← #12119072
Subgroups — #12119072
Order 1280.
862 conjugacy classes of subgroups.
The subgroup lattice has 324 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C4×D40 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10605891 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C5×#1846 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10601421 | C2 | 1 | 2 | yes | maximal |
| 5 | Dic5×#107 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10601362 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | C8×C4⋊Dic5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #10600349 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C22×D40 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C4×D20 | C22 | 1 | 2 | yes | — |
| 11 | C2×C5×#107 | C22 | 1 | 2 | yes | — |
| 12 | C2×C5×#106 | C22 | 1 | 2 | yes | — |
| 13 | C2×C4×C5×C8 | C22 | 1 | 1 | yes | characteristic |
| 14 | C5×#382 | C22 | 1 | 2 | yes | — |
| 15 | C2×#3953 | C22 | 1 | 2 | yes | — |
| 16 | C2×#3950 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×C8×Dic5 | C22 | 1 | 2 | yes | — |
| 18 | #65277 | C22 | 1 | 4 | yes | — |
| 19 | C4⋊C4×Dic5 | C22 | 1 | 4 | yes | — |
| 20 | C2×#3941 | C22 | 1 | 4 | yes | — |
| 21 | C4×C4⋊Dic5 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C4⋊#168 | C22 | 1 | 1 | yes | characteristic |
| 23 | C4×D40 | C4 | 1 | 8 | yes | complemented |
| 24 | #64959 | — | 2 | 4 | no | — |
| 25 | #64955 | — | 2 | 2 | no | — |
| 26 | #64953 | — | 2 | 2 | no | — |
| 27 | #64770 | C22 | 1 | 2 | yes | — |
| 28 | #64749 | C22 | 1 | 4 | yes | — |
| 29 | #22810 | — | 5 | 1 | no | maximal, Sylow |
| 30 | C22×D20 | C23 | 1 | 2 | yes | — |
| 31 | C22×C4×D5 | — | 2 | 2 | no | — |
| 32 | C22×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 33 | C22×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 34 | C2×C5×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 35 | C2×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 36 | C2×#685 | — | 2 | 2 | no | — |
| 37 | C2×#685 | — | 2 | 4 | no | — |
| 38 | C2×#685 | — | 2 | 4 | no | — |
| 39 | C2×C4⋊Dic5 | D4 | 1 | 2 | yes | — |
| 40 | C2×C4⋊Dic5 | C23 | 1 | 2 | yes | — |
| 41 | C2×C4×Dic5 | — | 2 | 4 | no | — |
| 42 | C2×C4×Dic5 | C23 | 1 | 2 | yes | — |
| 43 | C22×#168 | C23 | 1 | 2 | yes | — |
| 44 | C2×D40 | — | 2 | 2 | no | — |
| 45 | C2×D40 | — | 2 | 2 | no | — |
| 46 | C2×D40 | C2×C4 | 1 | 4 | yes | — |
| 47 | C4×D20 | — | 2 | 4 | no | — |
| 48 | C4×D20 | C2×C4 | 1 | 8 | yes | — |
| 49 | C5×#107 | — | 2 | 4 | no | — |
| 50 | C5×#106 | — | 4 | 2 | no | — |
| 51 | C4×C5×C8 | C2×C4 | 1 | 4 | yes | — |
| 52 | C5×#95 | — | 2 | 2 | no | — |
| 53 | C5×#95 | — | 2 | 2 | no | — |
| 54 | #3963 | — | 2 | 4 | no | — |
| 55 | #3963 | D4 | 1 | 1 | yes | characteristic |
| 56 | #3963 | D4 | 1 | 1 | yes | characteristic |
| 57 | #3953 | C2×C4 | 1 | 8 | yes | — |
| 58 | #3950 | C2×C4 | 1 | 4 | yes | — |
| 59 | C8×Dic5 | — | 2 | 4 | no | — |
| 60 | #3941 | — | 4 | 4 | no | — |
| 61 | C4⋊#168 | — | 2 | 2 | no | — |
| 62 | C2×C4×D8 | — | 5 | 1 | no | — |
| 63 | #2357 | — | 5 | 2 | no | — |
| 64 | #1970 | — | 5 | 2 | no | — |
| 65 | #1926 | — | 5 | 4 | no | — |
| 66 | #1846 | D5 | 1 | 1 | yes | characteristic, complemented |
| 67 | #1804 | — | 5 | 2 | no | — |
| 68 | C4×#107 | — | 5 | 2 | no | — |
| 69 | C8×C4⋊C4 | — | 5 | 1 | no | — |
| 70 | C23×D5 | — | 2 | 2 | no | — |
| 71 | C22×C4×C5 | — | 2 | 4 | no | — |
| 72 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 74 | C22×C4×C5 | C24 | 1 | 1 | yes | characteristic |
| 75 | C22×Dic5 | — | 2 | 2 | no | — |
| 76 | C22×Dic5 | C2×D4 | 1 | 2 | yes | — |
| 77 | C22×Dic5 | C2×D4 | 1 | 2 | yes | — |
| 78 | C2×D20 | — | 2 | 2 | no | — |
| 79 | C2×D20 | — | 2 | 2 | no | — |
| 80 | C2×D20 | — | 2 | 2 | no | — |
| 81 | C2×D20 | — | 2 | 2 | no | — |
| 82 | C2×D20 | — | 2 | 2 | no | — |
| 83 | C2×D20 | C22×C4 | 1 | 4 | yes | — |
| 84 | C2×C4×D5 | — | 2 | 8 | no | — |
| 85 | C2×C4×D5 | — | 8 | 4 | no | — |
| 86 | C2×C5×C8 | — | 2 | 1 | no | — |
| 87 | C2×C5×C8 | — | 2 | 1 | no | — |
| 88 | C2×C5×C8 | C22×C4 | 1 | 2 | yes | — |
| 89 | C2×C5×C8 | C22×C4 | 1 | 2 | yes | — |
| 90 | C2×C5×C8 | #41 | 1 | 2 | yes | — |
| 91 | C2×C5×C8 | #41 | 1 | 2 | yes | — |
| 92 | C5×C4⋊C4 | — | 2 | 4 | no | — |
| 93 | C5×C4⋊C4 | — | 2 | 4 | no | — |
| 94 | C42×C5 | C22×C4 | 1 | 2 | yes | — |
| 95 | C42×C5 | #41 | 1 | 2 | yes | — |
| 96 | #685 | — | 2 | 8 | no | — |
| 97 | #685 | — | 4 | 8 | no | — |
| 98 | #685 | — | 8 | 4 | no | — |
| 99 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 100 | C4⋊Dic5 | — | 2 | 4 | no | — |
| 101 | C4⋊Dic5 | C22×C4 | 1 | 4 | yes | — |
| 102 | C4×Dic5 | — | 2 | 2 | no | — |
| 103 | C4×Dic5 | — | 2 | 2 | no | — |
| 104 | C4×Dic5 | — | 8 | 4 | no | — |
| 105 | C2×#168 | — | 2 | 2 | no | — |
| 106 | C2×#168 | — | 2 | 2 | no | — |
| 107 | C2×#168 | — | 2 | 2 | no | — |
| 108 | D40 | — | 2 | 8 | no | — |
| 109 | C22×D8 | — | 5 | 1 | no | — |
| 110 | C2×C4×D4 | — | 5 | 2 | no | — |
| 111 | C4×D8 | — | 5 | 8 | no | — |
| 112 | C2×#107 | — | 5 | 1 | no | — |
| 113 | C2×#107 | C2×D5 | 1 | 2 | yes | — |
| 114 | C2×#106 | C2×D5 | 1 | 2 | yes | — |
| 115 | C2×C4⋊C8 | — | 5 | 1 | no | — |
| 116 | C2×#102 | — | 5 | 2 | no | — |
| 117 | C2×#102 | — | 5 | 4 | no | — |
| 118 | C2×C4×C8 | — | 5 | 2 | no | — |
| 119 | C2×C4×C8 | C2×D5 | 1 | 1 | yes | characteristic |
| 120 | #390 | — | 5 | 4 | no | — |
| 121 | #388 | — | 10 | 4 | no | — |
| 122 | #382 | C2×D5 | 1 | 2 | yes | — |
| 123 | #381 | — | 10 | 2 | no | — |
| 124 | C4×C4⋊C4 | — | 5 | 1 | no | — |
| 125 | C4×C4⋊C4 | — | 5 | 4 | no | — |
| 126 | C4×C22⋊C4 | — | 10 | 2 | no | — |
| 127 | #340 | — | 5 | 4 | no | — |
| 128 | #336 | — | 5 | 2 | no | — |
| 129 | C23×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 130 | C22×D5 | — | 2 | 4 | no | — |
| 131 | C22×D5 | — | 4 | 2 | no | — |
| 132 | C22×D5 | — | 4 | 2 | no | — |
| 133 | C22×D5 | — | 4 | 4 | no | — |
| 134 | C22×D5 | — | 4 | 4 | no | — |
| 135 | C2×C4×C5 | — | 2 | 2 | no | — |
| 136 | C2×C4×C5 | — | 2 | 2 | no | — |
| 137 | C2×C4×C5 | — | 4 | 4 | no | — |
| 138 | C2×C4×C5 | — | 4 | 4 | no | — |
| 139 | C2×C4×C5 | — | 4 | 4 | no | — |
| 140 | C2×C4×C5 | C4×D4 | 1 | 2 | yes | — |
| 141 | C2×C4×C5 | C4×D4 | 1 | 2 | yes | — |
| 142 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 143 | C2×C4×C5 | C2×#41 | 1 | 1 | yes | characteristic |
| 144 | C2×C4×C5 | C2×#41 | 1 | 1 | yes | characteristic |
| 145 | C2×C4×C5 | C2×#41 | 1 | 1 | yes | characteristic |
| 146 | C2×C4×C5 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 147 | C2×C4×C5 | 2−1+4 | 1 | 1 | yes | characteristic |
| 148 | C2×Dic5 | — | 2 | 2 | no | — |
| 149 | C2×Dic5 | — | 2 | 2 | no | — |
| 150 | C2×Dic5 | — | 2 | 4 | no | — |
| 151 | C2×Dic5 | — | 2 | 2 | no | — |
| 152 | C2×Dic5 | — | 2 | 2 | no | — |
| 153 | C2×Dic5 | — | 2 | 2 | no | — |
| 154 | C2×Dic5 | — | 4 | 4 | no | — |
| 155 | C2×Dic5 | #135 | 1 | 4 | yes | — |
| 156 | D20 | — | 2 | 4 | no | — |
| 157 | D20 | — | 2 | 4 | no | — |
| 158 | D20 | — | 4 | 2 | no | — |
| 159 | D20 | — | 4 | 2 | no | — |
| 160 | C4×D5 | — | 8 | 8 | no | — |
| 161 | C5×C8 | — | 2 | 2 | no | — |
| 162 | C5×C8 | #117 | 1 | 4 | yes | — |
| 163 | #168 | — | 4 | 2 | no | — |
| 164 | C22×D4 | — | 5 | 2 | no | — |
| 165 | C23×C4 | — | 10 | 2 | no | — |
| 166 | C2×D8 | — | 5 | 4 | no | — |
| 167 | C2×D8 | — | 10 | 2 | no | — |
| 168 | C2×D8 | — | 10 | 2 | no | — |
| 169 | C22×C8 | — | 5 | 2 | no | — |
| 170 | C22×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 171 | C22×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 172 | C4×D4 | — | 5 | 8 | no | — |
| 173 | C4×D4 | — | 10 | 4 | no | — |
| 174 | C2×C4⋊C4 | — | 5 | 2 | no | — |
| 175 | C2×C4⋊C4 | — | 5 | 2 | no | — |
| 176 | C2×C4⋊C4 | C22×D5 | 1 | 4 | yes | — |
| 177 | C2×C22⋊C4 | — | 10 | 2 | no | — |
| 178 | C2×C22⋊C4 | — | 10 | 4 | no | — |
| 179 | C2×C22⋊C4 | — | 10 | 4 | no | — |
| 180 | C2×C42 | — | 5 | 2 | no | — |
| 181 | C2×C42 | — | 10 | 4 | no | — |
| 182 | C2×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 183 | #107 | — | 2 | 4 | no | — |
| 184 | #107 | — | 5 | 4 | no | — |
| 185 | #106 | — | 4 | 2 | no | — |
| 186 | C4⋊C8 | — | 10 | 2 | no | — |
| 187 | #102 | — | 5 | 8 | no | — |
| 188 | #102 | — | 20 | 4 | no | — |
| 189 | C4×C8 | — | 10 | 4 | no | — |
| 190 | C4×C8 | C4×D5 | 1 | 4 | yes | — |
| 191 | #95 | — | 2 | 2 | no | — |
| 192 | #95 | — | 2 | 2 | no | — |
| 193 | #95 | — | 5 | 1 | no | — |
| 194 | #95 | — | 5 | 1 | no | — |
| 195 | #95 | — | 10 | 4 | no | — |
| 196 | C22×C5 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 197 | C22×C5 | #547 | 1 | 1 | yes | characteristic |
| 198 | C22×C5 | #550 | 1 | 1 | yes | characteristic |
| 199 | C22×C5 | C2×#135 | 1 | 2 | yes | — |
| 200 | C22×C5 | #576 | 1 | 2 | yes | — |
| 201 | C2×D5 | — | 4 | 4 | no | — |
| 202 | C2×D5 | — | 4 | 4 | no | — |
| 203 | C2×D5 | — | 4 | 4 | no | — |
| 204 | C2×D5 | — | 8 | 4 | no | — |
| 205 | C2×D5 | — | 8 | 4 | no | — |
| 206 | C4×C5 | — | 2 | 2 | no | — |
| 207 | C4×C5 | — | 2 | 2 | no | — |
| 208 | C4×C5 | — | 8 | 4 | no | — |
| 209 | C4×C5 | C2×#117 | 1 | 1 | yes | characteristic |
| 210 | C4×C5 | #520 | 1 | 1 | yes | characteristic |
| 211 | C4×C5 | #555 | 1 | 1 | yes | characteristic |
| 212 | C4×C5 | #556 | 1 | 1 | yes | characteristic |
| 213 | Dic5 | — | 2 | 4 | no | — |
| 214 | Dic5 | — | 4 | 2 | no | — |
| 215 | Dic5 | — | 4 | 4 | no | — |
| 216 | C24 | — | 10 | 2 | no | — |
| 217 | C2×D4 | — | 5 | 4 | no | — |
| 218 | C2×D4 | — | 10 | 2 | no | — |
| 219 | C2×D4 | — | 10 | 2 | no | — |
| 220 | C2×D4 | — | 10 | 2 | no | — |
| 221 | C2×D4 | — | 10 | 2 | no | — |
| 222 | C2×D4 | — | 10 | 2 | no | — |
| 223 | C22×C4 | — | 2 | 4 | no | — |
| 224 | C22×C4 | — | 5 | 2 | no | — |
| 225 | C22×C4 | — | 5 | 2 | no | — |
| 226 | C22×C4 | — | 10 | 2 | no | — |
| 227 | C22×C4 | — | 10 | 8 | no | — |
| 228 | C22×C4 | — | 40 | 4 | no | — |
| 229 | C22×C4 | D4×D5 | 1 | 1 | yes | characteristic |
| 230 | C22×C4 | D4×D5 | 1 | 1 | yes | characteristic |
| 231 | C22×C4 | C23×D5 | 1 | 1 | yes | characteristic |
| 232 | D8 | — | 10 | 8 | no | — |
| 233 | C2×C8 | — | 2 | 1 | no | — |
| 234 | C2×C8 | — | 2 | 1 | no | — |
| 235 | C2×C8 | — | 10 | 2 | no | — |
| 236 | C2×C8 | — | 10 | 2 | no | — |
| 237 | C2×C8 | — | 10 | 2 | no | — |
| 238 | C2×C8 | C2×C4×D5 | 1 | 2 | yes | — |
| 239 | C2×C8 | C2×C4×D5 | 1 | 2 | yes | — |
| 240 | C2×C8 | #711 | 1 | 2 | yes | — |
| 241 | C2×C8 | #713 | 1 | 2 | yes | — |
| 242 | C4⋊C4 | — | 2 | 4 | no | — |
| 243 | C4⋊C4 | — | 2 | 4 | no | — |
| 244 | C4⋊C4 | — | 5 | 4 | no | — |
| 245 | C4⋊C4 | — | 10 | 4 | no | — |
| 246 | C4⋊C4 | — | 10 | 2 | no | — |
| 247 | C22⋊C4 | — | 10 | 8 | no | — |
| 248 | C22⋊C4 | — | 20 | 8 | no | — |
| 249 | C22⋊C4 | — | 40 | 4 | no | — |
| 250 | C42 | — | 10 | 2 | no | — |
| 251 | C42 | — | 10 | 2 | no | — |
| 252 | C42 | — | 40 | 4 | no | — |
| 253 | C42 | C2×C4×D5 | 1 | 2 | yes | — |
| 254 | C42 | #713 | 1 | 2 | yes | — |
| 255 | C2×C5 | #2354 | 1 | 1 | yes | characteristic |
| 256 | C2×C5 | C4×#135 | 1 | 1 | yes | characteristic |
| 257 | C2×C5 | #2948 | 1 | 1 | yes | characteristic |
| 258 | C2×C5 | #3339 | 1 | 2 | yes | — |
| 259 | C2×C5 | #3396 | 1 | 2 | yes | — |
| 260 | D5 | — | 8 | 4 | no | — |
| 261 | C23 | — | 10 | 4 | no | — |
| 262 | C23 | — | 20 | 2 | no | — |
| 263 | C23 | — | 20 | 2 | no | — |
| 264 | C23 | — | 20 | 4 | no | — |
| 265 | C23 | — | 20 | 4 | no | — |
| 266 | C23 | C2×D4×D5 | 1 | 1 | yes | characteristic |
| 267 | D4 | — | 10 | 4 | no | — |
| 268 | D4 | — | 10 | 4 | no | — |
| 269 | D4 | — | 20 | 2 | no | — |
| 270 | D4 | — | 20 | 2 | no | — |
| 271 | C2×C4 | — | 2 | 2 | no | — |
| 272 | C2×C4 | — | 2 | 2 | no | — |
| 273 | C2×C4 | — | 4 | 4 | no | — |
| 274 | C2×C4 | — | 4 | 4 | no | — |
| 275 | C2×C4 | — | 4 | 4 | no | — |
| 276 | C2×C4 | — | 5 | 4 | no | — |
| 277 | C2×C4 | — | 10 | 2 | no | — |
| 278 | C2×C4 | — | 10 | 2 | no | — |
| 279 | C2×C4 | — | 10 | 2 | no | — |
| 280 | C2×C4 | — | 10 | 4 | no | — |
| 281 | C2×C4 | — | 10 | 2 | no | — |
| 282 | C2×C4 | — | 10 | 2 | no | — |
| 283 | C2×C4 | — | 20 | 4 | no | — |
| 284 | C2×C4 | — | 40 | 8 | no | — |
| 285 | C2×C4 | #4039 | 1 | 2 | yes | — |
| 286 | C2×C4 | #4039 | 1 | 2 | yes | — |
| 287 | C2×C4 | C22×C4×D5 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | C2×#711 | 1 | 1 | yes | characteristic |
| 289 | C2×C4 | #4144 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | C2×#713 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | C2×#713 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | #4147 | 1 | 1 | yes | characteristic |
| 293 | C8 | — | 2 | 2 | no | — |
| 294 | C8 | — | 20 | 2 | no | — |
| 295 | C8 | #4038 | 1 | 4 | yes | — |
| 296 | C5 | #22810 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 297 | C22 | — | 20 | 4 | no | — |
| 298 | C22 | — | 20 | 4 | no | — |
| 299 | C22 | — | 20 | 4 | no | — |
| 300 | C22 | — | 40 | 4 | no | — |
| 301 | C22 | — | 40 | 4 | no | — |
| 302 | C22 | C2×#4039 | 1 | 1 | yes | characteristic |
| 303 | C22 | #66052 | 1 | 1 | yes | characteristic |
| 304 | C22 | #66060 | 1 | 1 | yes | characteristic |
| 305 | C22 | D5×#135 | 1 | 2 | yes | — |
| 306 | C22 | #66103 | 1 | 2 | yes | — |
| 307 | C4 | — | 2 | 2 | no | — |
| 308 | C4 | — | 2 | 2 | no | — |
| 309 | C4 | — | 8 | 4 | no | — |
| 310 | C4 | — | 10 | 4 | no | — |
| 311 | C4 | — | 20 | 2 | no | — |
| 312 | C4 | — | 20 | 4 | no | — |
| 313 | C4 | C2×#4038 | 1 | 1 | yes | characteristic |
| 314 | C4 | #65838 | 1 | 1 | yes | characteristic |
| 315 | C4 | #66066 | 1 | 1 | yes | characteristic |
| 316 | C4 | #66068 | 1 | 1 | yes | characteristic |
| 317 | C2 | — | 40 | 4 | no | — |
| 318 | C2 | #10606055 | 1 | 1 | yes | characteristic |
| 319 | C2 | #10610896 | 1 | 1 | yes | characteristic |
| 320 | C2 | #10610898 | 1 | 1 | yes | characteristic |
| 321 | C2 | #10613546 | 1 | 2 | yes | — |
| 322 | C2 | #10613657 | 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