← #20287519
Subgroups — #20287519
Order 6784.
326 conjugacy classes of subgroups.
The subgroup lattice has 310 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #16811351 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #16811349 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #16810992 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #16810990 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #16810972 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #16810933 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #16810930 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16810907 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #16810833 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #16810831 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #16810831 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #16810751 | C2 | 1 | 1 | yes | maximal, characteristic |
| 13 | #16810727 | C2 | 1 | 1 | yes | maximal, characteristic |
| 14 | #16810715 | C2 | 1 | 1 | yes | maximal, characteristic |
| 15 | C53×#441 | C2 | 1 | 1 | yes | maximal, characteristic |
| 16 | C2×#11719925 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#11719925 | C22 | 1 | 1 | yes | characteristic |
| 18 | #13133217 | C22 | 1 | 1 | yes | characteristic |
| 19 | #13133216 | — | 2 | 1 | no | — |
| 20 | #13133216 | C22 | 1 | 1 | yes | characteristic |
| 21 | #13133215 | C22 | 1 | 1 | yes | characteristic |
| 22 | #13133215 | C22 | 1 | 1 | yes | characteristic |
| 23 | #13133209 | — | 2 | 1 | no | — |
| 24 | #13133209 | — | 2 | 1 | no | — |
| 25 | #13133209 | C22 | 1 | 1 | yes | characteristic |
| 26 | #13133208 | — | 2 | 1 | no | — |
| 27 | #13133208 | C22 | 1 | 1 | yes | characteristic |
| 28 | #13133207 | — | 2 | 1 | no | — |
| 29 | #13133207 | — | 2 | 1 | no | — |
| 30 | Q8×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 31 | Q8×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 32 | C4×Dic106 | — | 2 | 1 | no | — |
| 33 | C4×Dic106 | — | 2 | 1 | no | — |
| 34 | C4×Dic106 | C22 | 1 | 1 | yes | characteristic |
| 35 | #13133182 | C22 | 1 | 1 | yes | characteristic |
| 36 | #13133182 | C22 | 1 | 1 | yes | characteristic |
| 37 | #13133150 | C22 | 1 | 1 | yes | characteristic |
| 38 | #13133148 | C22 | 1 | 1 | yes | characteristic |
| 39 | C4⋊#88322 | C22 | 1 | 1 | yes | characteristic |
| 40 | #13133146 | C22 | 1 | 1 | yes | characteristic |
| 41 | #13133146 | C22 | 1 | 1 | yes | characteristic |
| 42 | #13133142 | C22 | 1 | 1 | yes | characteristic |
| 43 | #13133142 | C22 | 1 | 1 | yes | characteristic |
| 44 | #13133142 | C22 | 1 | 1 | yes | characteristic |
| 45 | #13133142 | C22 | 1 | 1 | yes | characteristic |
| 46 | #13133141 | C22 | 1 | 1 | yes | characteristic |
| 47 | #13133141 | C22 | 1 | 1 | yes | characteristic |
| 48 | #13133140 | C22 | 1 | 1 | yes | characteristic |
| 49 | #13133140 | C22 | 1 | 1 | yes | characteristic |
| 50 | #13133127 | C22 | 1 | 1 | yes | characteristic |
| 51 | C8×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 52 | C2×Q16×C53 | C22 | 1 | 1 | yes | characteristic |
| 53 | C4×Q8×C53 | C22 | 1 | 1 | yes | characteristic |
| 54 | C4×Q8×C53 | C22 | 1 | 1 | yes | characteristic |
| 55 | #106×C53 | C22 | 1 | 1 | yes | characteristic |
| 56 | #103×C53 | C22 | 1 | 1 | yes | characteristic |
| 57 | #103×C53 | C22 | 1 | 1 | yes | characteristic |
| 58 | #97×C53 | C22 | 1 | 1 | yes | characteristic |
| 59 | C2×Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 60 | C2×Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 61 | C2×Dic106 | — | 2 | 1 | no | — |
| 62 | C2×Dic106 | D4 | 1 | 1 | yes | characteristic |
| 63 | C2×Dic106 | C23 | 1 | 1 | yes | characteristic |
| 64 | C2×Dic106 | C23 | 1 | 1 | yes | characteristic |
| 65 | Q16×C53 | — | 2 | 2 | no | — |
| 66 | C2×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 67 | C2×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 68 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 69 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 70 | C4⋊C4×C53 | C23 | 1 | 1 | yes | characteristic |
| 71 | C4⋊C4×C53 | C23 | 1 | 1 | yes | characteristic |
| 72 | C42×C53 | — | 2 | 1 | no | — |
| 73 | C42×C53 | — | 2 | 1 | no | — |
| 74 | C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 75 | #11719925 | — | 2 | 2 | no | — |
| 76 | #11719925 | — | 2 | 2 | no | — |
| 77 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 78 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 79 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 80 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 81 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 82 | C4⋊Dic53 | D4 | 1 | 1 | yes | characteristic |
| 83 | C4⋊Dic53 | C23 | 1 | 1 | yes | characteristic |
| 84 | C4⋊Dic53 | C23 | 1 | 1 | yes | characteristic |
| 85 | #11719919 | — | 2 | 1 | no | — |
| 86 | #11719919 | — | 2 | 1 | no | — |
| 87 | #11719919 | — | 2 | 1 | no | — |
| 88 | #11719919 | — | 2 | 1 | no | — |
| 89 | #11719919 | — | 2 | 1 | no | — |
| 90 | #11719919 | — | 2 | 1 | no | — |
| 91 | #11719919 | — | 2 | 1 | no | — |
| 92 | #11719919 | — | 2 | 1 | no | — |
| 93 | #11719919 | — | 2 | 1 | no | — |
| 94 | #11719919 | — | 2 | 1 | no | — |
| 95 | #11719919 | D4 | 1 | 1 | yes | characteristic |
| 96 | #11719919 | D4 | 1 | 1 | yes | characteristic |
| 97 | C4×Dic53 | — | 2 | 1 | no | — |
| 98 | C4×Dic53 | — | 2 | 1 | no | — |
| 99 | C4×Dic53 | — | 2 | 1 | no | — |
| 100 | C4×Dic53 | — | 2 | 1 | no | — |
| 101 | C4×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 102 | C4×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 103 | C2×#88322 | C23 | 1 | 1 | yes | characteristic |
| 104 | C2×#88322 | C23 | 1 | 1 | yes | characteristic |
| 105 | Q8×C53 | — | 2 | 1 | no | — |
| 106 | Q8×C53 | — | 2 | 1 | no | — |
| 107 | Q8×C53 | — | 2 | 1 | no | — |
| 108 | Q8×C53 | #41 | 1 | 2 | yes | — |
| 109 | C2×C4×C53 | — | 2 | 1 | no | — |
| 110 | C2×C4×C53 | — | 2 | 1 | no | — |
| 111 | C2×C4×C53 | — | 2 | 1 | no | — |
| 112 | C2×C4×C53 | — | 2 | 1 | no | — |
| 113 | C2×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 114 | C2×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 116 | C2×Dic53 | — | 2 | 1 | no | — |
| 117 | C2×Dic53 | — | 2 | 1 | no | — |
| 118 | C2×Dic53 | — | 2 | 1 | no | — |
| 119 | C2×Dic53 | — | 2 | 1 | no | — |
| 120 | C2×Dic53 | C2×D4 | 1 | 1 | yes | characteristic |
| 121 | C2×Dic53 | C2×D4 | 1 | 1 | yes | characteristic |
| 122 | C2×Dic53 | C2×D4 | 1 | 1 | yes | characteristic |
| 123 | C2×Dic53 | C2×D4 | 1 | 1 | yes | characteristic |
| 124 | Dic106 | — | 2 | 2 | no | — |
| 125 | Dic106 | — | 2 | 1 | no | — |
| 126 | Dic106 | — | 2 | 1 | no | — |
| 127 | Dic106 | — | 2 | 1 | no | — |
| 128 | Dic106 | — | 2 | 1 | no | — |
| 129 | Dic106 | — | 4 | 2 | no | — |
| 130 | C8×C53 | — | 2 | 1 | no | — |
| 131 | C8×C53 | — | 2 | 1 | no | — |
| 132 | #88322 | — | 2 | 1 | no | — |
| 133 | #88322 | — | 2 | 1 | no | — |
| 134 | C22×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 135 | C4×C53 | — | 2 | 2 | no | — |
| 136 | C4×C53 | — | 2 | 1 | no | — |
| 137 | C4×C53 | — | 2 | 1 | no | — |
| 138 | C4×C53 | — | 4 | 1 | no | — |
| 139 | C4×C53 | — | 4 | 1 | no | — |
| 140 | C4×C53 | — | 4 | 1 | no | — |
| 141 | C4×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 142 | C4×C53 | 2−1+4 | 1 | 1 | yes | characteristic |
| 143 | Dic53 | — | 2 | 1 | no | — |
| 144 | Dic53 | — | 2 | 1 | no | — |
| 145 | Dic53 | — | 2 | 1 | no | — |
| 146 | Dic53 | — | 4 | 1 | no | — |
| 147 | Dic53 | — | 4 | 1 | no | — |
| 148 | Dic53 | — | 4 | 1 | no | — |
| 149 | Dic53 | — | 4 | 1 | no | — |
| 150 | Dic53 | #137 | 1 | 2 | yes | — |
| 151 | #3374 | — | 53 | 1 | no | maximal, Sylow |
| 152 | C2×C53 | #548 | 1 | 1 | yes | characteristic |
| 153 | C2×C53 | C2×#137 | 1 | 1 | yes | characteristic |
| 154 | C2×C53 | #578 | 1 | 1 | yes | characteristic |
| 155 | #557 | — | 53 | 1 | no | — |
| 156 | #557 | — | 53 | 1 | no | — |
| 157 | #491 | — | 53 | 1 | no | — |
| 158 | #491 | — | 53 | 1 | no | — |
| 159 | #487 | — | 53 | 1 | no | — |
| 160 | #479 | — | 53 | 1 | no | — |
| 161 | #479 | — | 53 | 1 | no | — |
| 162 | #475 | — | 53 | 1 | no | — |
| 163 | #462 | — | 53 | 1 | no | — |
| 164 | #462 | — | 53 | 1 | no | — |
| 165 | #462 | — | 53 | 1 | no | — |
| 166 | #446 | — | 53 | 1 | no | — |
| 167 | #441 | — | 53 | 1 | no | — |
| 168 | #441 | D53 | 1 | 1 | yes | characteristic |
| 169 | C4×Q16 | — | 53 | 1 | no | — |
| 170 | C53 | #3374 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 171 | C2×Q16 | — | 53 | 1 | no | — |
| 172 | C2×Q16 | — | 53 | 1 | no | — |
| 173 | C2×Q16 | C2×D53 | 1 | 1 | yes | characteristic |
| 174 | C4⋊Q8 | — | 53 | 1 | no | — |
| 175 | C4⋊Q8 | — | 53 | 1 | no | — |
| 176 | C4⋊Q8 | — | 53 | 1 | no | — |
| 177 | C4⋊Q8 | — | 53 | 1 | no | — |
| 178 | C4⋊Q8 | — | 106 | 1 | no | — |
| 179 | #125 | — | 53 | 1 | no | — |
| 180 | #125 | — | 53 | 1 | no | — |
| 181 | #125 | — | 106 | 1 | no | — |
| 182 | #125 | — | 106 | 1 | no | — |
| 183 | #125 | — | 106 | 1 | no | — |
| 184 | #125 | — | 106 | 1 | no | — |
| 185 | #125 | — | 106 | 1 | no | — |
| 186 | C4×Q8 | — | 53 | 1 | no | — |
| 187 | C4×Q8 | — | 53 | 1 | no | — |
| 188 | C4×Q8 | — | 53 | 1 | no | — |
| 189 | C4×Q8 | — | 53 | 1 | no | — |
| 190 | C4×Q8 | — | 53 | 1 | no | — |
| 191 | C4×Q8 | — | 106 | 1 | no | — |
| 192 | C4×Q8 | — | 106 | 1 | no | — |
| 193 | C4×Q8 | C2×D53 | 1 | 1 | yes | characteristic |
| 194 | C4×Q8 | C2×D53 | 1 | 1 | yes | characteristic |
| 195 | #107 | — | 53 | 1 | no | — |
| 196 | #106 | — | 53 | 1 | no | — |
| 197 | #106 | C2×D53 | 1 | 1 | yes | characteristic |
| 198 | C4⋊C8 | — | 53 | 1 | no | — |
| 199 | C4⋊C8 | — | 53 | 1 | no | — |
| 200 | C4⋊C8 | — | 53 | 1 | no | — |
| 201 | #103 | — | 53 | 1 | no | — |
| 202 | #103 | — | 53 | 1 | no | — |
| 203 | #103 | — | 53 | 1 | no | — |
| 204 | #103 | — | 53 | 1 | no | — |
| 205 | #103 | — | 53 | 1 | no | — |
| 206 | #103 | — | 53 | 1 | no | — |
| 207 | #103 | — | 53 | 1 | no | — |
| 208 | #103 | — | 53 | 1 | no | — |
| 209 | #103 | C2×D53 | 1 | 1 | yes | characteristic |
| 210 | #103 | C2×D53 | 1 | 1 | yes | characteristic |
| 211 | #97 | — | 53 | 1 | no | — |
| 212 | #97 | C2×D53 | 1 | 1 | yes | characteristic |
| 213 | C4×C8 | — | 53 | 1 | no | — |
| 214 | C2×Q8 | — | 53 | 1 | no | — |
| 215 | C2×Q8 | — | 53 | 1 | no | — |
| 216 | C2×Q8 | — | 53 | 1 | no | — |
| 217 | C2×Q8 | — | 106 | 1 | no | — |
| 218 | C2×Q8 | C22×D53 | 1 | 1 | yes | characteristic |
| 219 | C2×Q8 | C22×D53 | 1 | 1 | yes | characteristic |
| 220 | Q16 | — | 2 | 2 | no | — |
| 221 | Q16 | — | 106 | 2 | no | — |
| 222 | Q16 | — | 106 | 2 | no | — |
| 223 | C2×C8 | — | 53 | 1 | no | — |
| 224 | C2×C8 | — | 53 | 1 | no | — |
| 225 | C2×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 226 | C2×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 227 | C4⋊C4 | — | 2 | 1 | no | — |
| 228 | C4⋊C4 | — | 2 | 1 | no | — |
| 229 | C4⋊C4 | — | 53 | 1 | no | — |
| 230 | C4⋊C4 | — | 53 | 1 | no | — |
| 231 | C4⋊C4 | — | 53 | 1 | no | — |
| 232 | C4⋊C4 | — | 53 | 1 | no | — |
| 233 | C4⋊C4 | — | 53 | 1 | no | — |
| 234 | C4⋊C4 | — | 106 | 1 | no | — |
| 235 | C4⋊C4 | — | 106 | 1 | no | — |
| 236 | C4⋊C4 | — | 106 | 1 | no | — |
| 237 | C4⋊C4 | — | 106 | 1 | no | — |
| 238 | C4⋊C4 | — | 106 | 1 | no | — |
| 239 | C4⋊C4 | — | 106 | 1 | no | — |
| 240 | C4⋊C4 | — | 106 | 1 | no | — |
| 241 | C4⋊C4 | — | 106 | 1 | no | — |
| 242 | C4⋊C4 | — | 106 | 1 | no | — |
| 243 | C4⋊C4 | — | 106 | 1 | no | — |
| 244 | C4⋊C4 | — | 106 | 1 | no | — |
| 245 | C4⋊C4 | — | 106 | 1 | no | — |
| 246 | C4⋊C4 | — | 106 | 1 | no | — |
| 247 | C4⋊C4 | — | 106 | 1 | no | — |
| 248 | C4⋊C4 | — | 106 | 1 | no | — |
| 249 | C4⋊C4 | C22×D53 | 1 | 1 | yes | characteristic |
| 250 | C4⋊C4 | C22×D53 | 1 | 1 | yes | characteristic |
| 251 | C42 | — | 2 | 1 | no | — |
| 252 | C42 | — | 2 | 1 | no | — |
| 253 | C42 | — | 53 | 1 | no | — |
| 254 | C42 | — | 53 | 1 | no | — |
| 255 | C42 | — | 106 | 1 | no | — |
| 256 | C42 | — | 106 | 1 | no | — |
| 257 | C42 | — | 106 | 1 | no | — |
| 258 | C42 | — | 106 | 1 | no | — |
| 259 | C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 260 | Q8 | — | 2 | 1 | no | — |
| 261 | Q8 | — | 2 | 1 | no | — |
| 262 | Q8 | — | 2 | 1 | no | — |
| 263 | Q8 | — | 106 | 2 | no | — |
| 264 | Q8 | — | 106 | 1 | no | — |
| 265 | Q8 | — | 106 | 1 | no | — |
| 266 | Q8 | — | 106 | 1 | no | — |
| 267 | Q8 | — | 106 | 1 | no | — |
| 268 | Q8 | — | 212 | 2 | no | — |
| 269 | Q8 | #11719945 | 1 | 2 | yes | — |
| 270 | C2×C4 | — | 2 | 1 | no | — |
| 271 | C2×C4 | — | 2 | 1 | no | — |
| 272 | C2×C4 | — | 2 | 1 | no | — |
| 273 | C2×C4 | — | 2 | 1 | no | — |
| 274 | C2×C4 | — | 53 | 1 | no | — |
| 275 | C2×C4 | — | 53 | 1 | no | — |
| 276 | C2×C4 | — | 53 | 1 | no | — |
| 277 | C2×C4 | — | 53 | 1 | no | — |
| 278 | C2×C4 | — | 106 | 1 | no | — |
| 279 | C2×C4 | — | 106 | 1 | no | — |
| 280 | C2×C4 | — | 106 | 1 | no | — |
| 281 | C2×C4 | — | 106 | 1 | no | — |
| 282 | C2×C4 | D4×D53 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | D4×D53 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | C23×D53 | 1 | 1 | yes | characteristic |
| 285 | C8 | — | 2 | 1 | no | — |
| 286 | C8 | — | 2 | 1 | no | — |
| 287 | C8 | — | 106 | 1 | no | — |
| 288 | C8 | — | 106 | 1 | no | — |
| 289 | C22 | C2×D4×D53 | 1 | 1 | yes | characteristic |
| 290 | C4 | — | 2 | 2 | no | — |
| 291 | C4 | — | 2 | 1 | no | — |
| 292 | C4 | — | 2 | 1 | no | — |
| 293 | C4 | — | 4 | 1 | no | — |
| 294 | C4 | — | 4 | 1 | no | — |
| 295 | C4 | — | 4 | 1 | no | — |
| 296 | C4 | — | 53 | 2 | no | — |
| 297 | C4 | — | 106 | 1 | no | — |
| 298 | C4 | — | 106 | 1 | no | — |
| 299 | C4 | — | 106 | 1 | no | — |
| 300 | C4 | — | 212 | 1 | no | — |
| 301 | C4 | — | 212 | 1 | no | — |
| 302 | C4 | — | 212 | 1 | no | — |
| 303 | C4 | — | 212 | 1 | no | — |
| 304 | C4 | C2×#11719945 | 1 | 1 | yes | characteristic |
| 305 | C4 | #13133264 | 1 | 1 | yes | characteristic |
| 306 | C2 | #16811282 | 1 | 1 | yes | characteristic |
| 307 | C2 | #137×D53 | 1 | 1 | yes | characteristic |
| 308 | C2 | #16811439 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Finite Group Databasebeta