← #27300222
Subgroups — #27300222
Order 13568.
390 conjugacy classes of subgroups.
The subgroup lattice has 302 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20276481 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20276451 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20276443 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20274839 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20272822 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20272813 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C53×#2952 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#13133157 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#13133157 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16810661 | C22 | 1 | 1 | yes | characteristic |
| 12 | #16810444 | — | 2 | 2 | no | — |
| 13 | #16810433 | — | 2 | 2 | no | — |
| 14 | #16810430 | — | 2 | 2 | no | — |
| 15 | #16810394 | — | 2 | 1 | no | — |
| 16 | #16810379 | — | 2 | 1 | no | — |
| 17 | C2×#135×C53 | C22 | 1 | 1 | yes | characteristic |
| 18 | C53×#520 | — | 2 | 1 | no | — |
| 19 | C53×#442 | C4 | 1 | 2 | yes | complemented |
| 20 | C53×#441 | C4 | 1 | 2 | yes | complemented |
| 21 | C4×SD16×C53 | — | 2 | 2 | no | — |
| 22 | C2×#106×C53 | C22 | 1 | 1 | yes | characteristic |
| 23 | C53×#417 | — | 2 | 1 | no | — |
| 24 | C53×#405 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#11719917 | — | 2 | 1 | no | — |
| 26 | #13133176 | — | 2 | 1 | no | — |
| 27 | #13133157 | — | 2 | 1 | no | — |
| 28 | #13133157 | — | 2 | 1 | no | — |
| 29 | #13133157 | — | 2 | 1 | no | — |
| 30 | #13133157 | — | 2 | 1 | no | — |
| 31 | C2×#11719908 | — | 2 | 1 | no | — |
| 32 | C2×#11719908 | — | 2 | 1 | no | — |
| 33 | #13133153 | — | 2 | 1 | no | — |
| 34 | #13133153 | — | 2 | 1 | no | — |
| 35 | #13133151 | — | 2 | 1 | no | — |
| 36 | #13133149 | — | 2 | 1 | no | — |
| 37 | C2×#41×C53 | — | 2 | 1 | no | — |
| 38 | #135×C53 | — | 2 | 1 | no | — |
| 39 | #135×C53 | — | 2 | 1 | no | — |
| 40 | #135×C53 | D4 | 1 | 2 | yes | — |
| 41 | #135×C53 | Q8 | 1 | 2 | yes | — |
| 42 | C2×Q16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C2×SD16×C53 | — | 2 | 1 | no | — |
| 44 | C2×D8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | C2×M16×C53 | D4 | 1 | 1 | yes | characteristic |
| 46 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 47 | C4×Q8×C53 | — | 2 | 2 | no | — |
| 48 | C4×D4×C53 | — | 2 | 2 | no | — |
| 49 | C4×D4×C53 | — | 4 | 2 | no | — |
| 50 | #117×C53 | — | 4 | 1 | no | — |
| 51 | #117×C53 | D4 | 1 | 1 | yes | characteristic |
| 52 | C2×C4⋊C4×C53 | — | 2 | 1 | no | — |
| 53 | C2×C4⋊C4×C53 | — | 4 | 1 | no | — |
| 54 | #106×C53 | — | 2 | 1 | no | — |
| 55 | #106×C53 | C2×C4 | 1 | 2 | yes | — |
| 56 | #103×C53 | — | 2 | 2 | no | — |
| 57 | #102×C53 | — | 2 | 2 | no | — |
| 58 | #97×C53 | C2×C4 | 1 | 2 | yes | — |
| 59 | C4×C8×C53 | D4 | 1 | 2 | yes | — |
| 60 | #41×C53 | — | 2 | 2 | no | — |
| 61 | #41×C53 | — | 2 | 2 | no | — |
| 62 | #41×C53 | — | 4 | 1 | no | — |
| 63 | #41×C53 | — | 4 | 1 | no | — |
| 64 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 65 | C2×D4×C53 | — | 2 | 1 | no | — |
| 66 | C2×D4×C53 | — | 4 | 1 | no | — |
| 67 | C22×C4×C53 | — | 4 | 2 | no | — |
| 68 | C22×C4×C53 | — | 4 | 1 | no | — |
| 69 | C22×C4×C53 | — | 4 | 1 | no | — |
| 70 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 71 | Q16×C53 | — | 2 | 1 | no | — |
| 72 | Q16×C53 | C4⋊C4 | 1 | 2 | yes | — |
| 73 | SD16×C53 | — | 2 | 2 | no | — |
| 74 | SD16×C53 | — | 2 | 2 | no | — |
| 75 | D8×C53 | — | 2 | 1 | no | — |
| 76 | D8×C53 | C4⋊C4 | 1 | 2 | yes | — |
| 77 | M16×C53 | — | 2 | 1 | no | — |
| 78 | M16×C53 | — | 2 | 1 | no | — |
| 79 | C2×C8×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 80 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 81 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 84 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 85 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 86 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 87 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 88 | C4⋊C4×C53 | — | 4 | 2 | no | — |
| 89 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 90 | C22⋊C4×C53 | — | 2 | 1 | no | — |
| 91 | C22⋊C4×C53 | — | 4 | 2 | no | — |
| 92 | C42×C53 | — | 4 | 2 | no | — |
| 93 | C42×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 94 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 95 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 96 | C4×Dic53 | — | 4 | 1 | no | — |
| 97 | #11719917 | — | 4 | 1 | no | — |
| 98 | #11719917 | — | 4 | 1 | no | — |
| 99 | C2×#88322 | — | 4 | 1 | no | — |
| 100 | #11719908 | — | 4 | 1 | no | — |
| 101 | #11719908 | — | 4 | 1 | no | — |
| 102 | C23×C53 | — | 4 | 1 | no | — |
| 103 | C23×C53 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 104 | Q8×C53 | — | 2 | 2 | no | — |
| 105 | Q8×C53 | — | 4 | 1 | no | — |
| 106 | D4×C53 | — | 2 | 2 | no | — |
| 107 | D4×C53 | — | 4 | 2 | no | — |
| 108 | D4×C53 | — | 4 | 2 | no | — |
| 109 | D4×C53 | — | 4 | 1 | no | — |
| 110 | C2×C4×C53 | — | 2 | 2 | no | — |
| 111 | C2×C4×C53 | — | 4 | 1 | no | — |
| 112 | C2×C4×C53 | — | 4 | 2 | no | — |
| 113 | C2×C4×C53 | — | 4 | 2 | no | — |
| 114 | C2×C4×C53 | — | 4 | 2 | no | — |
| 115 | C2×C4×C53 | — | 16 | 2 | no | — |
| 116 | C2×C4×C53 | — | 16 | 1 | no | — |
| 117 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C53 | #121 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C53 | #135 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C53 | #135 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C53 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 123 | C2×Dic53 | — | 8 | 1 | no | — |
| 124 | C8×C53 | — | 2 | 2 | no | — |
| 125 | C8×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 126 | C8×C53 | #117 | 1 | 1 | yes | characteristic |
| 127 | C8×C53 | #121 | 1 | 1 | yes | characteristic |
| 128 | C8×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 129 | #88322 | — | 8 | 1 | no | — |
| 130 | #12828 | — | 53 | 1 | no | maximal, Sylow |
| 131 | C22×C53 | — | 4 | 1 | no | — |
| 132 | C22×C53 | — | 4 | 2 | no | — |
| 133 | C22×C53 | — | 8 | 1 | no | — |
| 134 | C22×C53 | — | 8 | 1 | no | — |
| 135 | C22×C53 | #418 | 1 | 1 | yes | characteristic |
| 136 | C22×C53 | #463 | 1 | 1 | yes | characteristic |
| 137 | C22×C53 | #478 | 1 | 1 | yes | characteristic |
| 138 | C4×C53 | — | 4 | 2 | no | — |
| 139 | C4×C53 | — | 4 | 2 | no | — |
| 140 | C4×C53 | — | 8 | 2 | no | — |
| 141 | C4×C53 | #380 | 1 | 1 | yes | characteristic |
| 142 | C4×C53 | #416 | 1 | 1 | yes | characteristic |
| 143 | C4×C53 | #463 | 1 | 1 | yes | characteristic |
| 144 | C4×C53 | #478 | 1 | 1 | yes | characteristic |
| 145 | Dic53 | — | 16 | 1 | no | — |
| 146 | #2952 | D53 | 1 | 1 | yes | characteristic |
| 147 | #2153 | — | 53 | 1 | no | — |
| 148 | #2145 | — | 53 | 1 | no | — |
| 149 | #2143 | — | 53 | 1 | no | — |
| 150 | #1833 | — | 53 | 1 | no | — |
| 151 | #1386 | — | 53 | 1 | no | — |
| 152 | #1383 | — | 53 | 1 | no | — |
| 153 | C2×C53 | — | 2 | 1 | no | — |
| 154 | C2×C53 | — | 2 | 1 | no | — |
| 155 | C2×C53 | — | 2 | 1 | no | — |
| 156 | C2×C53 | — | 8 | 2 | no | — |
| 157 | C2×C53 | #1804 | 1 | 1 | yes | characteristic |
| 158 | C2×#135 | C2×D53 | 1 | 1 | yes | characteristic |
| 159 | #520 | — | 2 | 1 | no | — |
| 160 | C2×M32 | — | 53 | 1 | no | — |
| 161 | C2×M32 | — | 53 | 1 | no | — |
| 162 | #442 | Dic53 | 1 | 2 | yes | complemented |
| 163 | #441 | Dic53 | 1 | 2 | yes | complemented |
| 164 | C4×SD16 | — | 2 | 2 | no | — |
| 165 | C2×#108 | — | 53 | 1 | no | — |
| 166 | #427 | — | 53 | 1 | no | — |
| 167 | C2×#106 | C2×D53 | 1 | 1 | yes | characteristic |
| 168 | #417 | — | 2 | 1 | no | — |
| 169 | #405 | C2×D53 | 1 | 1 | yes | characteristic |
| 170 | #364 | — | 106 | 2 | no | — |
| 171 | #360 | — | 106 | 2 | no | — |
| 172 | #359 | — | 106 | 2 | no | — |
| 173 | #344 | — | 106 | 1 | no | — |
| 174 | H(3,ℤ/4) | — | 106 | 1 | no | — |
| 175 | C53 | #12828 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 176 | C2×#41 | — | 2 | 1 | no | — |
| 177 | #135 | — | 2 | 1 | no | — |
| 178 | #135 | — | 2 | 1 | no | — |
| 179 | #135 | Dic106 | 1 | 2 | yes | — |
| 180 | #135 | D212 | 1 | 2 | yes | — |
| 181 | C2×Q16 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 182 | C2×SD16 | — | 2 | 1 | no | — |
| 183 | C2×D8 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 184 | C2×M16 | — | 106 | 1 | no | — |
| 185 | C2×M16 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 186 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 187 | C4×Q8 | — | 2 | 2 | no | — |
| 188 | C4×D4 | — | 2 | 2 | no | — |
| 189 | C4×D4 | — | 4 | 2 | no | — |
| 190 | #117 | — | 4 | 1 | no | — |
| 191 | #117 | — | 106 | 1 | no | — |
| 192 | #117 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 193 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 194 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 195 | M32 | — | 106 | 1 | no | — |
| 196 | M32 | — | 106 | 1 | no | — |
| 197 | M32 | — | 106 | 1 | no | — |
| 198 | M32 | — | 106 | 1 | no | — |
| 199 | C2×C16 | — | 106 | 1 | no | — |
| 200 | C2×C16 | — | 106 | 1 | no | — |
| 201 | #108 | — | 106 | 1 | no | — |
| 202 | #108 | — | 106 | 1 | no | — |
| 203 | #107 | — | 106 | 1 | no | — |
| 204 | #106 | — | 2 | 1 | no | — |
| 205 | #106 | — | 106 | 1 | no | — |
| 206 | #106 | C2×Dic53 | 1 | 2 | yes | — |
| 207 | #103 | — | 2 | 2 | no | — |
| 208 | #102 | — | 2 | 2 | no | — |
| 209 | #97 | C2×Dic53 | 1 | 2 | yes | — |
| 210 | C4×C8 | C22⋊D53 | 1 | 2 | yes | — |
| 211 | #41 | — | 2 | 2 | no | — |
| 212 | #41 | — | 2 | 2 | no | — |
| 213 | #41 | — | 4 | 1 | no | — |
| 214 | #41 | — | 4 | 1 | no | — |
| 215 | C2×Q8 | — | 2 | 1 | no | — |
| 216 | C2×D4 | — | 2 | 1 | no | — |
| 217 | C2×D4 | — | 4 | 1 | no | — |
| 218 | C22×C4 | — | 4 | 2 | no | — |
| 219 | C22×C4 | — | 4 | 1 | no | — |
| 220 | C22×C4 | — | 4 | 1 | no | — |
| 221 | C22×C4 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 222 | Q16 | — | 2 | 1 | no | — |
| 223 | Q16 | C4⋊Dic53 | 1 | 2 | yes | — |
| 224 | SD16 | — | 2 | 2 | no | — |
| 225 | SD16 | — | 2 | 2 | no | — |
| 226 | D8 | — | 2 | 1 | no | — |
| 227 | D8 | C4⋊Dic53 | 1 | 2 | yes | — |
| 228 | M16 | — | 2 | 1 | no | — |
| 229 | M16 | — | 2 | 1 | no | — |
| 230 | M16 | — | 212 | 1 | no | — |
| 231 | M16 | — | 212 | 1 | no | — |
| 232 | C2×C8 | — | 212 | 1 | no | — |
| 233 | C2×C8 | C22⋊Dic53 | 1 | 2 | yes | — |
| 234 | C2×C8 | C2×Dic106 | 1 | 1 | yes | characteristic |
| 235 | C2×C8 | C2×D212 | 1 | 1 | yes | characteristic |
| 236 | C2×C8 | #11719945 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | #11719945 | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | C22×Dic53 | 1 | 1 | yes | characteristic |
| 239 | C2×C8 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 240 | C4⋊C4 | — | 2 | 1 | no | — |
| 241 | C4⋊C4 | — | 2 | 2 | no | — |
| 242 | C4⋊C4 | — | 4 | 2 | no | — |
| 243 | C4⋊C4 | — | 4 | 1 | no | — |
| 244 | C4⋊C4 | — | 212 | 1 | no | — |
| 245 | C22⋊C4 | — | 2 | 1 | no | — |
| 246 | C22⋊C4 | — | 4 | 2 | no | — |
| 247 | C22⋊C4 | — | 212 | 1 | no | — |
| 248 | C42 | — | 4 | 2 | no | — |
| 249 | C42 | — | 212 | 1 | no | — |
| 250 | C42 | C22⋊Dic53 | 1 | 2 | yes | — |
| 251 | C16 | — | 212 | 1 | no | — |
| 252 | C16 | — | 212 | 1 | no | — |
| 253 | C23 | — | 4 | 1 | no | — |
| 254 | C23 | #13133243 | 1 | 1 | yes | characteristic |
| 255 | Q8 | — | 2 | 2 | no | — |
| 256 | Q8 | — | 4 | 1 | no | — |
| 257 | D4 | — | 2 | 2 | no | — |
| 258 | D4 | — | 4 | 2 | no | — |
| 259 | D4 | — | 4 | 2 | no | — |
| 260 | D4 | — | 4 | 1 | no | — |
| 261 | C2×C4 | — | 2 | 2 | no | — |
| 262 | C2×C4 | — | 4 | 1 | no | — |
| 263 | C2×C4 | — | 4 | 2 | no | — |
| 264 | C2×C4 | — | 4 | 2 | no | — |
| 265 | C2×C4 | — | 4 | 2 | no | — |
| 266 | C2×C4 | — | 16 | 2 | no | — |
| 267 | C2×C4 | — | 16 | 1 | no | — |
| 268 | C2×C4 | — | 424 | 1 | no | — |
| 269 | C2×C4 | C2×C22⋊Dic53 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | #13133191 | 1 | 1 | yes | characteristic |
| 271 | C2×C4 | #13133196 | 1 | 1 | yes | characteristic |
| 272 | C2×C4 | #13133237 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | #13133237 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | #13133243 | 1 | 1 | yes | characteristic |
| 275 | C8 | — | 2 | 2 | no | — |
| 276 | C8 | — | 424 | 1 | no | — |
| 277 | C8 | C2×C4⋊Dic53 | 1 | 1 | yes | characteristic |
| 278 | C8 | #13133176 | 1 | 1 | yes | characteristic |
| 279 | C8 | #13133191 | 1 | 1 | yes | characteristic |
| 280 | C8 | #13133196 | 1 | 1 | yes | characteristic |
| 281 | C22 | — | 4 | 1 | no | — |
| 282 | C22 | — | 4 | 2 | no | — |
| 283 | C22 | — | 8 | 1 | no | — |
| 284 | C22 | — | 8 | 1 | no | — |
| 285 | C22 | #16810628 | 1 | 1 | yes | characteristic |
| 286 | C22 | #16810838 | 1 | 1 | yes | characteristic |
| 287 | C22 | #16810923 | 1 | 1 | yes | characteristic |
| 288 | C4 | — | 4 | 2 | no | — |
| 289 | C4 | — | 4 | 2 | no | — |
| 290 | C4 | — | 8 | 2 | no | — |
| 291 | C4 | — | 848 | 1 | no | — |
| 292 | C4 | #16810483 | 1 | 1 | yes | characteristic |
| 293 | C4 | #16810619 | 1 | 1 | yes | characteristic |
| 294 | C4 | #16810838 | 1 | 1 | yes | characteristic |
| 295 | C4 | #16810923 | 1 | 1 | yes | characteristic |
| 296 | C2 | — | 2 | 1 | no | — |
| 297 | C2 | — | 2 | 1 | no | — |
| 298 | C2 | — | 2 | 1 | no | — |
| 299 | C2 | — | 8 | 2 | no | — |
| 300 | C2 | #20274725 | 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