |
| 176 | C2×#103 | — | 34 | 1 | no | — |
| 177 | C2×C4×C8 | — | 17 | 1 | no | — |
| 178 | C4⋊C16 | — | 34 | 1 | no | — |
| 179 | C4⋊C16 | — | 34 | 1 | no | — |
| 180 | #362 | — | 2 | 2 | no | — |
| 181 | #362 | — | 34 | 2 | no | — |
| 182 | C2×C17 | — | 2 | 1 | no | — |
| 183 | C2×C17 | — | 2 | 1 | no | — |
| 184 | C2×C17 | #1802 | 1 | 1 | yes | characteristic |
| 185 | C2×C17 | C2×#362 | 1 | 1 | yes | characteristic |
| 186 | C2×C17 | #2148 | 1 | 1 | yes | characteristic |
| 187 | C22×Q8 | — | 2 | 1 | no | — |
| 188 | C2×Q16 | — | 2 | 1 | no | — |
| 189 | C2×Q16 | — | 2 | 1 | no | — |
| 190 | C2×Q16 | — | 2 | 1 | no | — |
| 191 | C2×Q16 | Dic34 | 1 | 2 | yes | — |
| 192 | C2×Q16 | C4×D17 | 1 | 2 | yes | — |
| 193 | C2×Q16 | C22⋊D17 | 1 | 2 | yes | — |
| 194 | C2×M16 | — | 2 | 1 | no | — |
| 195 | C2×M16 | — | 34 | 1 | no | — |
| 196 | C22×C8 | — | 17 | 1 | no | — |
| 197 | C22×C8 | C22×D17 | 1 | 1 | yes | characteristic |
| 198 | C4×Q8 | — | 34 | 2 | no | — |
| 199 | C4×Q8 | — | 34 | 2 | no | — |
| 200 | C4×Q8 | — | 68 | 1 | no | — |
| 201 | C4×Q8 | — | 68 | 1 | no | — |
| 202 | C2×C4⋊C4 | — | 34 | 1 | no | — |
| 203 | C2×C4⋊C4 | — | 68 | 1 | no | — |
| 204 | C2×C42 | — | 17 | 1 | no | — |
| 205 | C2×C42 | — | 68 | 1 | no | — |
| 206 | M32 | — | 2 | 1 | no | — |
| 207 | M32 | — | 2 | 1 | no | — |
| 208 | M32 | — | 34 | 1 | no | — |
| 209 | M32 | — | 34 | 1 | no | — |
| 210 | C2×C16 | — | 2 | 1 | no | — |
| 211 | C2×C16 | — | 34 | 1 | no | — |
| 212 | #108 | — | 2 | 1 | no | — |
| 213 | #108 | — | 2 | 1 | no | — |
| 214 | #108 | — | 34 | 1 | no | — |
| 215 | #108 | — | 34 | 1 | no | — |
| 216 | #108 | — | 68 | 2 | no | — |
| 217 | #107 | — | 17 | 2 | no | — |
| 218 | #107 | — | 34 | 1 | no | — |
| 219 | C4⋊C8 | — | 68 | 1 | no | — |
| 220 | C4⋊C8 | — | 68 | 1 | no | — |
| 221 | #103 | — | 34 | 2 | no | — |
| 222 | #103 | — | 34 | 2 | no | — |
| 223 | C4×C8 | — | 17 | 1 | no | — |
| 224 | C4×C8 | — | 17 | 1 | no | — |
| 225 | C4×C8 | — | 17 | 1 | no | — |
| 226 | C4×C8 | — | 17 | 1 | no | — |
| 227 | C17 | #12819 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 228 | C2×Q8 | — | 2 | 2 | no | — |
| 229 | C2×Q8 | — | 2 | 2 | no | — |
| 230 | C2×Q8 | — | 2 | 2 | no | — |
| 231 | C2×Q8 | — | 4 | 1 | no | — |
| 232 | C2×Q8 | — | 4 | 1 | no | — |
| 233 | C2×Q8 | — | 4 | 1 | no | — |
| 234 | C22×C4 | — | 4 | 1 | no | — |
| 235 | C22×C4 | — | 17 | 1 | no | — |
| 236 | C22×C4 | — | 17 | 1 | no | — |
| 237 | C22×C4 | — | 68 | 1 | no | — |
| 238 | C22×C4 | D4×D17 | 1 | 1 | yes | characteristic |
| 239 | Q16 | — | 2 | 2 | no | — |
| 240 | Q16 | — | 2 | 2 | no | — |
| 241 | Q16 | — | 2 | 2 | no | — |
| 242 | Q16 | #63225 | 1 | 4 | yes | — |
| 243 | M16 | — | 4 | 1 | no | — |
| 244 | M16 | — | 4 | 1 | no | — |
| 245 | M16 | — | 68 | 1 | no | — |
| 246 | M16 | — | 68 | 1 | no | — |
| 247 | C2×C8 | — | 4 | 1 | no | — |
| 248 | C2×C8 | — | 17 | 1 | no | — |
| 249 | C2×C8 | — | 17 | 1 | no | — |
| 250 | C2×C8 | — | 34 | 1 | no | — |
| 251 | C2×C8 | — | 34 | 1 | no | — |
| 252 | C2×C8 | — | 68 | 1 | no | — |
| 253 | C2×C8 | C2×Dic34 | 1 | 1 | yes | characteristic |
| 254 | C2×C8 | C2×C4×D17 | 1 | 1 | yes | characteristic |
| 255 | C2×C8 | D4×D17 | 1 | 1 | yes | characteristic |
| 256 | C2×C8 | #63254 | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | #63254 | 1 | 1 | yes | characteristic |
| 258 | C2×C8 | C2×C22⋊D17 | 1 | 1 | yes | characteristic |
| 259 | C4⋊C4 | — | 34 | 2 | no | — |
| 260 | C4⋊C4 | — | 68 | 2 | no | — |
| 261 | C4⋊C4 | — | 68 | 2 | no | — |
| 262 | C4⋊C4 | — | 68 | 1 | no | — |
| 263 | C42 | — | 17 | 1 | no | — |
| 264 | C42 | — | 17 | 1 | no | — |
| 265 | C42 | — | 17 | 1 | no | — |
| 266 | C42 | — | 17 | 1 | no | — |
| 267 | C42 | — | 68 | 2 | no | — |
| 268 | C42 | — | 68 | 2 | no | — |
| 269 | C16 | — | 4 | 1 | no | — |
| 270 | C16 | — | 68 | 1 | no | — |
| 271 | C23 | SD16×D17 | 1 | 1 | yes | characteristic |
| 272 | Q8 | — | 2 | 4 | no | — |
| 273 | Q8 | — | 4 | 2 | no | — |
| 274 | Q8 | — | 4 | 2 | no | — |
| 275 | Q8 | — | 4 | 2 | no | — |
| 276 | C2×C4 | — | 4 | 2 | no | — |
| 277 | C2×C4 | — | 4 | 2 | no | — |
| 278 | C2×C4 | — | 4 | 2 | no | — |
| 279 | C2×C4 | — | 17 | 1 | no | — |
| 280 | C2×C4 | — | 17 | 1 | no | — |
| 281 | C2×C4 | — | 17 | 1 | no | — |
| 282 | C2×C4 | — | 17 | 1 | no | — |
| 283 | C2×C4 | — | 34 | 2 | no | — |
| 284 | C2×C4 | — | 68 | 1 | no | — |
| 285 | C2×C4 | — | 68 | 2 | no | — |
| 286 | C2×C4 | — | 136 | 2 | no | — |
| 287 | C2×C4 | #10587564 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | C22⋊C4×D17 | 1 | 1 | yes | characteristic |
| 289 | C2×C4 | D8×D17 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | #10587597 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | #10587601 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | #10587625 | 1 | 1 | yes | characteristic |
| 293 | C8 | — | 8 | 1 | no | — |
| 294 | C8 | — | 34 | 1 | no | — |
| 295 | C8 | — | 34 | 1 | no | — |
| 296 | C8 | — | 136 | 1 | no | — |
| 297 | C8 | #10587563 | 1 | 1 | yes | characteristic |
| 298 | C8 | #10587564 | 1 | 1 | yes | characteristic |
| 299 | C8 | C2×#63225 | 1 | 1 | yes | characteristic |
| 300 | C8 | #10587625 | 1 | 1 | yes | characteristic |
| 301 | C22 | — | 2 | 1 | no | — |
| 302 | C22 | — | 2 | 1 | no | — |
| 303 | C22 | #102×D17 | 1 | 1 | yes | characteristic |
| 304 | C22 | #11763012 | 1 | 1 | yes | characteristic |
| 305 | C22 | #11763097 | 1 | 1 | yes | characteristic |
| 306 | C4 | — | 4 | 4 | no | — |
| 307 | C4 | — | 17 | 2 | no | — |
| 308 | C4 | — | 34 | 1 | no | — |
| 309 | C4 | — | 34 | 1 | no | — |
| 310 | C4 | — | 34 | 1 | no | — |
| 311 | C4 | — | 136 | 2 | no | — |
| 312 | C4 | #11762667 | 1 | 1 | yes | characteristic |
| 313 | C4 | #11762807 | 1 | 1 | yes | characteristic |
| 314 | C4 | #11763002 | 1 | 1 | yes | characteristic |
| 315 | C4 | #11763087 | 1 | 1 | yes | characteristic |
| 316 | C2 | — | 2 | 1 | no | — |
| 317 | C2 | — | 2 | 1 | no | — |
| 318 | C2 | #14528782 | 1 | 1 | yes | characteristic |
| 319 | C2 | D17×#362 | 1 | 1 | yes | characteristic |
| 320 | C2 | #14530532 | 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