← #12199135
Subgroups — #12199135
Order 1280.
754 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 | C4×C5×#124 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10605038 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10602518 | C2 | 1 | 2 | yes | maximal |
| 4 | #10602419 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10602397 | C2 | 1 | 2 | yes | maximal |
| 6 | #10600462 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #10600432 | C2 | 1 | 2 | yes | maximal |
| 8 | #10600340 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C4×C4⋊#168 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C2×C5×#124 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C4×C5×Q8 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C4×C5×D4 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#4016 | C22 | 1 | 2 | yes | — |
| 14 | C5×#391 | C22 | 1 | 1 | yes | characteristic |
| 15 | C5×#390 | C22 | 1 | 1 | yes | characteristic |
| 16 | C5×#388 | — | 2 | 2 | no | — |
| 17 | C5×#382 | C22 | 1 | 1 | yes | characteristic |
| 18 | C4×C5×C22⋊C4 | — | 2 | 2 | no | — |
| 19 | C43×C5 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#3967 | C22 | 1 | 4 | yes | — |
| 21 | C2×#3964 | C22 | 1 | 4 | yes | — |
| 22 | #65225 | C22 | 1 | 2 | yes | — |
| 23 | #65224 | C22 | 1 | 2 | yes | — |
| 24 | #65217 | — | 2 | 2 | no | — |
| 25 | #65217 | — | 2 | 2 | no | — |
| 26 | C2×C4⋊#168 | C22 | 1 | 2 | yes | — |
| 27 | C2×C4×#168 | C22 | 1 | 2 | yes | — |
| 28 | #64749 | C22 | 1 | 4 | yes | — |
| 29 | #64749 | C22 | 1 | 4 | yes | — |
| 30 | #64744 | C22 | 1 | 2 | yes | — |
| 31 | #29140 | — | 5 | 1 | no | maximal, Sylow |
| 32 | C22×C5×Q8 | C23 | 1 | 1 | yes | characteristic |
| 33 | C22×C5×D4 | C23 | 1 | 1 | yes | characteristic |
| 34 | C23×C4×C5 | — | 2 | 1 | no | — |
| 35 | C5×#124 | — | 2 | 2 | no | — |
| 36 | C5×#124 | Q8 | 1 | 4 | yes | — |
| 37 | C4×C5×Q8 | — | 2 | 4 | no | — |
| 38 | C4×C5×D4 | — | 2 | 4 | no | — |
| 39 | C2×C5×C4⋊C4 | — | 2 | 1 | no | — |
| 40 | C2×C5×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 41 | C2×C5×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 42 | C2×C5×C22⋊C4 | — | 2 | 2 | no | — |
| 43 | C2×C5×C22⋊C4 | — | 2 | 2 | no | — |
| 44 | C2×C5×C22⋊C4 | — | 2 | 1 | no | — |
| 45 | C2×C42×C5 | — | 2 | 1 | no | — |
| 46 | C2×C42×C5 | D4 | 1 | 2 | yes | — |
| 47 | C2×C42×C5 | D4 | 1 | 2 | yes | — |
| 48 | C2×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 49 | C2×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×C4⋊Dic5 | — | 2 | 2 | no | — |
| 52 | C2×C4⋊Dic5 | — | 2 | 2 | no | — |
| 53 | C2×C4⋊Dic5 | C23 | 1 | 4 | yes | — |
| 54 | C2×#683 | — | 2 | 4 | no | — |
| 55 | C22×#168 | C23 | 1 | 4 | yes | — |
| 56 | #4016 | — | 4 | 4 | no | — |
| 57 | C5×#95 | — | 2 | 2 | no | — |
| 58 | C5×#95 | — | 2 | 2 | no | — |
| 59 | #3967 | — | 4 | 4 | no | — |
| 60 | #3964 | — | 4 | 4 | no | — |
| 61 | #3963 | — | 2 | 4 | no | — |
| 62 | C4⋊#168 | — | 2 | 4 | no | — |
| 63 | C4×#168 | — | 2 | 4 | no | — |
| 64 | #2859 | — | 5 | 1 | no | — |
| 65 | C4×#124 | D5 | 1 | 1 | yes | characteristic |
| 66 | #2026 | — | 5 | 2 | no | — |
| 67 | #2025 | — | 5 | 2 | no | — |
| 68 | #1958 | — | 5 | 2 | no | — |
| 69 | #1835 | — | 5 | 2 | no | — |
| 70 | #1803 | — | 5 | 2 | no | — |
| 71 | #1802 | — | 5 | 2 | no | — |
| 72 | C4×C4⋊C8 | — | 5 | 1 | no | — |
| 73 | C24×C5 | — | 2 | 1 | no | — |
| 74 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 75 | C2×C5×Q8 | — | 2 | 2 | no | — |
| 76 | C2×C5×Q8 | — | 2 | 2 | no | — |
| 77 | C2×C5×Q8 | C2×Q8 | 1 | 2 | yes | — |
| 78 | C2×C5×D4 | — | 2 | 2 | no | — |
| 79 | C2×C5×D4 | — | 2 | 1 | no | — |
| 80 | C2×C5×D4 | — | 2 | 2 | no | — |
| 81 | C2×C5×D4 | C2×Q8 | 1 | 2 | yes | — |
| 82 | C22×C4×C5 | — | 2 | 1 | no | — |
| 83 | C22×C4×C5 | — | 2 | 1 | no | — |
| 84 | C22×C4×C5 | — | 2 | 1 | no | — |
| 85 | C22×C4×C5 | — | 4 | 2 | no | — |
| 86 | C22×C4×C5 | — | 4 | 4 | no | — |
| 87 | C22×C4×C5 | C2×D4 | 1 | 2 | yes | — |
| 88 | C22×C4×C5 | C2×D4 | 1 | 2 | yes | — |
| 89 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C22×C4×C5 | C24 | 1 | 1 | yes | characteristic |
| 92 | C22×Dic5 | — | 2 | 4 | no | — |
| 93 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 94 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 95 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 96 | C5×C22⋊C4 | — | 2 | 8 | no | — |
| 97 | C5×C22⋊C4 | — | 4 | 4 | no | — |
| 98 | C5×C22⋊C4 | — | 4 | 2 | no | — |
| 99 | C42×C5 | — | 2 | 4 | no | — |
| 100 | C42×C5 | — | 2 | 2 | no | — |
| 101 | C42×C5 | — | 4 | 2 | no | — |
| 102 | C42×C5 | — | 4 | 2 | no | — |
| 103 | C42×C5 | C2×Q8 | 1 | 2 | yes | — |
| 104 | C42×C5 | #41 | 1 | 2 | yes | — |
| 105 | C42×C5 | #41 | 1 | 4 | yes | — |
| 106 | C4⋊Dic5 | — | 2 | 4 | no | — |
| 107 | C4⋊Dic5 | — | 2 | 4 | no | — |
| 108 | C4⋊Dic5 | — | 4 | 4 | no | — |
| 109 | C4⋊Dic5 | — | 4 | 4 | no | — |
| 110 | #683 | — | 4 | 8 | no | — |
| 111 | C2×#168 | — | 2 | 4 | no | — |
| 112 | C2×#168 | — | 2 | 4 | no | — |
| 113 | C2×#168 | — | 2 | 4 | no | — |
| 114 | C2×#125 | — | 5 | 2 | no | — |
| 115 | C2×#124 | C2×D5 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×Q8 | C2×D5 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×D4 | C2×D5 | 1 | 1 | yes | characteristic |
| 118 | C2×C4⋊C8 | — | 5 | 2 | no | — |
| 119 | C2×#103 | — | 5 | 4 | no | — |
| 120 | C2×#102 | — | 5 | 4 | no | — |
| 121 | C2×C4×C8 | — | 5 | 2 | no | — |
| 122 | #391 | C2×D5 | 1 | 1 | yes | characteristic |
| 123 | #390 | C2×D5 | 1 | 1 | yes | characteristic |
| 124 | #389 | — | 10 | 2 | no | — |
| 125 | #389 | — | 10 | 2 | no | — |
| 126 | #388 | — | 2 | 2 | no | — |
| 127 | #384 | — | 5 | 2 | no | — |
| 128 | #382 | — | 5 | 2 | no | — |
| 129 | #382 | C2×D5 | 1 | 1 | yes | characteristic |
| 130 | C4×C22⋊C4 | — | 2 | 2 | no | — |
| 131 | C43 | C2×D5 | 1 | 1 | yes | characteristic |
| 132 | #340 | — | 5 | 4 | no | — |
| 133 | #340 | — | 5 | 4 | no | — |
| 134 | #336 | — | 5 | 2 | no | — |
| 135 | C23×C5 | — | 2 | 2 | no | — |
| 136 | C23×C5 | — | 4 | 1 | no | — |
| 137 | C23×C5 | — | 4 | 1 | no | — |
| 138 | C23×C5 | — | 4 | 2 | no | — |
| 139 | C23×C5 | — | 4 | 2 | no | — |
| 140 | C23×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 141 | C5×Q8 | — | 2 | 4 | no | — |
| 142 | C5×Q8 | — | 4 | 2 | no | — |
| 143 | C5×D4 | — | 2 | 4 | no | — |
| 144 | C5×D4 | — | 4 | 2 | no | — |
| 145 | C2×C4×C5 | — | 2 | 2 | no | — |
| 146 | C2×C4×C5 | — | 2 | 2 | no | — |
| 147 | C2×C4×C5 | — | 2 | 2 | no | — |
| 148 | C2×C4×C5 | — | 2 | 2 | no | — |
| 149 | C2×C4×C5 | — | 2 | 2 | no | — |
| 150 | C2×C4×C5 | — | 2 | 2 | no | — |
| 151 | C2×C4×C5 | — | 2 | 2 | no | — |
| 152 | C2×C4×C5 | — | 4 | 1 | no | — |
| 153 | C2×C4×C5 | — | 4 | 1 | no | — |
| 154 | C2×C4×C5 | — | 4 | 1 | no | — |
| 155 | C2×C4×C5 | — | 4 | 1 | no | — |
| 156 | C2×C4×C5 | — | 4 | 2 | no | — |
| 157 | C2×C4×C5 | — | 4 | 2 | no | — |
| 158 | C2×C4×C5 | — | 8 | 4 | no | — |
| 159 | C2×C4×C5 | C22⋊Q8 | 1 | 4 | yes | — |
| 160 | C2×C4×C5 | #124 | 1 | 2 | yes | — |
| 161 | C2×C4×C5 | #124 | 1 | 2 | yes | — |
| 162 | C2×C4×C5 | #135 | 1 | 4 | yes | — |
| 163 | C2×C4×C5 | C22×Q8 | 1 | 1 | yes | characteristic |
| 164 | C2×C4×C5 | C2×#41 | 1 | 1 | yes | characteristic |
| 165 | C2×C4×C5 | C2×#41 | 1 | 2 | yes | — |
| 166 | C2×C4×C5 | H(5,𝔽2) | 1 | 2 | yes | — |
| 167 | C2×Dic5 | — | 4 | 4 | no | — |
| 168 | C2×Dic5 | — | 4 | 4 | no | — |
| 169 | C2×Dic5 | — | 4 | 4 | no | — |
| 170 | #168 | — | 4 | 4 | no | — |
| 171 | C22×Q8 | C22×D5 | 1 | 1 | yes | characteristic |
| 172 | C22×D4 | C22×D5 | 1 | 1 | yes | characteristic |
| 173 | C23×C4 | — | 2 | 1 | no | — |
| 174 | C22×C8 | — | 5 | 4 | no | — |
| 175 | #125 | — | 20 | 4 | no | — |
| 176 | #124 | — | 2 | 2 | no | — |
| 177 | #124 | Dic10 | 1 | 4 | yes | — |
| 178 | C4×Q8 | — | 2 | 4 | no | — |
| 179 | C4×D4 | — | 2 | 4 | no | — |
| 180 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 181 | C2×C4⋊C4 | — | 5 | 4 | no | — |
| 182 | C2×C4⋊C4 | — | 10 | 4 | no | — |
| 183 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 184 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 185 | C2×C4⋊C4 | C22×D5 | 1 | 1 | yes | characteristic |
| 186 | C2×C4⋊C4 | C22×D5 | 1 | 1 | yes | characteristic |
| 187 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 188 | C2×C22⋊C4 | — | 2 | 1 | no | — |
| 189 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 190 | C2×C42 | — | 2 | 1 | no | — |
| 191 | C2×C42 | C22⋊D5 | 1 | 2 | yes | — |
| 192 | C2×C42 | C22⋊D5 | 1 | 2 | yes | — |
| 193 | C2×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 194 | C2×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 195 | C2×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 196 | C4⋊C8 | — | 10 | 4 | no | — |
| 197 | #103 | — | 20 | 4 | no | — |
| 198 | #102 | — | 20 | 4 | no | — |
| 199 | C4×C8 | — | 10 | 4 | no | — |
| 200 | #95 | — | 2 | 2 | no | — |
| 201 | #95 | — | 2 | 2 | no | — |
| 202 | #95 | — | 10 | 4 | no | — |
| 203 | C22×C5 | — | 4 | 2 | no | — |
| 204 | C22×C5 | — | 4 | 4 | no | — |
| 205 | C22×C5 | — | 8 | 1 | no | — |
| 206 | C22×C5 | — | 8 | 1 | no | — |
| 207 | C22×C5 | — | 8 | 2 | no | — |
| 208 | C22×C5 | C2×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 209 | C22×C5 | C2×#124 | 1 | 1 | yes | characteristic |
| 210 | C22×C5 | #535 | 1 | 1 | yes | characteristic |
| 211 | C22×C5 | C2×#135 | 1 | 2 | yes | — |
| 212 | C22×C5 | #575 | 1 | 2 | yes | — |
| 213 | C4×C5 | — | 2 | 4 | no | — |
| 214 | C4×C5 | — | 2 | 2 | no | — |
| 215 | C4×C5 | — | 4 | 2 | no | — |
| 216 | C4×C5 | — | 4 | 2 | no | — |
| 217 | C4×C5 | — | 8 | 1 | no | — |
| 218 | C4×C5 | — | 8 | 1 | no | — |
| 219 | C4×C5 | #488 | 1 | 4 | yes | — |
| 220 | C4×C5 | #554 | 1 | 2 | yes | — |
| 221 | C4×C5 | #555 | 1 | 2 | yes | — |
| 222 | Dic5 | — | 8 | 4 | no | — |
| 223 | C24 | — | 2 | 1 | no | — |
| 224 | C2×Q8 | — | 2 | 1 | no | — |
| 225 | C2×Q8 | — | 2 | 2 | no | — |
| 226 | C2×Q8 | — | 2 | 2 | no | — |
| 227 | C2×Q8 | C2×Dic10 | 1 | 2 | yes | — |
| 228 | C2×D4 | — | 2 | 1 | no | — |
| 229 | C2×D4 | — | 2 | 2 | no | — |
| 230 | C2×D4 | — | 2 | 2 | no | — |
| 231 | C2×D4 | C2×Dic10 | 1 | 2 | yes | — |
| 232 | C22×C4 | — | 2 | 1 | no | — |
| 233 | C22×C4 | — | 2 | 1 | no | — |
| 234 | C22×C4 | — | 2 | 1 | no | — |
| 235 | C22×C4 | — | 4 | 4 | no | — |
| 236 | C22×C4 | — | 4 | 2 | no | — |
| 237 | C22×C4 | — | 10 | 4 | no | — |
| 238 | C22×C4 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 239 | C22×C4 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 240 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 241 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 242 | C22×C4 | C23×D5 | 1 | 1 | yes | characteristic |
| 243 | C2×C8 | — | 10 | 4 | no | — |
| 244 | C2×C8 | — | 10 | 4 | no | — |
| 245 | C2×C8 | — | 10 | 4 | no | — |
| 246 | C4⋊C4 | — | 2 | 2 | no | — |
| 247 | C4⋊C4 | — | 2 | 2 | no | — |
| 248 | C4⋊C4 | — | 4 | 2 | no | — |
| 249 | C4⋊C4 | — | 10 | 4 | no | — |
| 250 | C4⋊C4 | — | 10 | 4 | no | — |
| 251 | C4⋊C4 | — | 20 | 8 | no | — |
| 252 | C4⋊C4 | — | 20 | 4 | no | — |
| 253 | C4⋊C4 | — | 20 | 4 | no | — |
| 254 | C22⋊C4 | — | 2 | 8 | no | — |
| 255 | C22⋊C4 | — | 4 | 4 | no | — |
| 256 | C22⋊C4 | — | 4 | 2 | no | — |
| 257 | C42 | — | 2 | 4 | no | — |
| 258 | C42 | — | 2 | 2 | no | — |
| 259 | C42 | — | 4 | 2 | no | — |
| 260 | C42 | — | 4 | 2 | no | — |
| 261 | C42 | C2×Dic10 | 1 | 2 | yes | — |
| 262 | C42 | #709 | 1 | 2 | yes | — |
| 263 | C42 | #711 | 1 | 4 | yes | — |
| 264 | C2×C5 | — | 8 | 2 | no | — |
| 265 | C2×C5 | #2563 | 1 | 1 | yes | characteristic |
| 266 | C2×C5 | #3076 | 1 | 2 | yes | — |
| 267 | C2×C5 | C2×#488 | 1 | 1 | yes | characteristic |
| 268 | C2×C5 | #3142 | 1 | 1 | yes | characteristic |
| 269 | C2×C5 | #3212 | 1 | 2 | yes | — |
| 270 | C23 | — | 2 | 2 | no | — |
| 271 | C23 | — | 4 | 1 | no | — |
| 272 | C23 | — | 4 | 2 | no | — |
| 273 | C23 | — | 4 | 2 | no | — |
| 274 | C23 | — | 4 | 1 | no | — |
| 275 | C23 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 276 | Q8 | — | 2 | 4 | no | — |
| 277 | Q8 | — | 4 | 2 | no | — |
| 278 | D4 | — | 2 | 4 | no | — |
| 279 | D4 | — | 4 | 2 | no | — |
| 280 | C2×C4 | — | 2 | 2 | no | — |
| 281 | C2×C4 | — | 2 | 2 | no | — |
| 282 | C2×C4 | — | 2 | 2 | no | — |
| 283 | C2×C4 | — | 2 | 2 | no | — |
| 284 | C2×C4 | — | 2 | 2 | no | — |
| 285 | C2×C4 | — | 2 | 2 | no | — |
| 286 | C2×C4 | — | 2 | 2 | no | — |
| 287 | C2×C4 | — | 4 | 1 | no | — |
| 288 | C2×C4 | — | 4 | 1 | no | — |
| 289 | C2×C4 | — | 4 | 1 | no | — |
| 290 | C2×C4 | — | 4 | 1 | no | — |
| 291 | C2×C4 | — | 4 | 2 | no | — |
| 292 | C2×C4 | — | 4 | 2 | no | — |
| 293 | C2×C4 | — | 8 | 4 | no | — |
| 294 | C2×C4 | — | 20 | 4 | no | — |
| 295 | C2×C4 | — | 20 | 4 | no | — |
| 296 | C2×C4 | — | 20 | 4 | no | — |
| 297 | C2×C4 | #4070 | 1 | 4 | yes | — |
| 298 | C2×C4 | #4082 | 1 | 2 | yes | — |
| 299 | C2×C4 | #4082 | 1 | 2 | yes | — |
| 300 | C2×C4 | #4096 | 1 | 4 | yes | — |
| 301 | C2×C4 | C22×Dic10 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | C2×#709 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | C2×#711 | 1 | 2 | yes | — |
| 304 | C2×C4 | #4149 | 1 | 2 | yes | — |
| 305 | C8 | — | 20 | 4 | no | — |
| 306 | C5 | #29140 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 307 | C22 | — | 4 | 2 | no | — |
| 308 | C22 | — | 4 | 4 | no | — |
| 309 | C22 | — | 8 | 2 | no | — |
| 310 | C22 | — | 8 | 1 | no | — |
| 311 | C22 | — | 8 | 1 | no | — |
| 312 | C22 | C2×#4070 | 1 | 1 | yes | characteristic |
| 313 | C22 | C2×#4082 | 1 | 1 | yes | characteristic |
| 314 | C22 | C2×#4096 | 1 | 2 | yes | — |
| 315 | C22 | #66156 | 1 | 2 | yes | — |
| 316 | C22 | #66164 | 1 | 1 | yes | characteristic |
| 317 | C4 | — | 2 | 4 | no | — |
| 318 | C4 | — | 2 | 2 | no | — |
| 319 | C4 | — | 4 | 2 | no | — |
| 320 | C4 | — | 4 | 2 | no | — |
| 321 | C4 | — | 8 | 1 | no | — |
| 322 | C4 | — | 8 | 1 | no | — |
| 323 | C4 | — | 40 | 4 | no | — |
| 324 | C4 | #65345 | 1 | 4 | yes | — |
| 325 | C4 | #65877 | 1 | 2 | yes | — |
| 326 | C4 | #65897 | 1 | 2 | yes | — |
| 327 | C2 | — | 8 | 2 | no | — |
| 328 | C2 | #10614115 | 1 | 1 | yes | characteristic |
| 329 | C2 | #10614238 | 1 | 2 | yes | — |
| 330 | C2 | C2×#65345 | 1 | 1 | yes | characteristic |
| 331 | C2 | #10614876 | 1 | 2 | yes | — |
| 332 | C2 | #10615015 | 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