← #17189696
Subgroups — #17189696
Order 4352.
622 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#11763214 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #14539212 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #14531021 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #14531012 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #14531008 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #14530497 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | C4×C17×M32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#10587590 | — | 2 | 1 | no | — |
| 9 | C2×#10587590 | C22 | 1 | 2 | yes | — |
| 10 | #11763359 | — | 2 | 1 | no | — |
| 11 | C2×#10587555 | — | 2 | 1 | no | — |
| 12 | #11763214 | — | 2 | 1 | no | — |
| 13 | #11763214 | — | 2 | 1 | no | — |
| 14 | #11763214 | C22 | 1 | 2 | yes | — |
| 15 | #11763214 | C22 | 1 | 2 | yes | — |
| 16 | #11763193 | — | 2 | 1 | no | — |
| 17 | #11763193 | C22 | 1 | 2 | yes | — |
| 18 | #11763189 | C22 | 1 | 1 | yes | characteristic |
| 19 | #11763186 | C22 | 1 | 1 | yes | characteristic |
| 20 | #11763182 | — | 2 | 1 | no | — |
| 21 | C2×#10587489 | — | 2 | 1 | no | — |
| 22 | C2×#10587489 | C22 | 1 | 2 | yes | — |
| 23 | #11762614 | C22 | 1 | 8 | yes | — |
| 24 | #11762611 | C22 | 1 | 8 | yes | — |
| 25 | C2×C17×M32 | C22 | 1 | 2 | yes | — |
| 26 | C2×C4×C8×C17 | C22 | 1 | 1 | yes | characteristic |
| 27 | C17×#346 | C22 | 1 | 2 | yes | — |
| 28 | C4×C16×C17 | C22 | 1 | 2 | yes | — |
| 29 | C2×#63252 | — | 2 | 2 | no | — |
| 30 | C22×Dic34 | — | 4 | 1 | no | — |
| 31 | C22×C8×C17 | D4 | 1 | 1 | yes | characteristic |
| 32 | C22×C8×C17 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×C42×C17 | D4 | 1 | 1 | yes | characteristic |
| 34 | #10587616 | — | 4 | 1 | no | — |
| 35 | #10587616 | — | 4 | 1 | no | — |
| 36 | C2×C4⋊Dic17 | — | 2 | 2 | no | — |
| 37 | C2×C4⋊Dic17 | — | 4 | 1 | no | — |
| 38 | C2×Dic68 | — | 2 | 1 | no | — |
| 39 | C2×Dic68 | C23 | 1 | 2 | yes | — |
| 40 | #10587590 | — | 2 | 8 | no | — |
| 41 | #10587590 | — | 4 | 4 | no | — |
| 42 | C2×D136 | — | 2 | 1 | no | — |
| 43 | C2×D136 | C23 | 1 | 2 | yes | — |
| 44 | C2×#63219 | — | 2 | 2 | no | — |
| 45 | C2×#63219 | — | 2 | 2 | no | — |
| 46 | #10587561 | — | 2 | 2 | no | — |
| 47 | #10587560 | — | 2 | 1 | no | — |
| 48 | C4×D68 | — | 4 | 1 | no | — |
| 49 | C4×D68 | — | 4 | 1 | no | — |
| 50 | #10587555 | — | 2 | 1 | no | — |
| 51 | #10587555 | — | 2 | 2 | no | — |
| 52 | #10587555 | — | 2 | 2 | no | — |
| 53 | #10587555 | — | 4 | 1 | no | — |
| 54 | #10587555 | — | 4 | 1 | no | — |
| 55 | C17×M32 | — | 2 | 4 | no | — |
| 56 | C2×C16×C17 | C23 | 1 | 4 | yes | — |
| 57 | C4×C8×C17 | D4 | 1 | 1 | yes | characteristic |
| 58 | C4×C8×C17 | D4 | 1 | 1 | yes | characteristic |
| 59 | C4×C8×C17 | C23 | 1 | 1 | yes | characteristic |
| 60 | C4×C8×C17 | C23 | 1 | 1 | yes | characteristic |
| 61 | #10587489 | — | 2 | 2 | no | — |
| 62 | #10587489 | — | 2 | 2 | no | — |
| 63 | #10587489 | — | 2 | 2 | no | — |
| 64 | #10587489 | C23 | 1 | 4 | yes | — |
| 65 | C22×C4×C17 | D8 | 1 | 1 | yes | characteristic |
| 66 | C22×C4×C17 | D8 | 1 | 1 | yes | characteristic |
| 67 | C22×C4×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | C2×C22⋊D17 | — | 4 | 2 | no | — |
| 69 | C22×Dic17 | — | 4 | 2 | no | — |
| 70 | #63252 | — | 4 | 4 | no | — |
| 71 | #63252 | — | 4 | 4 | no | — |
| 72 | C2×D68 | — | 2 | 2 | no | — |
| 73 | C2×D68 | — | 4 | 1 | no | — |
| 74 | C2×D68 | — | 4 | 1 | no | — |
| 75 | C2×C4×D17 | — | 4 | 2 | no | — |
| 76 | C2×Dic34 | — | 2 | 2 | no | — |
| 77 | C2×Dic34 | — | 4 | 2 | no | — |
| 78 | C2×Dic34 | — | 4 | 2 | no | — |
| 79 | C2×Dic34 | — | 8 | 4 | no | — |
| 80 | C2×C8×C17 | SD16 | 1 | 4 | yes | — |
| 81 | C2×C8×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C2×C8×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C2×C8×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C8×C17 | #41 | 1 | 4 | yes | — |
| 85 | C2×C8×C17 | C24 | 1 | 1 | yes | characteristic |
| 86 | C42×C17 | D8 | 1 | 1 | yes | characteristic |
| 87 | C42×C17 | D8 | 1 | 1 | yes | characteristic |
| 88 | C42×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C42×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | #63227 | — | 4 | 2 | no | — |
| 91 | #63227 | — | 4 | 2 | no | — |
| 92 | C4⋊Dic17 | — | 2 | 4 | no | — |
| 93 | C4⋊Dic17 | — | 4 | 2 | no | — |
| 94 | C4⋊Dic17 | — | 4 | 2 | no | — |
| 95 | C4⋊Dic17 | — | 4 | 1 | no | — |
| 96 | C4⋊Dic17 | — | 4 | 1 | no | — |
| 97 | C4⋊Dic17 | — | 4 | 2 | no | — |
| 98 | Dic68 | — | 2 | 4 | no | — |
| 99 | Dic68 | — | 4 | 2 | no | — |
| 100 | D136 | — | 2 | 4 | no | — |
| 101 | D136 | — | 4 | 2 | no | — |
| 102 | #63219 | — | 4 | 4 | no | — |
| 103 | #63219 | — | 4 | 4 | no | — |
| 104 | C16×C17 | — | 2 | 4 | no | — |
| 105 | #33707 | — | 17 | 1 | no | maximal, Sylow |
| 106 | C23×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 107 | C22×D17 | — | 4 | 2 | no | — |
| 108 | C2×C4×C17 | — | 2 | 2 | no | — |
| 109 | C2×C4×C17 | — | 2 | 2 | no | — |
| 110 | C2×C4×C17 | #124 | 1 | 2 | yes | — |
| 111 | C2×C4×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 112 | C2×C4×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 113 | C2×C4×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 114 | C2×C4×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C17 | C2×SD16 | 1 | 2 | yes | — |
| 117 | C2×C4×C17 | C22×D4 | 1 | 1 | yes | characteristic |
| 118 | C22⋊D17 | — | 8 | 4 | no | — |
| 119 | C2×Dic17 | — | 4 | 2 | no | — |
| 120 | C2×Dic17 | — | 4 | 4 | no | — |
| 121 | C2×Dic17 | — | 8 | 4 | no | — |
| 122 | D68 | — | 4 | 2 | no | — |
| 123 | D68 | — | 4 | 2 | no | — |
| 124 | D68 | — | 8 | 2 | no | — |
| 125 | D68 | — | 8 | 2 | no | — |
| 126 | C4×D17 | — | 8 | 4 | no | — |
| 127 | Dic34 | — | 4 | 2 | no | — |
| 128 | Dic34 | — | 4 | 2 | no | — |
| 129 | Dic34 | — | 8 | 4 | no | — |
| 130 | Dic34 | — | 8 | 4 | no | — |
| 131 | C8×C17 | #124 | 1 | 2 | yes | — |
| 132 | C8×C17 | C2×SD16 | 1 | 2 | yes | — |
| 133 | C8×C17 | C2×SD16 | 1 | 2 | yes | — |
| 134 | C8×C17 | C2×#41 | 1 | 2 | yes | — |
| 135 | C2×#500 | — | 17 | 1 | no | — |
| 136 | #3147 | — | 17 | 1 | no | — |
| 137 | #2243 | — | 17 | 4 | no | — |
| 138 | #2241 | — | 17 | 2 | no | — |
| 139 | #2240 | — | 17 | 2 | no | — |
| 140 | #2139 | — | 17 | 4 | no | — |
| 141 | C4×M32 | D17 | 1 | 1 | yes | characteristic, complemented |
| 142 | C22×C17 | — | 2 | 1 | no | — |
| 143 | C22×C17 | — | 2 | 1 | no | — |
| 144 | C22×C17 | #486 | 1 | 2 | yes | — |
| 145 | C22×C17 | C22×D8 | 1 | 1 | yes | characteristic |
| 146 | C2×D17 | — | 8 | 2 | no | — |
| 147 | C2×D17 | — | 8 | 2 | no | — |
| 148 | C2×D17 | — | 8 | 2 | no | — |
| 149 | C4×C17 | — | 2 | 1 | no | — |
| 150 | C4×C17 | — | 2 | 1 | no | — |
| 151 | C4×C17 | #486 | 1 | 2 | yes | — |
| 152 | C4×C17 | #509 | 1 | 2 | yes | — |
| 153 | C4×C17 | #510 | 1 | 2 | yes | — |
| 154 | C4×C17 | C2×#124 | 1 | 1 | yes | characteristic |
| 155 | C4×C17 | C22×SD16 | 1 | 1 | yes | characteristic |
| 156 | Dic17 | — | 8 | 2 | no | — |
| 157 | Dic17 | — | 8 | 4 | no | — |
| 158 | C2×#135 | — | 17 | 2 | no | — |
| 159 | C2×#135 | — | 34 | 1 | no | — |
| 160 | #532 | — | 34 | 1 | no | — |
| 161 | C2×C4⋊Q8 | — | 34 | 1 | no | — |
| 162 | C2×M32 | C2×D17 | 1 | 2 | yes | — |
| 163 | #500 | — | 17 | 2 | no | — |
| 164 | #500 | — | 17 | 2 | no | — |
| 165 | #500 | — | 34 | 1 | no | — |
| 166 | #500 | — | 34 | 1 | no | — |
| 167 | #495 | — | 17 | 2 | no | — |
| 168 | #495 | — | 34 | 1 | no | — |
| 169 | #494 | — | 17 | 1 | no | — |
| 170 | #493 | — | 17 | 1 | no | — |
| 171 | #492 | — | 34 | 1 | no | — |
| 172 | C2×#107 | — | 17 | 2 | no | — |
| 173 | C2×#107 | — | 34 | 1 | no | — |
| 174 | C2×C4×C8 | C2×D17 | 1 | 1 | yes | characteristic |
| 175 | #358 | — | 17 | 8 | no | — |
| 176 | #357 | — | 17 | 8 | no | — |
| 177 | #346 | C2×D17 | 1 | 2 | yes | — |
| 178 | C4×C16 | C2×D17 | 1 | 2 | yes | — |
| 179 | C2×C17 | — | 2 | 2 | no | — |
| 180 | C2×C17 | C2×#486 | 1 | 1 | yes | characteristic |
| 181 | C2×C17 | C2×#509 | 1 | 1 | yes | characteristic |
| 182 | C2×C17 | C2×#510 | 1 | 1 | yes | characteristic |
| 183 | D17 | — | 16 | 2 | no | — |
| 184 | C2×#41 | — | 34 | 2 | no | — |
| 185 | C22×Q8 | — | 68 | 1 | no | — |
| 186 | #135 | — | 34 | 8 | no | — |
| 187 | #135 | — | 68 | 4 | no | — |
| 188 | C2×Q16 | — | 17 | 2 | no | — |
| 189 | C2×Q16 | — | 34 | 1 | no | — |
| 190 | C2×SD16 | — | 34 | 2 | no | — |
| 191 | C2×SD16 | — | 34 | 2 | no | — |
| 192 | C2×D8 | — | 17 | 2 | no | — |
| 193 | C2×D8 | — | 34 | 1 | no | — |
| 194 | C22×C8 | D68 | 1 | 1 | yes | characteristic |
| 195 | C22×C8 | C22×D17 | 1 | 1 | yes | characteristic |
| 196 | C4⋊Q8 | — | 34 | 2 | no | — |
| 197 | C4⋊Q8 | — | 34 | 2 | no | — |
| 198 | C4⋊Q8 | — | 34 | 1 | no | — |
| 199 | C4⋊Q8 | — | 68 | 1 | no | — |
| 200 | C4⋊Q8 | — | 68 | 1 | no | — |
| 201 | #127 | — | 34 | 1 | no | — |
| 202 | #124 | — | 34 | 2 | no | — |
| 203 | #121 | — | 68 | 1 | no | — |
| 204 | #121 | — | 68 | 1 | no | — |
| 205 | C4×D4 | — | 68 | 1 | no | — |
| 206 | C4×D4 | — | 68 | 1 | no | — |
| 207 | C2×C4⋊C4 | — | 34 | 2 | no | — |
| 208 | C2×C4⋊C4 | — | 68 | 1 | no | — |
| 209 | C2×C42 | D68 | 1 | 1 | yes | characteristic |
| 210 | M32 | — | 2 | 4 | no | — |
| 211 | C2×C16 | C22×D17 | 1 | 4 | yes | — |
| 212 | #107 | — | 17 | 4 | no | — |
| 213 | #107 | — | 34 | 2 | no | — |
| 214 | #107 | — | 34 | 2 | no | — |
| 215 | #107 | — | 34 | 2 | no | — |
| 216 | C4×C8 | D68 | 1 | 1 | yes | characteristic |
| 217 | C4×C8 | D68 | 1 | 1 | yes | characteristic |
| 218 | C4×C8 | C22×D17 | 1 | 1 | yes | characteristic |
| 219 | C4×C8 | C22×D17 | 1 | 1 | yes | characteristic |
| 220 | C17 | #33707 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 221 | #41 | — | 68 | 4 | no | — |
| 222 | #41 | — | 68 | 4 | no | — |
| 223 | C2×Q8 | — | 34 | 2 | no | — |
| 224 | C2×Q8 | — | 68 | 2 | no | — |
| 225 | C2×Q8 | — | 68 | 2 | no | — |
| 226 | C2×Q8 | — | 136 | 4 | no | — |
| 227 | C2×D4 | — | 34 | 2 | no | — |
| 228 | C2×D4 | — | 68 | 2 | no | — |
| 229 | C2×D4 | — | 68 | 1 | no | — |
| 230 | C2×D4 | — | 68 | 1 | no | — |
| 231 | C22×C4 | — | 68 | 2 | no | — |
| 232 | C22×C4 | — | 68 | 2 | no | — |
| 233 | C22×C4 | D136 | 1 | 1 | yes | characteristic |
| 234 | C22×C4 | D136 | 1 | 1 | yes | characteristic |
| 235 | C22×C4 | C2×D68 | 1 | 1 | yes | characteristic |
| 236 | Q16 | — | 34 | 4 | no | — |
| 237 | Q16 | — | 68 | 2 | no | — |
| 238 | SD16 | — | 68 | 4 | no | — |
| 239 | SD16 | — | 68 | 4 | no | — |
| 240 | D8 | — | 34 | 4 | no | — |
| 241 | D8 | — | 68 | 2 | no | — |
| 242 | C2×C8 | #63219 | 1 | 4 | yes | — |
| 243 | C2×C8 | C2×D68 | 1 | 1 | yes | characteristic |
| 244 | C2×C8 | C2×D68 | 1 | 1 | yes | characteristic |
| 245 | C2×C8 | C2×D68 | 1 | 1 | yes | characteristic |
| 246 | C2×C8 | #63252 | 1 | 4 | yes | — |
| 247 | C2×C8 | C23×D17 | 1 | 1 | yes | characteristic |
| 248 | C4⋊C4 | — | 34 | 4 | no | — |
| 249 | C4⋊C4 | — | 68 | 2 | no | — |
| 250 | C4⋊C4 | — | 68 | 2 | no | — |
| 251 | C4⋊C4 | — | 68 | 2 | no | — |
| 252 | C4⋊C4 | — | 68 | 1 | no | — |
| 253 | C4⋊C4 | — | 68 | 1 | no | — |
| 254 | C22⋊C4 | — | 68 | 2 | no | — |
| 255 | C22⋊C4 | — | 68 | 2 | no | — |
| 256 | C42 | D136 | 1 | 1 | yes | characteristic |
| 257 | C42 | D136 | 1 | 1 | yes | characteristic |
| 258 | C42 | C2×D68 | 1 | 1 | yes | characteristic |
| 259 | C42 | C2×D68 | 1 | 1 | yes | characteristic |
| 260 | C16 | — | 2 | 4 | no | — |
| 261 | C23 | — | 68 | 2 | no | — |
| 262 | C23 | C2×D136 | 1 | 1 | yes | characteristic |
| 263 | Q8 | — | 68 | 2 | no | — |
| 264 | Q8 | — | 68 | 2 | no | — |
| 265 | Q8 | — | 136 | 4 | no | — |
| 266 | Q8 | — | 136 | 4 | no | — |
| 267 | D4 | — | 68 | 2 | no | — |
| 268 | D4 | — | 68 | 2 | no | — |
| 269 | D4 | — | 136 | 4 | no | — |
| 270 | D4 | — | 136 | 2 | no | — |
| 271 | D4 | — | 136 | 2 | no | — |
| 272 | C2×C4 | — | 2 | 2 | no | — |
| 273 | C2×C4 | — | 2 | 2 | no | — |
| 274 | C2×C4 | — | 68 | 2 | no | — |
| 275 | C2×C4 | — | 68 | 4 | no | — |
| 276 | C2×C4 | — | 136 | 4 | no | — |
| 277 | C2×C4 | — | 136 | 4 | no | — |
| 278 | C2×C4 | #10587561 | 1 | 2 | yes | — |
| 279 | C2×C4 | C2×#63219 | 1 | 2 | yes | — |
| 280 | C2×C4 | C2×D136 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | C2×D136 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | C2×D136 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | C2×D136 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | C2×D136 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | C22×D68 | 1 | 1 | yes | characteristic |
| 286 | C8 | #10587561 | 1 | 2 | yes | — |
| 287 | C8 | C2×#63219 | 1 | 2 | yes | — |
| 288 | C8 | C2×#63219 | 1 | 2 | yes | — |
| 289 | C8 | C2×#63252 | 1 | 2 | yes | — |
| 290 | C22 | — | 2 | 1 | no | — |
| 291 | C22 | — | 2 | 1 | no | — |
| 292 | C22 | — | 136 | 2 | no | — |
| 293 | C22 | — | 136 | 2 | no | — |
| 294 | C22 | — | 136 | 2 | no | — |
| 295 | C22 | #11763156 | 1 | 2 | yes | — |
| 296 | C22 | C22×D136 | 1 | 1 | yes | characteristic |
| 297 | C4 | — | 2 | 1 | no | — |
| 298 | C4 | — | 2 | 1 | no | — |
| 299 | C4 | — | 136 | 2 | no | — |
| 300 | C4 | — | 136 | 4 | no | — |
| 301 | C4 | #11763156 | 1 | 2 | yes | — |
| 302 | C4 | #11763247 | 1 | 2 | yes | — |
| 303 | C4 | #11763252 | 1 | 2 | yes | — |
| 304 | C4 | C2×#10587561 | 1 | 1 | yes | characteristic |
| 305 | C4 | C22×#63219 | 1 | 1 | yes | characteristic |
| 306 | C2 | — | 2 | 2 | no | — |
| 307 | C2 | — | 272 | 2 | no | — |
| 308 | C2 | C2×#11763156 | 1 | 1 | yes | characteristic |
| 309 | C2 | C2×#11763247 | 1 | 1 | yes | characteristic |
| 310 | C2 | C2×#11763252 | 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