← #27299196
Subgroups — #27299196
Order 13568.
426 conjugacy classes of subgroups.
The subgroup lattice has 294 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16811040 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16810407 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20276312 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×C4×#11719908 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20272804 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20272797 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C53×#1764 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16811040 | — | 2 | 2 | no | — |
| 9 | #16811040 | C4 | 1 | 4 | yes | complemented |
| 10 | C2×#13133157 | — | 2 | 1 | no | — |
| 11 | C2×#13133157 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#13133157 | C22 | 1 | 1 | yes | characteristic |
| 13 | C22×#11719908 | — | 2 | 1 | no | — |
| 14 | C22×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 15 | C22×#11719908 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16810407 | C4 | 1 | 4 | yes | complemented |
| 17 | #16810399 | — | 2 | 1 | no | — |
| 18 | C4×#11719908 | — | 2 | 2 | no | — |
| 19 | C4×#11719908 | — | 2 | 1 | no | — |
| 20 | C2×#131×C53 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#104×C53 | C4 | 1 | 1 | yes | characteristic |
| 22 | C53×#416 | C4 | 1 | 1 | yes | characteristic |
| 23 | C53×#405 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 25 | C53×#341 | C4 | 1 | 1 | yes | characteristic |
| 26 | C53×#339 | C4 | 1 | 1 | yes | characteristic |
| 27 | #13133157 | — | 2 | 1 | no | — |
| 28 | #13133157 | — | 2 | 1 | no | — |
| 29 | #13133157 | — | 2 | 4 | no | — |
| 30 | #13133157 | C2×C4 | 1 | 2 | yes | — |
| 31 | #13133157 | C2×C4 | 1 | 2 | yes | — |
| 32 | C2×#11719908 | — | 2 | 1 | no | — |
| 33 | C2×#11719908 | — | 2 | 1 | no | — |
| 34 | C2×#11719908 | — | 2 | 1 | no | — |
| 35 | C2×#11719908 | — | 2 | 1 | no | — |
| 36 | C2×#11719908 | — | 2 | 4 | no | — |
| 37 | C2×#11719908 | C2×C4 | 1 | 2 | yes | — |
| 38 | C2×#11719908 | C2×C4 | 1 | 2 | yes | — |
| 39 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 40 | C2×#11719908 | D4 | 1 | 1 | yes | characteristic |
| 41 | C2×#11719908 | D4 | 1 | 2 | yes | — |
| 42 | C2×#41×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | #131×C53 | — | 2 | 1 | no | — |
| 44 | #131×C53 | — | 2 | 1 | no | — |
| 45 | #131×C53 | C2×C4 | 1 | 2 | yes | — |
| 46 | #131×C53 | C2×C4 | 1 | 2 | yes | — |
| 47 | C2×M16×C53 | — | 2 | 1 | no | — |
| 48 | C2×M16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C2×M16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | C22×C8×C53 | — | 2 | 1 | no | — |
| 51 | C22×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 53 | #117×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C2×C42×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | #104×C53 | C8 | 1 | 4 | yes | — |
| 56 | #103×C53 | C8 | 1 | 2 | yes | — |
| 57 | #102×C53 | C8 | 1 | 2 | yes | — |
| 58 | #97×C53 | — | 2 | 1 | no | — |
| 59 | C4×C8×C53 | — | 2 | 1 | no | — |
| 60 | C4×C8×C53 | — | 2 | 1 | no | — |
| 61 | C4×C8×C53 | — | 2 | 1 | no | — |
| 62 | #41×C53 | — | 2 | 1 | no | — |
| 63 | #41×C53 | — | 2 | 1 | no | — |
| 64 | #41×C53 | C42 | 1 | 2 | yes | — |
| 65 | #41×C53 | C2×C8 | 1 | 2 | yes | — |
| 66 | C2×Q8×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 67 | C2×D4×C53 | — | 2 | 1 | no | — |
| 68 | C2×D4×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 69 | C22×C4×C53 | — | 2 | 1 | no | — |
| 70 | C22×C4×C53 | — | 2 | 1 | no | — |
| 71 | C22×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 72 | M16×C53 | — | 2 | 1 | no | — |
| 73 | M16×C53 | — | 2 | 1 | no | — |
| 74 | M16×C53 | — | 2 | 2 | no | — |
| 75 | M16×C53 | — | 2 | 2 | no | — |
| 76 | M16×C53 | C42 | 1 | 2 | yes | — |
| 77 | M16×C53 | C2×C8 | 1 | 2 | yes | — |
| 78 | C2×C8×C53 | — | 2 | 1 | no | — |
| 79 | C2×C8×C53 | — | 2 | 1 | no | — |
| 80 | C2×C8×C53 | — | 2 | 1 | no | — |
| 81 | C2×C8×C53 | — | 2 | 1 | no | — |
| 82 | C2×C8×C53 | — | 2 | 2 | no | — |
| 83 | C2×C8×C53 | — | 2 | 2 | no | — |
| 84 | C2×C8×C53 | C2×C8 | 1 | 2 | yes | — |
| 85 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 86 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 87 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 90 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 91 | C4⋊C4×C53 | C2×C8 | 1 | 2 | yes | — |
| 92 | C22⋊C4×C53 | — | 2 | 1 | no | — |
| 93 | C42×C53 | — | 2 | 1 | no | — |
| 94 | C42×C53 | — | 2 | 1 | no | — |
| 95 | C42×C53 | C2×C8 | 1 | 2 | yes | — |
| 96 | #11719908 | — | 2 | 2 | no | — |
| 97 | #11719908 | — | 2 | 2 | no | — |
| 98 | #11719908 | — | 2 | 2 | no | — |
| 99 | #11719908 | C22⋊C4 | 1 | 2 | yes | — |
| 100 | #11719908 | C22⋊C4 | 1 | 2 | yes | — |
| 101 | C23×C53 | — | 2 | 1 | no | — |
| 102 | C23×C53 | #131 | 1 | 1 | yes | characteristic |
| 103 | Q8×C53 | — | 2 | 1 | no | — |
| 104 | Q8×C53 | C4×C8 | 1 | 2 | yes | — |
| 105 | D4×C53 | — | 2 | 2 | no | — |
| 106 | D4×C53 | — | 2 | 2 | no | — |
| 107 | D4×C53 | — | 2 | 1 | no | — |
| 108 | D4×C53 | C4×C8 | 1 | 2 | yes | — |
| 109 | C2×C4×C53 | — | 2 | 4 | no | — |
| 110 | C2×C4×C53 | — | 2 | 1 | no | — |
| 111 | C2×C4×C53 | — | 2 | 2 | no | — |
| 112 | C2×C4×C53 | — | 2 | 2 | no | — |
| 113 | C2×C4×C53 | — | 2 | 2 | no | — |
| 114 | C2×C4×C53 | — | 2 | 2 | no | — |
| 115 | C2×C4×C53 | C2×C42 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C53 | #131 | 1 | 1 | yes | characteristic |
| 121 | C8×C53 | — | 2 | 2 | no | — |
| 122 | C8×C53 | — | 2 | 2 | no | — |
| 123 | C8×C53 | — | 2 | 2 | no | — |
| 124 | C8×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 125 | C8×C53 | #117 | 1 | 1 | yes | characteristic |
| 126 | C8×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 127 | C8×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 128 | #12614 | — | 53 | 1 | no | maximal, Sylow |
| 129 | C22×C53 | — | 2 | 2 | no | — |
| 130 | C22×C53 | — | 2 | 2 | no | — |
| 131 | C22×C53 | — | 4 | 2 | no | — |
| 132 | C22×C53 | #405 | 1 | 1 | yes | characteristic |
| 133 | C22×C53 | C8×D4 | 1 | 1 | yes | characteristic |
| 134 | C22×C53 | C8×D4 | 1 | 1 | yes | characteristic |
| 135 | C4×C53 | — | 2 | 4 | no | — |
| 136 | C4×C53 | — | 2 | 2 | no | — |
| 137 | C4×C53 | — | 4 | 2 | no | — |
| 138 | C4×C53 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 139 | C4×C53 | C2×C4×C8 | 1 | 1 | yes | characteristic |
| 140 | C4×C53 | C8×D4 | 1 | 1 | yes | characteristic |
| 141 | C4×C53 | C8×D4 | 1 | 1 | yes | characteristic |
| 142 | C2×#504 | — | 53 | 1 | no | — |
| 143 | C2×#350 | — | 53 | 1 | no | — |
| 144 | #2108 | — | 53 | 1 | no | — |
| 145 | C2×C4×C16 | — | 53 | 1 | no | — |
| 146 | #1764 | D53 | 1 | 1 | yes | characteristic |
| 147 | #1379 | — | 53 | 1 | no | — |
| 148 | #1376 | — | 53 | 1 | no | — |
| 149 | C2×C53 | — | 2 | 1 | no | — |
| 150 | C2×C53 | — | 2 | 1 | no | — |
| 151 | C2×C53 | — | 4 | 2 | no | — |
| 152 | C2×C53 | C8×C22⋊C4 | 1 | 1 | yes | characteristic |
| 153 | C2×C53 | #2170 | 1 | 2 | yes | — |
| 154 | C2×#131 | C2×D53 | 1 | 1 | yes | characteristic |
| 155 | #504 | — | 53 | 4 | no | — |
| 156 | #504 | — | 106 | 2 | no | — |
| 157 | C2×M32 | — | 53 | 1 | no | — |
| 158 | C2×M32 | — | 53 | 1 | no | — |
| 159 | C2×M32 | — | 106 | 1 | no | — |
| 160 | C22×C16 | — | 53 | 1 | no | — |
| 161 | C22×C16 | — | 53 | 1 | no | — |
| 162 | C22×C16 | — | 106 | 1 | no | — |
| 163 | C2×#104 | Dic53 | 1 | 1 | yes | characteristic |
| 164 | #416 | Dic53 | 1 | 1 | yes | characteristic |
| 165 | #405 | C2×D53 | 1 | 1 | yes | characteristic |
| 166 | C2×C4×C8 | C2×D53 | 1 | 1 | yes | characteristic |
| 167 | #350 | — | 53 | 4 | no | — |
| 168 | #346 | — | 106 | 1 | no | — |
| 169 | C4×C16 | — | 106 | 2 | no | — |
| 170 | C4×C16 | — | 106 | 1 | no | — |
| 171 | #341 | Dic53 | 1 | 1 | yes | characteristic |
| 172 | #339 | Dic53 | 1 | 1 | yes | characteristic |
| 173 | C53 | #12614 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 174 | C2×#41 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 175 | #131 | — | 2 | 1 | no | — |
| 176 | #131 | — | 2 | 1 | no | — |
| 177 | #131 | C4×D53 | 1 | 2 | yes | — |
| 178 | #131 | C4×D53 | 1 | 2 | yes | — |
| 179 | C2×M16 | — | 2 | 1 | no | — |
| 180 | C2×M16 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 181 | C2×M16 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 182 | C22×C8 | — | 2 | 1 | no | — |
| 183 | C22×C8 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 184 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 185 | #117 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 186 | C2×C42 | C2×Dic53 | 1 | 1 | yes | characteristic |
| 187 | M32 | — | 53 | 2 | no | — |
| 188 | M32 | — | 53 | 2 | no | — |
| 189 | M32 | — | 106 | 4 | no | — |
| 190 | M32 | — | 106 | 1 | no | — |
| 191 | M32 | — | 106 | 1 | no | — |
| 192 | C2×C16 | — | 53 | 2 | no | — |
| 193 | C2×C16 | — | 53 | 2 | no | — |
| 194 | C2×C16 | — | 53 | 1 | no | — |
| 195 | C2×C16 | — | 53 | 1 | no | — |
| 196 | C2×C16 | — | 53 | 2 | no | — |
| 197 | C2×C16 | — | 106 | 4 | no | — |
| 198 | C2×C16 | — | 106 | 1 | no | — |
| 199 | C2×C16 | — | 106 | 1 | no | — |
| 200 | C2×C16 | — | 106 | 1 | no | — |
| 201 | C2×C16 | — | 106 | 1 | no | — |
| 202 | #104 | #88322 | 1 | 4 | yes | — |
| 203 | #103 | #88322 | 1 | 2 | yes | — |
| 204 | #102 | #88322 | 1 | 2 | yes | — |
| 205 | #97 | — | 2 | 1 | no | — |
| 206 | C4×C8 | — | 2 | 1 | no | — |
| 207 | C4×C8 | — | 2 | 1 | no | — |
| 208 | C4×C8 | — | 2 | 1 | no | — |
| 209 | #41 | — | 2 | 1 | no | — |
| 210 | #41 | — | 2 | 1 | no | — |
| 211 | #41 | C2×#88322 | 1 | 2 | yes | — |
| 212 | #41 | C4×Dic53 | 1 | 2 | yes | — |
| 213 | C2×Q8 | C2×#88322 | 1 | 1 | yes | characteristic |
| 214 | C2×D4 | — | 2 | 1 | no | — |
| 215 | C2×D4 | C2×#88322 | 1 | 1 | yes | characteristic |
| 216 | C22×C4 | — | 2 | 1 | no | — |
| 217 | C22×C4 | — | 2 | 1 | no | — |
| 218 | C22×C4 | C22×Dic53 | 1 | 1 | yes | characteristic |
| 219 | M16 | — | 2 | 1 | no | — |
| 220 | M16 | — | 2 | 1 | no | — |
| 221 | M16 | — | 2 | 2 | no | — |
| 222 | M16 | — | 2 | 2 | no | — |
| 223 | M16 | C2×#88322 | 1 | 2 | yes | — |
| 224 | M16 | C4×Dic53 | 1 | 2 | yes | — |
| 225 | C2×C8 | — | 2 | 1 | no | — |
| 226 | C2×C8 | — | 2 | 1 | no | — |
| 227 | C2×C8 | — | 2 | 1 | no | — |
| 228 | C2×C8 | — | 2 | 1 | no | — |
| 229 | C2×C8 | — | 2 | 2 | no | — |
| 230 | C2×C8 | — | 2 | 2 | no | — |
| 231 | C2×C8 | C2×#88322 | 1 | 2 | yes | — |
| 232 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 233 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 234 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 235 | C2×C8 | D4×D53 | 1 | 1 | yes | characteristic |
| 236 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | #11719947 | 1 | 1 | yes | characteristic |
| 238 | C4⋊C4 | C2×#88322 | 1 | 2 | yes | — |
| 239 | C22⋊C4 | — | 2 | 1 | no | — |
| 240 | C42 | — | 2 | 1 | no | — |
| 241 | C42 | — | 2 | 1 | no | — |
| 242 | C42 | C2×#88322 | 1 | 2 | yes | — |
| 243 | C16 | — | 53 | 2 | no | — |
| 244 | C16 | — | 53 | 2 | no | — |
| 245 | C16 | — | 106 | 2 | no | — |
| 246 | C16 | — | 106 | 2 | no | — |
| 247 | C16 | — | 106 | 2 | no | — |
| 248 | C23 | — | 2 | 1 | no | — |
| 249 | C23 | #13133224 | 1 | 1 | yes | characteristic |
| 250 | Q8 | — | 2 | 1 | no | — |
| 251 | Q8 | C4×#88322 | 1 | 2 | yes | — |
| 252 | D4 | — | 2 | 2 | no | — |
| 253 | D4 | — | 2 | 2 | no | — |
| 254 | D4 | — | 2 | 1 | no | — |
| 255 | D4 | C4×#88322 | 1 | 2 | yes | — |
| 256 | C2×C4 | — | 2 | 4 | no | — |
| 257 | C2×C4 | — | 2 | 2 | no | — |
| 258 | C2×C4 | — | 2 | 1 | no | — |
| 259 | C2×C4 | — | 2 | 2 | no | — |
| 260 | C2×C4 | — | 2 | 2 | no | — |
| 261 | C2×C4 | — | 2 | 2 | no | — |
| 262 | C2×C4 | C2×C4×Dic53 | 1 | 1 | yes | characteristic |
| 263 | C2×C4 | D4×Dic53 | 1 | 1 | yes | characteristic |
| 264 | C2×C4 | D4×Dic53 | 1 | 1 | yes | characteristic |
| 265 | C2×C4 | C22×#88322 | 1 | 1 | yes | characteristic |
| 266 | C2×C4 | C22×#88322 | 1 | 1 | yes | characteristic |
| 267 | C2×C4 | #13133224 | 1 | 1 | yes | characteristic |
| 268 | C8 | — | 2 | 2 | no | — |
| 269 | C8 | — | 2 | 2 | no | — |
| 270 | C8 | — | 2 | 2 | no | — |
| 271 | C8 | C22⋊C4×D53 | 1 | 1 | yes | characteristic |
| 272 | C8 | #13133173 | 1 | 1 | yes | characteristic |
| 273 | C8 | #13133177 | 1 | 1 | yes | characteristic |
| 274 | C8 | #13133177 | 1 | 1 | yes | characteristic |
| 275 | C22 | — | 2 | 2 | no | — |
| 276 | C22 | — | 2 | 2 | no | — |
| 277 | C22 | — | 4 | 2 | no | — |
| 278 | C22 | #16810579 | 1 | 1 | yes | characteristic |
| 279 | C22 | D4×#88322 | 1 | 1 | yes | characteristic |
| 280 | C22 | D4×#88322 | 1 | 1 | yes | characteristic |
| 281 | C4 | — | 2 | 4 | no | — |
| 282 | C4 | — | 2 | 2 | no | — |
| 283 | C4 | — | 4 | 2 | no | — |
| 284 | C4 | C22⋊C4×Dic53 | 1 | 1 | yes | characteristic |
| 285 | C4 | C2×C4×#88322 | 1 | 1 | yes | characteristic |
| 286 | C4 | D4×#88322 | 1 | 1 | yes | characteristic |
| 287 | C4 | D4×#88322 | 1 | 1 | yes | characteristic |
| 288 | C2 | — | 2 | 1 | no | — |
| 289 | C2 | — | 2 | 1 | no | — |
| 290 | C2 | — | 4 | 2 | no | — |
| 291 | C2 | C22⋊C4×#88322 | 1 | 1 | yes | characteristic |
| 292 | C2 | #20276552 | 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