← #27538593
Subgroups — #27538593
Order 13568.
650 conjugacy classes of subgroups.
The subgroup lattice has 336 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×#13133163 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#16811040 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20284370 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20276821 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #20276804 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #20276497 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #20276435 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #20276434 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C53×#3059 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C22×Dic212 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16811194 | — | 2 | 1 | no | — |
| 12 | C2×#13133163 | — | 2 | 4 | no | — |
| 13 | C2×#13133163 | C22 | 1 | 2 | yes | — |
| 14 | C2×#13133163 | C22 | 1 | 2 | yes | — |
| 15 | #16811040 | — | 2 | 4 | no | — |
| 16 | C2×#13133157 | C22 | 1 | 2 | yes | — |
| 17 | C2×#13133157 | C22 | 1 | 1 | yes | characteristic |
| 18 | C22×#11719908 | C22 | 1 | 2 | yes | — |
| 19 | C22×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 20 | #16810881 | — | 2 | 2 | no | — |
| 21 | #16810864 | C22 | 1 | 4 | yes | — |
| 22 | #16810661 | C22 | 1 | 1 | yes | characteristic |
| 23 | #16810633 | — | 2 | 1 | no | — |
| 24 | #16810450 | C22 | 1 | 2 | yes | — |
| 25 | #16810448 | C22 | 1 | 2 | yes | — |
| 26 | #16810427 | C22 | 1 | 4 | yes | — |
| 27 | #16810426 | C22 | 1 | 4 | yes | — |
| 28 | C22×Q16×C53 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×#131×C53 | C22 | 1 | 1 | yes | characteristic |
| 30 | C53×#536 | — | 2 | 1 | no | — |
| 31 | C53×#470 | — | 2 | 2 | no | — |
| 32 | C53×#467 | C22 | 1 | 4 | yes | — |
| 33 | C53×#427 | C22 | 1 | 1 | yes | characteristic |
| 34 | C53×#419 | — | 2 | 1 | no | — |
| 35 | C22×Dic106 | — | 2 | 1 | no | — |
| 36 | C2×Dic212 | — | 2 | 1 | no | — |
| 37 | C2×Dic212 | — | 2 | 2 | no | — |
| 38 | C2×Dic212 | — | 2 | 2 | no | — |
| 39 | C2×Dic212 | C23 | 1 | 2 | yes | — |
| 40 | #13133215 | — | 4 | 1 | no | — |
| 41 | #13133204 | — | 4 | 1 | no | — |
| 42 | #13133198 | — | 4 | 2 | no | — |
| 43 | #13133196 | — | 2 | 4 | no | — |
| 44 | #13133176 | — | 2 | 1 | no | — |
| 45 | #13133163 | — | 2 | 4 | no | — |
| 46 | #13133163 | — | 2 | 4 | no | — |
| 47 | #13133157 | — | 2 | 2 | no | — |
| 48 | #13133157 | — | 2 | 4 | no | — |
| 49 | C2×#11719908 | — | 2 | 4 | no | — |
| 50 | C2×#11719908 | D4 | 1 | 4 | yes | — |
| 51 | C2×#11719908 | C23 | 1 | 2 | yes | — |
| 52 | C2×#11719908 | C23 | 1 | 1 | yes | characteristic |
| 53 | C2×#11719908 | C23 | 1 | 1 | yes | characteristic |
| 54 | #13133151 | C23 | 1 | 2 | yes | — |
| 55 | #13133149 | — | 2 | 1 | no | — |
| 56 | #13133142 | — | 2 | 4 | no | — |
| 57 | C2×#41×C53 | D4 | 1 | 1 | yes | characteristic |
| 58 | C22×Q8×C53 | — | 2 | 1 | no | — |
| 59 | C2×Q16×C53 | — | 2 | 1 | no | — |
| 60 | C2×Q16×C53 | — | 2 | 2 | no | — |
| 61 | C2×Q16×C53 | — | 2 | 2 | no | — |
| 62 | C2×Q16×C53 | C23 | 1 | 2 | yes | — |
| 63 | #131×C53 | — | 2 | 2 | no | — |
| 64 | #131×C53 | — | 2 | 2 | no | — |
| 65 | C2×M16×C53 | D4 | 1 | 2 | yes | — |
| 66 | C2×M16×C53 | D4 | 1 | 1 | yes | characteristic |
| 67 | C22×C8×C53 | C23 | 1 | 2 | yes | — |
| 68 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 69 | C4⋊Q8×C53 | — | 4 | 1 | no | — |
| 70 | #124×C53 | — | 4 | 1 | no | — |
| 71 | #123×C53 | — | 4 | 2 | no | — |
| 72 | C22⋊Q8×C53 | — | 2 | 4 | no | — |
| 73 | #117×C53 | — | 2 | 1 | no | — |
| 74 | #107×C53 | C23 | 1 | 2 | yes | — |
| 75 | #106×C53 | — | 2 | 1 | no | — |
| 76 | #103×C53 | — | 2 | 4 | no | — |
| 77 | #41×C53 | — | 2 | 2 | no | — |
| 78 | #41×C53 | — | 2 | 2 | no | — |
| 79 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 80 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 81 | C2×Q8×C53 | — | 4 | 2 | no | — |
| 82 | C2×Q8×C53 | — | 4 | 2 | no | — |
| 83 | C2×Q8×C53 | D8 | 1 | 1 | yes | characteristic |
| 84 | C2×D4×C53 | D8 | 1 | 2 | yes | — |
| 85 | C2×D4×C53 | D8 | 1 | 1 | yes | characteristic |
| 86 | C22×C4×C53 | — | 4 | 1 | no | — |
| 87 | C22×C4×C53 | C2×D4 | 1 | 2 | yes | — |
| 88 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C22×Dic53 | — | 4 | 1 | no | — |
| 90 | C2×Dic106 | — | 2 | 2 | no | — |
| 91 | C2×Dic106 | — | 4 | 2 | no | — |
| 92 | C2×Dic106 | — | 4 | 2 | no | — |
| 93 | C2×Dic106 | — | 4 | 1 | no | — |
| 94 | Q16×C53 | — | 2 | 2 | no | — |
| 95 | Q16×C53 | — | 2 | 2 | no | — |
| 96 | Q16×C53 | — | 4 | 2 | no | — |
| 97 | M16×C53 | — | 2 | 2 | no | — |
| 98 | M16×C53 | — | 2 | 4 | no | — |
| 99 | C2×C8×C53 | — | 2 | 2 | no | — |
| 100 | C2×C8×C53 | — | 2 | 2 | no | — |
| 101 | C2×C8×C53 | C2×D4 | 1 | 2 | yes | — |
| 102 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C2×C8×C53 | C2×D4 | 1 | 2 | yes | — |
| 104 | C2×C8×C53 | #41 | 1 | 2 | yes | — |
| 105 | C2×C8×C53 | C24 | 1 | 1 | yes | characteristic |
| 106 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 107 | C4⋊C4×C53 | — | 4 | 4 | no | — |
| 108 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 109 | C22⋊C4×C53 | — | 4 | 4 | no | — |
| 110 | C42×C53 | — | 4 | 1 | no | — |
| 111 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 112 | C22⋊Dic53 | — | 4 | 4 | no | — |
| 113 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 114 | #11719919 | — | 4 | 4 | no | — |
| 115 | C4×Dic53 | — | 4 | 1 | no | — |
| 116 | Dic212 | — | 2 | 2 | no | — |
| 117 | Dic212 | — | 2 | 2 | no | — |
| 118 | Dic212 | — | 4 | 2 | no | — |
| 119 | #11719908 | — | 2 | 2 | no | — |
| 120 | #11719908 | C2×D4 | 1 | 2 | yes | — |
| 121 | #11719908 | C2×D4 | 1 | 2 | yes | — |
| 122 | C23×C53 | C2×D8 | 1 | 2 | yes | — |
| 123 | C23×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 124 | Q8×C53 | — | 2 | 2 | no | — |
| 125 | Q8×C53 | — | 4 | 2 | no | — |
| 126 | Q8×C53 | — | 4 | 2 | no | — |
| 127 | Q8×C53 | — | 8 | 2 | no | — |
| 128 | D4×C53 | — | 2 | 4 | no | — |
| 129 | D4×C53 | — | 2 | 2 | no | — |
| 130 | C2×C4×C53 | — | 2 | 2 | no | — |
| 131 | C2×C4×C53 | — | 2 | 2 | no | — |
| 132 | C2×C4×C53 | — | 4 | 2 | no | — |
| 133 | C2×C4×C53 | — | 4 | 2 | no | — |
| 134 | C2×C4×C53 | — | 8 | 2 | no | — |
| 135 | C2×C4×C53 | #121 | 1 | 2 | yes | — |
| 136 | C2×C4×C53 | C2×D8 | 1 | 2 | yes | — |
| 137 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 138 | C2×C4×C53 | #135 | 1 | 2 | yes | — |
| 139 | C2×C4×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 140 | C2×Dic53 | — | 4 | 2 | no | — |
| 141 | C2×Dic53 | — | 4 | 2 | no | — |
| 142 | C2×Dic53 | — | 8 | 2 | no | — |
| 143 | Dic106 | — | 4 | 2 | no | — |
| 144 | Dic106 | — | 4 | 2 | no | — |
| 145 | Dic106 | — | 8 | 2 | no | — |
| 146 | C8×C53 | — | 2 | 2 | no | — |
| 147 | C8×C53 | #121 | 1 | 2 | yes | — |
| 148 | C8×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 149 | C8×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 150 | #33608 | — | 53 | 1 | no | maximal, Sylow |
| 151 | C22×C53 | — | 2 | 2 | no | — |
| 152 | C22×C53 | — | 2 | 4 | no | — |
| 153 | C22×C53 | — | 2 | 2 | no | — |
| 154 | C22×C53 | #466 | 1 | 2 | yes | — |
| 155 | C22×C53 | C22×D8 | 1 | 1 | yes | characteristic |
| 156 | C4×C53 | — | 2 | 2 | no | — |
| 157 | C4×C53 | — | 8 | 2 | no | — |
| 158 | C4×C53 | — | 8 | 2 | no | — |
| 159 | C4×C53 | #466 | 1 | 2 | yes | — |
| 160 | C4×C53 | C2×#121 | 1 | 1 | yes | characteristic |
| 161 | C4×C53 | C2×#135 | 1 | 1 | yes | characteristic |
| 162 | Dic53 | — | 8 | 2 | no | — |
| 163 | Dic53 | — | 8 | 2 | no | — |
| 164 | C22×Q32 | — | 53 | 1 | no | — |
| 165 | C2×#504 | — | 53 | 1 | no | — |
| 166 | #3059 | — | 53 | 1 | no | — |
| 167 | #3059 | D53 | 1 | 1 | yes | characteristic |
| 168 | #2219 | — | 53 | 4 | no | — |
| 169 | #2216 | — | 53 | 4 | no | — |
| 170 | #2158 | — | 53 | 1 | no | — |
| 171 | #2141 | — | 53 | 1 | no | — |
| 172 | #2141 | — | 53 | 1 | no | — |
| 173 | C2×C53 | — | 2 | 2 | no | — |
| 174 | C2×C53 | — | 4 | 2 | no | — |
| 175 | C2×C53 | C2×#466 | 1 | 1 | yes | characteristic |
| 176 | C2×C53 | #3417 | 1 | 2 | yes | — |
| 177 | C22×Q16 | — | 53 | 1 | no | — |
| 178 | C22×Q16 | C2×D53 | 1 | 1 | yes | characteristic |
| 179 | C2×#131 | C2×D53 | 1 | 1 | yes | characteristic |
| 180 | #536 | — | 2 | 1 | no | — |
| 181 | #536 | — | 106 | 1 | no | — |
| 182 | C2×Q32 | — | 53 | 2 | no | — |
| 183 | C2×Q32 | — | 53 | 2 | no | — |
| 184 | C2×Q32 | — | 106 | 4 | no | — |
| 185 | #504 | — | 106 | 4 | no | — |
| 186 | C2×M32 | — | 53 | 1 | no | — |
| 187 | C2×M32 | — | 53 | 2 | no | — |
| 188 | C22×C16 | — | 53 | 1 | no | — |
| 189 | C22×C16 | — | 53 | 2 | no | — |
| 190 | #470 | — | 2 | 2 | no | — |
| 191 | #470 | — | 106 | 2 | no | — |
| 192 | #467 | — | 53 | 4 | no | — |
| 193 | #467 | C2×D53 | 1 | 4 | yes | — |
| 194 | #427 | — | 53 | 1 | no | — |
| 195 | #427 | C2×D53 | 1 | 1 | yes | characteristic |
| 196 | #419 | — | 2 | 1 | no | — |
| 197 | #419 | — | 106 | 1 | no | — |
| 198 | #367 | — | 53 | 2 | no | — |
| 199 | #366 | — | 53 | 2 | no | — |
| 200 | #358 | — | 53 | 4 | no | — |
| 201 | #358 | — | 53 | 4 | no | — |
| 202 | C53 | #33608 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 203 | C2×#41 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 204 | C22×Q8 | — | 2 | 1 | no | — |
| 205 | C22×Q8 | — | 106 | 1 | no | — |
| 206 | C2×Q16 | — | 2 | 2 | no | — |
| 207 | C2×Q16 | — | 2 | 2 | no | — |
| 208 | C2×Q16 | — | 2 | 1 | no | — |
| 209 | C2×Q16 | — | 53 | 2 | no | — |
| 210 | C2×Q16 | — | 106 | 2 | no | — |
| 211 | C2×Q16 | — | 106 | 2 | no | — |
| 212 | C2×Q16 | — | 106 | 1 | no | — |
| 213 | C2×Q16 | C22×D53 | 1 | 2 | yes | — |
| 214 | #131 | — | 2 | 2 | no | — |
| 215 | #131 | — | 2 | 2 | no | — |
| 216 | C2×M16 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 217 | C2×M16 | C22⋊D53 | 1 | 2 | yes | — |
| 218 | C22×C8 | C22×D53 | 1 | 2 | yes | — |
| 219 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 220 | C4⋊Q8 | — | 4 | 1 | no | — |
| 221 | C4⋊Q8 | — | 212 | 1 | no | — |
| 222 | #124 | — | 4 | 1 | no | — |
| 223 | #124 | — | 212 | 1 | no | — |
| 224 | #123 | — | 4 | 2 | no | — |
| 225 | #123 | — | 212 | 2 | no | — |
| 226 | C22⋊Q8 | — | 2 | 4 | no | — |
| 227 | C22⋊Q8 | — | 106 | 4 | no | — |
| 228 | #117 | — | 2 | 1 | no | — |
| 229 | #117 | — | 106 | 1 | no | — |
| 230 | Q32 | — | 106 | 4 | no | — |
| 231 | Q32 | — | 106 | 4 | no | — |
| 232 | M32 | — | 106 | 4 | no | — |
| 233 | M32 | — | 106 | 2 | no | — |
| 234 | C2×C16 | — | 53 | 4 | no | — |
| 235 | C2×C16 | — | 53 | 1 | no | — |
| 236 | C2×C16 | — | 53 | 1 | no | — |
| 237 | C2×C16 | — | 53 | 2 | no | — |
| 238 | C2×C16 | — | 106 | 4 | no | — |
| 239 | #107 | — | 53 | 2 | no | — |
| 240 | #107 | C22×D53 | 1 | 2 | yes | — |
| 241 | #106 | — | 2 | 1 | no | — |
| 242 | #106 | — | 106 | 1 | no | — |
| 243 | #103 | — | 2 | 4 | no | — |
| 244 | #103 | — | 106 | 4 | no | — |
| 245 | #41 | — | 2 | 2 | no | — |
| 246 | #41 | — | 2 | 2 | no | — |
| 247 | C2×Q8 | — | 2 | 2 | no | — |
| 248 | C2×Q8 | — | 4 | 1 | no | — |
| 249 | C2×Q8 | — | 4 | 2 | no | — |
| 250 | C2×Q8 | — | 4 | 2 | no | — |
| 251 | C2×Q8 | — | 106 | 2 | no | — |
| 252 | C2×Q8 | — | 212 | 2 | no | — |
| 253 | C2×Q8 | — | 212 | 2 | no | — |
| 254 | C2×Q8 | — | 212 | 1 | no | — |
| 255 | C2×Q8 | #11719922 | 1 | 1 | yes | characteristic |
| 256 | C2×D4 | #11719922 | 1 | 2 | yes | — |
| 257 | C2×D4 | #11719922 | 1 | 1 | yes | characteristic |
| 258 | C22×C4 | — | 4 | 1 | no | — |
| 259 | C22×C4 | — | 212 | 1 | no | — |
| 260 | C22×C4 | C2×C22⋊D53 | 1 | 2 | yes | — |
| 261 | C22×C4 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 262 | Q16 | — | 2 | 2 | no | — |
| 263 | Q16 | — | 2 | 2 | no | — |
| 264 | Q16 | — | 4 | 2 | no | — |
| 265 | Q16 | — | 106 | 2 | no | — |
| 266 | Q16 | — | 106 | 2 | no | — |
| 267 | Q16 | — | 212 | 2 | no | — |
| 268 | M16 | — | 2 | 2 | no | — |
| 269 | M16 | — | 2 | 4 | no | — |
| 270 | C2×C8 | — | 2 | 2 | no | — |
| 271 | C2×C8 | — | 2 | 2 | no | — |
| 272 | C2×C8 | D4×D53 | 1 | 2 | yes | — |
| 273 | C2×C8 | #11719947 | 1 | 2 | yes | — |
| 274 | C2×C8 | C2×C22⋊D53 | 1 | 2 | yes | — |
| 275 | C2×C8 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 276 | C2×C8 | C23×D53 | 1 | 1 | yes | characteristic |
| 277 | C4⋊C4 | — | 2 | 2 | no | — |
| 278 | C4⋊C4 | — | 4 | 4 | no | — |
| 279 | C4⋊C4 | — | 106 | 2 | no | — |
| 280 | C4⋊C4 | — | 212 | 4 | no | — |
| 281 | C22⋊C4 | — | 4 | 1 | no | — |
| 282 | C22⋊C4 | — | 4 | 4 | no | — |
| 283 | C22⋊C4 | — | 212 | 1 | no | — |
| 284 | C22⋊C4 | — | 212 | 4 | no | — |
| 285 | C42 | — | 4 | 1 | no | — |
| 286 | C42 | — | 212 | 1 | no | — |
| 287 | C16 | — | 53 | 2 | no | — |
| 288 | C16 | — | 53 | 2 | no | — |
| 289 | C16 | — | 106 | 2 | no | — |
| 290 | C23 | C2×#11719922 | 1 | 2 | yes | — |
| 291 | C23 | C2×#11719922 | 1 | 1 | yes | characteristic |
| 292 | Q8 | — | 2 | 2 | no | — |
| 293 | Q8 | — | 4 | 2 | no | — |
| 294 | Q8 | — | 4 | 2 | no | — |
| 295 | Q8 | — | 8 | 2 | no | — |
| 296 | Q8 | — | 424 | 2 | no | — |
| 297 | Q8 | — | 212 | 2 | no | — |
| 298 | Q8 | — | 212 | 2 | no | — |
| 299 | D4 | — | 2 | 4 | no | — |
| 300 | D4 | — | 2 | 2 | no | — |
| 301 | C2×C4 | — | 2 | 2 | no | — |
| 302 | C2×C4 | — | 2 | 2 | no | — |
| 303 | C2×C4 | — | 4 | 2 | no | — |
| 304 | C2×C4 | — | 4 | 2 | no | — |
| 305 | C2×C4 | — | 8 | 2 | no | — |
| 306 | C2×C4 | — | 424 | 2 | no | — |
| 307 | C2×C4 | — | 212 | 2 | no | — |
| 308 | C2×C4 | — | 212 | 2 | no | — |
| 309 | C2×C4 | #13133189 | 1 | 2 | yes | — |
| 310 | C2×C4 | C2×#11719922 | 1 | 2 | yes | — |
| 311 | C2×C4 | C2×#11719922 | 1 | 1 | yes | characteristic |
| 312 | C2×C4 | #13133237 | 1 | 2 | yes | — |
| 313 | C2×C4 | C22×C22⋊D53 | 1 | 1 | yes | characteristic |
| 314 | C8 | — | 2 | 2 | no | — |
| 315 | C8 | #13133189 | 1 | 2 | yes | — |
| 316 | C8 | C2×D4×D53 | 1 | 1 | yes | characteristic |
| 317 | C8 | C2×#11719947 | 1 | 1 | yes | characteristic |
| 318 | C22 | — | 2 | 2 | no | — |
| 319 | C22 | — | 2 | 2 | no | — |
| 320 | C22 | — | 2 | 4 | no | — |
| 321 | C22 | #16810857 | 1 | 2 | yes | — |
| 322 | C22 | C22×#11719922 | 1 | 1 | yes | characteristic |
| 323 | C4 | — | 2 | 2 | no | — |
| 324 | C4 | — | 8 | 2 | no | — |
| 325 | C4 | — | 8 | 2 | no | — |
| 326 | C4 | — | 424 | 2 | no | — |
| 327 | C4 | — | 424 | 2 | no | — |
| 328 | C4 | #16810857 | 1 | 2 | yes | — |
| 329 | C4 | C2×#13133189 | 1 | 1 | yes | characteristic |
| 330 | C4 | C2×#13133237 | 1 | 1 | yes | characteristic |
| 331 | C2 | — | 2 | 2 | no | — |
| 332 | C2 | — | 4 | 2 | no | — |
| 333 | C2 | C2×#16810857 | 1 | 1 | yes | characteristic |
| 334 | C2 | #20287866 | 1 | 2 | yes | — |
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