← #27300626
Subgroups — #27300626
Order 13568.
400 conjugacy classes of subgroups.
The subgroup lattice has 316 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20283315 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20276503 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#16810449 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #20274779 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #20272824 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20272811 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C2×C53×#350 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16811106 | — | 2 | 1 | no | — |
| 9 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 10 | #16810665 | — | 2 | 2 | no | — |
| 11 | #16810665 | — | 2 | 1 | no | — |
| 12 | #16810661 | — | 2 | 1 | no | — |
| 13 | #16810661 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#13133151 | — | 2 | 1 | no | — |
| 15 | C2×#13133151 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#13133151 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16810624 | — | 2 | 1 | no | — |
| 18 | #16810451 | — | 2 | 1 | no | — |
| 19 | #16810449 | — | 2 | 2 | no | — |
| 20 | #16810449 | — | 2 | 1 | no | — |
| 21 | #16810391 | — | 2 | 1 | no | — |
| 22 | C2×#131×C53 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×M32×C53 | C22 | 1 | 1 | yes | characteristic |
| 24 | C22×C16×C53 | C22 | 1 | 1 | yes | characteristic |
| 25 | C53×#350 | C4 | 1 | 4 | yes | complemented |
| 26 | C2×#11719917 | — | 2 | 1 | no | — |
| 27 | Q8×Dic53 | — | 4 | 1 | no | — |
| 28 | D4×Dic53 | — | 4 | 2 | no | — |
| 29 | D4×Dic53 | — | 4 | 1 | no | — |
| 30 | #13133176 | — | 2 | 2 | no | — |
| 31 | #13133176 | — | 2 | 1 | no | — |
| 32 | C2×C4⋊Dic53 | — | 2 | 2 | no | — |
| 33 | C2×C4⋊Dic53 | — | 2 | 1 | no | — |
| 34 | C2×C4⋊Dic53 | — | 2 | 1 | no | — |
| 35 | #13133153 | — | 2 | 1 | no | — |
| 36 | #13133153 | — | 2 | 1 | no | — |
| 37 | #13133151 | — | 2 | 2 | no | — |
| 38 | #13133151 | — | 2 | 1 | no | — |
| 39 | #13133151 | — | 2 | 1 | no | — |
| 40 | #13133151 | — | 2 | 1 | no | — |
| 41 | #13133151 | D4 | 1 | 1 | yes | characteristic |
| 42 | #13133151 | D4 | 1 | 2 | yes | — |
| 43 | #13133151 | D4 | 1 | 1 | yes | characteristic |
| 44 | #13133149 | — | 2 | 2 | no | — |
| 45 | #13133149 | — | 2 | 1 | no | — |
| 46 | #13133141 | — | 4 | 1 | no | — |
| 47 | #13133138 | — | 4 | 1 | no | — |
| 48 | C2×#41×C53 | D4 | 1 | 1 | yes | characteristic |
| 49 | #131×C53 | — | 2 | 1 | no | — |
| 50 | #131×C53 | — | 2 | 1 | no | — |
| 51 | #131×C53 | C2×C4 | 1 | 2 | yes | — |
| 52 | #131×C53 | Q8 | 1 | 2 | yes | — |
| 53 | C2×M16×C53 | — | 2 | 1 | no | — |
| 54 | C2×M16×C53 | D4 | 1 | 1 | yes | characteristic |
| 55 | C22×C8×C53 | — | 2 | 1 | no | — |
| 56 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 57 | M32×C53 | — | 2 | 1 | no | — |
| 58 | M32×C53 | C2×C4 | 1 | 2 | yes | complemented |
| 59 | C2×C16×C53 | — | 2 | 1 | no | — |
| 60 | C2×C16×C53 | — | 2 | 1 | no | — |
| 61 | C2×C16×C53 | — | 2 | 1 | no | — |
| 62 | C2×C16×C53 | C2×C4 | 1 | 2 | yes | complemented |
| 63 | #41×C53 | — | 2 | 1 | no | — |
| 64 | #41×C53 | — | 2 | 1 | no | — |
| 65 | #41×C53 | C4⋊C4 | 1 | 2 | yes | — |
| 66 | #41×C53 | Q16 | 1 | 2 | yes | — |
| 67 | C2×Q8×C53 | D8 | 1 | 1 | yes | characteristic |
| 68 | C2×D4×C53 | — | 2 | 1 | no | — |
| 69 | C2×D4×C53 | D8 | 1 | 1 | yes | characteristic |
| 70 | C22×C4×C53 | — | 2 | 1 | no | — |
| 71 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C22×Dic53 | — | 4 | 2 | no | — |
| 73 | C22×Dic53 | — | 4 | 1 | no | — |
| 74 | C22×Dic53 | — | 4 | 1 | no | — |
| 75 | M16×C53 | — | 2 | 1 | no | — |
| 76 | M16×C53 | — | 2 | 2 | no | — |
| 77 | M16×C53 | — | 2 | 2 | no | — |
| 78 | M16×C53 | C4⋊C4 | 1 | 2 | yes | — |
| 79 | C2×C8×C53 | — | 2 | 1 | no | — |
| 80 | C2×C8×C53 | — | 2 | 2 | no | — |
| 81 | C2×C8×C53 | — | 2 | 2 | no | — |
| 82 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 86 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 87 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 88 | C22⋊Dic53 | — | 4 | 2 | no | — |
| 89 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 90 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 91 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 92 | C4⋊Dic53 | — | 2 | 1 | no | — |
| 93 | C4⋊Dic53 | — | 4 | 2 | no | — |
| 94 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 95 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 96 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 97 | C4×Dic53 | — | 4 | 2 | no | — |
| 98 | C4×Dic53 | — | 4 | 1 | no | — |
| 99 | #11719917 | — | 4 | 1 | no | — |
| 100 | #11719917 | — | 4 | 1 | no | — |
| 101 | C2×#88322 | — | 4 | 1 | no | — |
| 102 | C16×C53 | — | 2 | 2 | no | — |
| 103 | C16×C53 | — | 2 | 2 | no | — |
| 104 | C23×C53 | — | 2 | 1 | no | — |
| 105 | C23×C53 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 106 | Q8×C53 | — | 2 | 1 | no | — |
| 107 | Q8×C53 | #107 | 1 | 2 | yes | — |
| 108 | D4×C53 | — | 2 | 2 | no | — |
| 109 | D4×C53 | — | 2 | 2 | no | — |
| 110 | D4×C53 | — | 2 | 1 | no | — |
| 111 | D4×C53 | #107 | 1 | 2 | yes | — |
| 112 | C2×C4×C53 | — | 2 | 1 | no | — |
| 113 | C2×C4×C53 | — | 2 | 2 | no | — |
| 114 | C2×C4×C53 | — | 2 | 2 | no | — |
| 115 | C2×C4×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C53 | #123 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C53 | C2×Q16 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C53 | #137 | 1 | 1 | yes | characteristic |
| 121 | C2×Dic53 | — | 4 | 1 | no | — |
| 122 | C2×Dic53 | — | 4 | 2 | no | — |
| 123 | C2×Dic53 | — | 4 | 1 | no | — |
| 124 | C2×Dic53 | — | 8 | 1 | no | — |
| 125 | C2×Dic53 | — | 16 | 1 | no | — |
| 126 | C2×Dic53 | — | 16 | 2 | no | — |
| 127 | C2×Dic53 | — | 16 | 1 | no | — |
| 128 | C8×C53 | — | 2 | 2 | no | — |
| 129 | C8×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 130 | C8×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 131 | C8×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 132 | C8×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 133 | #88322 | — | 8 | 1 | no | — |
| 134 | #12925 | — | 53 | 1 | no | maximal, Sylow |
| 135 | C22×C53 | — | 2 | 1 | no | — |
| 136 | C22×C53 | — | 2 | 1 | no | — |
| 137 | C22×C53 | — | 2 | 2 | no | — |
| 138 | C22×C53 | — | 4 | 1 | no | — |
| 139 | C22×C53 | — | 4 | 1 | no | — |
| 140 | C22×C53 | #428 | 1 | 1 | yes | characteristic |
| 141 | C22×C53 | #447 | 1 | 1 | yes | characteristic |
| 142 | C22×C53 | #484 | 1 | 1 | yes | characteristic |
| 143 | C4×C53 | — | 2 | 2 | no | — |
| 144 | C4×C53 | #385 | 1 | 1 | yes | characteristic |
| 145 | C4×C53 | C2×#107 | 1 | 1 | yes | characteristic |
| 146 | C4×C53 | #451 | 1 | 1 | yes | characteristic |
| 147 | C4×C53 | #480 | 1 | 1 | yes | characteristic |
| 148 | Dic53 | — | 8 | 2 | no | — |
| 149 | Dic53 | — | 8 | 1 | no | — |
| 150 | Dic53 | — | 8 | 1 | no | — |
| 151 | Dic53 | — | 16 | 1 | no | — |
| 152 | #2913 | — | 53 | 1 | no | — |
| 153 | #2159 | — | 53 | 1 | no | — |
| 154 | C2×#366 | — | 53 | 1 | no | — |
| 155 | C2×#350 | D53 | 1 | 1 | yes | characteristic |
| 156 | #1816 | — | 53 | 1 | no | — |
| 157 | #1387 | — | 53 | 1 | no | — |
| 158 | #1382 | — | 53 | 1 | no | — |
| 159 | C2×C53 | — | 2 | 1 | no | — |
| 160 | C2×C53 | — | 2 | 1 | no | — |
| 161 | C2×C53 | — | 4 | 2 | no | — |
| 162 | C2×C53 | #1852 | 1 | 1 | yes | characteristic |
| 163 | C2×C53 | #2192 | 1 | 1 | yes | characteristic |
| 164 | C2×C53 | #2195 | 1 | 1 | yes | characteristic |
| 165 | C2×#131 | C2×D53 | 1 | 1 | yes | characteristic |
| 166 | #520 | — | 106 | 1 | no | — |
| 167 | C2×M32 | C2×D53 | 1 | 1 | yes | characteristic |
| 168 | C22×C16 | C2×D53 | 1 | 1 | yes | characteristic |
| 169 | C2×#108 | — | 53 | 1 | no | — |
| 170 | #428 | — | 106 | 2 | no | — |
| 171 | #428 | — | 106 | 1 | no | — |
| 172 | #427 | — | 53 | 1 | no | — |
| 173 | #427 | — | 106 | 1 | no | — |
| 174 | C2×#107 | — | 53 | 1 | no | — |
| 175 | C2×#107 | — | 53 | 1 | no | — |
| 176 | C2×#107 | — | 106 | 1 | no | — |
| 177 | #417 | — | 106 | 1 | no | — |
| 178 | #367 | — | 106 | 1 | no | — |
| 179 | #366 | — | 106 | 2 | no | — |
| 180 | #366 | — | 106 | 1 | no | — |
| 181 | #350 | Dic53 | 1 | 4 | yes | complemented |
| 182 | #343 | — | 106 | 1 | no | — |
| 183 | C53 | #12925 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 184 | C2×#41 | D212 | 1 | 1 | yes | characteristic |
| 185 | #131 | — | 2 | 1 | no | — |
| 186 | #131 | — | 2 | 1 | no | — |
| 187 | #131 | Dic106 | 1 | 2 | yes | — |
| 188 | #131 | C2×Dic53 | 1 | 2 | yes | — |
| 189 | C2×M16 | — | 2 | 1 | no | — |
| 190 | C2×M16 | — | 106 | 1 | no | — |
| 191 | C2×M16 | D212 | 1 | 1 | yes | characteristic |
| 192 | C22×C8 | — | 2 | 1 | no | — |
| 193 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 194 | C4×Q8 | — | 212 | 1 | no | — |
| 195 | C4×D4 | — | 212 | 2 | no | — |
| 196 | C4×D4 | — | 212 | 1 | no | — |
| 197 | #117 | — | 106 | 2 | no | — |
| 198 | #117 | — | 106 | 1 | no | — |
| 199 | C2×C4⋊C4 | — | 106 | 2 | no | — |
| 200 | C2×C4⋊C4 | — | 106 | 1 | no | — |
| 201 | C2×C4⋊C4 | — | 106 | 1 | no | — |
| 202 | M32 | — | 2 | 1 | no | — |
| 203 | M32 | C2×Dic53 | 1 | 2 | yes | complemented |
| 204 | C2×C16 | — | 2 | 1 | no | — |
| 205 | C2×C16 | — | 2 | 1 | no | — |
| 206 | C2×C16 | — | 2 | 1 | no | — |
| 207 | C2×C16 | C2×Dic53 | 1 | 2 | yes | complemented |
| 208 | #108 | — | 106 | 1 | no | — |
| 209 | #108 | — | 106 | 1 | no | — |
| 210 | #107 | — | 53 | 2 | no | — |
| 211 | #107 | — | 53 | 1 | no | — |
| 212 | #107 | — | 53 | 1 | no | — |
| 213 | #107 | — | 106 | 2 | no | — |
| 214 | #107 | — | 106 | 1 | no | — |
| 215 | #107 | — | 106 | 1 | no | — |
| 216 | #107 | — | 106 | 1 | no | — |
| 217 | #106 | — | 106 | 2 | no | — |
| 218 | #106 | — | 106 | 1 | no | — |
| 219 | #103 | — | 212 | 1 | no | — |
| 220 | #102 | — | 212 | 1 | no | — |
| 221 | #41 | — | 2 | 1 | no | — |
| 222 | #41 | — | 2 | 1 | no | — |
| 223 | #41 | Dic212 | 1 | 2 | yes | — |
| 224 | #41 | C4⋊Dic53 | 1 | 2 | yes | — |
| 225 | C2×Q8 | D424 | 1 | 1 | yes | characteristic |
| 226 | C2×D4 | — | 2 | 1 | no | — |
| 227 | C2×D4 | D424 | 1 | 1 | yes | characteristic |
| 228 | C22×C4 | — | 2 | 1 | no | — |
| 229 | C22×C4 | — | 212 | 1 | no | — |
| 230 | C22×C4 | — | 212 | 2 | no | — |
| 231 | C22×C4 | — | 212 | 1 | no | — |
| 232 | C22×C4 | C2×D212 | 1 | 1 | yes | characteristic |
| 233 | M16 | — | 2 | 1 | no | — |
| 234 | M16 | — | 2 | 2 | no | — |
| 235 | M16 | — | 2 | 2 | no | — |
| 236 | M16 | — | 212 | 1 | no | — |
| 237 | M16 | — | 212 | 1 | no | — |
| 238 | M16 | C4⋊Dic53 | 1 | 2 | yes | — |
| 239 | C2×C8 | — | 2 | 1 | no | — |
| 240 | C2×C8 | — | 2 | 2 | no | — |
| 241 | C2×C8 | — | 2 | 2 | no | — |
| 242 | C2×C8 | — | 212 | 1 | no | — |
| 243 | C2×C8 | C2×Dic106 | 1 | 1 | yes | characteristic |
| 244 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 245 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 246 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 247 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 248 | C2×C8 | C22×Dic53 | 1 | 1 | yes | characteristic |
| 249 | C4⋊C4 | — | 106 | 1 | no | — |
| 250 | C4⋊C4 | — | 106 | 2 | no | — |
| 251 | C4⋊C4 | — | 106 | 1 | no | — |
| 252 | C4⋊C4 | — | 212 | 2 | no | — |
| 253 | C4⋊C4 | — | 212 | 1 | no | — |
| 254 | C4⋊C4 | — | 212 | 1 | no | — |
| 255 | C4⋊C4 | — | 212 | 1 | no | — |
| 256 | C22⋊C4 | — | 212 | 2 | no | — |
| 257 | C22⋊C4 | — | 212 | 1 | no | — |
| 258 | C42 | — | 212 | 2 | no | — |
| 259 | C42 | — | 212 | 1 | no | — |
| 260 | C16 | — | 2 | 2 | no | — |
| 261 | C16 | — | 2 | 2 | no | — |
| 262 | C23 | — | 2 | 1 | no | — |
| 263 | C23 | #13133245 | 1 | 1 | yes | characteristic |
| 264 | Q8 | — | 2 | 1 | no | — |
| 265 | Q8 | #13133151 | 1 | 2 | yes | — |
| 266 | D4 | — | 2 | 2 | no | — |
| 267 | D4 | — | 2 | 2 | no | — |
| 268 | D4 | — | 2 | 1 | no | — |
| 269 | D4 | #13133151 | 1 | 2 | yes | — |
| 270 | C2×C4 | — | 2 | 1 | no | — |
| 271 | C2×C4 | — | 2 | 2 | no | — |
| 272 | C2×C4 | — | 2 | 2 | no | — |
| 273 | C2×C4 | — | 848 | 1 | no | — |
| 274 | C2×C4 | — | 848 | 1 | no | — |
| 275 | C2×C4 | — | 848 | 2 | no | — |
| 276 | C2×C4 | — | 424 | 1 | no | — |
| 277 | C2×C4 | — | 212 | 1 | no | — |
| 278 | C2×C4 | — | 212 | 2 | no | — |
| 279 | C2×C4 | — | 212 | 1 | no | — |
| 280 | C2×C4 | C2×C4⋊Dic53 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | #13133187 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | #13133201 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | C2×D424 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | C2×Dic212 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | #13133250 | 1 | 1 | yes | characteristic |
| 286 | C8 | — | 2 | 2 | no | — |
| 287 | C8 | — | 424 | 1 | no | — |
| 288 | C8 | D4×Dic53 | 1 | 1 | yes | characteristic |
| 289 | C8 | D4×Dic53 | 1 | 1 | yes | characteristic |
| 290 | C8 | #13133194 | 1 | 1 | yes | characteristic |
| 291 | C8 | #13133194 | 1 | 1 | yes | characteristic |
| 292 | C22 | — | 2 | 1 | no | — |
| 293 | C22 | — | 2 | 1 | no | — |
| 294 | C22 | — | 2 | 2 | no | — |
| 295 | C22 | — | 4 | 1 | no | — |
| 296 | C22 | — | 4 | 1 | no | — |
| 297 | C22 | #16810665 | 1 | 1 | yes | characteristic |
| 298 | C22 | #16810759 | 1 | 1 | yes | characteristic |
| 299 | C22 | #16810959 | 1 | 1 | yes | characteristic |
| 300 | C4 | — | 2 | 2 | no | — |
| 301 | C4 | — | 848 | 1 | no | — |
| 302 | C4 | — | 424 | 2 | no | — |
| 303 | C4 | — | 424 | 1 | no | — |
| 304 | C4 | — | 424 | 1 | no | — |
| 305 | C4 | #16810498 | 1 | 1 | yes | characteristic |
| 306 | C4 | C2×#13133151 | 1 | 1 | yes | characteristic |
| 307 | C4 | #16810781 | 1 | 1 | yes | characteristic |
| 308 | C4 | #16810937 | 1 | 1 | yes | characteristic |
| 309 | C2 | — | 2 | 1 | no | — |
| 310 | C2 | — | 2 | 1 | no | — |
| 311 | C2 | — | 4 | 2 | no | — |
| 312 | C2 | #20274905 | 1 | 1 | yes | characteristic |
| 313 | C2 | #20276674 | 1 | 1 | yes | characteristic |
| 314 | C2 | #20276695 | 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