← #27300201
Subgroups — #27300201
Order 13568.
400 conjugacy classes of subgroups.
The subgroup lattice has 332 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16810740 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16810443 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20276464 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20274885 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20272819 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20272818 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C53×#2148 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#13133224 | — | 2 | 1 | no | — |
| 9 | C2×#13133157 | C22 | 1 | 1 | yes | characteristic |
| 10 | #16810740 | — | 2 | 1 | no | — |
| 11 | #16810740 | — | 2 | 1 | no | — |
| 12 | #16810740 | C4 | 1 | 2 | yes | — |
| 13 | #16810740 | C4 | 1 | 2 | yes | — |
| 14 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#13133153 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#13133152 | — | 2 | 1 | no | — |
| 17 | #16810646 | — | 2 | 1 | no | — |
| 18 | C2×#13133144 | — | 2 | 1 | no | — |
| 19 | C2×C8×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 20 | #16810443 | — | 2 | 1 | no | — |
| 21 | #16810443 | — | 2 | 1 | no | — |
| 22 | #16810439 | — | 2 | 1 | no | — |
| 23 | #16810436 | — | 2 | 1 | no | — |
| 24 | C2×#135×C53 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×M32×C53 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#108×C53 | C22 | 1 | 1 | yes | characteristic |
| 27 | C53×#362 | — | 2 | 1 | no | — |
| 28 | C53×#361 | — | 2 | 1 | no | — |
| 29 | #13133224 | — | 2 | 2 | no | — |
| 30 | #13133224 | — | 2 | 2 | no | — |
| 31 | #13133224 | — | 4 | 1 | no | — |
| 32 | #13133224 | — | 4 | 1 | no | — |
| 33 | C2×#11719917 | — | 2 | 1 | no | — |
| 34 | C2×#11719917 | — | 2 | 1 | no | — |
| 35 | C2×#11719917 | — | 4 | 1 | no | — |
| 36 | C22×#88322 | — | 4 | 1 | no | — |
| 37 | C22×#88322 | D4 | 1 | 1 | yes | characteristic |
| 38 | C2×C4×Dic53 | D4 | 1 | 1 | yes | characteristic |
| 39 | #13133157 | — | 2 | 1 | no | — |
| 40 | #13133157 | — | 2 | 1 | no | — |
| 41 | C2×#11719908 | — | 2 | 1 | no | — |
| 42 | #13133153 | — | 2 | 1 | no | — |
| 43 | #13133153 | — | 2 | 1 | no | — |
| 44 | #13133153 | — | 2 | 1 | no | — |
| 45 | #13133153 | C2×C4 | 1 | 2 | yes | — |
| 46 | #13133152 | — | 4 | 1 | no | — |
| 47 | #13133152 | — | 4 | 1 | no | — |
| 48 | #13133146 | — | 4 | 2 | no | — |
| 49 | #13133144 | — | 2 | 2 | no | — |
| 50 | #13133144 | — | 2 | 2 | no | — |
| 51 | C8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C8×Dic53 | D4 | 1 | 1 | yes | characteristic |
| 54 | C8×Dic53 | D4 | 1 | 1 | yes | characteristic |
| 55 | C2×#41×C53 | — | 2 | 1 | no | — |
| 56 | #135×C53 | — | 2 | 1 | no | — |
| 57 | #135×C53 | — | 2 | 1 | no | — |
| 58 | #135×C53 | C2×C4 | 1 | 2 | yes | — |
| 59 | #135×C53 | Q8 | 1 | 2 | yes | — |
| 60 | C2×Q16×C53 | D4 | 1 | 1 | yes | characteristic |
| 61 | C2×SD16×C53 | — | 2 | 1 | no | — |
| 62 | C2×D8×C53 | D4 | 1 | 1 | yes | characteristic |
| 63 | C2×M16×C53 | — | 2 | 1 | no | — |
| 64 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 65 | M32×C53 | — | 2 | 1 | no | — |
| 66 | M32×C53 | — | 2 | 1 | no | — |
| 67 | C2×C16×C53 | — | 2 | 1 | no | — |
| 68 | #108×C53 | — | 2 | 1 | no | — |
| 69 | #108×C53 | — | 2 | 1 | no | — |
| 70 | #41×C53 | — | 2 | 2 | no | — |
| 71 | #41×C53 | — | 2 | 2 | no | — |
| 72 | #41×C53 | — | 4 | 1 | no | — |
| 73 | #41×C53 | — | 4 | 1 | no | — |
| 74 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 75 | C2×D4×C53 | — | 2 | 1 | no | — |
| 76 | C2×D4×C53 | — | 4 | 1 | no | — |
| 77 | C22×C4×C53 | — | 4 | 1 | no | — |
| 78 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | C22×Dic53 | — | 2 | 1 | no | — |
| 80 | Q16×C53 | — | 2 | 1 | no | — |
| 81 | Q16×C53 | C4⋊C4 | 1 | 2 | yes | — |
| 82 | SD16×C53 | — | 2 | 2 | no | — |
| 83 | SD16×C53 | — | 2 | 2 | no | — |
| 84 | D8×C53 | — | 2 | 1 | no | — |
| 85 | D8×C53 | C4⋊C4 | 1 | 2 | yes | — |
| 86 | M16×C53 | — | 4 | 1 | no | — |
| 87 | M16×C53 | — | 4 | 1 | no | — |
| 88 | C2×C8×C53 | — | 4 | 1 | no | — |
| 89 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 90 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 93 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 94 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 95 | C4×Dic53 | — | 2 | 1 | no | — |
| 96 | C4×Dic53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 97 | C4×Dic53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 98 | #11719917 | — | 2 | 2 | no | — |
| 99 | #11719917 | — | 4 | 1 | no | — |
| 100 | #11719917 | — | 4 | 2 | no | — |
| 101 | #11719917 | — | 4 | 2 | no | — |
| 102 | #11719917 | — | 4 | 1 | no | — |
| 103 | #11719917 | — | 4 | 1 | no | — |
| 104 | C2×#88322 | — | 4 | 1 | no | — |
| 105 | C2×#88322 | — | 4 | 2 | no | — |
| 106 | C2×#88322 | — | 4 | 2 | no | — |
| 107 | C2×#88322 | — | 4 | 1 | no | — |
| 108 | C2×#88322 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 109 | C2×#88322 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 110 | C2×#88322 | D8 | 1 | 1 | yes | characteristic |
| 111 | C2×#88322 | D8 | 1 | 1 | yes | characteristic |
| 112 | C2×#88322 | SD16 | 1 | 1 | yes | characteristic |
| 113 | C2×#88322 | SD16 | 1 | 1 | yes | characteristic |
| 114 | C16×C53 | — | 4 | 1 | no | — |
| 115 | #11719908 | — | 4 | 1 | no | — |
| 116 | C23×C53 | — | 4 | 1 | no | — |
| 117 | C23×C53 | #137 | 1 | 1 | yes | characteristic |
| 118 | Q8×C53 | — | 2 | 2 | no | — |
| 119 | Q8×C53 | — | 4 | 1 | no | — |
| 120 | D4×C53 | — | 2 | 2 | no | — |
| 121 | D4×C53 | — | 4 | 2 | no | — |
| 122 | D4×C53 | — | 4 | 2 | no | — |
| 123 | D4×C53 | — | 4 | 1 | no | — |
| 124 | C2×C4×C53 | — | 4 | 1 | no | — |
| 125 | C2×C4×C53 | — | 4 | 2 | no | — |
| 126 | C2×C4×C53 | — | 4 | 2 | no | — |
| 127 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C53 | #121 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C53 | C2×D8 | 1 | 1 | yes | characteristic |
| 131 | C2×C4×C53 | C2×SD16 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C53 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 133 | C2×Dic53 | — | 2 | 1 | no | — |
| 134 | C2×Dic53 | — | 2 | 1 | no | — |
| 135 | C2×Dic53 | — | 4 | 2 | no | — |
| 136 | C8×C53 | — | 8 | 1 | no | — |
| 137 | C8×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 138 | C8×C53 | #117 | 1 | 1 | yes | characteristic |
| 139 | C8×C53 | #121 | 1 | 1 | yes | characteristic |
| 140 | C8×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 141 | #88322 | — | 4 | 2 | no | — |
| 142 | #88322 | — | 8 | 1 | no | — |
| 143 | #88322 | #102 | 1 | 1 | yes | characteristic |
| 144 | #88322 | #102 | 1 | 1 | yes | characteristic |
| 145 | #88322 | #102 | 1 | 1 | yes | characteristic |
| 146 | #88322 | #102 | 1 | 1 | yes | characteristic |
| 147 | #12825 | — | 53 | 1 | no | maximal, Sylow |
| 148 | C22×C53 | — | 4 | 1 | no | — |
| 149 | C22×C53 | — | 4 | 2 | no | — |
| 150 | C22×C53 | — | 8 | 2 | no | — |
| 151 | C22×C53 | #417 | 1 | 1 | yes | characteristic |
| 152 | C22×C53 | #461 | 1 | 1 | yes | characteristic |
| 153 | C22×C53 | #474 | 1 | 1 | yes | characteristic |
| 154 | C4×C53 | — | 4 | 2 | no | — |
| 155 | C4×C53 | #380 | 1 | 1 | yes | characteristic |
| 156 | C4×C53 | C2×#102 | 1 | 1 | yes | characteristic |
| 157 | C4×C53 | #459 | 1 | 1 | yes | characteristic |
| 158 | C4×C53 | #476 | 1 | 1 | yes | characteristic |
| 159 | Dic53 | — | 2 | 2 | no | — |
| 160 | Dic53 | — | 4 | 1 | no | — |
| 161 | C2×#443 | — | 53 | 1 | no | — |
| 162 | C2×#364 | — | 53 | 1 | no | — |
| 163 | #2148 | — | 53 | 1 | no | — |
| 164 | #2148 | D53 | 1 | 1 | yes | characteristic |
| 165 | #1848 | — | 53 | 1 | no | — |
| 166 | #1385 | — | 53 | 1 | no | — |
| 167 | #1385 | — | 53 | 1 | no | — |
| 168 | C2×C53 | — | 2 | 1 | no | — |
| 169 | C2×C53 | — | 2 | 1 | no | — |
| 170 | C2×C53 | — | 2 | 1 | no | — |
| 171 | C2×C53 | — | 8 | 2 | no | — |
| 172 | C2×C53 | #1802 | 1 | 1 | yes | characteristic |
| 173 | C2×#135 | C2×D53 | 1 | 1 | yes | characteristic |
| 174 | C2×#131 | — | 106 | 1 | no | — |
| 175 | C2×M32 | — | 53 | 1 | no | — |
| 176 | C2×M32 | C2×D53 | 1 | 1 | yes | characteristic |
| 177 | #443 | — | 53 | 2 | no | — |
| 178 | #443 | — | 53 | 2 | no | — |
| 179 | #443 | — | 106 | 1 | no | — |
| 180 | #443 | — | 106 | 1 | no | — |
| 181 | C2×#108 | — | 53 | 1 | no | — |
| 182 | C2×#108 | — | 53 | 1 | no | — |
| 183 | C2×#108 | — | 106 | 1 | no | — |
| 184 | C2×#108 | C2×D53 | 1 | 1 | yes | characteristic |
| 185 | #423 | — | 106 | 1 | no | — |
| 186 | C2×#104 | — | 106 | 1 | no | — |
| 187 | C2×C4×C8 | — | 53 | 1 | no | — |
| 188 | #364 | — | 106 | 1 | no | — |
| 189 | #364 | — | 106 | 1 | no | — |
| 190 | #362 | — | 2 | 1 | no | — |
| 191 | #362 | — | 106 | 1 | no | — |
| 192 | #361 | — | 2 | 1 | no | — |
| 193 | #361 | — | 106 | 1 | no | — |
| 194 | C53 | #12825 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 195 | C2×#41 | — | 2 | 1 | no | — |
| 196 | #135 | — | 2 | 1 | no | — |
| 197 | #135 | — | 2 | 1 | no | — |
| 198 | #135 | Dic106 | 1 | 2 | yes | — |
| 199 | #135 | C4×D53 | 1 | 2 | yes | — |
| 200 | C2×Q16 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 201 | C2×SD16 | — | 2 | 1 | no | — |
| 202 | C2×D8 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 203 | #131 | — | 106 | 2 | no | — |
| 204 | #131 | — | 106 | 2 | no | — |
| 205 | #131 | — | 212 | 1 | no | — |
| 206 | #131 | — | 212 | 1 | no | — |
| 207 | C2×M16 | — | 2 | 1 | no | — |
| 208 | C2×M16 | — | 106 | 1 | no | — |
| 209 | C2×M16 | — | 106 | 1 | no | — |
| 210 | C2×M16 | — | 212 | 1 | no | — |
| 211 | C22×C8 | — | 53 | 1 | no | — |
| 212 | C22×C8 | — | 212 | 1 | no | — |
| 213 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 214 | C2×C42 | — | 53 | 1 | no | — |
| 215 | M32 | — | 2 | 1 | no | — |
| 216 | M32 | — | 2 | 1 | no | — |
| 217 | M32 | — | 106 | 1 | no | — |
| 218 | M32 | — | 106 | 1 | no | — |
| 219 | C2×C16 | — | 2 | 1 | no | — |
| 220 | C2×C16 | — | 106 | 1 | no | — |
| 221 | #108 | — | 2 | 1 | no | — |
| 222 | #108 | — | 2 | 1 | no | — |
| 223 | #108 | — | 53 | 2 | no | — |
| 224 | #108 | — | 106 | 1 | no | — |
| 225 | #108 | — | 106 | 1 | no | — |
| 226 | #108 | — | 106 | 1 | no | — |
| 227 | #108 | — | 212 | 1 | no | — |
| 228 | #108 | — | 212 | 1 | no | — |
| 229 | C4⋊C8 | — | 212 | 2 | no | — |
| 230 | #104 | — | 106 | 2 | no | — |
| 231 | #104 | — | 106 | 2 | no | — |
| 232 | C4×C8 | — | 53 | 1 | no | — |
| 233 | C4×C8 | — | 53 | 1 | no | — |
| 234 | C4×C8 | — | 53 | 1 | no | — |
| 235 | C4×C8 | — | 53 | 1 | no | — |
| 236 | #41 | — | 2 | 2 | no | — |
| 237 | #41 | — | 2 | 2 | no | — |
| 238 | #41 | — | 4 | 1 | no | — |
| 239 | #41 | — | 4 | 1 | no | — |
| 240 | C2×Q8 | — | 2 | 1 | no | — |
| 241 | C2×D4 | — | 2 | 1 | no | — |
| 242 | C2×D4 | — | 4 | 1 | no | — |
| 243 | C22×C4 | — | 4 | 1 | no | — |
| 244 | C22×C4 | — | 106 | 1 | no | — |
| 245 | C22×C4 | D4×D53 | 1 | 1 | yes | characteristic |
| 246 | Q16 | — | 2 | 1 | no | — |
| 247 | Q16 | #11719919 | 1 | 2 | yes | — |
| 248 | SD16 | — | 2 | 2 | no | — |
| 249 | SD16 | — | 2 | 2 | no | — |
| 250 | D8 | — | 2 | 1 | no | — |
| 251 | D8 | #11719919 | 1 | 2 | yes | — |
| 252 | M16 | — | 4 | 1 | no | — |
| 253 | M16 | — | 4 | 1 | no | — |
| 254 | M16 | — | 106 | 2 | no | — |
| 255 | M16 | — | 212 | 2 | no | — |
| 256 | M16 | — | 212 | 2 | no | — |
| 257 | M16 | — | 212 | 1 | no | — |
| 258 | M16 | — | 212 | 1 | no | — |
| 259 | M16 | — | 212 | 1 | no | — |
| 260 | C2×C8 | — | 4 | 1 | no | — |
| 261 | C2×C8 | — | 53 | 1 | no | — |
| 262 | C2×C8 | — | 53 | 1 | no | — |
| 263 | C2×C8 | — | 53 | 1 | no | — |
| 264 | C2×C8 | — | 53 | 1 | no | — |
| 265 | C2×C8 | — | 53 | 1 | no | — |
| 266 | C2×C8 | — | 53 | 1 | no | — |
| 267 | C2×C8 | — | 212 | 2 | no | — |
| 268 | C2×C8 | — | 212 | 2 | no | — |
| 269 | C2×C8 | — | 212 | 1 | no | — |
| 270 | C2×C8 | — | 212 | 1 | no | — |
| 271 | C2×C8 | C2×Dic106 | 1 | 1 | yes | characteristic |
| 272 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 273 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 274 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 275 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 276 | C2×C8 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 277 | C42 | — | 53 | 1 | no | — |
| 278 | C42 | — | 53 | 1 | no | — |
| 279 | C42 | — | 106 | 1 | no | — |
| 280 | C16 | — | 4 | 1 | no | — |
| 281 | C16 | — | 212 | 1 | no | — |
| 282 | C23 | — | 4 | 1 | no | — |
| 283 | C23 | #13133246 | 1 | 1 | yes | characteristic |
| 284 | Q8 | — | 2 | 2 | no | — |
| 285 | Q8 | — | 4 | 1 | no | — |
| 286 | D4 | — | 2 | 2 | no | — |
| 287 | D4 | — | 4 | 2 | no | — |
| 288 | D4 | — | 4 | 2 | no | — |
| 289 | D4 | — | 4 | 1 | no | — |
| 290 | C2×C4 | — | 4 | 1 | no | — |
| 291 | C2×C4 | — | 4 | 2 | no | — |
| 292 | C2×C4 | — | 4 | 2 | no | — |
| 293 | C2×C4 | — | 106 | 1 | no | — |
| 294 | C2×C4 | — | 106 | 1 | no | — |
| 295 | C2×C4 | — | 212 | 2 | no | — |
| 296 | C2×C4 | C22⋊C4×D53 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | #13133189 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #13133194 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | D8×D53 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | SD16×D53 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | #13133241 | 1 | 1 | yes | characteristic |
| 302 | C8 | — | 8 | 1 | no | — |
| 303 | C8 | — | 53 | 1 | no | — |
| 304 | C8 | — | 53 | 1 | no | — |
| 305 | C8 | — | 53 | 1 | no | — |
| 306 | C8 | — | 53 | 1 | no | — |
| 307 | C8 | — | 424 | 1 | no | — |
| 308 | C8 | — | 212 | 2 | no | — |
| 309 | C8 | C2×#11719919 | 1 | 1 | yes | characteristic |
| 310 | C8 | #13133173 | 1 | 1 | yes | characteristic |
| 311 | C8 | #13133189 | 1 | 1 | yes | characteristic |
| 312 | C8 | #13133194 | 1 | 1 | yes | characteristic |
| 313 | C22 | — | 4 | 1 | no | — |
| 314 | C22 | — | 4 | 2 | no | — |
| 315 | C22 | — | 8 | 2 | no | — |
| 316 | C22 | #16810621 | 1 | 1 | yes | characteristic |
| 317 | C22 | #16810826 | 1 | 1 | yes | characteristic |
| 318 | C22 | #16810901 | 1 | 1 | yes | characteristic |
| 319 | C4 | — | 4 | 2 | no | — |
| 320 | C4 | — | 106 | 2 | no | — |
| 321 | C4 | — | 212 | 1 | no | — |
| 322 | C4 | #16810481 | 1 | 1 | yes | characteristic |
| 323 | C4 | #102×D53 | 1 | 1 | yes | characteristic |
| 324 | C4 | #16810816 | 1 | 1 | yes | characteristic |
| 325 | C4 | #16810911 | 1 | 1 | yes | characteristic |
| 326 | C2 | — | 2 | 1 | no | — |
| 327 | C2 | — | 2 | 1 | no | — |
| 328 | C2 | — | 2 | 1 | no | — |
| 329 | C2 | — | 8 | 2 | no | — |
| 330 | C2 | #20274713 | 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