← #27381597
Subgroups — #27381597
Order 13568.
908 conjugacy classes of subgroups.
The subgroup lattice has 250 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20283131 | C2 | 1 | 3 | yes | maximal |
| 2 | #20274471 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20274470 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #20274469 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #338×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | H(3,ℤ/4)×D53 | C2 | 1 | 3 | yes | maximal |
| 7 | C53×#1747 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×#11719912 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×M16×D53 | C22 | 1 | 3 | yes | — |
| 10 | #117×D53 | C22 | 1 | 3 | yes | — |
| 11 | #16810581 | C22 | 1 | 3 | yes | — |
| 12 | #16810580 | C4 | 1 | 6 | yes | complemented |
| 13 | #16810579 | C22 | 1 | 3 | yes | — |
| 14 | #16810578 | C22 | 1 | 3 | yes | — |
| 15 | #16810577 | C4 | 1 | 6 | yes | complemented |
| 16 | #16810575 | C4 | 1 | 6 | yes | complemented |
| 17 | C4×#11719912 | C4 | 1 | 6 | yes | complemented |
| 18 | #16810381 | C22 | 1 | 3 | yes | — |
| 19 | #16810380 | C22 | 1 | 3 | yes | — |
| 20 | #16810379 | C22 | 1 | 6 | yes | — |
| 21 | C53×#405 | C22 | 1 | 3 | yes | — |
| 22 | C53×#338 | C22 | 1 | 1 | yes | characteristic |
| 23 | C53×H(3,ℤ/4) | C22 | 1 | 3 | yes | — |
| 24 | C22×C4×D53 | C23 | 1 | 1 | yes | characteristic |
| 25 | C2×#11719917 | C23 | 1 | 3 | yes | — |
| 26 | C2×#11719912 | C2×C4 | 1 | 6 | yes | — |
| 27 | C2×#11719912 | C2×C4 | 1 | 3 | yes | — |
| 28 | C2×#11719912 | C2×C4 | 1 | 3 | yes | — |
| 29 | C2×#11719912 | C2×C4 | 1 | 3 | yes | — |
| 30 | C2×#11719912 | C2×C4 | 1 | 3 | yes | — |
| 31 | M16×D53 | — | 2 | 6 | no | — |
| 32 | M16×D53 | — | 2 | 6 | no | — |
| 33 | C22×#88322 | C23 | 1 | 1 | yes | characteristic |
| 34 | C2×C8×D53 | C2×C4 | 1 | 6 | yes | — |
| 35 | #13133176 | C23 | 1 | 3 | yes | — |
| 36 | #13133175 | — | 2 | 3 | no | — |
| 37 | #13133174 | C2×C4 | 1 | 6 | yes | — |
| 38 | #13133173 | — | 2 | 3 | no | — |
| 39 | C4⋊C4×D53 | — | 2 | 3 | no | — |
| 40 | C22⋊C4×D53 | — | 2 | 3 | no | — |
| 41 | C42×D53 | C2×C4 | 1 | 6 | yes | — |
| 42 | #13133128 | C2×C4 | 1 | 6 | yes | complemented |
| 43 | #13133127 | C2×C4 | 1 | 6 | yes | complemented |
| 44 | #13133127 | C2×C4 | 1 | 6 | yes | complemented |
| 45 | C4×#88322 | C2×C4 | 1 | 6 | yes | complemented |
| 46 | C8×Dic53 | C2×C4 | 1 | 6 | yes | complemented |
| 47 | C8×Dic53 | C2×C4 | 1 | 6 | yes | complemented |
| 48 | C2×M16×C53 | C23 | 1 | 3 | yes | — |
| 49 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 50 | #117×C53 | C23 | 1 | 3 | yes | — |
| 51 | #97×C53 | C2×C4 | 1 | 6 | yes | complemented |
| 52 | C4×C8×C53 | C2×C4 | 1 | 6 | yes | complemented |
| 53 | C23×D53 | #41 | 1 | 1 | yes | characteristic |
| 54 | C22×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 55 | C22×Dic53 | #41 | 1 | 1 | yes | characteristic |
| 56 | C2×C4×D53 | — | 2 | 6 | no | — |
| 57 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 58 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 59 | C2×C4×D53 | #41 | 1 | 3 | yes | — |
| 60 | C2×C4×D53 | #41 | 1 | 3 | yes | — |
| 61 | M16×C53 | — | 2 | 6 | no | — |
| 62 | C2×C8×C53 | C22×C4 | 1 | 6 | yes | — |
| 63 | C2×C8×C53 | C22×C4 | 1 | 3 | yes | — |
| 64 | C2×C8×C53 | C22×C4 | 1 | 3 | yes | — |
| 65 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 66 | C22⋊C4×C53 | — | 2 | 3 | no | — |
| 67 | C42×C53 | C22×C4 | 1 | 6 | yes | — |
| 68 | C22⋊Dic53 | — | 2 | 3 | no | — |
| 69 | #11719921 | — | 2 | 6 | no | — |
| 70 | C4⋊Dic53 | — | 2 | 3 | no | — |
| 71 | #11719919 | — | 2 | 6 | no | — |
| 72 | C4×Dic53 | — | 2 | 6 | no | — |
| 73 | C4×Dic53 | C22×C4 | 1 | 6 | yes | — |
| 74 | #11719917 | — | 2 | 6 | no | — |
| 75 | C2×#88322 | C22×C4 | 1 | 6 | yes | — |
| 76 | C2×#88322 | C22×C4 | 1 | 3 | yes | — |
| 77 | C2×#88322 | C22×C4 | 1 | 3 | yes | — |
| 78 | #11719912 | — | 2 | 12 | no | — |
| 79 | #11719912 | C42 | 1 | 3 | yes | — |
| 80 | #11719912 | C42 | 1 | 6 | yes | — |
| 81 | #11719912 | C42 | 1 | 3 | yes | — |
| 82 | #11719912 | C42 | 1 | 1 | yes | characteristic |
| 83 | #11719912 | C42 | 1 | 3 | yes | — |
| 84 | C8×D53 | — | 2 | 6 | no | — |
| 85 | C8×D53 | — | 2 | 6 | no | — |
| 86 | C23×C53 | C2×#41 | 1 | 1 | yes | characteristic |
| 87 | C22×D53 | — | 8 | 1 | no | — |
| 88 | C22×D53 | #117 | 1 | 3 | yes | — |
| 89 | C22×D53 | #117 | 1 | 3 | yes | — |
| 90 | C2×C4×C53 | — | 2 | 6 | no | — |
| 91 | C2×C4×C53 | C23×C4 | 1 | 3 | yes | — |
| 92 | C2×C4×C53 | C2×#41 | 1 | 3 | yes | — |
| 93 | C2×Dic53 | — | 2 | 6 | no | — |
| 94 | C2×Dic53 | #117 | 1 | 3 | yes | — |
| 95 | C2×Dic53 | #117 | 1 | 3 | yes | — |
| 96 | C4×D53 | — | 8 | 6 | no | — |
| 97 | C4×D53 | C2×C42 | 1 | 3 | yes | — |
| 98 | C4×D53 | C2×C42 | 1 | 1 | yes | characteristic |
| 99 | C4×D53 | #117 | 1 | 3 | yes | — |
| 100 | C4×D53 | #117 | 1 | 3 | yes | — |
| 101 | C4×D53 | #117 | 1 | 6 | yes | — |
| 102 | C8×C53 | — | 2 | 6 | no | — |
| 103 | C8×C53 | C2×C42 | 1 | 1 | yes | characteristic |
| 104 | C8×C53 | C2×C42 | 1 | 3 | yes | — |
| 105 | #88322 | — | 2 | 6 | no | — |
| 106 | #88322 | C2×C42 | 1 | 1 | yes | characteristic |
| 107 | #88322 | C2×C42 | 1 | 3 | yes | — |
| 108 | #20370 | — | 53 | 1 | no | maximal, Sylow |
| 109 | C22×C53 | — | 4 | 1 | no | — |
| 110 | C22×C53 | C2×#117 | 1 | 3 | yes | — |
| 111 | C2×D53 | — | 4 | 3 | no | — |
| 112 | C2×D53 | — | 4 | 3 | no | — |
| 113 | C2×D53 | #376 | 1 | 1 | yes | characteristic |
| 114 | C2×D53 | #376 | 1 | 3 | yes | — |
| 115 | C4×C53 | — | 4 | 6 | no | — |
| 116 | C4×C53 | C22×C42 | 1 | 1 | yes | characteristic |
| 117 | C4×C53 | C2×#117 | 1 | 3 | yes | — |
| 118 | Dic53 | — | 4 | 6 | no | — |
| 119 | Dic53 | #376 | 1 | 1 | yes | characteristic |
| 120 | Dic53 | #376 | 1 | 3 | yes | — |
| 121 | #2873 | — | 53 | 3 | no | — |
| 122 | #1747 | — | 53 | 3 | no | — |
| 123 | #1747 | — | 53 | 1 | no | — |
| 124 | #1747 | — | 53 | 3 | no | — |
| 125 | #1747 | D53 | 1 | 1 | yes | characteristic, complemented |
| 126 | C2×#338 | — | 53 | 1 | no | — |
| 127 | C2×H(3,ℤ/4) | — | 53 | 3 | no | — |
| 128 | C2×C53 | — | 2 | 3 | no | — |
| 129 | C2×C53 | C2×#376 | 1 | 1 | yes | characteristic |
| 130 | D53 | — | 2 | 3 | no | — |
| 131 | D53 | — | 2 | 1 | no | — |
| 132 | C22×M16 | — | 53 | 3 | no | — |
| 133 | C22×M16 | — | 53 | 1 | no | — |
| 134 | C2×#117 | — | 53 | 3 | no | — |
| 135 | #405 | — | 53 | 3 | no | — |
| 136 | #405 | — | 53 | 3 | no | — |
| 137 | #405 | — | 53 | 3 | no | — |
| 138 | #405 | — | 53 | 6 | no | — |
| 139 | #405 | — | 53 | 6 | no | — |
| 140 | #405 | C2×D53 | 1 | 3 | yes | — |
| 141 | C4×M16 | — | 53 | 6 | no | — |
| 142 | C4×M16 | — | 53 | 6 | no | — |
| 143 | #338 | — | 53 | 3 | no | — |
| 144 | #338 | C2×D53 | 1 | 1 | yes | characteristic |
| 145 | H(3,ℤ/4) | — | 53 | 3 | no | — |
| 146 | H(3,ℤ/4) | — | 53 | 6 | no | — |
| 147 | H(3,ℤ/4) | C2×D53 | 1 | 3 | yes | — |
| 148 | C53 | #20370 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 149 | C23×C4 | — | 53 | 1 | no | — |
| 150 | C2×M16 | — | 53 | 3 | no | — |
| 151 | C2×M16 | — | 53 | 3 | no | — |
| 152 | C2×M16 | — | 53 | 3 | no | — |
| 153 | C2×M16 | — | 53 | 3 | no | — |
| 154 | C2×M16 | — | 53 | 3 | no | — |
| 155 | C2×M16 | — | 53 | 6 | no | — |
| 156 | C2×M16 | — | 106 | 6 | no | — |
| 157 | C2×M16 | — | 106 | 6 | no | — |
| 158 | C2×M16 | C22×D53 | 1 | 3 | yes | — |
| 159 | C22×C8 | — | 53 | 1 | no | — |
| 160 | C22×C8 | — | 53 | 6 | no | — |
| 161 | C22×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 162 | #117 | — | 53 | 3 | no | — |
| 163 | #117 | — | 53 | 6 | no | — |
| 164 | #117 | — | 106 | 3 | no | — |
| 165 | #117 | — | 106 | 3 | no | — |
| 166 | #117 | C22×D53 | 1 | 3 | yes | — |
| 167 | C2×C4⋊C4 | — | 106 | 3 | no | — |
| 168 | C2×C22⋊C4 | — | 106 | 3 | no | — |
| 169 | C2×C42 | — | 53 | 6 | no | — |
| 170 | #97 | — | 53 | 6 | no | — |
| 171 | #97 | — | 53 | 6 | no | — |
| 172 | #97 | — | 53 | 6 | no | — |
| 173 | #97 | C4×D53 | 1 | 6 | yes | complemented |
| 174 | C4×C8 | — | 53 | 6 | no | — |
| 175 | C4×C8 | — | 53 | 6 | no | — |
| 176 | C4×C8 | — | 53 | 6 | no | — |
| 177 | C4×C8 | C4×D53 | 1 | 6 | yes | complemented |
| 178 | C24 | — | 53 | 1 | no | — |
| 179 | C22×C4 | — | 53 | 1 | no | — |
| 180 | C22×C4 | — | 53 | 3 | no | — |
| 181 | C22×C4 | — | 53 | 3 | no | — |
| 182 | C22×C4 | — | 53 | 3 | no | — |
| 183 | C22×C4 | — | 53 | 3 | no | — |
| 184 | C22×C4 | — | 106 | 6 | no | — |
| 185 | C22×C4 | C23×D53 | 1 | 1 | yes | characteristic |
| 186 | M16 | — | 2 | 6 | no | — |
| 187 | M16 | — | 53 | 3 | no | — |
| 188 | M16 | — | 53 | 6 | no | — |
| 189 | M16 | — | 53 | 3 | no | — |
| 190 | M16 | — | 53 | 3 | no | — |
| 191 | M16 | — | 53 | 1 | no | — |
| 192 | M16 | — | 106 | 12 | no | — |
| 193 | M16 | — | 106 | 6 | no | — |
| 194 | C2×C8 | — | 53 | 3 | no | — |
| 195 | C2×C8 | — | 53 | 3 | no | — |
| 196 | C2×C8 | — | 53 | 6 | no | — |
| 197 | C2×C8 | — | 106 | 6 | no | — |
| 198 | C2×C8 | — | 106 | 6 | no | — |
| 199 | C2×C8 | C2×C4×D53 | 1 | 6 | yes | — |
| 200 | C2×C8 | C2×C4×D53 | 1 | 3 | yes | — |
| 201 | C2×C8 | C2×C4×D53 | 1 | 3 | yes | — |
| 202 | C4⋊C4 | — | 2 | 3 | no | — |
| 203 | C4⋊C4 | — | 106 | 6 | no | — |
| 204 | C4⋊C4 | — | 106 | 3 | no | — |
| 205 | C22⋊C4 | — | 2 | 3 | no | — |
| 206 | C22⋊C4 | — | 106 | 6 | no | — |
| 207 | C22⋊C4 | — | 106 | 3 | no | — |
| 208 | C42 | — | 53 | 6 | no | — |
| 209 | C42 | — | 106 | 6 | no | — |
| 210 | C42 | C2×C4×D53 | 1 | 6 | yes | — |
| 211 | C23 | — | 53 | 3 | no | — |
| 212 | C23 | — | 53 | 3 | no | — |
| 213 | C23 | — | 424 | 1 | no | — |
| 214 | C23 | #41×D53 | 1 | 1 | yes | characteristic |
| 215 | C2×C4 | — | 2 | 6 | no | — |
| 216 | C2×C4 | — | 53 | 3 | no | — |
| 217 | C2×C4 | — | 53 | 3 | no | — |
| 218 | C2×C4 | — | 53 | 3 | no | — |
| 219 | C2×C4 | — | 53 | 1 | no | — |
| 220 | C2×C4 | — | 53 | 3 | no | — |
| 221 | C2×C4 | — | 53 | 3 | no | — |
| 222 | C2×C4 | — | 53 | 6 | no | — |
| 223 | C2×C4 | — | 106 | 6 | no | — |
| 224 | C2×C4 | — | 424 | 6 | no | — |
| 225 | C2×C4 | C22×C4×D53 | 1 | 3 | yes | — |
| 226 | C2×C4 | #41×D53 | 1 | 3 | yes | — |
| 227 | C8 | — | 2 | 6 | no | — |
| 228 | C8 | — | 53 | 1 | no | — |
| 229 | C8 | — | 53 | 3 | no | — |
| 230 | C8 | — | 106 | 6 | no | — |
| 231 | C8 | C42×D53 | 1 | 3 | yes | — |
| 232 | C8 | C42×D53 | 1 | 1 | yes | characteristic |
| 233 | C22 | — | 4 | 1 | no | — |
| 234 | C22 | — | 53 | 1 | no | — |
| 235 | C22 | — | 53 | 3 | no | — |
| 236 | C22 | — | 212 | 3 | no | — |
| 237 | C22 | — | 212 | 3 | no | — |
| 238 | C22 | #117×D53 | 1 | 3 | yes | — |
| 239 | C4 | — | 4 | 6 | no | — |
| 240 | C4 | — | 53 | 3 | no | — |
| 241 | C4 | — | 53 | 1 | no | — |
| 242 | C4 | — | 212 | 6 | no | — |
| 243 | C4 | C2×C42×D53 | 1 | 1 | yes | characteristic |
| 244 | C4 | #117×D53 | 1 | 3 | yes | — |
| 245 | C2 | — | 2 | 3 | no | — |
| 246 | C2 | — | 106 | 3 | no | — |
| 247 | C2 | — | 106 | 1 | no | — |
| 248 | C2 | #376×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