← #27537718
Subgroups — #27537718
Order 13568.
572 conjugacy classes of subgroups.
The subgroup lattice has 284 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20283428 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20276548 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #20276546 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C16×D212 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #20276534 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #20276514 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #20276327 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | C53×#2162 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#13133223 | C22 | 1 | 2 | yes | — |
| 10 | #16811173 | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×#13133156 | C22 | 1 | 4 | yes | — |
| 12 | C2×C16×D53 | C22 | 1 | 4 | yes | — |
| 13 | #16810702 | C4 | 1 | 2 | yes | — |
| 14 | #16810702 | C4 | 1 | 2 | yes | — |
| 15 | C8×D212 | C22 | 1 | 2 | yes | — |
| 16 | C8×D212 | C22 | 1 | 2 | yes | — |
| 17 | #16810647 | C4 | 1 | 1 | yes | characteristic |
| 18 | #16810589 | C4 | 1 | 2 | yes | — |
| 19 | C4⋊#11719908 | C22 | 1 | 1 | yes | characteristic |
| 20 | C4⋊#11719908 | C22 | 1 | 1 | yes | characteristic |
| 21 | C22⋊#11719908 | C22 | 1 | 2 | yes | — |
| 22 | #16810402 | C22 | 1 | 4 | yes | — |
| 23 | #16810402 | C22 | 1 | 4 | yes | — |
| 24 | C4×#11719908 | C22 | 1 | 2 | yes | — |
| 25 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 26 | C53×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 27 | C53×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 28 | C53×C22⋊C16 | C22 | 1 | 2 | yes | — |
| 29 | C4×C16×C53 | C22 | 1 | 2 | yes | — |
| 30 | C2×#11719945 | C2×C4 | 1 | 2 | yes | — |
| 31 | #13133223 | — | 4 | 2 | no | — |
| 32 | #13133223 | — | 4 | 2 | no | — |
| 33 | C2×#11719917 | C2×C4 | 1 | 2 | yes | — |
| 34 | C2×#11719912 | C2×C4 | 1 | 4 | yes | — |
| 35 | C2×C8×D53 | C23 | 1 | 4 | yes | — |
| 36 | #13133217 | C8 | 1 | 1 | yes | characteristic |
| 37 | #13133214 | C8 | 1 | 1 | yes | characteristic |
| 38 | #13133205 | C8 | 1 | 2 | yes | — |
| 39 | #13133191 | C8 | 1 | 2 | yes | — |
| 40 | #13133191 | C8 | 1 | 2 | yes | — |
| 41 | C4×D212 | C2×C4 | 1 | 2 | yes | — |
| 42 | C4×D212 | C2×C4 | 1 | 2 | yes | — |
| 43 | #13133156 | — | 4 | 4 | no | — |
| 44 | C2×#11719908 | C23 | 1 | 2 | yes | — |
| 45 | C2×#11719908 | C23 | 1 | 2 | yes | — |
| 46 | C16×D53 | — | 4 | 4 | no | — |
| 47 | C4⋊#88322 | C2×C4 | 1 | 2 | yes | — |
| 48 | C4⋊#88322 | C2×C4 | 1 | 2 | yes | — |
| 49 | #13133129 | C2×C4 | 1 | 4 | yes | — |
| 50 | #13133129 | C2×C4 | 1 | 4 | yes | — |
| 51 | C22×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 53 | C2×C42×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C2×C16×C53 | C23 | 1 | 2 | yes | — |
| 55 | C2×C16×C53 | C23 | 1 | 2 | yes | — |
| 56 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 57 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 58 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 59 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 60 | C22×C4×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 61 | C22×C4×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 62 | C22×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 63 | C2×C22⋊D53 | C2×C8 | 1 | 4 | yes | — |
| 64 | #11719945 | — | 4 | 2 | no | — |
| 65 | #11719945 | — | 4 | 2 | no | — |
| 66 | C2×D212 | C2×C8 | 1 | 2 | yes | — |
| 67 | C2×D212 | C2×C8 | 1 | 2 | yes | — |
| 68 | C2×D212 | C2×C8 | 1 | 2 | yes | — |
| 69 | C2×C4×D53 | C22×C4 | 1 | 4 | yes | — |
| 70 | C2×Dic106 | C2×C8 | 1 | 2 | yes | — |
| 71 | C2×C8×C53 | — | 2 | 1 | no | — |
| 72 | C2×C8×C53 | — | 2 | 1 | no | — |
| 73 | C2×C8×C53 | — | 4 | 1 | no | — |
| 74 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 76 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | C2×C8×C53 | C24 | 1 | 1 | yes | characteristic |
| 78 | C42×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 79 | C42×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 80 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 81 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | #11719921 | C2×C8 | 1 | 4 | yes | — |
| 83 | #11719921 | C2×C8 | 1 | 4 | yes | — |
| 84 | C4⋊Dic53 | C2×C8 | 1 | 2 | yes | — |
| 85 | C4⋊Dic53 | C2×C8 | 1 | 2 | yes | — |
| 86 | #11719917 | — | 4 | 2 | no | — |
| 87 | C2×#88322 | C22×C4 | 1 | 4 | yes | — |
| 88 | #11719912 | — | 4 | 4 | no | — |
| 89 | C8×D53 | — | 2 | 4 | no | — |
| 90 | C8×D53 | — | 2 | 4 | no | — |
| 91 | C16×C53 | — | 2 | 2 | no | — |
| 92 | C16×C53 | — | 2 | 2 | no | — |
| 93 | #11719908 | — | 2 | 2 | no | — |
| 94 | #11719908 | — | 2 | 2 | no | — |
| 95 | C23×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 96 | C22×D53 | C22×C8 | 1 | 4 | yes | — |
| 97 | C2×C4×C53 | — | 2 | 1 | no | — |
| 98 | C2×C4×C53 | — | 2 | 1 | no | — |
| 99 | C2×C4×C53 | — | 2 | 2 | no | — |
| 100 | C2×C4×C53 | — | 2 | 2 | no | — |
| 101 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 102 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 103 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 104 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 106 | C2×C4×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 107 | C22⋊D53 | — | 4 | 4 | no | — |
| 108 | C2×Dic53 | C22×C8 | 1 | 4 | yes | — |
| 109 | D212 | — | 2 | 4 | no | — |
| 110 | D212 | — | 2 | 4 | no | — |
| 111 | D212 | — | 4 | 2 | no | — |
| 112 | C4×D53 | — | 2 | 4 | no | — |
| 113 | C4×D53 | — | 2 | 4 | no | — |
| 114 | Dic106 | — | 4 | 2 | no | — |
| 115 | C8×C53 | — | 2 | 1 | no | — |
| 116 | C8×C53 | — | 2 | 1 | no | — |
| 117 | C8×C53 | — | 2 | 1 | no | — |
| 118 | C8×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 119 | C8×C53 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 120 | #88322 | — | 2 | 4 | no | — |
| 121 | #33493 | — | 53 | 1 | no | maximal, Sylow |
| 122 | C22×C53 | — | 2 | 1 | no | — |
| 123 | C22×C53 | — | 2 | 1 | no | — |
| 124 | C22×C53 | — | 2 | 1 | no | — |
| 125 | C22×C53 | C23×C8 | 1 | 1 | yes | characteristic |
| 126 | C2×D53 | — | 2 | 4 | no | — |
| 127 | C2×D53 | — | 2 | 4 | no | — |
| 128 | C2×D53 | — | 2 | 4 | no | — |
| 129 | C4×C53 | — | 2 | 1 | no | — |
| 130 | C4×C53 | — | 2 | 1 | no | — |
| 131 | C4×C53 | — | 2 | 1 | no | — |
| 132 | C4×C53 | #504 | 1 | 2 | yes | — |
| 133 | C4×C53 | #504 | 1 | 2 | yes | — |
| 134 | C4×C53 | #518 | 1 | 1 | yes | characteristic |
| 135 | C4×C53 | #568 | 1 | 1 | yes | characteristic |
| 136 | Dic53 | — | 2 | 4 | no | — |
| 137 | #2931 | — | 53 | 1 | no | — |
| 138 | #2169 | — | 53 | 2 | no | — |
| 139 | #2169 | — | 53 | 2 | no | — |
| 140 | D4×C16 | — | 53 | 2 | no | — |
| 141 | D4×C16 | — | 53 | 2 | no | — |
| 142 | #2162 | — | 53 | 1 | no | — |
| 143 | #2162 | D53 | 1 | 1 | yes | characteristic, complemented |
| 144 | #2113 | — | 53 | 4 | no | — |
| 145 | C2×C53 | — | 4 | 1 | no | — |
| 146 | C2×C53 | #2972 | 1 | 1 | yes | characteristic |
| 147 | C2×C53 | C2×#504 | 1 | 1 | yes | characteristic |
| 148 | C2×C53 | C2×#504 | 1 | 1 | yes | characteristic |
| 149 | D53 | — | 4 | 4 | no | — |
| 150 | C2×#131 | — | 53 | 2 | no | — |
| 151 | #532 | — | 53 | 1 | no | — |
| 152 | C2×M32 | — | 53 | 4 | no | — |
| 153 | C22×C16 | — | 53 | 4 | no | — |
| 154 | #436 | — | 53 | 2 | no | — |
| 155 | #436 | — | 53 | 2 | no | — |
| 156 | C8×D4 | — | 53 | 2 | no | — |
| 157 | C8×D4 | — | 53 | 2 | no | — |
| 158 | #423 | — | 53 | 1 | no | — |
| 159 | #408 | — | 53 | 2 | no | — |
| 160 | C2×C4×C8 | C2×D53 | 1 | 1 | yes | characteristic |
| 161 | C4⋊C16 | — | 53 | 1 | no | — |
| 162 | C4⋊C16 | — | 53 | 1 | no | — |
| 163 | C4⋊C16 | C2×D53 | 1 | 1 | yes | characteristic |
| 164 | C4⋊C16 | C2×D53 | 1 | 1 | yes | characteristic |
| 165 | C22⋊C16 | — | 53 | 2 | no | — |
| 166 | C22⋊C16 | — | 53 | 4 | no | — |
| 167 | C22⋊C16 | — | 53 | 4 | no | — |
| 168 | C22⋊C16 | C2×D53 | 1 | 2 | yes | — |
| 169 | C4×C16 | — | 53 | 2 | no | — |
| 170 | C4×C16 | C2×D53 | 1 | 2 | yes | — |
| 171 | C53 | #33493 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 172 | C2×#41 | — | 53 | 2 | no | — |
| 173 | #131 | — | 212 | 2 | no | — |
| 174 | #131 | — | 212 | 2 | no | — |
| 175 | C2×M16 | — | 53 | 2 | no | — |
| 176 | C2×M16 | — | 53 | 4 | no | — |
| 177 | C22×C8 | — | 53 | 4 | no | — |
| 178 | C22×C8 | C4×D53 | 1 | 1 | yes | characteristic |
| 179 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 180 | C4⋊Q8 | — | 53 | 1 | no | — |
| 181 | #127 | — | 53 | 1 | no | — |
| 182 | #124 | — | 53 | 2 | no | — |
| 183 | #121 | — | 53 | 2 | no | — |
| 184 | #121 | — | 53 | 2 | no | — |
| 185 | C4×D4 | — | 53 | 2 | no | — |
| 186 | C4×D4 | — | 53 | 2 | no | — |
| 187 | C2×C42 | C4×D53 | 1 | 1 | yes | characteristic |
| 188 | M32 | — | 212 | 4 | no | — |
| 189 | C2×C16 | — | 53 | 2 | no | — |
| 190 | C2×C16 | — | 53 | 2 | no | — |
| 191 | C2×C16 | — | 212 | 4 | no | — |
| 192 | C2×C16 | C22×D53 | 1 | 2 | yes | — |
| 193 | C2×C16 | C22×D53 | 1 | 2 | yes | — |
| 194 | C4⋊C8 | — | 53 | 2 | no | — |
| 195 | C4⋊C8 | — | 53 | 2 | no | — |
| 196 | C22⋊C8 | — | 53 | 4 | no | — |
| 197 | C22⋊C8 | — | 53 | 4 | no | — |
| 198 | C4×C8 | C4×D53 | 1 | 1 | yes | characteristic |
| 199 | C4×C8 | C4×D53 | 1 | 1 | yes | characteristic |
| 200 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 201 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 202 | #41 | — | 212 | 2 | no | — |
| 203 | #41 | — | 212 | 2 | no | — |
| 204 | C2×Q8 | — | 53 | 2 | no | — |
| 205 | C2×D4 | — | 53 | 4 | no | — |
| 206 | C2×D4 | — | 53 | 2 | no | — |
| 207 | C2×D4 | — | 53 | 2 | no | — |
| 208 | C2×D4 | — | 53 | 2 | no | — |
| 209 | C22×C4 | — | 53 | 4 | no | — |
| 210 | C22×C4 | C8×D53 | 1 | 1 | yes | characteristic |
| 211 | C22×C4 | C8×D53 | 1 | 1 | yes | characteristic |
| 212 | C22×C4 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 213 | M16 | — | 212 | 4 | no | — |
| 214 | M16 | — | 212 | 2 | no | — |
| 215 | C2×C8 | — | 2 | 1 | no | — |
| 216 | C2×C8 | — | 2 | 1 | no | — |
| 217 | C2×C8 | — | 4 | 1 | no | — |
| 218 | C2×C8 | — | 53 | 4 | no | — |
| 219 | C2×C8 | — | 106 | 4 | no | — |
| 220 | C2×C8 | — | 106 | 4 | no | — |
| 221 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 222 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 223 | C2×C8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 224 | C2×C8 | C23×D53 | 1 | 1 | yes | characteristic |
| 225 | C4⋊C4 | — | 53 | 2 | no | — |
| 226 | C4⋊C4 | — | 53 | 2 | no | — |
| 227 | C22⋊C4 | — | 53 | 4 | no | — |
| 228 | C22⋊C4 | — | 53 | 4 | no | — |
| 229 | C42 | C8×D53 | 1 | 1 | yes | characteristic |
| 230 | C42 | C8×D53 | 1 | 1 | yes | characteristic |
| 231 | C42 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 232 | C42 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 233 | C16 | — | 2 | 2 | no | — |
| 234 | C16 | — | 2 | 2 | no | — |
| 235 | C16 | — | 106 | 2 | no | — |
| 236 | C16 | — | 106 | 2 | no | — |
| 237 | C23 | — | 53 | 4 | no | — |
| 238 | C23 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 239 | Q8 | — | 212 | 2 | no | — |
| 240 | D4 | — | 106 | 4 | no | — |
| 241 | D4 | — | 106 | 4 | no | — |
| 242 | D4 | — | 212 | 2 | no | — |
| 243 | D4 | — | 212 | 4 | no | — |
| 244 | C2×C4 | — | 2 | 2 | no | — |
| 245 | C2×C4 | — | 2 | 2 | no | — |
| 246 | C2×C4 | — | 2 | 1 | no | — |
| 247 | C2×C4 | — | 2 | 1 | no | — |
| 248 | C2×C4 | — | 53 | 4 | no | — |
| 249 | C2×C4 | — | 106 | 4 | no | — |
| 250 | C2×C4 | — | 106 | 4 | no | — |
| 251 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 252 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 253 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 254 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 255 | C2×C4 | C2×C8×D53 | 1 | 1 | yes | characteristic |
| 256 | C2×C4 | C22×C4×D53 | 1 | 1 | yes | characteristic |
| 257 | C8 | — | 2 | 1 | no | — |
| 258 | C8 | — | 2 | 1 | no | — |
| 259 | C8 | — | 2 | 1 | no | — |
| 260 | C8 | — | 106 | 4 | no | — |
| 261 | C8 | #13133263 | 1 | 1 | yes | characteristic |
| 262 | C8 | #13133263 | 1 | 1 | yes | characteristic |
| 263 | C22 | — | 2 | 1 | no | — |
| 264 | C22 | — | 2 | 1 | no | — |
| 265 | C22 | — | 2 | 1 | no | — |
| 266 | C22 | — | 106 | 4 | no | — |
| 267 | C22 | — | 106 | 4 | no | — |
| 268 | C22 | — | 106 | 4 | no | — |
| 269 | C22 | C22×C8×D53 | 1 | 1 | yes | characteristic |
| 270 | C4 | — | 2 | 1 | no | — |
| 271 | C4 | — | 2 | 1 | no | — |
| 272 | C4 | — | 2 | 1 | no | — |
| 273 | C4 | — | 106 | 4 | no | — |
| 274 | C4 | #16811039 | 1 | 2 | yes | — |
| 275 | C4 | #16811041 | 1 | 2 | yes | — |
| 276 | C4 | #16811097 | 1 | 1 | yes | characteristic |
| 277 | C4 | #16811389 | 1 | 1 | yes | characteristic |
| 278 | C2 | — | 4 | 1 | no | — |
| 279 | C2 | — | 212 | 4 | no | — |
| 280 | C2 | #20283738 | 1 | 1 | yes | characteristic |
| 281 | C2 | C2×#16811039 | 1 | 1 | yes | characteristic |
| 282 | C2 | C2×#16811041 | 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