← #28302183
Subgroups — #28302183
Order 13568.
342 conjugacy classes of subgroups.
The subgroup lattice has 272 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20290262 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20289922 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | M16×#88324 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#88324⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #20289548 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #20289548 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #20274616 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16811639 | C22 | 1 | 1 | yes | characteristic |
| 9 | #16811639 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C4×#88324 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C4×#88324 | C22 | 1 | 1 | yes | characteristic |
| 12 | #16811486 | C4 | 1 | 2 | yes | — |
| 13 | #16811484 | C4 | 1 | 2 | yes | — |
| 14 | #88324⋊C8 | — | 2 | 1 | no | — |
| 15 | #88324⋊C8 | — | 2 | 1 | no | — |
| 16 | #88324⋊C8 | C4 | 1 | 2 | yes | — |
| 17 | #88324⋊C8 | C4 | 1 | 2 | yes | — |
| 18 | #16811464 | — | 2 | 1 | no | — |
| 19 | C8×#88324 | — | 2 | 1 | no | — |
| 20 | C2×#13133152 | C4 | 1 | 1 | yes | characteristic |
| 21 | C2×#13133152 | C4 | 1 | 1 | yes | characteristic |
| 22 | M16×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 23 | M16×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×C8×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 25 | #16810388 | C4 | 1 | 1 | yes | characteristic |
| 26 | C53×#341 | C4 | 1 | 1 | yes | characteristic |
| 27 | C22×#88324 | — | 2 | 1 | no | — |
| 28 | C22×#88324 | Q8 | 1 | 2 | yes | — |
| 29 | C4⋊#88324 | C2×C4 | 1 | 2 | yes | — |
| 30 | C4⋊#88324 | C2×C4 | 1 | 2 | yes | — |
| 31 | C22⋊#88324 | — | 2 | 1 | no | — |
| 32 | C22⋊#88324 | — | 2 | 1 | no | — |
| 33 | C4×#88324 | — | 2 | 1 | no | — |
| 34 | C4×#88324 | — | 2 | 1 | no | — |
| 35 | C4×#88324 | — | 2 | 1 | no | — |
| 36 | C4×#88324 | C2×C4 | 1 | 2 | yes | — |
| 37 | C4×#88324 | C2×C4 | 1 | 2 | yes | — |
| 38 | C4×#88324 | D4 | 1 | 1 | yes | characteristic |
| 39 | C4×#88324 | D4 | 1 | 1 | yes | characteristic |
| 40 | C2×#11719917 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C2×#11719917 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C22×#88322 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C2×C4×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 44 | #13133152 | — | 2 | 2 | no | — |
| 45 | #13133152 | — | 2 | 2 | no | — |
| 46 | #13133127 | — | 2 | 1 | no | — |
| 47 | #13133127 | — | 2 | 1 | no | — |
| 48 | C8×Dic53 | — | 2 | 1 | no | — |
| 49 | C8×Dic53 | — | 2 | 1 | no | — |
| 50 | C8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C2×M16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | C2×M16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C22×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 57 | C22×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 58 | C22×Dic53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 59 | C22×Dic53 | #41 | 1 | 1 | yes | characteristic |
| 60 | C2×#88324 | — | 2 | 4 | no | — |
| 61 | C2×#88324 | — | 2 | 2 | no | — |
| 62 | C2×#88324 | — | 2 | 2 | no | — |
| 63 | C2×#88324 | — | 2 | 2 | no | — |
| 64 | C2×#88324 | — | 4 | 2 | no | — |
| 65 | C2×#88324 | C4⋊C4 | 1 | 2 | yes | — |
| 66 | C2×#88324 | C4⋊C4 | 1 | 2 | yes | — |
| 67 | M16×C53 | — | 2 | 1 | no | — |
| 68 | M16×C53 | — | 2 | 1 | no | — |
| 69 | M16×C53 | — | 2 | 1 | no | — |
| 70 | M16×C53 | M16 | 1 | 2 | yes | — |
| 71 | C2×C8×C53 | — | 2 | 1 | no | — |
| 72 | C2×C8×C53 | — | 2 | 1 | no | — |
| 73 | C2×C8×C53 | — | 2 | 1 | no | — |
| 74 | C2×C8×C53 | — | 2 | 1 | no | — |
| 75 | C2×C8×C53 | C42 | 1 | 1 | yes | characteristic |
| 76 | C2×C8×C53 | C42 | 1 | 1 | yes | characteristic |
| 77 | C4×Dic53 | C22×C4 | 1 | 1 | yes | characteristic |
| 78 | C4×Dic53 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | C4×Dic53 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C4×Dic53 | #41 | 1 | 1 | yes | characteristic |
| 81 | #11719917 | — | 2 | 1 | no | — |
| 82 | #11719917 | — | 2 | 1 | no | — |
| 83 | #11719917 | — | 2 | 1 | no | — |
| 84 | #11719917 | M16 | 1 | 2 | yes | — |
| 85 | C2×#88322 | — | 2 | 1 | no | — |
| 86 | C2×#88322 | — | 2 | 1 | no | — |
| 87 | C2×#88322 | — | 2 | 1 | no | — |
| 88 | C2×#88322 | — | 2 | 1 | no | — |
| 89 | C2×#88322 | C42 | 1 | 1 | yes | characteristic |
| 90 | C2×#88322 | C42 | 1 | 1 | yes | characteristic |
| 91 | C23×C53 | C4×Q8 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×C53 | C2×C42 | 1 | 1 | yes | characteristic |
| 93 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×C4×C53 | C2×M16 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C53 | C2×M16 | 1 | 1 | yes | characteristic |
| 96 | C2×C4×C53 | #131 | 1 | 1 | yes | characteristic |
| 97 | C2×C4×C53 | #131 | 1 | 1 | yes | characteristic |
| 98 | C2×Dic53 | — | 2 | 2 | no | — |
| 99 | C2×Dic53 | — | 2 | 2 | no | — |
| 100 | C2×Dic53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 101 | C2×Dic53 | #117 | 1 | 1 | yes | characteristic |
| 102 | C2×Dic53 | C4×D4 | 1 | 1 | yes | characteristic |
| 103 | C2×Dic53 | C4×Q8 | 1 | 1 | yes | characteristic |
| 104 | #88324 | — | 2 | 4 | no | — |
| 105 | #88324 | — | 2 | 2 | no | — |
| 106 | #88324 | — | 2 | 2 | no | — |
| 107 | #88324 | — | 4 | 2 | no | — |
| 108 | #88324 | — | 4 | 2 | no | — |
| 109 | C8×C53 | — | 2 | 1 | no | — |
| 110 | C8×C53 | — | 2 | 1 | no | — |
| 111 | C8×C53 | — | 2 | 2 | no | — |
| 112 | C8×C53 | — | 4 | 1 | no | — |
| 113 | #88322 | — | 2 | 1 | no | — |
| 114 | #88322 | — | 2 | 1 | no | — |
| 115 | #88322 | — | 2 | 2 | no | — |
| 116 | #88322 | — | 4 | 1 | no | — |
| 117 | #8296 | — | 53 | 1 | no | maximal, Sylow |
| 118 | C22×C53 | — | 2 | 1 | no | — |
| 119 | C22×C53 | — | 2 | 1 | no | — |
| 120 | C22×C53 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 121 | C22×C53 | #446 | 1 | 1 | yes | characteristic |
| 122 | C22×C53 | #446 | 1 | 1 | yes | characteristic |
| 123 | C4×C53 | C4×M16 | 1 | 1 | yes | characteristic |
| 124 | C4×C53 | #405 | 1 | 1 | yes | characteristic |
| 125 | C4×C53 | #435 | 1 | 1 | yes | characteristic |
| 126 | C4×C53 | #435 | 1 | 1 | yes | characteristic |
| 127 | Dic53 | — | 2 | 1 | no | — |
| 128 | Dic53 | — | 2 | 1 | no | — |
| 129 | Dic53 | #430 | 1 | 2 | yes | — |
| 130 | Dic53 | #443 | 1 | 2 | yes | — |
| 131 | C4×#108 | — | 53 | 1 | no | — |
| 132 | #1564 | — | 53 | 1 | no | — |
| 133 | #1451 | — | 53 | 1 | no | — |
| 134 | C8×M16 | — | 53 | 1 | no | — |
| 135 | C2×#322 | — | 53 | 1 | no | — |
| 136 | #1278 | — | 53 | 1 | no | — |
| 137 | #1278 | — | 53 | 1 | no | — |
| 138 | C2×C53 | — | 2 | 1 | no | — |
| 139 | C2×C53 | — | 2 | 1 | no | — |
| 140 | C2×C53 | #1771 | 1 | 1 | yes | characteristic |
| 141 | C2×C53 | #1772 | 1 | 1 | yes | characteristic |
| 142 | C2×C53 | #1778 | 1 | 1 | yes | characteristic |
| 143 | #431 | — | 53 | 1 | no | — |
| 144 | #431 | — | 53 | 1 | no | — |
| 145 | C2×#108 | — | 53 | 1 | no | — |
| 146 | C2×#108 | — | 53 | 1 | no | — |
| 147 | C4×M16 | — | 53 | 1 | no | — |
| 148 | C4×M16 | — | 53 | 1 | no | — |
| 149 | C2×C4×C8 | — | 53 | 1 | no | — |
| 150 | C2×C4×C8 | — | 53 | 1 | no | — |
| 151 | C2×C4×C8 | — | 53 | 1 | no | — |
| 152 | #341 | — | 53 | 1 | no | — |
| 153 | #341 | #6157 | 1 | 1 | yes | characteristic |
| 154 | #335 | — | 53 | 2 | no | — |
| 155 | #334 | — | 53 | 2 | no | — |
| 156 | #322 | — | 53 | 2 | no | — |
| 157 | #322 | — | 53 | 2 | no | — |
| 158 | #322 | — | 106 | 1 | no | — |
| 159 | #322 | — | 106 | 1 | no | — |
| 160 | #322 | — | 106 | 1 | no | — |
| 161 | C82 | — | 106 | 1 | no | — |
| 162 | C53 | #8296 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 163 | C2×M16 | — | 53 | 1 | no | — |
| 164 | C2×M16 | — | 53 | 1 | no | — |
| 165 | C2×M16 | C2×#6157 | 1 | 1 | yes | characteristic |
| 166 | C2×M16 | C2×#6157 | 1 | 1 | yes | characteristic |
| 167 | C22×C8 | — | 53 | 1 | no | — |
| 168 | C22×C8 | — | 53 | 2 | no | — |
| 169 | C22×C8 | — | 106 | 1 | no | — |
| 170 | C22×C8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 171 | C2×C42 | — | 53 | 1 | no | — |
| 172 | #108 | — | 106 | 2 | no | — |
| 173 | #108 | — | 106 | 2 | no | — |
| 174 | C4⋊C8 | — | 53 | 2 | no | — |
| 175 | C4⋊C8 | — | 53 | 2 | no | — |
| 176 | C22⋊C8 | — | 106 | 1 | no | — |
| 177 | C22⋊C8 | — | 106 | 1 | no | — |
| 178 | #97 | — | 106 | 1 | no | — |
| 179 | #97 | — | 106 | 1 | no | — |
| 180 | C4×C8 | — | 53 | 1 | no | — |
| 181 | C4×C8 | — | 53 | 2 | no | — |
| 182 | C4×C8 | — | 53 | 1 | no | — |
| 183 | C4×C8 | — | 53 | 1 | no | — |
| 184 | C4×C8 | — | 53 | 1 | no | — |
| 185 | C4×C8 | — | 53 | 1 | no | — |
| 186 | C4×C8 | — | 53 | 1 | no | — |
| 187 | C4×C8 | — | 53 | 2 | no | — |
| 188 | C4×C8 | — | 106 | 1 | no | — |
| 189 | C4×C8 | — | 106 | 1 | no | — |
| 190 | C4×C8 | — | 106 | 1 | no | — |
| 191 | C4×C8 | — | 106 | 1 | no | — |
| 192 | C4×C8 | — | 106 | 1 | no | — |
| 193 | C22×C4 | — | 53 | 1 | no | — |
| 194 | C22×C4 | — | 53 | 1 | no | — |
| 195 | C22×C4 | C22×#6157 | 1 | 1 | yes | characteristic |
| 196 | M16 | — | 2 | 1 | no | — |
| 197 | M16 | — | 2 | 1 | no | — |
| 198 | M16 | — | 2 | 1 | no | — |
| 199 | M16 | — | 53 | 2 | no | — |
| 200 | M16 | — | 106 | 1 | no | — |
| 201 | M16 | — | 106 | 1 | no | — |
| 202 | M16 | — | 106 | 1 | no | — |
| 203 | M16 | #11719936 | 1 | 2 | yes | — |
| 204 | C2×C8 | — | 2 | 1 | no | — |
| 205 | C2×C8 | — | 2 | 1 | no | — |
| 206 | C2×C8 | — | 2 | 1 | no | — |
| 207 | C2×C8 | — | 2 | 1 | no | — |
| 208 | C2×C8 | — | 53 | 2 | no | — |
| 209 | C2×C8 | — | 53 | 2 | no | — |
| 210 | C2×C8 | — | 53 | 1 | no | — |
| 211 | C2×C8 | — | 53 | 1 | no | — |
| 212 | C2×C8 | — | 106 | 2 | no | — |
| 213 | C2×C8 | — | 106 | 4 | no | — |
| 214 | C2×C8 | — | 106 | 1 | no | — |
| 215 | C2×C8 | — | 106 | 1 | no | — |
| 216 | C2×C8 | — | 106 | 1 | no | — |
| 217 | C2×C8 | — | 106 | 2 | no | — |
| 218 | C2×C8 | — | 106 | 2 | no | — |
| 219 | C2×C8 | — | 106 | 1 | no | — |
| 220 | C2×C8 | — | 212 | 2 | no | — |
| 221 | C2×C8 | C4×#6157 | 1 | 1 | yes | characteristic |
| 222 | C2×C8 | C4×#6157 | 1 | 1 | yes | characteristic |
| 223 | C42 | — | 53 | 1 | no | — |
| 224 | C42 | — | 53 | 1 | no | — |
| 225 | C42 | — | 53 | 1 | no | — |
| 226 | C42 | — | 53 | 1 | no | — |
| 227 | C23 | Q8×#6157 | 1 | 1 | yes | characteristic |
| 228 | C2×C4 | — | 53 | 1 | no | — |
| 229 | C2×C4 | — | 53 | 1 | no | — |
| 230 | C2×C4 | — | 53 | 1 | no | — |
| 231 | C2×C4 | — | 53 | 1 | no | — |
| 232 | C2×C4 | — | 106 | 2 | no | — |
| 233 | C2×C4 | — | 106 | 2 | no | — |
| 234 | C2×C4 | C2×C4×#6157 | 1 | 1 | yes | characteristic |
| 235 | C2×C4 | D4×#6157 | 1 | 1 | yes | characteristic |
| 236 | C2×C4 | D53⋊M16 | 1 | 1 | yes | characteristic |
| 237 | C2×C4 | C2×#11719936 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | #13133304 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | #13133304 | 1 | 1 | yes | characteristic |
| 240 | C8 | — | 2 | 1 | no | — |
| 241 | C8 | — | 2 | 1 | no | — |
| 242 | C8 | — | 2 | 2 | no | — |
| 243 | C8 | — | 4 | 1 | no | — |
| 244 | C8 | — | 106 | 2 | no | — |
| 245 | C8 | — | 106 | 2 | no | — |
| 246 | C8 | — | 106 | 4 | no | — |
| 247 | C8 | — | 106 | 1 | no | — |
| 248 | C8 | — | 106 | 1 | no | — |
| 249 | C8 | — | 106 | 2 | no | — |
| 250 | C8 | — | 212 | 2 | no | — |
| 251 | C8 | — | 212 | 2 | no | — |
| 252 | C8 | — | 212 | 1 | no | — |
| 253 | C22 | — | 2 | 1 | no | — |
| 254 | C22 | — | 2 | 1 | no | — |
| 255 | C22 | C4⋊C4×#6157 | 1 | 1 | yes | characteristic |
| 256 | C22 | #16811673 | 1 | 1 | yes | characteristic |
| 257 | C22 | #16811675 | 1 | 1 | yes | characteristic |
| 258 | C4 | — | 53 | 2 | no | — |
| 259 | C4 | — | 53 | 2 | no | — |
| 260 | C4 | — | 106 | 1 | no | — |
| 261 | C4 | — | 106 | 1 | no | — |
| 262 | C4 | C4×#11719936 | 1 | 1 | yes | characteristic |
| 263 | C4 | #16811587 | 1 | 1 | yes | characteristic |
| 264 | C4 | #16811652 | 1 | 1 | yes | characteristic |
| 265 | C4 | #16811655 | 1 | 1 | yes | characteristic |
| 266 | C2 | — | 2 | 1 | no | — |
| 267 | C2 | — | 2 | 1 | no | — |
| 268 | C2 | #20290514 | 1 | 1 | yes | characteristic |
| 269 | C2 | #20290520 | 1 | 1 | yes | characteristic |
| 270 | C2 | #20290536 | 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