← #12254674
Subgroups — #12254674
Order 1280.
622 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C5×#500 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10610198 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10602176 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #10602175 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10602174 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10601391 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | C4×#3944 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×C5×C4⋊Q8 | — | 2 | 1 | no | — |
| 9 | C2×#4050 | — | 2 | 1 | no | — |
| 10 | C2×#4050 | C22 | 1 | 2 | yes | — |
| 11 | #65811 | — | 2 | 1 | no | — |
| 12 | C5×#500 | — | 2 | 1 | no | — |
| 13 | C5×#500 | — | 2 | 1 | no | — |
| 14 | C5×#500 | C22 | 1 | 2 | yes | — |
| 15 | C5×#500 | C22 | 1 | 2 | yes | — |
| 16 | C2×C5×#107 | — | 2 | 1 | no | — |
| 17 | C2×C5×#107 | C22 | 1 | 2 | yes | — |
| 18 | C2×C4×C5×C8 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#3944 | C22 | 1 | 2 | yes | — |
| 20 | #64988 | C22 | 1 | 1 | yes | characteristic |
| 21 | #64985 | — | 2 | 1 | no | — |
| 22 | #64985 | C22 | 1 | 2 | yes | — |
| 23 | #64984 | C22 | 1 | 1 | yes | characteristic |
| 24 | #64982 | — | 2 | 1 | no | — |
| 25 | #64712 | C22 | 1 | 8 | yes | — |
| 26 | #64711 | C22 | 1 | 8 | yes | — |
| 27 | #64681 | C22 | 1 | 2 | yes | — |
| 28 | C4×#672 | C22 | 1 | 2 | yes | — |
| 29 | #33707 | — | 5 | 1 | no | maximal, Sylow |
| 30 | C22×C5×Q8 | — | 4 | 1 | no | — |
| 31 | C2×#709 | — | 2 | 2 | no | — |
| 32 | C22×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 33 | C22×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 34 | C5×C4⋊Q8 | — | 2 | 2 | no | — |
| 35 | C5×C4⋊Q8 | — | 2 | 2 | no | — |
| 36 | C5×C4⋊Q8 | — | 4 | 1 | no | — |
| 37 | C5×C4⋊Q8 | — | 4 | 1 | no | — |
| 38 | C2×C5×C4⋊C4 | — | 2 | 2 | no | — |
| 39 | C2×C5×C4⋊C4 | — | 4 | 1 | no | — |
| 40 | C2×C42×C5 | D4 | 1 | 1 | yes | characteristic |
| 41 | #4076 | — | 4 | 1 | no | — |
| 42 | #4076 | — | 4 | 1 | no | — |
| 43 | C2×Dic20 | — | 2 | 1 | no | — |
| 44 | C2×Dic20 | C23 | 1 | 2 | yes | — |
| 45 | #4050 | — | 2 | 8 | no | — |
| 46 | #4050 | — | 4 | 4 | no | — |
| 47 | C2×D40 | — | 2 | 1 | no | — |
| 48 | C2×D40 | C23 | 1 | 2 | yes | — |
| 49 | C2×#677 | — | 2 | 2 | no | — |
| 50 | C2×#677 | — | 2 | 2 | no | — |
| 51 | #4021 | — | 2 | 2 | no | — |
| 52 | #4020 | — | 2 | 1 | no | — |
| 53 | C4×D20 | — | 4 | 1 | no | — |
| 54 | C4×D20 | — | 4 | 1 | no | — |
| 55 | #4015 | — | 2 | 1 | no | — |
| 56 | C5×#107 | — | 2 | 2 | no | — |
| 57 | C5×#107 | — | 2 | 2 | no | — |
| 58 | C5×#107 | — | 2 | 2 | no | — |
| 59 | C5×#107 | C23 | 1 | 4 | yes | — |
| 60 | C4×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 61 | C4×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 62 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 63 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 64 | #3944 | — | 2 | 4 | no | — |
| 65 | C2×#672 | C23 | 1 | 4 | yes | — |
| 66 | C2×#500 | D5 | 1 | 1 | yes | characteristic, complemented |
| 67 | #3147 | — | 5 | 1 | no | — |
| 68 | #2243 | — | 5 | 4 | no | — |
| 69 | #2241 | — | 5 | 2 | no | — |
| 70 | #2240 | — | 5 | 2 | no | — |
| 71 | #2139 | — | 5 | 4 | no | — |
| 72 | C4×M32 | — | 5 | 1 | no | — |
| 73 | C2×C5×Q8 | — | 4 | 2 | no | — |
| 74 | C2×C5×Q8 | — | 4 | 2 | no | — |
| 75 | C2×C5×Q8 | — | 8 | 4 | no | — |
| 76 | C22×C4×C5 | — | 4 | 2 | no | — |
| 77 | C22×C4×C5 | D8 | 1 | 1 | yes | characteristic |
| 78 | C22×C4×C5 | D8 | 1 | 1 | yes | characteristic |
| 79 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C2×C22⋊D5 | — | 4 | 2 | no | — |
| 81 | #709 | — | 4 | 4 | no | — |
| 82 | #709 | — | 4 | 4 | no | — |
| 83 | C2×D20 | — | 2 | 2 | no | — |
| 84 | C2×D20 | — | 4 | 1 | no | — |
| 85 | C2×D20 | — | 4 | 1 | no | — |
| 86 | C2×C4×D5 | — | 4 | 2 | no | — |
| 87 | C2×Dic10 | — | 2 | 2 | no | — |
| 88 | C2×C5×C8 | SD16 | 1 | 4 | yes | — |
| 89 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C2×C5×C8 | #41 | 1 | 4 | yes | — |
| 93 | C2×C5×C8 | C24 | 1 | 1 | yes | characteristic |
| 94 | C5×C4⋊C4 | — | 2 | 4 | no | — |
| 95 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 96 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 97 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 98 | C42×C5 | D8 | 1 | 1 | yes | characteristic |
| 99 | C42×C5 | D8 | 1 | 1 | yes | characteristic |
| 100 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | #685 | — | 4 | 2 | no | — |
| 103 | #685 | — | 4 | 2 | no | — |
| 104 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 105 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 106 | Dic20 | — | 2 | 4 | no | — |
| 107 | Dic20 | — | 4 | 2 | no | — |
| 108 | D40 | — | 2 | 4 | no | — |
| 109 | D40 | — | 4 | 2 | no | — |
| 110 | #677 | — | 4 | 4 | no | — |
| 111 | #677 | — | 4 | 4 | no | — |
| 112 | #672 | — | 2 | 4 | no | — |
| 113 | C2×#135 | — | 5 | 2 | no | — |
| 114 | C2×#135 | — | 10 | 1 | no | — |
| 115 | #532 | — | 10 | 1 | no | — |
| 116 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 117 | C2×M32 | — | 5 | 2 | no | — |
| 118 | #500 | — | 2 | 1 | no | — |
| 119 | #500 | — | 2 | 1 | no | — |
| 120 | #500 | C2×D5 | 1 | 2 | yes | — |
| 121 | #500 | C2×D5 | 1 | 2 | yes | — |
| 122 | #495 | — | 5 | 2 | no | — |
| 123 | #495 | — | 10 | 1 | no | — |
| 124 | #494 | — | 5 | 1 | no | — |
| 125 | #493 | — | 5 | 1 | no | — |
| 126 | #492 | — | 10 | 1 | no | — |
| 127 | C2×#107 | — | 2 | 1 | no | — |
| 128 | C2×#107 | C2×D5 | 1 | 2 | yes | — |
| 129 | C2×C4×C8 | C2×D5 | 1 | 1 | yes | characteristic |
| 130 | #358 | — | 5 | 8 | no | — |
| 131 | #357 | — | 5 | 8 | no | — |
| 132 | #346 | — | 5 | 2 | no | — |
| 133 | C4×C16 | — | 5 | 2 | no | — |
| 134 | C23×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 135 | C22×D5 | — | 4 | 2 | no | — |
| 136 | C5×Q8 | — | 8 | 4 | no | — |
| 137 | C5×Q8 | — | 8 | 4 | no | — |
| 138 | C2×C4×C5 | — | 2 | 2 | no | — |
| 139 | C2×C4×C5 | — | 2 | 2 | no | — |
| 140 | C2×C4×C5 | — | 4 | 4 | no | — |
| 141 | C2×C4×C5 | — | 8 | 4 | no | — |
| 142 | C2×C4×C5 | #124 | 1 | 2 | yes | — |
| 143 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 144 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 145 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 146 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 147 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 148 | C2×C4×C5 | C2×SD16 | 1 | 2 | yes | — |
| 149 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 150 | C22⋊D5 | — | 8 | 4 | no | — |
| 151 | C2×Dic5 | — | 4 | 2 | no | — |
| 152 | D20 | — | 4 | 2 | no | — |
| 153 | D20 | — | 4 | 2 | no | — |
| 154 | D20 | — | 8 | 2 | no | — |
| 155 | D20 | — | 8 | 2 | no | — |
| 156 | C4×D5 | — | 8 | 4 | no | — |
| 157 | Dic10 | — | 4 | 2 | no | — |
| 158 | Dic10 | — | 4 | 2 | no | — |
| 159 | C5×C8 | #124 | 1 | 2 | yes | — |
| 160 | C5×C8 | C2×SD16 | 1 | 2 | yes | — |
| 161 | C5×C8 | C2×SD16 | 1 | 2 | yes | — |
| 162 | C5×C8 | C2×#41 | 1 | 2 | yes | — |
| 163 | C2×#41 | — | 10 | 2 | no | — |
| 164 | C22×Q8 | — | 4 | 1 | no | — |
| 165 | #135 | — | 10 | 8 | no | — |
| 166 | #135 | — | 20 | 4 | no | — |
| 167 | C2×Q16 | — | 5 | 2 | no | — |
| 168 | C2×Q16 | — | 10 | 1 | no | — |
| 169 | C2×SD16 | — | 10 | 2 | no | — |
| 170 | C2×SD16 | — | 10 | 2 | no | — |
| 171 | C2×D8 | — | 5 | 2 | no | — |
| 172 | C2×D8 | — | 10 | 1 | no | — |
| 173 | C22×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 174 | C22×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 175 | C4⋊Q8 | — | 2 | 2 | no | — |
| 176 | C4⋊Q8 | — | 2 | 2 | no | — |
| 177 | C4⋊Q8 | — | 4 | 1 | no | — |
| 178 | C4⋊Q8 | — | 4 | 1 | no | — |
| 179 | C4⋊Q8 | — | 10 | 1 | no | — |
| 180 | #127 | — | 10 | 1 | no | — |
| 181 | #124 | — | 10 | 2 | no | — |
| 182 | #121 | — | 20 | 1 | no | — |
| 183 | #121 | — | 20 | 1 | no | — |
| 184 | C4×D4 | — | 20 | 1 | no | — |
| 185 | C4×D4 | — | 20 | 1 | no | — |
| 186 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 187 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 188 | C2×C42 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 189 | M32 | — | 10 | 4 | no | — |
| 190 | C2×C16 | — | 5 | 4 | no | — |
| 191 | #107 | — | 2 | 2 | no | — |
| 192 | #107 | — | 2 | 2 | no | — |
| 193 | #107 | — | 2 | 2 | no | — |
| 194 | #107 | C22×D5 | 1 | 4 | yes | — |
| 195 | C4×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 196 | C4×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 197 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 198 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 199 | C22×C5 | — | 2 | 1 | no | — |
| 200 | C22×C5 | — | 2 | 1 | no | — |
| 201 | C22×C5 | #486 | 1 | 2 | yes | — |
| 202 | C22×C5 | C22×D8 | 1 | 1 | yes | characteristic |
| 203 | C2×D5 | — | 8 | 2 | no | — |
| 204 | C2×D5 | — | 8 | 2 | no | — |
| 205 | C2×D5 | — | 8 | 2 | no | — |
| 206 | C4×C5 | — | 2 | 1 | no | — |
| 207 | C4×C5 | — | 2 | 1 | no | — |
| 208 | C4×C5 | — | 8 | 4 | no | — |
| 209 | C4×C5 | #486 | 1 | 2 | yes | — |
| 210 | C4×C5 | #509 | 1 | 2 | yes | — |
| 211 | C4×C5 | #510 | 1 | 2 | yes | — |
| 212 | C4×C5 | C2×#124 | 1 | 1 | yes | characteristic |
| 213 | C4×C5 | C22×SD16 | 1 | 1 | yes | characteristic |
| 214 | Dic5 | — | 8 | 2 | no | — |
| 215 | #41 | — | 20 | 4 | no | — |
| 216 | #41 | — | 20 | 4 | no | — |
| 217 | C2×Q8 | — | 4 | 2 | no | — |
| 218 | C2×Q8 | — | 4 | 2 | no | — |
| 219 | C2×Q8 | — | 8 | 4 | no | — |
| 220 | C2×Q8 | — | 10 | 2 | no | — |
| 221 | C2×D4 | — | 10 | 2 | no | — |
| 222 | C2×D4 | — | 20 | 2 | no | — |
| 223 | C2×D4 | — | 20 | 1 | no | — |
| 224 | C2×D4 | — | 20 | 1 | no | — |
| 225 | C22×C4 | — | 4 | 2 | no | — |
| 226 | C22×C4 | — | 20 | 2 | no | — |
| 227 | C22×C4 | #686 | 1 | 1 | yes | characteristic |
| 228 | C22×C4 | #686 | 1 | 1 | yes | characteristic |
| 229 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 230 | Q16 | — | 10 | 4 | no | — |
| 231 | Q16 | — | 20 | 2 | no | — |
| 232 | SD16 | — | 20 | 4 | no | — |
| 233 | SD16 | — | 20 | 4 | no | — |
| 234 | D8 | — | 10 | 4 | no | — |
| 235 | D8 | — | 20 | 2 | no | — |
| 236 | C2×C8 | #688 | 1 | 4 | yes | — |
| 237 | C2×C8 | #713 | 1 | 4 | yes | — |
| 238 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 239 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 240 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 241 | C2×C8 | C23×D5 | 1 | 1 | yes | characteristic |
| 242 | C4⋊C4 | — | 2 | 4 | no | — |
| 243 | C4⋊C4 | — | 4 | 2 | no | — |
| 244 | C4⋊C4 | — | 4 | 2 | no | — |
| 245 | C4⋊C4 | — | 4 | 2 | no | — |
| 246 | C4⋊C4 | — | 20 | 1 | no | — |
| 247 | C4⋊C4 | — | 20 | 1 | no | — |
| 248 | C22⋊C4 | — | 20 | 2 | no | — |
| 249 | C22⋊C4 | — | 20 | 2 | no | — |
| 250 | C42 | #686 | 1 | 1 | yes | characteristic |
| 251 | C42 | #686 | 1 | 1 | yes | characteristic |
| 252 | C42 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 253 | C42 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 254 | C16 | — | 10 | 4 | no | — |
| 255 | C2×C5 | — | 2 | 2 | no | — |
| 256 | C2×C5 | C2×#486 | 1 | 1 | yes | characteristic |
| 257 | C2×C5 | C2×#509 | 1 | 1 | yes | characteristic |
| 258 | C2×C5 | C2×#510 | 1 | 1 | yes | characteristic |
| 259 | D5 | — | 16 | 2 | no | — |
| 260 | C23 | — | 20 | 2 | no | — |
| 261 | C23 | C2×#686 | 1 | 1 | yes | characteristic |
| 262 | Q8 | — | 8 | 4 | no | — |
| 263 | Q8 | — | 8 | 4 | no | — |
| 264 | Q8 | — | 20 | 2 | no | — |
| 265 | Q8 | — | 20 | 2 | no | — |
| 266 | D4 | — | 20 | 2 | no | — |
| 267 | D4 | — | 20 | 2 | no | — |
| 268 | D4 | — | 40 | 4 | no | — |
| 269 | D4 | — | 40 | 2 | no | — |
| 270 | D4 | — | 40 | 2 | no | — |
| 271 | C2×C4 | — | 2 | 2 | no | — |
| 272 | C2×C4 | — | 2 | 2 | no | — |
| 273 | C2×C4 | — | 4 | 4 | no | — |
| 274 | C2×C4 | — | 8 | 4 | no | — |
| 275 | C2×C4 | — | 20 | 2 | no | — |
| 276 | C2×C4 | — | 40 | 4 | no | — |
| 277 | C2×C4 | C2×#686 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | C2×#686 | 1 | 1 | yes | characteristic |
| 279 | C2×C4 | C2×#686 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | C2×#686 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | C2×#686 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | C2×#688 | 1 | 2 | yes | — |
| 283 | C2×C4 | #4093 | 1 | 2 | yes | — |
| 284 | C2×C4 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 285 | C8 | C2×#688 | 1 | 2 | yes | — |
| 286 | C8 | C2×#688 | 1 | 2 | yes | — |
| 287 | C8 | #4093 | 1 | 2 | yes | — |
| 288 | C8 | C2×#713 | 1 | 2 | yes | — |
| 289 | C5 | #33707 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 290 | C22 | — | 2 | 1 | no | — |
| 291 | C22 | — | 2 | 1 | no | — |
| 292 | C22 | — | 40 | 2 | no | — |
| 293 | C22 | — | 40 | 2 | no | — |
| 294 | C22 | — | 40 | 2 | no | — |
| 295 | C22 | #65377 | 1 | 2 | yes | — |
| 296 | C22 | C22×#686 | 1 | 1 | yes | characteristic |
| 297 | C4 | — | 2 | 1 | no | — |
| 298 | C4 | — | 2 | 1 | no | — |
| 299 | C4 | — | 8 | 4 | no | — |
| 300 | C4 | — | 40 | 2 | no | — |
| 301 | C4 | #65377 | 1 | 2 | yes | — |
| 302 | C4 | #65436 | 1 | 2 | yes | — |
| 303 | C4 | #65468 | 1 | 2 | yes | — |
| 304 | C4 | C22×#688 | 1 | 1 | yes | characteristic |
| 305 | C4 | C2×#4093 | 1 | 1 | yes | characteristic |
| 306 | C2 | — | 2 | 2 | no | — |
| 307 | C2 | — | 80 | 2 | no | — |
| 308 | C2 | C2×#65377 | 1 | 1 | yes | characteristic |
| 309 | C2 | C2×#65436 | 1 | 1 | yes | characteristic |
| 310 | C2 | C2×#65468 | 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