← #12254722
Subgroups — #12254722
Order 1280.
826 conjugacy classes of subgroups.
The subgroup lattice has 334 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C5×#3147 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#65483 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #10610198 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10602115 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10602114 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10602107 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #10602085 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C2×#64681 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×C5×#135 | — | 2 | 1 | no | — |
| 10 | C2×C5×#135 | C22 | 1 | 2 | yes | — |
| 11 | C5×#532 | — | 2 | 1 | no | — |
| 12 | C2×#4050 | — | 2 | 1 | no | — |
| 13 | C2×#4050 | C22 | 1 | 2 | yes | — |
| 14 | #65811 | — | 2 | 1 | no | — |
| 15 | C5×#495 | — | 2 | 1 | no | — |
| 16 | C5×#495 | C22 | 1 | 2 | yes | — |
| 17 | C5×#494 | C22 | 1 | 1 | yes | characteristic |
| 18 | C5×#493 | C22 | 1 | 1 | yes | characteristic |
| 19 | C5×#492 | — | 2 | 1 | no | — |
| 20 | C2×C4×C5×C8 | C22 | 1 | 1 | yes | characteristic |
| 21 | #65483 | — | 2 | 16 | no | — |
| 22 | C2×#3961 | C22 | 1 | 4 | yes | — |
| 23 | C2×#3960 | C22 | 1 | 4 | yes | — |
| 24 | C2×#3959 | C22 | 1 | 4 | yes | — |
| 25 | C2×#3958 | C22 | 1 | 4 | yes | — |
| 26 | C22×#672 | C22 | 1 | 2 | yes | — |
| 27 | #64988 | C22 | 1 | 1 | yes | characteristic |
| 28 | #64985 | — | 2 | 1 | no | — |
| 29 | #64985 | C22 | 1 | 2 | yes | — |
| 30 | #64984 | C22 | 1 | 1 | yes | characteristic |
| 31 | #64982 | — | 2 | 1 | no | — |
| 32 | #64681 | C22 | 1 | 2 | yes | — |
| 33 | #64681 | C22 | 1 | 2 | yes | — |
| 34 | #33716 | — | 5 | 1 | no | maximal, Sylow |
| 35 | C2×C5×#41 | — | 2 | 2 | no | — |
| 36 | C2×#709 | — | 2 | 2 | no | — |
| 37 | C5×#135 | — | 2 | 8 | no | — |
| 38 | C5×#135 | — | 4 | 4 | no | — |
| 39 | C2×C5×Q16 | — | 2 | 1 | no | — |
| 40 | C2×C5×Q16 | C23 | 1 | 2 | yes | — |
| 41 | C2×C5×SD16 | — | 2 | 2 | no | — |
| 42 | C2×C5×SD16 | — | 2 | 2 | no | — |
| 43 | C2×C5×D8 | — | 2 | 1 | no | — |
| 44 | C2×C5×D8 | C23 | 1 | 2 | yes | — |
| 45 | C22×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 46 | C22×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 47 | C5×C4⋊Q8 | — | 2 | 1 | no | — |
| 48 | C5×#127 | — | 2 | 1 | no | — |
| 49 | C5×#124 | — | 2 | 2 | no | — |
| 50 | C5×#121 | — | 4 | 2 | no | — |
| 51 | C4×C5×D4 | — | 4 | 2 | no | — |
| 52 | C2×C42×C5 | D4 | 1 | 1 | yes | characteristic |
| 53 | #4076 | — | 4 | 2 | no | — |
| 54 | C2×Dic20 | — | 2 | 1 | no | — |
| 55 | C2×Dic20 | C23 | 1 | 2 | yes | — |
| 56 | #4050 | — | 2 | 8 | no | — |
| 57 | #4050 | — | 4 | 4 | no | — |
| 58 | C2×D40 | — | 2 | 1 | no | — |
| 59 | C2×D40 | C23 | 1 | 2 | yes | — |
| 60 | C2×#677 | — | 2 | 2 | no | — |
| 61 | C2×#677 | — | 2 | 2 | no | — |
| 62 | #4021 | — | 2 | 2 | no | — |
| 63 | #4020 | — | 2 | 1 | no | — |
| 64 | C4×D20 | — | 4 | 2 | no | — |
| 65 | #4015 | — | 2 | 1 | no | — |
| 66 | C4×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 67 | C4×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 68 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 69 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 70 | #3961 | — | 2 | 8 | no | — |
| 71 | #3960 | — | 2 | 8 | no | — |
| 72 | #3959 | — | 2 | 8 | no | — |
| 73 | #3958 | — | 2 | 8 | no | — |
| 74 | C2×#672 | D4 | 1 | 8 | yes | complemented |
| 75 | C2×#672 | C23 | 1 | 4 | yes | — |
| 76 | C2×#508 | — | 5 | 4 | no | — |
| 77 | #3147 | — | 5 | 1 | no | — |
| 78 | #3147 | D5 | 1 | 1 | yes | characteristic, complemented |
| 79 | #2251 | — | 5 | 2 | no | — |
| 80 | #2251 | — | 5 | 2 | no | — |
| 81 | #2250 | — | 5 | 2 | no | — |
| 82 | #2250 | — | 5 | 2 | no | — |
| 83 | C2×#346 | — | 5 | 1 | no | — |
| 84 | C5×#41 | — | 4 | 4 | no | — |
| 85 | C5×#41 | — | 4 | 4 | no | — |
| 86 | C2×C5×Q8 | — | 2 | 2 | no | — |
| 87 | C2×C5×D4 | — | 2 | 2 | no | — |
| 88 | C2×C5×D4 | — | 4 | 2 | no | — |
| 89 | C2×C5×D4 | — | 4 | 2 | no | — |
| 90 | C22×C4×C5 | — | 4 | 2 | no | — |
| 91 | C22×C4×C5 | D8 | 1 | 2 | yes | — |
| 92 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C2×C22⋊D5 | — | 4 | 2 | no | — |
| 94 | #709 | — | 4 | 4 | no | — |
| 95 | #709 | — | 4 | 4 | no | — |
| 96 | C2×D20 | — | 2 | 2 | no | — |
| 97 | C2×D20 | — | 4 | 2 | no | — |
| 98 | C2×C4×D5 | — | 4 | 2 | no | — |
| 99 | C2×Dic10 | — | 2 | 2 | no | — |
| 100 | C5×Q16 | — | 2 | 4 | no | — |
| 101 | C5×Q16 | — | 4 | 2 | no | — |
| 102 | C5×SD16 | — | 4 | 4 | no | — |
| 103 | C5×SD16 | — | 4 | 4 | no | — |
| 104 | C5×D8 | — | 2 | 4 | no | — |
| 105 | C5×D8 | — | 4 | 2 | no | — |
| 106 | C2×C5×C8 | D8 | 1 | 4 | yes | — |
| 107 | C2×C5×C8 | C2×D4 | 1 | 4 | yes | — |
| 108 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C2×C5×C8 | C24 | 1 | 1 | yes | characteristic |
| 112 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 113 | C5×C22⋊C4 | — | 4 | 4 | no | — |
| 114 | C42×C5 | D8 | 1 | 2 | yes | — |
| 115 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 117 | #685 | — | 4 | 4 | no | — |
| 118 | C4⋊Dic5 | — | 4 | 2 | no | — |
| 119 | Dic20 | — | 2 | 4 | no | — |
| 120 | Dic20 | — | 4 | 2 | no | — |
| 121 | D40 | — | 2 | 4 | no | — |
| 122 | D40 | — | 4 | 2 | no | — |
| 123 | #677 | — | 4 | 4 | no | — |
| 124 | #677 | — | 4 | 4 | no | — |
| 125 | #672 | C2×D4 | 1 | 8 | yes | — |
| 126 | C2×#135 | — | 2 | 1 | no | — |
| 127 | C2×#135 | — | 5 | 2 | no | — |
| 128 | C2×#135 | — | 10 | 1 | no | — |
| 129 | C2×#135 | C2×D5 | 1 | 2 | yes | — |
| 130 | #532 | — | 2 | 1 | no | — |
| 131 | #532 | — | 10 | 1 | no | — |
| 132 | #508 | — | 10 | 16 | no | — |
| 133 | C2×Q32 | — | 5 | 4 | no | — |
| 134 | C2×SD32 | — | 5 | 4 | no | — |
| 135 | C2×SD32 | — | 5 | 4 | no | — |
| 136 | C2×D16 | — | 5 | 4 | no | — |
| 137 | C22×C16 | — | 5 | 2 | no | — |
| 138 | #495 | — | 2 | 1 | no | — |
| 139 | #495 | — | 5 | 2 | no | — |
| 140 | #495 | — | 10 | 1 | no | — |
| 141 | #495 | C2×D5 | 1 | 2 | yes | — |
| 142 | #494 | — | 5 | 1 | no | — |
| 143 | #494 | C2×D5 | 1 | 1 | yes | characteristic |
| 144 | #493 | — | 5 | 1 | no | — |
| 145 | #493 | C2×D5 | 1 | 1 | yes | characteristic |
| 146 | #492 | — | 2 | 1 | no | — |
| 147 | #492 | — | 10 | 1 | no | — |
| 148 | C2×C4×C8 | C2×D5 | 1 | 1 | yes | characteristic |
| 149 | #346 | — | 5 | 2 | no | — |
| 150 | #346 | — | 5 | 2 | no | — |
| 151 | C23×C5 | — | 4 | 2 | no | — |
| 152 | C23×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 153 | C22×D5 | — | 4 | 2 | no | — |
| 154 | C5×Q8 | — | 4 | 2 | no | — |
| 155 | C5×Q8 | — | 4 | 2 | no | — |
| 156 | C5×D4 | — | 4 | 2 | no | — |
| 157 | C5×D4 | — | 4 | 2 | no | — |
| 158 | C5×D4 | — | 8 | 4 | no | — |
| 159 | C5×D4 | — | 8 | 4 | no | — |
| 160 | C2×C4×C5 | — | 4 | 2 | no | — |
| 161 | C2×C4×C5 | — | 4 | 2 | no | — |
| 162 | C2×C4×C5 | — | 8 | 4 | no | — |
| 163 | C2×C4×C5 | #127 | 1 | 2 | yes | — |
| 164 | C2×C4×C5 | C2×D8 | 1 | 2 | yes | — |
| 165 | C2×C4×C5 | C2×D8 | 1 | 2 | yes | — |
| 166 | C2×C4×C5 | C2×D8 | 1 | 2 | yes | — |
| 167 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 168 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 169 | C22⋊D5 | — | 8 | 4 | no | — |
| 170 | C2×Dic5 | — | 4 | 2 | no | — |
| 171 | D20 | — | 4 | 2 | no | — |
| 172 | D20 | — | 4 | 2 | no | — |
| 173 | D20 | — | 8 | 4 | no | — |
| 174 | C4×D5 | — | 8 | 4 | no | — |
| 175 | Dic10 | — | 4 | 2 | no | — |
| 176 | Dic10 | — | 4 | 2 | no | — |
| 177 | C5×C8 | #127 | 1 | 2 | yes | — |
| 178 | C5×C8 | C2×D8 | 1 | 2 | yes | — |
| 179 | C5×C8 | C2×D8 | 1 | 2 | yes | — |
| 180 | C5×C8 | C22×D4 | 1 | 2 | yes | — |
| 181 | C2×#41 | — | 2 | 2 | no | — |
| 182 | C2×#41 | — | 10 | 2 | no | — |
| 183 | #135 | — | 2 | 8 | no | — |
| 184 | #135 | — | 4 | 4 | no | — |
| 185 | #135 | — | 10 | 8 | no | — |
| 186 | #135 | — | 20 | 4 | no | — |
| 187 | C2×Q16 | — | 2 | 1 | no | — |
| 188 | C2×Q16 | — | 5 | 2 | no | — |
| 189 | C2×Q16 | — | 10 | 1 | no | — |
| 190 | C2×Q16 | C22×D5 | 1 | 2 | yes | — |
| 191 | C2×SD16 | — | 2 | 2 | no | — |
| 192 | C2×SD16 | — | 2 | 2 | no | — |
| 193 | C2×SD16 | — | 10 | 2 | no | — |
| 194 | C2×SD16 | — | 10 | 2 | no | — |
| 195 | C2×D8 | — | 2 | 1 | no | — |
| 196 | C2×D8 | — | 5 | 2 | no | — |
| 197 | C2×D8 | — | 10 | 1 | no | — |
| 198 | C2×D8 | C22×D5 | 1 | 2 | yes | — |
| 199 | C22×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 200 | C22×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 201 | C4⋊Q8 | — | 2 | 1 | no | — |
| 202 | C4⋊Q8 | — | 10 | 1 | no | — |
| 203 | #127 | — | 2 | 1 | no | — |
| 204 | #127 | — | 10 | 1 | no | — |
| 205 | #124 | — | 2 | 2 | no | — |
| 206 | #124 | — | 10 | 2 | no | — |
| 207 | #121 | — | 4 | 2 | no | — |
| 208 | #121 | — | 20 | 2 | no | — |
| 209 | C4×D4 | — | 4 | 2 | no | — |
| 210 | C4×D4 | — | 20 | 2 | no | — |
| 211 | C2×C42 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 212 | Q32 | — | 10 | 8 | no | — |
| 213 | SD32 | — | 10 | 8 | no | — |
| 214 | SD32 | — | 10 | 8 | no | — |
| 215 | D16 | — | 10 | 8 | no | — |
| 216 | C2×C16 | — | 5 | 8 | no | — |
| 217 | C2×C16 | — | 5 | 4 | no | — |
| 218 | C4×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 219 | C4×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 220 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 221 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 222 | C22×C5 | — | 2 | 2 | no | — |
| 223 | C22×C5 | — | 8 | 2 | no | — |
| 224 | C22×C5 | — | 8 | 4 | no | — |
| 225 | C22×C5 | #493 | 1 | 2 | yes | — |
| 226 | C22×C5 | C22×D8 | 1 | 1 | yes | characteristic |
| 227 | C2×D5 | — | 8 | 2 | no | — |
| 228 | C2×D5 | — | 8 | 4 | no | — |
| 229 | C4×C5 | — | 2 | 2 | no | — |
| 230 | C4×C5 | — | 2 | 2 | no | — |
| 231 | C4×C5 | — | 8 | 2 | no | — |
| 232 | C4×C5 | #493 | 1 | 2 | yes | — |
| 233 | C4×C5 | C2×#127 | 1 | 1 | yes | characteristic |
| 234 | C4×C5 | C22×D8 | 1 | 1 | yes | characteristic |
| 235 | Dic5 | — | 8 | 2 | no | — |
| 236 | #41 | — | 4 | 4 | no | — |
| 237 | #41 | — | 4 | 4 | no | — |
| 238 | #41 | — | 20 | 4 | no | — |
| 239 | #41 | — | 20 | 4 | no | — |
| 240 | C2×Q8 | — | 2 | 2 | no | — |
| 241 | C2×Q8 | — | 10 | 2 | no | — |
| 242 | C2×D4 | — | 2 | 2 | no | — |
| 243 | C2×D4 | — | 4 | 2 | no | — |
| 244 | C2×D4 | — | 4 | 2 | no | — |
| 245 | C2×D4 | — | 10 | 2 | no | — |
| 246 | C2×D4 | — | 20 | 2 | no | — |
| 247 | C2×D4 | — | 20 | 2 | no | — |
| 248 | C22×C4 | — | 4 | 2 | no | — |
| 249 | C22×C4 | — | 20 | 2 | no | — |
| 250 | C22×C4 | #686 | 1 | 2 | yes | — |
| 251 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 252 | Q16 | — | 2 | 4 | no | — |
| 253 | Q16 | — | 4 | 2 | no | — |
| 254 | Q16 | — | 10 | 4 | no | — |
| 255 | Q16 | — | 20 | 2 | no | — |
| 256 | SD16 | — | 4 | 4 | no | — |
| 257 | SD16 | — | 4 | 4 | no | — |
| 258 | SD16 | — | 20 | 4 | no | — |
| 259 | SD16 | — | 20 | 4 | no | — |
| 260 | D8 | — | 2 | 4 | no | — |
| 261 | D8 | — | 4 | 2 | no | — |
| 262 | D8 | — | 10 | 4 | no | — |
| 263 | D8 | — | 20 | 2 | no | — |
| 264 | C2×C8 | #686 | 1 | 4 | yes | — |
| 265 | C2×C8 | D4×D5 | 1 | 4 | yes | — |
| 266 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 267 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 268 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 269 | C2×C8 | C23×D5 | 1 | 1 | yes | characteristic |
| 270 | C4⋊C4 | — | 4 | 2 | no | — |
| 271 | C4⋊C4 | — | 20 | 2 | no | — |
| 272 | C22⋊C4 | — | 4 | 4 | no | — |
| 273 | C22⋊C4 | — | 20 | 4 | no | — |
| 274 | C42 | #686 | 1 | 2 | yes | — |
| 275 | C42 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 276 | C42 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 277 | C16 | — | 5 | 8 | no | — |
| 278 | C2×C5 | — | 2 | 2 | no | — |
| 279 | C2×C5 | — | 16 | 2 | no | — |
| 280 | C2×C5 | C2×#493 | 1 | 1 | yes | characteristic |
| 281 | C2×C5 | #3414 | 1 | 2 | yes | — |
| 282 | D5 | — | 16 | 2 | no | — |
| 283 | C23 | — | 4 | 2 | no | — |
| 284 | C23 | — | 20 | 2 | no | — |
| 285 | C23 | C2×#686 | 1 | 1 | yes | characteristic |
| 286 | Q8 | — | 4 | 2 | no | — |
| 287 | Q8 | — | 4 | 2 | no | — |
| 288 | Q8 | — | 20 | 2 | no | — |
| 289 | Q8 | — | 20 | 2 | no | — |
| 290 | D4 | — | 4 | 2 | no | — |
| 291 | D4 | — | 4 | 2 | no | — |
| 292 | D4 | — | 8 | 4 | no | — |
| 293 | D4 | — | 8 | 4 | no | — |
| 294 | D4 | — | 20 | 2 | no | — |
| 295 | D4 | — | 20 | 2 | no | — |
| 296 | D4 | — | 40 | 4 | no | — |
| 297 | D4 | — | 40 | 4 | no | — |
| 298 | C2×C4 | — | 4 | 2 | no | — |
| 299 | C2×C4 | — | 4 | 2 | no | — |
| 300 | C2×C4 | — | 8 | 4 | no | — |
| 301 | C2×C4 | — | 20 | 2 | no | — |
| 302 | C2×C4 | — | 40 | 4 | no | — |
| 303 | C2×C4 | C2×#686 | 1 | 2 | yes | — |
| 304 | C2×C4 | C2×#686 | 1 | 2 | yes | — |
| 305 | C2×C4 | C2×#686 | 1 | 2 | yes | — |
| 306 | C2×C4 | C2×#686 | 1 | 1 | yes | characteristic |
| 307 | C2×C4 | #4086 | 1 | 2 | yes | — |
| 308 | C2×C4 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 309 | C8 | C2×#686 | 1 | 2 | yes | — |
| 310 | C8 | C2×#686 | 1 | 2 | yes | — |
| 311 | C8 | #4086 | 1 | 2 | yes | — |
| 312 | C8 | C2×D4×D5 | 1 | 2 | yes | — |
| 313 | C5 | #33716 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 314 | C22 | — | 2 | 2 | no | — |
| 315 | C22 | — | 8 | 4 | no | — |
| 316 | C22 | — | 8 | 2 | no | — |
| 317 | C22 | — | 40 | 4 | no | — |
| 318 | C22 | — | 40 | 2 | no | — |
| 319 | C22 | #65362 | 1 | 2 | yes | — |
| 320 | C22 | C22×#686 | 1 | 1 | yes | characteristic |
| 321 | C4 | — | 2 | 2 | no | — |
| 322 | C4 | — | 2 | 2 | no | — |
| 323 | C4 | — | 8 | 2 | no | — |
| 324 | C4 | — | 40 | 2 | no | — |
| 325 | C4 | #65362 | 1 | 2 | yes | — |
| 326 | C4 | C22×#686 | 1 | 1 | yes | characteristic |
| 327 | C4 | C2×#4086 | 1 | 1 | yes | characteristic |
| 328 | C2 | — | 2 | 2 | no | — |
| 329 | C2 | — | 16 | 2 | no | — |
| 330 | C2 | — | 80 | 2 | no | — |
| 331 | C2 | C2×#65362 | 1 | 1 | yes | characteristic |
| 332 | C2 | #10615625 | 1 | 2 | yes | — |
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