← #28297963
Subgroups — #28297963
Order 13568.
10698 conjugacy classes of subgroups.
The subgroup lattice has 286 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20289528 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20289502 | C2 | 1 | 15 | yes | maximal, complemented |
| 3 | #20289501 | C2 | 1 | 15 | yes | maximal, complemented |
| 4 | #20289486 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | C2×#16811438 | C2 | 1 | 15 | yes | maximal, complemented |
| 6 | #20289427 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C53×2−1+6 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #16811459 | C22 | 1 | 15 | yes | complemented |
| 9 | #16811458 | — | 2 | 10 | no | — |
| 10 | C2×#13133264 | C22 | 1 | 15 | yes | complemented |
| 11 | 2−1+4×D53 | — | 2 | 6 | no | — |
| 12 | #16811438 | C22 | 1 | 60 | yes | complemented |
| 13 | #16811438 | C22 | 1 | 120 | yes | complemented |
| 14 | #16811434 | C22 | 1 | 60 | yes | complemented |
| 15 | #16811431 | C22 | 1 | 60 | yes | complemented |
| 16 | #16811426 | C22 | 1 | 20 | yes | complemented |
| 17 | #16811421 | C22 | 1 | 90 | yes | complemented |
| 18 | C2×#13133248 | C22 | 1 | 45 | yes | complemented |
| 19 | C2×#13133247 | C22 | 1 | 45 | yes | complemented |
| 20 | C2×#13133237 | C22 | 1 | 45 | yes | complemented |
| 21 | C22×#11719925 | C22 | 1 | 15 | yes | complemented |
| 22 | #16811390 | C22 | 1 | 15 | yes | complemented |
| 23 | C2×#13133224 | C22 | 1 | 15 | yes | complemented |
| 24 | C53×#585 | — | 2 | 10 | no | — |
| 25 | C53×#585 | C22 | 1 | 1 | yes | characteristic |
| 26 | C53×#585 | C22 | 1 | 15 | yes | complemented |
| 27 | C2×2−1+4×C53 | — | 2 | 6 | no | — |
| 28 | C2×2−1+4×C53 | C22 | 1 | 15 | yes | complemented |
| 29 | #13133265 | — | 2 | 15 | no | — |
| 30 | #13133264 | — | 2 | 60 | no | — |
| 31 | #13133264 | C23 | 1 | 60 | yes | — |
| 32 | #13133263 | C23 | 1 | 20 | yes | — |
| 33 | #13133262 | — | 2 | 45 | no | — |
| 34 | C2×#11719945 | C23 | 1 | 45 | yes | — |
| 35 | C2×#11719947 | — | 2 | 45 | no | — |
| 36 | #41×D53 | — | 2 | 60 | no | — |
| 37 | C22×Dic106 | C23 | 1 | 15 | yes | — |
| 38 | C2×Q8×D53 | — | 2 | 15 | no | — |
| 39 | #13133248 | C23 | 1 | 180 | yes | — |
| 40 | #13133247 | C23 | 1 | 180 | yes | — |
| 41 | #13133243 | C23 | 1 | 60 | yes | — |
| 42 | #13133242 | C23 | 1 | 60 | yes | — |
| 43 | #13133237 | C23 | 1 | 60 | yes | — |
| 44 | #13133237 | C23 | 1 | 180 | yes | — |
| 45 | C2×#11719925 | C23 | 1 | 45 | yes | — |
| 46 | C2×#11719925 | C23 | 1 | 90 | yes | — |
| 47 | C2×#11719923 | C23 | 1 | 45 | yes | — |
| 48 | C2×#11719924 | C23 | 1 | 45 | yes | — |
| 49 | C2×#11719922 | C23 | 1 | 15 | yes | — |
| 50 | #13133224 | C23 | 1 | 20 | yes | — |
| 51 | #13133224 | C23 | 1 | 60 | yes | — |
| 52 | C2×#11719917 | C23 | 1 | 45 | yes | — |
| 53 | C22×#88322 | C23 | 1 | 15 | yes | — |
| 54 | 2−1+4×C53 | — | 2 | 60 | no | — |
| 55 | 2−1+4×C53 | — | 2 | 15 | no | — |
| 56 | 2−1+4×C53 | D4 | 1 | 6 | yes | — |
| 57 | 2−1+4×C53 | C23 | 1 | 60 | yes | — |
| 58 | H(5,𝔽2)×C53 | — | 2 | 45 | no | — |
| 59 | H(5,𝔽2)×C53 | D4 | 1 | 10 | yes | — |
| 60 | H(5,𝔽2)×C53 | C23 | 1 | 20 | yes | — |
| 61 | C2×#41×C53 | — | 2 | 60 | no | — |
| 62 | C2×#41×C53 | — | 2 | 45 | no | — |
| 63 | C2×#41×C53 | C23 | 1 | 45 | yes | — |
| 64 | C2×#41×C53 | C23 | 1 | 15 | yes | — |
| 65 | C22×Q8×C53 | — | 2 | 15 | no | — |
| 66 | C22×Q8×C53 | C23 | 1 | 15 | yes | — |
| 67 | #41×C53 | — | 2 | 20 | no | — |
| 68 | #41×C53 | — | 2 | 180 | no | — |
| 69 | #41×C53 | — | 2 | 90 | no | — |
| 70 | #41×C53 | C2×D4 | 1 | 60 | yes | — |
| 71 | #41×C53 | C24 | 1 | 60 | yes | — |
| 72 | #41×C53 | C24 | 1 | 20 | yes | — |
| 73 | C2×Q8×C53 | — | 2 | 60 | no | — |
| 74 | C2×Q8×C53 | — | 2 | 45 | no | — |
| 75 | C2×Q8×C53 | C2×D4 | 1 | 15 | yes | — |
| 76 | C2×Q8×C53 | C24 | 1 | 45 | yes | — |
| 77 | C2×D4×C53 | — | 2 | 45 | no | — |
| 78 | C2×D4×C53 | — | 2 | 60 | no | — |
| 79 | C2×D4×C53 | C2×D4 | 1 | 45 | yes | — |
| 80 | C2×D4×C53 | C24 | 1 | 15 | yes | — |
| 81 | C22×C4×C53 | — | 2 | 15 | no | — |
| 82 | C22×C4×C53 | — | 2 | 45 | no | — |
| 83 | C22×C4×C53 | C24 | 1 | 15 | yes | — |
| 84 | C2×C22⋊D53 | — | 2 | 45 | no | — |
| 85 | C22×Dic53 | — | 2 | 15 | no | — |
| 86 | #11719949 | — | 2 | 20 | no | — |
| 87 | Q8×D53 | — | 2 | 60 | no | — |
| 88 | #11719947 | — | 2 | 180 | no | — |
| 89 | D4×D53 | — | 2 | 60 | no | — |
| 90 | #11719945 | — | 2 | 90 | no | — |
| 91 | #11719945 | C24 | 1 | 60 | yes | — |
| 92 | C2×D212 | C24 | 1 | 15 | yes | — |
| 93 | C2×C4×D53 | — | 2 | 45 | no | — |
| 94 | C2×Dic106 | — | 2 | 45 | no | — |
| 95 | C2×Dic106 | C24 | 1 | 45 | yes | — |
| 96 | #11719925 | C24 | 1 | 10 | yes | — |
| 97 | #11719925 | C24 | 1 | 90 | yes | — |
| 98 | #11719924 | C24 | 1 | 60 | yes | — |
| 99 | #11719923 | C24 | 1 | 60 | yes | — |
| 100 | #11719922 | C24 | 1 | 6 | yes | — |
| 101 | #11719922 | C24 | 1 | 30 | yes | — |
| 102 | #11719917 | C24 | 1 | 60 | yes | — |
| 103 | C2×#88322 | C24 | 1 | 15 | yes | — |
| 104 | C2×#88322 | C24 | 1 | 45 | yes | — |
| 105 | C23×C53 | — | 2 | 15 | no | — |
| 106 | C23×C53 | C22×D4 | 1 | 15 | yes | — |
| 107 | C22×D53 | — | 2 | 15 | no | — |
| 108 | Q8×C53 | — | 2 | 45 | no | — |
| 109 | Q8×C53 | C22×D4 | 1 | 20 | yes | — |
| 110 | Q8×C53 | C25 | 1 | 10 | yes | — |
| 111 | D4×C53 | — | 2 | 60 | no | — |
| 112 | D4×C53 | — | 2 | 15 | no | — |
| 113 | D4×C53 | C22×D4 | 1 | 60 | yes | — |
| 114 | D4×C53 | C25 | 1 | 6 | yes | — |
| 115 | C2×C4×C53 | — | 2 | 45 | no | — |
| 116 | C2×C4×C53 | — | 2 | 60 | no | — |
| 117 | C2×C4×C53 | C22×D4 | 1 | 45 | yes | — |
| 118 | C2×C4×C53 | C25 | 1 | 15 | yes | — |
| 119 | C22⋊D53 | — | 2 | 60 | no | — |
| 120 | C2×Dic53 | — | 2 | 45 | no | — |
| 121 | D212 | — | 2 | 15 | no | — |
| 122 | D212 | C25 | 1 | 6 | yes | — |
| 123 | C4×D53 | — | 2 | 60 | no | — |
| 124 | Dic106 | — | 2 | 45 | no | — |
| 125 | Dic106 | C25 | 1 | 10 | yes | — |
| 126 | #88322 | C25 | 1 | 1 | yes | characteristic |
| 127 | #88322 | C25 | 1 | 15 | yes | — |
| 128 | #63093 | — | 53 | 1 | no | maximal, Sylow |
| 129 | C22×C53 | — | 2 | 6 | no | — |
| 130 | C22×C53 | — | 4 | 15 | no | — |
| 131 | C22×C53 | — | 8 | 15 | no | — |
| 132 | C22×C53 | C23×D4 | 1 | 15 | yes | — |
| 133 | C2×D53 | — | 2 | 6 | no | — |
| 134 | C2×D53 | — | 8 | 15 | no | — |
| 135 | C4×C53 | — | 2 | 10 | no | — |
| 136 | C4×C53 | C23×D4 | 1 | 15 | yes | — |
| 137 | C4×C53 | C26 | 1 | 1 | yes | characteristic |
| 138 | Dic53 | — | 2 | 10 | no | — |
| 139 | 2−1+6 | — | 53 | 1 | no | — |
| 140 | 2−1+6 | D53 | 1 | 1 | yes | characteristic, complemented |
| 141 | #3586 | — | 53 | 15 | no | — |
| 142 | #3586 | — | 53 | 15 | no | — |
| 143 | #3584 | — | 53 | 15 | no | — |
| 144 | C2×#578 | — | 53 | 15 | no | — |
| 145 | #3573 | — | 53 | 1 | no | — |
| 146 | C2×C53 | — | 2 | 15 | no | — |
| 147 | C2×C53 | — | 4 | 6 | no | — |
| 148 | C2×C53 | C24×D4 | 1 | 1 | yes | characteristic |
| 149 | D53 | — | 4 | 6 | no | — |
| 150 | #585 | — | 2 | 10 | no | — |
| 151 | #585 | — | 53 | 15 | no | — |
| 152 | #585 | — | 106 | 10 | no | — |
| 153 | #585 | C2×D53 | 1 | 1 | yes | characteristic |
| 154 | #585 | C2×D53 | 1 | 15 | yes | complemented |
| 155 | C2×2−1+4 | — | 2 | 6 | no | — |
| 156 | C2×2−1+4 | — | 53 | 15 | no | — |
| 157 | C2×2−1+4 | — | 106 | 6 | no | — |
| 158 | C2×2−1+4 | C2×D53 | 1 | 15 | yes | complemented |
| 159 | #578 | — | 53 | 60 | no | — |
| 160 | #578 | — | 53 | 120 | no | — |
| 161 | #577 | — | 53 | 60 | no | — |
| 162 | #577 | — | 53 | 60 | no | — |
| 163 | #576 | — | 53 | 20 | no | — |
| 164 | #575 | — | 53 | 90 | no | — |
| 165 | C2×#137 | — | 53 | 45 | no | — |
| 166 | C2×#137 | — | 53 | 45 | no | — |
| 167 | C2×#135 | — | 53 | 45 | no | — |
| 168 | C22×Q16 | — | 53 | 15 | no | — |
| 169 | #568 | — | 53 | 15 | no | — |
| 170 | C2×#131 | — | 53 | 15 | no | — |
| 171 | C53 | #63093 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 172 | 2−1+4 | — | 2 | 60 | no | — |
| 173 | 2−1+4 | — | 2 | 15 | no | — |
| 174 | 2−1+4 | — | 53 | 60 | no | — |
| 175 | 2−1+4 | — | 106 | 60 | no | — |
| 176 | 2−1+4 | — | 106 | 15 | no | — |
| 177 | 2−1+4 | C22⋊D53 | 1 | 6 | yes | — |
| 178 | 2−1+4 | C22×D53 | 1 | 60 | yes | — |
| 179 | H(5,𝔽2) | — | 2 | 45 | no | — |
| 180 | H(5,𝔽2) | — | 53 | 20 | no | — |
| 181 | H(5,𝔽2) | — | 106 | 45 | no | — |
| 182 | H(5,𝔽2) | C22⋊D53 | 1 | 10 | yes | — |
| 183 | H(5,𝔽2) | C22×D53 | 1 | 20 | yes | — |
| 184 | C2×#41 | — | 2 | 60 | no | — |
| 185 | C2×#41 | — | 2 | 45 | no | — |
| 186 | C2×#41 | — | 53 | 45 | no | — |
| 187 | C2×#41 | — | 106 | 60 | no | — |
| 188 | C2×#41 | — | 106 | 45 | no | — |
| 189 | C2×#41 | C22×D53 | 1 | 45 | yes | — |
| 190 | C2×#41 | C22×D53 | 1 | 15 | yes | — |
| 191 | C22×Q8 | — | 2 | 15 | no | — |
| 192 | C22×Q8 | — | 53 | 15 | no | — |
| 193 | C22×Q8 | — | 106 | 15 | no | — |
| 194 | C22×Q8 | C22×D53 | 1 | 15 | yes | — |
| 195 | #137 | — | 53 | 180 | no | — |
| 196 | #137 | — | 53 | 180 | no | — |
| 197 | AGL(1,ℤ/8) | — | 53 | 60 | no | — |
| 198 | AGL(1,ℤ/8) | — | 53 | 60 | no | — |
| 199 | #135 | — | 53 | 60 | no | — |
| 200 | #135 | — | 53 | 180 | no | — |
| 201 | C2×Q16 | — | 53 | 45 | no | — |
| 202 | C2×Q16 | — | 53 | 90 | no | — |
| 203 | C2×SD16 | — | 53 | 45 | no | — |
| 204 | C2×SD16 | — | 53 | 45 | no | — |
| 205 | C2×D8 | — | 53 | 15 | no | — |
| 206 | #131 | — | 53 | 20 | no | — |
| 207 | #131 | — | 53 | 60 | no | — |
| 208 | C2×M16 | — | 53 | 45 | no | — |
| 209 | C22×C8 | — | 53 | 15 | no | — |
| 210 | #41 | — | 2 | 20 | no | — |
| 211 | #41 | — | 2 | 180 | no | — |
| 212 | #41 | — | 2 | 90 | no | — |
| 213 | #41 | — | 53 | 60 | no | — |
| 214 | #41 | — | 106 | 20 | no | — |
| 215 | #41 | — | 106 | 180 | no | — |
| 216 | #41 | — | 106 | 90 | no | — |
| 217 | #41 | C2×C22⋊D53 | 1 | 60 | yes | — |
| 218 | #41 | C23×D53 | 1 | 60 | yes | — |
| 219 | #41 | C23×D53 | 1 | 20 | yes | — |
| 220 | C2×Q8 | — | 2 | 60 | no | — |
| 221 | C2×Q8 | — | 2 | 45 | no | — |
| 222 | C2×Q8 | — | 53 | 45 | no | — |
| 223 | C2×Q8 | — | 106 | 60 | no | — |
| 224 | C2×Q8 | — | 106 | 45 | no | — |
| 225 | C2×Q8 | C2×C22⋊D53 | 1 | 15 | yes | — |
| 226 | C2×Q8 | C23×D53 | 1 | 45 | yes | — |
| 227 | C2×D4 | — | 2 | 45 | no | — |
| 228 | C2×D4 | — | 2 | 60 | no | — |
| 229 | C2×D4 | — | 53 | 15 | no | — |
| 230 | C2×D4 | — | 106 | 45 | no | — |
| 231 | C2×D4 | — | 106 | 60 | no | — |
| 232 | C2×D4 | C2×C22⋊D53 | 1 | 45 | yes | — |
| 233 | C2×D4 | C23×D53 | 1 | 15 | yes | — |
| 234 | C22×C4 | — | 2 | 15 | no | — |
| 235 | C22×C4 | — | 2 | 45 | no | — |
| 236 | C22×C4 | — | 106 | 15 | no | — |
| 237 | C22×C4 | — | 106 | 45 | no | — |
| 238 | C22×C4 | C23×D53 | 1 | 15 | yes | — |
| 239 | Q16 | — | 53 | 10 | no | — |
| 240 | Q16 | — | 53 | 90 | no | — |
| 241 | SD16 | — | 53 | 60 | no | — |
| 242 | SD16 | — | 53 | 60 | no | — |
| 243 | D8 | — | 53 | 6 | no | — |
| 244 | D8 | — | 53 | 30 | no | — |
| 245 | M16 | — | 53 | 60 | no | — |
| 246 | C2×C8 | — | 53 | 15 | no | — |
| 247 | C2×C8 | — | 53 | 45 | no | — |
| 248 | C23 | — | 2 | 15 | no | — |
| 249 | C23 | — | 106 | 15 | no | — |
| 250 | C23 | C22×C22⋊D53 | 1 | 15 | yes | — |
| 251 | Q8 | — | 2 | 45 | no | — |
| 252 | Q8 | — | 53 | 10 | no | — |
| 253 | Q8 | — | 106 | 45 | no | — |
| 254 | Q8 | C22×C22⋊D53 | 1 | 20 | yes | — |
| 255 | Q8 | C24×D53 | 1 | 10 | yes | — |
| 256 | D4 | — | 2 | 60 | no | — |
| 257 | D4 | — | 2 | 15 | no | — |
| 258 | D4 | — | 53 | 6 | no | — |
| 259 | D4 | — | 106 | 60 | no | — |
| 260 | D4 | — | 106 | 15 | no | — |
| 261 | D4 | C22×C22⋊D53 | 1 | 60 | yes | — |
| 262 | D4 | C24×D53 | 1 | 6 | yes | — |
| 263 | C2×C4 | — | 2 | 45 | no | — |
| 264 | C2×C4 | — | 2 | 60 | no | — |
| 265 | C2×C4 | — | 106 | 45 | no | — |
| 266 | C2×C4 | — | 106 | 60 | no | — |
| 267 | C2×C4 | C22×C22⋊D53 | 1 | 45 | yes | — |
| 268 | C2×C4 | C24×D53 | 1 | 15 | yes | — |
| 269 | C8 | — | 53 | 1 | no | — |
| 270 | C8 | — | 53 | 15 | no | — |
| 271 | C22 | — | 2 | 6 | no | — |
| 272 | C22 | — | 4 | 15 | no | — |
| 273 | C22 | — | 8 | 15 | no | — |
| 274 | C22 | — | 106 | 6 | no | — |
| 275 | C22 | — | 424 | 15 | no | — |
| 276 | C22 | C23×C22⋊D53 | 1 | 15 | yes | — |
| 277 | C4 | — | 2 | 10 | no | — |
| 278 | C4 | — | 106 | 10 | no | — |
| 279 | C4 | C23×C22⋊D53 | 1 | 15 | yes | — |
| 280 | C4 | C25×D53 | 1 | 1 | yes | characteristic |
| 281 | C2 | — | 2 | 15 | no | — |
| 282 | C2 | — | 4 | 6 | no | — |
| 283 | C2 | — | 212 | 6 | no | — |
| 284 | C2 | C24×C22⋊D53 | 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