← #28290637
Subgroups — #28290637
Order 13568.
2026 conjugacy classes of subgroups.
The subgroup lattice has 336 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16811440 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20287866 | C2 | 1 | 8 | yes | maximal, complemented |
| 3 | #20287862 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #20287849 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×#16811063 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #20287818 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C53×#3584 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#13133264 | — | 2 | 1 | no | — |
| 9 | #16811440 | — | 2 | 4 | no | — |
| 10 | #16811440 | C22 | 1 | 8 | yes | complemented |
| 11 | C2×#13133250 | — | 2 | 3 | no | — |
| 12 | C2×#13133239 | C22 | 1 | 3 | yes | complemented |
| 13 | C22×Dic212 | C22 | 1 | 3 | yes | complemented |
| 14 | #16811064 | C22 | 1 | 12 | yes | complemented |
| 15 | #16811063 | C22 | 1 | 12 | yes | complemented |
| 16 | #16811063 | C22 | 1 | 24 | yes | complemented |
| 17 | #16811059 | C22 | 1 | 12 | yes | complemented |
| 18 | #16811058 | C22 | 1 | 4 | yes | complemented |
| 19 | #16811053 | C22 | 1 | 24 | yes | complemented |
| 20 | C2×#13133163 | C22 | 1 | 12 | yes | complemented |
| 21 | C2×#13133161 | C22 | 1 | 12 | yes | complemented |
| 22 | #16811040 | C22 | 1 | 8 | yes | complemented |
| 23 | C2×#13133157 | C22 | 1 | 6 | yes | complemented |
| 24 | C53×#585 | — | 2 | 1 | no | — |
| 25 | C53×#578 | C22 | 1 | 4 | yes | complemented |
| 26 | C53×#577 | — | 2 | 4 | no | — |
| 27 | C53×#576 | C22 | 1 | 4 | yes | complemented |
| 28 | C53×#575 | — | 2 | 3 | no | — |
| 29 | C2×#135×C53 | C22 | 1 | 6 | yes | — |
| 30 | C2×#131×C53 | C22 | 1 | 1 | yes | characteristic |
| 31 | #13133264 | — | 2 | 8 | no | — |
| 32 | #13133264 | — | 4 | 4 | no | — |
| 33 | C2×#11719945 | — | 2 | 3 | no | — |
| 34 | C2×#11719947 | — | 4 | 3 | no | — |
| 35 | C22×Dic106 | — | 2 | 3 | no | — |
| 36 | C2×Q8×D53 | — | 4 | 1 | no | — |
| 37 | #13133250 | — | 2 | 24 | no | — |
| 38 | #13133239 | — | 2 | 6 | no | — |
| 39 | #13133239 | C23 | 1 | 12 | yes | — |
| 40 | C2×Dic212 | — | 2 | 3 | no | — |
| 41 | C2×Dic212 | — | 2 | 6 | no | — |
| 42 | C2×Dic212 | C23 | 1 | 12 | yes | — |
| 43 | C2×Dic212 | C23 | 1 | 3 | yes | — |
| 44 | C2×#11719913 | — | 2 | 3 | no | — |
| 45 | C2×D424 | C23 | 1 | 1 | yes | characteristic |
| 46 | #13133163 | C23 | 1 | 24 | yes | — |
| 47 | #13133161 | C23 | 1 | 24 | yes | — |
| 48 | #13133160 | C23 | 1 | 8 | yes | — |
| 49 | #13133158 | C23 | 1 | 8 | yes | — |
| 50 | #13133157 | C23 | 1 | 4 | yes | — |
| 51 | #13133157 | C23 | 1 | 12 | yes | — |
| 52 | C2×#11719908 | C23 | 1 | 12 | yes | — |
| 53 | 2−1+4×C53 | — | 2 | 4 | no | — |
| 54 | 2−1+4×C53 | — | 4 | 1 | no | — |
| 55 | H(5,𝔽2)×C53 | — | 2 | 4 | no | — |
| 56 | H(5,𝔽2)×C53 | — | 4 | 3 | no | — |
| 57 | C2×#41×C53 | — | 2 | 6 | no | — |
| 58 | C2×#41×C53 | — | 4 | 4 | no | — |
| 59 | C2×#41×C53 | D4 | 1 | 1 | yes | characteristic |
| 60 | #137×C53 | — | 2 | 12 | no | — |
| 61 | AGL(1,ℤ/8)×C53 | — | 2 | 12 | no | — |
| 62 | #135×C53 | — | 2 | 6 | no | — |
| 63 | #135×C53 | — | 2 | 6 | no | — |
| 64 | #135×C53 | C23 | 1 | 12 | yes | — |
| 65 | #135×C53 | C23 | 1 | 4 | yes | — |
| 66 | C2×Q16×C53 | C23 | 1 | 6 | yes | — |
| 67 | C2×SD16×C53 | — | 2 | 6 | no | — |
| 68 | C2×D8×C53 | C23 | 1 | 6 | yes | — |
| 69 | #131×C53 | D4 | 1 | 4 | yes | — |
| 70 | #131×C53 | C23 | 1 | 4 | yes | — |
| 71 | C2×M16×C53 | D4 | 1 | 3 | yes | — |
| 72 | C22×C8×C53 | C23 | 1 | 3 | yes | — |
| 73 | #41×C53 | — | 2 | 4 | no | — |
| 74 | #41×C53 | — | 2 | 12 | no | — |
| 75 | #41×C53 | — | 4 | 4 | no | — |
| 76 | #41×C53 | — | 4 | 12 | no | — |
| 77 | #41×C53 | — | 4 | 6 | no | — |
| 78 | #41×C53 | — | 4 | 6 | no | — |
| 79 | #41×C53 | D8 | 1 | 4 | yes | — |
| 80 | #41×C53 | C2×D4 | 1 | 4 | yes | — |
| 81 | C2×Q8×C53 | — | 2 | 6 | no | — |
| 82 | C2×Q8×C53 | — | 4 | 4 | no | — |
| 83 | C2×Q8×C53 | D8 | 1 | 1 | yes | characteristic |
| 84 | C2×D4×C53 | — | 2 | 6 | no | — |
| 85 | C2×D4×C53 | — | 4 | 6 | no | — |
| 86 | C2×D4×C53 | — | 4 | 12 | no | — |
| 87 | C2×D4×C53 | D8 | 1 | 3 | yes | — |
| 88 | C22×C4×C53 | — | 4 | 6 | no | — |
| 89 | C22×C4×C53 | C2×D4 | 1 | 3 | yes | — |
| 90 | C2×C22⋊D53 | — | 4 | 3 | no | — |
| 91 | C22×Dic53 | — | 4 | 3 | no | — |
| 92 | Q8×D53 | — | 4 | 8 | no | — |
| 93 | #11719947 | — | 4 | 24 | no | — |
| 94 | #11719945 | — | 2 | 12 | no | — |
| 95 | #11719945 | — | 4 | 6 | no | — |
| 96 | C2×D212 | — | 2 | 1 | no | — |
| 97 | C2×C4×D53 | — | 4 | 3 | no | — |
| 98 | C2×Dic106 | — | 2 | 3 | no | — |
| 99 | C2×Dic106 | — | 2 | 12 | no | — |
| 100 | C2×Dic106 | — | 4 | 3 | no | — |
| 101 | C2×Dic106 | — | 4 | 6 | no | — |
| 102 | Q16×C53 | — | 2 | 6 | no | — |
| 103 | Q16×C53 | C24 | 1 | 4 | yes | — |
| 104 | SD16×C53 | — | 2 | 4 | no | — |
| 105 | SD16×C53 | — | 2 | 12 | no | — |
| 106 | D8×C53 | — | 2 | 6 | no | — |
| 107 | D8×C53 | C24 | 1 | 4 | yes | — |
| 108 | M16×C53 | C2×D4 | 1 | 12 | yes | — |
| 109 | C2×C8×C53 | C2×D4 | 1 | 3 | yes | — |
| 110 | C2×C8×C53 | C2×D4 | 1 | 6 | yes | — |
| 111 | C2×C8×C53 | C24 | 1 | 1 | yes | characteristic |
| 112 | C2×C8×C53 | C24 | 1 | 6 | yes | — |
| 113 | Dic212 | — | 2 | 3 | no | — |
| 114 | Dic212 | — | 2 | 12 | no | — |
| 115 | Dic212 | C24 | 1 | 6 | yes | — |
| 116 | D424 | — | 2 | 1 | no | — |
| 117 | D424 | C24 | 1 | 2 | yes | — |
| 118 | #11719913 | — | 2 | 12 | no | — |
| 119 | #11719908 | C24 | 1 | 8 | yes | — |
| 120 | C23×C53 | — | 4 | 6 | no | — |
| 121 | C23×C53 | C2×D8 | 1 | 3 | yes | — |
| 122 | C22×D53 | — | 4 | 1 | no | — |
| 123 | Q8×C53 | — | 2 | 4 | no | — |
| 124 | Q8×C53 | — | 4 | 6 | no | — |
| 125 | Q8×C53 | C2×D8 | 1 | 4 | yes | — |
| 126 | D4×C53 | — | 2 | 4 | no | — |
| 127 | D4×C53 | — | 4 | 4 | no | — |
| 128 | D4×C53 | — | 4 | 12 | no | — |
| 129 | D4×C53 | — | 4 | 6 | no | — |
| 130 | D4×C53 | C2×D8 | 1 | 12 | yes | — |
| 131 | C2×C4×C53 | — | 4 | 6 | no | — |
| 132 | C2×C4×C53 | — | 4 | 4 | no | — |
| 133 | C2×C4×C53 | — | 4 | 12 | no | — |
| 134 | C2×C4×C53 | C2×D8 | 1 | 3 | yes | — |
| 135 | C2×C4×C53 | C2×D8 | 1 | 6 | yes | — |
| 136 | C2×C4×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 137 | C2×C4×C53 | C22×D4 | 1 | 6 | yes | — |
| 138 | C22⋊D53 | — | 4 | 12 | no | — |
| 139 | C2×Dic53 | — | 4 | 3 | no | — |
| 140 | C2×Dic53 | — | 4 | 12 | no | — |
| 141 | D212 | — | 2 | 2 | no | — |
| 142 | D212 | — | 4 | 1 | no | — |
| 143 | C4×D53 | — | 4 | 12 | no | — |
| 144 | Dic106 | — | 2 | 6 | no | — |
| 145 | Dic106 | — | 4 | 3 | no | — |
| 146 | Dic106 | — | 4 | 12 | no | — |
| 147 | C8×C53 | C22×D4 | 1 | 6 | yes | — |
| 148 | C8×C53 | C22×D4 | 1 | 1 | yes | characteristic |
| 149 | C8×C53 | C25 | 1 | 1 | yes | characteristic |
| 150 | #62637 | — | 53 | 1 | no | maximal, Sylow |
| 151 | C22×C53 | — | 4 | 3 | no | — |
| 152 | C22×C53 | — | 4 | 4 | no | — |
| 153 | C22×C53 | — | 16 | 6 | no | — |
| 154 | C22×C53 | C22×D8 | 1 | 1 | yes | characteristic |
| 155 | C22×C53 | C22×D8 | 1 | 6 | yes | — |
| 156 | C2×D53 | — | 4 | 2 | no | — |
| 157 | C2×D53 | — | 8 | 2 | no | — |
| 158 | C4×C53 | — | 4 | 4 | no | — |
| 159 | C4×C53 | C22×D8 | 1 | 1 | yes | characteristic |
| 160 | C4×C53 | C22×D8 | 1 | 6 | yes | — |
| 161 | C4×C53 | C23×D4 | 1 | 1 | yes | characteristic |
| 162 | Dic53 | — | 4 | 6 | no | — |
| 163 | #3584 | D53 | 1 | 1 | yes | characteristic, complemented |
| 164 | C2×#578 | — | 53 | 1 | no | — |
| 165 | #3417 | — | 53 | 8 | no | — |
| 166 | #3416 | — | 53 | 8 | no | — |
| 167 | #3414 | — | 53 | 6 | no | — |
| 168 | C2×#510 | — | 53 | 6 | no | — |
| 169 | #3407 | — | 53 | 1 | no | — |
| 170 | C2×C53 | — | 2 | 1 | no | — |
| 171 | C2×C53 | — | 2 | 6 | no | — |
| 172 | C2×C53 | — | 8 | 4 | no | — |
| 173 | C2×C53 | C23×D8 | 1 | 1 | yes | characteristic |
| 174 | D53 | — | 8 | 2 | no | — |
| 175 | #585 | — | 2 | 1 | no | — |
| 176 | C2×2−1+4 | — | 106 | 1 | no | — |
| 177 | #578 | — | 53 | 8 | no | — |
| 178 | #578 | — | 106 | 4 | no | — |
| 179 | #578 | C2×D53 | 1 | 4 | yes | complemented |
| 180 | #577 | — | 2 | 4 | no | — |
| 181 | #576 | C2×D53 | 1 | 4 | yes | complemented |
| 182 | #575 | — | 2 | 3 | no | — |
| 183 | C2×#137 | — | 106 | 3 | no | — |
| 184 | C2×#135 | — | 53 | 3 | no | — |
| 185 | C2×#135 | C2×D53 | 1 | 6 | yes | — |
| 186 | C22×Q16 | — | 53 | 3 | no | — |
| 187 | C2×#131 | C2×D53 | 1 | 1 | yes | characteristic |
| 188 | #510 | — | 53 | 12 | no | — |
| 189 | #510 | — | 53 | 24 | no | — |
| 190 | #510 | — | 53 | 12 | no | — |
| 191 | #509 | — | 53 | 4 | no | — |
| 192 | #509 | — | 53 | 12 | no | — |
| 193 | #508 | — | 53 | 24 | no | — |
| 194 | C2×Q32 | — | 53 | 12 | no | — |
| 195 | C2×SD32 | — | 53 | 12 | no | — |
| 196 | #504 | — | 53 | 8 | no | — |
| 197 | C2×M32 | — | 53 | 6 | no | — |
| 198 | C53 | #62637 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 199 | 2−1+4 | — | 2 | 4 | no | — |
| 200 | 2−1+4 | — | 4 | 1 | no | — |
| 201 | 2−1+4 | — | 106 | 8 | no | — |
| 202 | 2−1+4 | — | 212 | 4 | no | — |
| 203 | H(5,𝔽2) | — | 2 | 4 | no | — |
| 204 | H(5,𝔽2) | — | 4 | 3 | no | — |
| 205 | C2×#41 | — | 2 | 6 | no | — |
| 206 | C2×#41 | — | 4 | 4 | no | — |
| 207 | C2×#41 | — | 106 | 3 | no | — |
| 208 | C2×#41 | — | 212 | 3 | no | — |
| 209 | C2×#41 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 210 | C22×Q8 | — | 106 | 3 | no | — |
| 211 | C22×Q8 | — | 212 | 1 | no | — |
| 212 | #137 | — | 2 | 12 | no | — |
| 213 | #137 | — | 106 | 24 | no | — |
| 214 | AGL(1,ℤ/8) | — | 2 | 12 | no | — |
| 215 | #135 | — | 2 | 6 | no | — |
| 216 | #135 | — | 2 | 6 | no | — |
| 217 | #135 | — | 53 | 12 | no | — |
| 218 | #135 | — | 106 | 6 | no | — |
| 219 | #135 | C22×D53 | 1 | 4 | yes | — |
| 220 | #135 | C22×D53 | 1 | 12 | yes | — |
| 221 | C2×Q16 | — | 53 | 3 | no | — |
| 222 | C2×Q16 | — | 53 | 12 | no | — |
| 223 | C2×Q16 | — | 106 | 3 | no | — |
| 224 | C2×Q16 | — | 106 | 6 | no | — |
| 225 | C2×Q16 | C22×D53 | 1 | 6 | yes | — |
| 226 | C2×SD16 | — | 2 | 6 | no | — |
| 227 | C2×SD16 | — | 106 | 3 | no | — |
| 228 | C2×D8 | — | 53 | 1 | no | — |
| 229 | C2×D8 | C22×D53 | 1 | 6 | yes | — |
| 230 | #131 | C22⋊D53 | 1 | 4 | yes | — |
| 231 | #131 | C22×D53 | 1 | 4 | yes | — |
| 232 | C2×M16 | C22⋊D53 | 1 | 3 | yes | — |
| 233 | C22×C8 | C22×D53 | 1 | 3 | yes | — |
| 234 | Q32 | — | 53 | 24 | no | — |
| 235 | SD32 | — | 53 | 8 | no | — |
| 236 | SD32 | — | 53 | 24 | no | — |
| 237 | D16 | — | 53 | 8 | no | — |
| 238 | M32 | — | 53 | 4 | no | — |
| 239 | M32 | — | 53 | 12 | no | — |
| 240 | C2×C16 | — | 53 | 12 | no | — |
| 241 | #41 | — | 2 | 4 | no | — |
| 242 | #41 | — | 2 | 12 | no | — |
| 243 | #41 | — | 4 | 4 | no | — |
| 244 | #41 | — | 4 | 12 | no | — |
| 245 | #41 | — | 4 | 6 | no | — |
| 246 | #41 | — | 4 | 6 | no | — |
| 247 | #41 | — | 106 | 12 | no | — |
| 248 | #41 | — | 212 | 24 | no | — |
| 249 | #41 | — | 212 | 6 | no | — |
| 250 | #41 | #11719922 | 1 | 4 | yes | — |
| 251 | #41 | C2×C22⋊D53 | 1 | 4 | yes | — |
| 252 | C2×Q8 | — | 2 | 6 | no | — |
| 253 | C2×Q8 | — | 4 | 4 | no | — |
| 254 | C2×Q8 | — | 106 | 3 | no | — |
| 255 | C2×Q8 | — | 106 | 12 | no | — |
| 256 | C2×Q8 | — | 212 | 8 | no | — |
| 257 | C2×Q8 | — | 212 | 3 | no | — |
| 258 | C2×Q8 | — | 212 | 6 | no | — |
| 259 | C2×Q8 | #11719922 | 1 | 1 | yes | characteristic |
| 260 | C2×D4 | — | 2 | 6 | no | — |
| 261 | C2×D4 | — | 4 | 6 | no | — |
| 262 | C2×D4 | — | 4 | 12 | no | — |
| 263 | C2×D4 | — | 106 | 1 | no | — |
| 264 | C2×D4 | — | 212 | 3 | no | — |
| 265 | C2×D4 | #11719922 | 1 | 3 | yes | — |
| 266 | C22×C4 | — | 4 | 6 | no | — |
| 267 | C22×C4 | — | 212 | 3 | no | — |
| 268 | C22×C4 | — | 212 | 3 | no | — |
| 269 | C22×C4 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 270 | Q16 | — | 2 | 6 | no | — |
| 271 | Q16 | — | 53 | 6 | no | — |
| 272 | Q16 | — | 106 | 3 | no | — |
| 273 | Q16 | — | 106 | 12 | no | — |
| 274 | Q16 | C23×D53 | 1 | 4 | yes | — |
| 275 | SD16 | — | 2 | 4 | no | — |
| 276 | SD16 | — | 2 | 12 | no | — |
| 277 | SD16 | — | 106 | 12 | no | — |
| 278 | D8 | — | 2 | 6 | no | — |
| 279 | D8 | — | 53 | 2 | no | — |
| 280 | D8 | — | 106 | 1 | no | — |
| 281 | D8 | C23×D53 | 1 | 4 | yes | — |
| 282 | M16 | C2×C22⋊D53 | 1 | 12 | yes | — |
| 283 | C2×C8 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 284 | C2×C8 | C2×C22⋊D53 | 1 | 6 | yes | — |
| 285 | C2×C8 | C23×D53 | 1 | 6 | yes | — |
| 286 | C2×C8 | C23×D53 | 1 | 1 | yes | characteristic |
| 287 | C16 | — | 53 | 8 | no | — |
| 288 | C23 | — | 4 | 6 | no | — |
| 289 | C23 | — | 212 | 1 | no | — |
| 290 | C23 | C2×#11719922 | 1 | 3 | yes | — |
| 291 | Q8 | — | 2 | 4 | no | — |
| 292 | Q8 | — | 4 | 6 | no | — |
| 293 | Q8 | — | 106 | 6 | no | — |
| 294 | Q8 | — | 212 | 3 | no | — |
| 295 | Q8 | — | 212 | 12 | no | — |
| 296 | Q8 | C2×#11719922 | 1 | 4 | yes | — |
| 297 | D4 | — | 2 | 4 | no | — |
| 298 | D4 | — | 4 | 12 | no | — |
| 299 | D4 | — | 4 | 4 | no | — |
| 300 | D4 | — | 4 | 6 | no | — |
| 301 | D4 | — | 106 | 2 | no | — |
| 302 | D4 | — | 212 | 12 | no | — |
| 303 | D4 | — | 212 | 1 | no | — |
| 304 | D4 | C2×#11719922 | 1 | 12 | yes | — |
| 305 | C2×C4 | — | 4 | 6 | no | — |
| 306 | C2×C4 | — | 4 | 4 | no | — |
| 307 | C2×C4 | — | 4 | 12 | no | — |
| 308 | C2×C4 | — | 212 | 3 | no | — |
| 309 | C2×C4 | — | 212 | 12 | no | — |
| 310 | C2×C4 | — | 212 | 12 | no | — |
| 311 | C2×C4 | C2×#11719922 | 1 | 6 | yes | — |
| 312 | C2×C4 | C2×#11719922 | 1 | 3 | yes | — |
| 313 | C2×C4 | C22×C22⋊D53 | 1 | 1 | yes | characteristic |
| 314 | C2×C4 | C22×C22⋊D53 | 1 | 6 | yes | — |
| 315 | C8 | C22×C22⋊D53 | 1 | 1 | yes | characteristic |
| 316 | C8 | C22×C22⋊D53 | 1 | 6 | yes | — |
| 317 | C8 | C24×D53 | 1 | 1 | yes | characteristic |
| 318 | C22 | — | 4 | 3 | no | — |
| 319 | C22 | — | 4 | 4 | no | — |
| 320 | C22 | — | 16 | 6 | no | — |
| 321 | C22 | — | 424 | 2 | no | — |
| 322 | C22 | — | 212 | 2 | no | — |
| 323 | C22 | C22×#11719922 | 1 | 6 | yes | — |
| 324 | C22 | C22×#11719922 | 1 | 1 | yes | characteristic |
| 325 | C4 | — | 4 | 4 | no | — |
| 326 | C4 | — | 212 | 6 | no | — |
| 327 | C4 | C22×#11719922 | 1 | 1 | yes | characteristic |
| 328 | C4 | C22×#11719922 | 1 | 6 | yes | — |
| 329 | C4 | C23×C22⋊D53 | 1 | 1 | yes | characteristic |
| 330 | C2 | — | 2 | 1 | no | — |
| 331 | C2 | — | 2 | 6 | no | — |
| 332 | C2 | — | 8 | 4 | no | — |
| 333 | C2 | — | 424 | 2 | no | — |
| 334 | C2 | C23×#11719922 | 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