← #28330813
Subgroups — #28330813
Order 13568.
11222 conjugacy classes of subgroups.
The subgroup lattice has 254 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | 2−1+4×#6157 | C2 | 1 | 12 | yes | maximal, complemented |
| 2 | H(5,𝔽2)×#6157 | C2 | 1 | 20 | yes | maximal, complemented |
| 3 | #20291695 | C2 | 1 | 15 | yes | maximal, complemented |
| 4 | C2×#16811692 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | #585×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #16811692 | C22 | 1 | 60 | yes | complemented |
| 7 | #16811692 | C22 | 1 | 120 | yes | complemented |
| 8 | #16811692 | C22 | 1 | 20 | yes | complemented |
| 9 | #16811691 | C22 | 1 | 30 | yes | complemented |
| 10 | #16811690 | C22 | 1 | 90 | yes | complemented |
| 11 | #41×#6157 | C22 | 1 | 120 | yes | complemented |
| 12 | C2×Q8×#6157 | C22 | 1 | 30 | yes | complemented |
| 13 | C2×D4×#6157 | C22 | 1 | 90 | yes | complemented |
| 14 | C2×#13133296 | C22 | 1 | 45 | yes | complemented |
| 15 | C22×C4⋊#6157 | C22 | 1 | 15 | yes | complemented |
| 16 | #16811460 | C4 | 1 | 6 | yes | complemented |
| 17 | #16811459 | C4 | 1 | 15 | yes | complemented |
| 18 | #16811458 | C4 | 1 | 10 | yes | complemented |
| 19 | 2−1+4×D53 | C22 | 1 | 6 | yes | — |
| 20 | H(5,𝔽2)×D53 | C22 | 1 | 10 | yes | — |
| 21 | C2×#41×D53 | C22 | 1 | 15 | yes | — |
| 22 | C53×#585 | C4 | 1 | 1 | yes | characteristic, complemented |
| 23 | C23×#6157 | C23 | 1 | 30 | yes | — |
| 24 | Q8×#6157 | C23 | 1 | 120 | yes | — |
| 25 | Q8×#6157 | C23 | 1 | 40 | yes | — |
| 26 | D4×#6157 | C23 | 1 | 360 | yes | — |
| 27 | D4×#6157 | C23 | 1 | 120 | yes | — |
| 28 | #13133296 | C23 | 1 | 60 | yes | — |
| 29 | #13133296 | C23 | 1 | 180 | yes | — |
| 30 | C2×C4⋊#6157 | C23 | 1 | 45 | yes | — |
| 31 | C2×C4⋊#6157 | C23 | 1 | 90 | yes | — |
| 32 | C2×C4⋊#6157 | C23 | 1 | 15 | yes | — |
| 33 | C2×C22⋊#6157 | C23 | 1 | 90 | yes | — |
| 34 | C2×C4×#6157 | C23 | 1 | 90 | yes | — |
| 35 | #13133265 | C2×C4 | 1 | 30 | yes | complemented |
| 36 | #13133264 | C2×C4 | 1 | 60 | yes | complemented |
| 37 | #13133263 | C2×C4 | 1 | 60 | yes | complemented |
| 38 | #13133262 | C2×C4 | 1 | 90 | yes | complemented |
| 39 | C2×#11719949 | C2×C4 | 1 | 15 | yes | complemented |
| 40 | C2×#11719945 | C2×C4 | 1 | 45 | yes | complemented |
| 41 | C2×#11719947 | C2×C4 | 1 | 45 | yes | complemented |
| 42 | #41×D53 | C2×C4 | 1 | 60 | yes | complemented |
| 43 | #41×D53 | C2×C4 | 1 | 60 | yes | complemented |
| 44 | #41×D53 | C23 | 1 | 60 | yes | — |
| 45 | #41×D53 | C23 | 1 | 20 | yes | — |
| 46 | C2×Q8×D53 | C23 | 1 | 15 | yes | — |
| 47 | C2×D4×D53 | C23 | 1 | 45 | yes | — |
| 48 | C22×C4×D53 | C23 | 1 | 15 | yes | — |
| 49 | 2−1+4×C53 | C2×C4 | 1 | 6 | yes | complemented |
| 50 | H(5,𝔽2)×C53 | C2×C4 | 1 | 10 | yes | complemented |
| 51 | C2×#41×C53 | C2×C4 | 1 | 15 | yes | complemented |
| 52 | C23×D53 | C24 | 1 | 15 | yes | — |
| 53 | C22×#6157 | — | 8 | 30 | no | — |
| 54 | C22×#6157 | C24 | 1 | 30 | yes | — |
| 55 | C22×#6157 | C24 | 1 | 90 | yes | — |
| 56 | #41×C53 | C22×C4 | 1 | 20 | yes | complemented |
| 57 | #41×C53 | C22×C4 | 1 | 60 | yes | complemented |
| 58 | C2×Q8×C53 | C22×C4 | 1 | 15 | yes | complemented |
| 59 | C2×D4×C53 | C22×C4 | 1 | 45 | yes | complemented |
| 60 | C22×C4×C53 | C22×C4 | 1 | 15 | yes | complemented |
| 61 | C2×C22⋊D53 | C22×C4 | 1 | 90 | yes | complemented |
| 62 | C22×Dic53 | C22×C4 | 1 | 15 | yes | complemented |
| 63 | #11719949 | C22×C4 | 1 | 20 | yes | complemented |
| 64 | #11719949 | C22×C4 | 1 | 60 | yes | complemented |
| 65 | Q8×D53 | C22×C4 | 1 | 60 | yes | complemented |
| 66 | Q8×D53 | C24 | 1 | 20 | yes | — |
| 67 | #11719947 | C22×C4 | 1 | 60 | yes | complemented |
| 68 | #11719947 | C22×C4 | 1 | 180 | yes | complemented |
| 69 | D4×D53 | C22×C4 | 1 | 180 | yes | complemented |
| 70 | D4×D53 | C24 | 1 | 60 | yes | — |
| 71 | #11719945 | C22×C4 | 1 | 60 | yes | complemented |
| 72 | #11719945 | C22×C4 | 1 | 180 | yes | complemented |
| 73 | C2×D212 | C22×C4 | 1 | 45 | yes | complemented |
| 74 | C2×C4×D53 | C22×C4 | 1 | 45 | yes | complemented |
| 75 | C2×C4×D53 | C22×C4 | 1 | 45 | yes | complemented |
| 76 | C2×C4×D53 | C24 | 1 | 15 | yes | — |
| 77 | C2×C4×D53 | C24 | 1 | 45 | yes | — |
| 78 | C2×Dic106 | C22×C4 | 1 | 45 | yes | complemented |
| 79 | C22⋊#6157 | C24 | 1 | 120 | yes | — |
| 80 | C4⋊#6157 | C24 | 1 | 10 | yes | — |
| 81 | C4⋊#6157 | C24 | 1 | 6 | yes | — |
| 82 | C4⋊#6157 | C24 | 1 | 90 | yes | — |
| 83 | C4⋊#6157 | C24 | 1 | 30 | yes | — |
| 84 | C4×#6157 | C24 | 1 | 120 | yes | — |
| 85 | C23×C53 | C23×C4 | 1 | 15 | yes | — |
| 86 | C22×D53 | — | 4 | 15 | no | — |
| 87 | C22×D53 | — | 4 | 15 | no | — |
| 88 | C22×D53 | C23×C4 | 1 | 45 | yes | — |
| 89 | C22×D53 | C25 | 1 | 15 | yes | — |
| 90 | C2×#6157 | — | 4 | 90 | no | — |
| 91 | C2×#6157 | — | 4 | 30 | no | — |
| 92 | C2×#6157 | C25 | 1 | 12 | yes | — |
| 93 | C2×#6157 | C25 | 1 | 20 | yes | — |
| 94 | Q8×C53 | C23×C4 | 1 | 20 | yes | — |
| 95 | D4×C53 | C23×C4 | 1 | 60 | yes | — |
| 96 | C2×C4×C53 | C23×C4 | 1 | 45 | yes | — |
| 97 | C2×C4×C53 | C23×C4 | 1 | 15 | yes | — |
| 98 | C22⋊D53 | C23×C4 | 1 | 120 | yes | — |
| 99 | C2×Dic53 | C23×C4 | 1 | 45 | yes | — |
| 100 | C2×Dic53 | C23×C4 | 1 | 15 | yes | — |
| 101 | D212 | C23×C4 | 1 | 60 | yes | — |
| 102 | C4×D53 | C23×C4 | 1 | 15 | yes | — |
| 103 | C4×D53 | C23×C4 | 1 | 15 | yes | — |
| 104 | C4×D53 | C23×C4 | 1 | 90 | yes | — |
| 105 | C4×D53 | C25 | 1 | 15 | yes | — |
| 106 | C4×D53 | C25 | 1 | 1 | yes | characteristic |
| 107 | Dic106 | C23×C4 | 1 | 60 | yes | — |
| 108 | #62654 | — | 53 | 1 | no | maximal, Sylow |
| 109 | C22×C53 | — | 4 | 15 | no | — |
| 110 | C22×C53 | C24×C4 | 1 | 15 | yes | — |
| 111 | C2×D53 | — | 2 | 15 | no | — |
| 112 | C2×D53 | — | 2 | 15 | no | — |
| 113 | C2×D53 | — | 4 | 45 | no | — |
| 114 | C2×D53 | C24×C4 | 1 | 15 | yes | — |
| 115 | C2×D53 | C26 | 1 | 1 | yes | characteristic |
| 116 | #6157 | — | 2 | 20 | no | — |
| 117 | #6157 | — | 2 | 12 | no | — |
| 118 | C4×C53 | C24×C4 | 1 | 15 | yes | — |
| 119 | C4×C53 | C24×C4 | 1 | 1 | yes | characteristic |
| 120 | Dic53 | C24×C4 | 1 | 15 | yes | — |
| 121 | Dic53 | C24×C4 | 1 | 1 | yes | characteristic |
| 122 | C2×#585 | — | 53 | 1 | no | — |
| 123 | C4×2−1+4 | — | 53 | 12 | no | — |
| 124 | C4×H(5,𝔽2) | — | 53 | 20 | no | — |
| 125 | #3428 | — | 53 | 15 | no | — |
| 126 | C2×#520 | — | 53 | 15 | no | — |
| 127 | C2×C53 | — | 2 | 15 | no | — |
| 128 | C2×C53 | C25×C4 | 1 | 1 | yes | characteristic |
| 129 | D53 | — | 2 | 15 | no | — |
| 130 | D53 | H(7,𝔽2) | 1 | 1 | yes | characteristic |
| 131 | D53 | 2−1+6 | 1 | 1 | yes | characteristic |
| 132 | #585 | — | 53 | 6 | no | — |
| 133 | #585 | — | 53 | 10 | no | — |
| 134 | #585 | — | 53 | 15 | no | — |
| 135 | #585 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 136 | C2×2−1+4 | — | 53 | 6 | no | — |
| 137 | C2×H(5,𝔽2) | — | 53 | 10 | no | — |
| 138 | C22×#41 | — | 53 | 15 | no | — |
| 139 | #520 | — | 53 | 60 | no | — |
| 140 | #520 | — | 53 | 120 | no | — |
| 141 | #520 | — | 53 | 20 | no | — |
| 142 | #519 | — | 53 | 30 | no | — |
| 143 | #518 | — | 53 | 90 | no | — |
| 144 | C4×#41 | — | 53 | 120 | no | — |
| 145 | C2×C4×Q8 | — | 53 | 30 | no | — |
| 146 | C2×C4×D4 | — | 53 | 90 | no | — |
| 147 | C2×#117 | — | 53 | 45 | no | — |
| 148 | C22×C4⋊C4 | — | 53 | 15 | no | — |
| 149 | C53 | #62654 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 150 | 2−1+4 | — | 53 | 60 | no | — |
| 151 | 2−1+4 | — | 53 | 30 | no | — |
| 152 | 2−1+4 | C2×#6157 | 1 | 6 | yes | complemented |
| 153 | H(5,𝔽2) | — | 53 | 60 | no | — |
| 154 | H(5,𝔽2) | — | 53 | 90 | no | — |
| 155 | H(5,𝔽2) | C2×#6157 | 1 | 10 | yes | complemented |
| 156 | C2×#41 | — | 53 | 60 | no | — |
| 157 | C2×#41 | — | 53 | 20 | no | — |
| 158 | C2×#41 | — | 53 | 60 | no | — |
| 159 | C2×#41 | — | 53 | 60 | no | — |
| 160 | C2×#41 | — | 53 | 15 | no | — |
| 161 | C2×#41 | — | 53 | 45 | no | — |
| 162 | C2×#41 | — | 53 | 45 | no | — |
| 163 | C2×#41 | C2×#6157 | 1 | 15 | yes | complemented |
| 164 | C22×Q8 | — | 53 | 15 | no | — |
| 165 | C22×D4 | — | 53 | 45 | no | — |
| 166 | C23×C4 | — | 53 | 15 | no | — |
| 167 | C23×C4 | — | 53 | 30 | no | — |
| 168 | C4×Q8 | — | 53 | 120 | no | — |
| 169 | C4×Q8 | — | 53 | 40 | no | — |
| 170 | C4×D4 | — | 53 | 360 | no | — |
| 171 | C4×D4 | — | 53 | 120 | no | — |
| 172 | #117 | — | 53 | 60 | no | — |
| 173 | #117 | — | 53 | 180 | no | — |
| 174 | C2×C4⋊C4 | — | 53 | 45 | no | — |
| 175 | C2×C4⋊C4 | — | 53 | 90 | no | — |
| 176 | C2×C4⋊C4 | — | 53 | 15 | no | — |
| 177 | C2×C22⋊C4 | — | 53 | 90 | no | — |
| 178 | C2×C42 | — | 53 | 90 | no | — |
| 179 | C24 | — | 53 | 15 | no | — |
| 180 | #41 | — | 53 | 20 | no | — |
| 181 | #41 | — | 53 | 60 | no | — |
| 182 | #41 | — | 53 | 60 | no | — |
| 183 | #41 | — | 53 | 180 | no | — |
| 184 | #41 | — | 53 | 60 | no | — |
| 185 | #41 | — | 53 | 180 | no | — |
| 186 | #41 | C22×#6157 | 1 | 20 | yes | complemented |
| 187 | #41 | C22×#6157 | 1 | 60 | yes | complemented |
| 188 | C2×Q8 | — | 53 | 60 | no | — |
| 189 | C2×Q8 | — | 53 | 20 | no | — |
| 190 | C2×Q8 | — | 53 | 45 | no | — |
| 191 | C2×Q8 | C22×#6157 | 1 | 15 | yes | complemented |
| 192 | C2×D4 | — | 53 | 90 | no | — |
| 193 | C2×D4 | — | 53 | 180 | no | — |
| 194 | C2×D4 | — | 53 | 60 | no | — |
| 195 | C2×D4 | — | 53 | 45 | no | — |
| 196 | C2×D4 | C22×#6157 | 1 | 45 | yes | complemented |
| 197 | C22×C4 | — | 53 | 15 | no | — |
| 198 | C22×C4 | — | 53 | 45 | no | — |
| 199 | C22×C4 | — | 53 | 45 | no | — |
| 200 | C22×C4 | — | 53 | 45 | no | — |
| 201 | C22×C4 | — | 53 | 15 | no | — |
| 202 | C22×C4 | — | 53 | 30 | no | — |
| 203 | C22×C4 | — | 53 | 90 | no | — |
| 204 | C22×C4 | — | 424 | 30 | no | — |
| 205 | C22×C4 | C22×#6157 | 1 | 15 | yes | complemented |
| 206 | C4⋊C4 | — | 53 | 10 | no | — |
| 207 | C4⋊C4 | — | 53 | 6 | no | — |
| 208 | C4⋊C4 | — | 53 | 90 | no | — |
| 209 | C4⋊C4 | — | 53 | 30 | no | — |
| 210 | C22⋊C4 | — | 53 | 120 | no | — |
| 211 | C42 | — | 53 | 120 | no | — |
| 212 | C23 | — | 53 | 15 | no | — |
| 213 | C23 | — | 53 | 45 | no | — |
| 214 | C23 | — | 212 | 15 | no | — |
| 215 | C23 | — | 212 | 15 | no | — |
| 216 | C23 | C23×#6157 | 1 | 15 | yes | — |
| 217 | Q8 | — | 53 | 60 | no | — |
| 218 | Q8 | C23×#6157 | 1 | 20 | yes | — |
| 219 | D4 | — | 53 | 120 | no | — |
| 220 | D4 | — | 53 | 60 | no | — |
| 221 | D4 | C23×#6157 | 1 | 60 | yes | — |
| 222 | C2×C4 | — | 53 | 15 | no | — |
| 223 | C2×C4 | — | 53 | 45 | no | — |
| 224 | C2×C4 | — | 53 | 15 | no | — |
| 225 | C2×C4 | — | 53 | 15 | no | — |
| 226 | C2×C4 | — | 53 | 90 | no | — |
| 227 | C2×C4 | — | 53 | 15 | no | — |
| 228 | C2×C4 | — | 53 | 1 | no | — |
| 229 | C2×C4 | — | 53 | 12 | no | — |
| 230 | C2×C4 | — | 53 | 20 | no | — |
| 231 | C2×C4 | — | 212 | 30 | no | — |
| 232 | C2×C4 | — | 212 | 90 | no | — |
| 233 | C2×C4 | C23×#6157 | 1 | 15 | yes | — |
| 234 | C2×C4 | C23×#6157 | 1 | 45 | yes | — |
| 235 | C22 | — | 4 | 15 | no | — |
| 236 | C22 | — | 53 | 1 | no | — |
| 237 | C22 | — | 53 | 15 | no | — |
| 238 | C22 | — | 106 | 15 | no | — |
| 239 | C22 | — | 106 | 15 | no | — |
| 240 | C22 | — | 212 | 45 | no | — |
| 241 | C22 | C24×#6157 | 1 | 15 | yes | — |
| 242 | C4 | — | 53 | 1 | no | — |
| 243 | C4 | — | 53 | 15 | no | — |
| 244 | C4 | — | 106 | 20 | no | — |
| 245 | C4 | — | 106 | 12 | no | — |
| 246 | C4 | C24×#6157 | 1 | 1 | yes | characteristic |
| 247 | C4 | C24×#6157 | 1 | 15 | yes | — |
| 248 | C2 | — | 2 | 15 | no | — |
| 249 | C2 | — | 53 | 1 | no | — |
| 250 | C2 | — | 53 | 1 | no | — |
| 251 | C2 | — | 106 | 15 | no | — |
| 252 | C2 | C25×#6157 | 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