← #12252431
Subgroups — #12252431
Order 1280.
520 conjugacy classes of subgroups.
The subgroup lattice has 302 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #10610217 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C5×#347 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10602228 | C2 | 1 | 2 | yes | maximal |
| 4 | #10602216 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C2×#64686 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10599882 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #10599881 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | C2×#4047 | C4 | 1 | 1 | yes | characteristic |
| 9 | C2×#4047 | C22 | 1 | 1 | yes | characteristic |
| 10 | C4×#709 | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×C5×M32 | C22 | 1 | 2 | yes | — |
| 12 | C2×C5×#97 | C22 | 1 | 1 | yes | characteristic |
| 13 | #65390 | C4 | 1 | 1 | yes | characteristic |
| 14 | #65390 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#3944 | C22 | 1 | 2 | yes | — |
| 16 | C4×#681 | C4 | 1 | 1 | yes | characteristic |
| 17 | #64997 | C4 | 1 | 2 | yes | — |
| 18 | #64997 | C22 | 1 | 2 | yes | — |
| 19 | #64996 | C22 | 1 | 2 | yes | — |
| 20 | D5×#97 | C4 | 1 | 2 | yes | — |
| 21 | C5×#347 | C22 | 1 | 2 | yes | — |
| 22 | C5×#347 | C22 | 1 | 2 | yes | — |
| 23 | #64734 | C22 | 1 | 8 | yes | — |
| 24 | #64731 | C22 | 1 | 8 | yes | — |
| 25 | #64686 | C22 | 1 | 2 | yes | — |
| 26 | #64686 | C22 | 1 | 2 | yes | — |
| 27 | #33426 | — | 5 | 1 | no | maximal, Sylow |
| 28 | C2×#709 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C22×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C22×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 31 | C2×C42×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C4×C22⋊D5 | — | 2 | 2 | no | — |
| 33 | #4072 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C2×#681 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C2×#681 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | #4047 | — | 2 | 2 | no | — |
| 37 | #4047 | — | 2 | 2 | no | — |
| 38 | #4047 | — | 2 | 2 | no | — |
| 39 | #4047 | — | 2 | 2 | no | — |
| 40 | C2×#676 | C2×C4 | 1 | 2 | yes | — |
| 41 | C2×#676 | C23 | 1 | 2 | yes | — |
| 42 | C2×C8×D5 | C2×C4 | 1 | 2 | yes | — |
| 43 | C2×C8×D5 | C2×C4 | 1 | 2 | yes | — |
| 44 | C4×D20 | — | 2 | 1 | no | — |
| 45 | #4018 | C2×C4 | 1 | 2 | yes | — |
| 46 | C42×D5 | C2×C4 | 1 | 2 | yes | — |
| 47 | C4×Dic10 | — | 2 | 1 | no | — |
| 48 | C5×M32 | — | 2 | 2 | no | — |
| 49 | C5×M32 | C23 | 1 | 4 | yes | — |
| 50 | C2×C5×C16 | — | 2 | 2 | no | — |
| 51 | C5×#97 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C5×#97 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C5×#97 | C23 | 1 | 1 | yes | characteristic |
| 54 | C5×#97 | C23 | 1 | 1 | yes | characteristic |
| 55 | #3947 | C2×C4 | 1 | 2 | yes | — |
| 56 | #3947 | C2×C4 | 1 | 2 | yes | — |
| 57 | C8×Dic5 | C2×C4 | 1 | 2 | yes | — |
| 58 | C8×Dic5 | C23 | 1 | 2 | yes | — |
| 59 | #3944 | — | 2 | 2 | no | — |
| 60 | #3944 | C23 | 1 | 4 | yes | — |
| 61 | C2×#672 | — | 2 | 2 | no | — |
| 62 | #3935 | C2×C4 | 1 | 2 | yes | — |
| 63 | C4×#168 | C2×C4 | 1 | 2 | yes | — |
| 64 | #2875 | — | 5 | 1 | no | — |
| 65 | #2166 | — | 5 | 4 | no | — |
| 66 | #2165 | — | 5 | 4 | no | — |
| 67 | #2153 | — | 5 | 2 | no | — |
| 68 | #2115 | — | 5 | 2 | no | — |
| 69 | C2×#347 | — | 5 | 1 | no | — |
| 70 | C2×#347 | D5 | 1 | 1 | yes | characteristic, complemented |
| 71 | C22×C4×C5 | — | 2 | 1 | no | — |
| 72 | C22×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 73 | C2×C22⋊D5 | M16 | 1 | 2 | yes | — |
| 74 | #709 | — | 2 | 1 | no | — |
| 75 | #709 | — | 2 | 1 | no | — |
| 76 | #709 | — | 2 | 1 | no | — |
| 77 | #709 | — | 2 | 1 | no | — |
| 78 | C2×D20 | M16 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×D5 | — | 2 | 2 | no | — |
| 80 | C2×C4×D5 | C22×C4 | 1 | 2 | yes | — |
| 81 | C2×Dic10 | M16 | 1 | 1 | yes | characteristic |
| 82 | C2×C5×C8 | — | 2 | 1 | no | — |
| 83 | C2×C5×C8 | — | 2 | 1 | no | — |
| 84 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 85 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 86 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 87 | C2×C5×C8 | #41 | 1 | 2 | yes | — |
| 88 | C2×C5×C8 | #41 | 1 | 2 | yes | — |
| 89 | C2×C5×C8 | C24 | 1 | 1 | yes | characteristic |
| 90 | C42×C5 | — | 2 | 1 | no | — |
| 91 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 92 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 93 | C22⋊Dic5 | — | 2 | 1 | no | — |
| 94 | #685 | — | 2 | 2 | no | — |
| 95 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 96 | #683 | — | 2 | 2 | no | — |
| 97 | C4×Dic5 | — | 2 | 2 | no | — |
| 98 | C4×Dic5 | C22×C4 | 1 | 2 | yes | — |
| 99 | #681 | — | 2 | 1 | no | — |
| 100 | #681 | — | 2 | 1 | no | — |
| 101 | #681 | M16 | 1 | 2 | yes | — |
| 102 | #681 | M16 | 1 | 2 | yes | — |
| 103 | C2×#168 | C22×C4 | 1 | 2 | yes | — |
| 104 | C2×#168 | C22×C4 | 1 | 2 | yes | — |
| 105 | #676 | — | 2 | 4 | no | — |
| 106 | #676 | — | 2 | 4 | no | — |
| 107 | C8×D5 | — | 2 | 4 | no | — |
| 108 | C8×D5 | — | 2 | 4 | no | — |
| 109 | C5×C16 | — | 2 | 4 | no | — |
| 110 | #672 | — | 2 | 4 | no | — |
| 111 | C2×#131 | — | 5 | 1 | no | — |
| 112 | C2×#131 | — | 5 | 1 | no | — |
| 113 | C4×#41 | — | 5 | 1 | no | — |
| 114 | C2×M32 | — | 5 | 2 | no | — |
| 115 | C2×M32 | C2×D5 | 1 | 2 | yes | — |
| 116 | #405 | — | 5 | 1 | no | — |
| 117 | #405 | — | 5 | 1 | no | — |
| 118 | #405 | — | 5 | 2 | no | — |
| 119 | #405 | — | 5 | 2 | no | — |
| 120 | C4×M16 | — | 5 | 1 | no | — |
| 121 | C4×M16 | — | 5 | 2 | no | — |
| 122 | C2×#97 | — | 5 | 2 | no | — |
| 123 | C2×#97 | C2×D5 | 1 | 1 | yes | characteristic |
| 124 | #364 | — | 5 | 8 | no | — |
| 125 | #349 | — | 5 | 8 | no | — |
| 126 | #347 | — | 5 | 2 | no | — |
| 127 | #347 | — | 5 | 2 | no | — |
| 128 | #347 | C2×D5 | 1 | 2 | yes | — |
| 129 | #347 | C2×D5 | 1 | 2 | yes | — |
| 130 | C23×C5 | C2×M16 | 1 | 1 | yes | characteristic |
| 131 | C22×D5 | C2×M16 | 1 | 2 | yes | — |
| 132 | C2×C4×C5 | — | 2 | 1 | no | — |
| 133 | C2×C4×C5 | — | 2 | 1 | no | — |
| 134 | C2×C4×C5 | — | 8 | 1 | no | — |
| 135 | C2×C4×C5 | #117 | 1 | 1 | yes | characteristic |
| 136 | C2×C4×C5 | #117 | 1 | 1 | yes | characteristic |
| 137 | C2×C4×C5 | C2×M16 | 1 | 1 | yes | characteristic |
| 138 | C2×C4×C5 | C2×M16 | 1 | 1 | yes | characteristic |
| 139 | C2×C4×C5 | C2×M16 | 1 | 1 | yes | characteristic |
| 140 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 141 | C22⋊D5 | — | 2 | 2 | no | — |
| 142 | C22⋊D5 | — | 2 | 2 | no | — |
| 143 | C2×Dic5 | — | 2 | 2 | no | — |
| 144 | C2×Dic5 | C2×M16 | 1 | 2 | yes | — |
| 145 | D20 | — | 2 | 1 | no | — |
| 146 | D20 | — | 2 | 1 | no | — |
| 147 | C4×D5 | — | 2 | 2 | no | — |
| 148 | C4×D5 | — | 2 | 2 | no | — |
| 149 | C4×D5 | — | 8 | 2 | no | — |
| 150 | Dic10 | — | 2 | 1 | no | — |
| 151 | Dic10 | — | 2 | 1 | no | — |
| 152 | C5×C8 | — | 2 | 1 | no | — |
| 153 | C5×C8 | — | 2 | 1 | no | — |
| 154 | C5×C8 | #117 | 1 | 2 | yes | — |
| 155 | C5×C8 | C2×#41 | 1 | 2 | yes | — |
| 156 | #168 | — | 2 | 2 | no | — |
| 157 | #168 | C2×M16 | 1 | 4 | yes | — |
| 158 | C2×#41 | — | 5 | 1 | no | — |
| 159 | #131 | — | 10 | 2 | no | — |
| 160 | #131 | — | 10 | 2 | no | — |
| 161 | #131 | — | 10 | 2 | no | — |
| 162 | #131 | — | 10 | 2 | no | — |
| 163 | C2×M16 | — | 5 | 1 | no | — |
| 164 | C2×M16 | — | 5 | 1 | no | — |
| 165 | C2×M16 | — | 5 | 2 | no | — |
| 166 | C2×M16 | — | 5 | 2 | no | — |
| 167 | C22×C8 | — | 5 | 2 | no | — |
| 168 | C22×C8 | — | 5 | 2 | no | — |
| 169 | C22×C8 | C4×D5 | 1 | 1 | yes | characteristic |
| 170 | C22×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 171 | C4×Q8 | — | 10 | 1 | no | — |
| 172 | C4×D4 | — | 10 | 1 | no | — |
| 173 | C4×D4 | — | 10 | 2 | no | — |
| 174 | #117 | — | 5 | 1 | no | — |
| 175 | #117 | — | 5 | 2 | no | — |
| 176 | C2×C42 | — | 5 | 2 | no | — |
| 177 | C2×C42 | C4×D5 | 1 | 1 | yes | characteristic |
| 178 | M32 | — | 2 | 2 | no | — |
| 179 | M32 | — | 5 | 4 | no | — |
| 180 | M32 | — | 10 | 2 | no | — |
| 181 | M32 | C22×D5 | 1 | 4 | yes | — |
| 182 | C2×C16 | — | 2 | 2 | no | — |
| 183 | C2×C16 | — | 10 | 2 | no | — |
| 184 | #97 | — | 5 | 2 | no | — |
| 185 | #97 | — | 5 | 2 | no | — |
| 186 | #97 | — | 5 | 2 | no | — |
| 187 | #97 | C4×D5 | 1 | 1 | yes | characteristic |
| 188 | #97 | C4×D5 | 1 | 1 | yes | characteristic |
| 189 | #97 | C22×D5 | 1 | 1 | yes | characteristic |
| 190 | #97 | C22×D5 | 1 | 1 | yes | characteristic |
| 191 | C4×C8 | — | 5 | 2 | no | — |
| 192 | C4×C8 | — | 5 | 2 | no | — |
| 193 | C4×C8 | — | 5 | 2 | no | — |
| 194 | C22×C5 | — | 4 | 1 | no | — |
| 195 | C22×C5 | #432 | 1 | 1 | yes | characteristic |
| 196 | C22×C5 | #432 | 1 | 1 | yes | characteristic |
| 197 | C22×C5 | C22×M16 | 1 | 1 | yes | characteristic |
| 198 | C2×D5 | — | 2 | 2 | no | — |
| 199 | C2×D5 | — | 4 | 2 | no | — |
| 200 | C4×C5 | — | 4 | 1 | no | — |
| 201 | C4×C5 | — | 4 | 1 | no | — |
| 202 | C4×C5 | #432 | 1 | 1 | yes | characteristic |
| 203 | C4×C5 | #432 | 1 | 1 | yes | characteristic |
| 204 | C4×C5 | C2×#117 | 1 | 1 | yes | characteristic |
| 205 | C4×C5 | C22×M16 | 1 | 1 | yes | characteristic |
| 206 | Dic5 | — | 2 | 2 | no | — |
| 207 | Dic5 | — | 4 | 2 | no | — |
| 208 | #41 | — | 10 | 1 | no | — |
| 209 | #41 | — | 10 | 1 | no | — |
| 210 | #41 | — | 10 | 1 | no | — |
| 211 | #41 | — | 10 | 1 | no | — |
| 212 | C2×Q8 | — | 5 | 1 | no | — |
| 213 | C2×D4 | — | 5 | 2 | no | — |
| 214 | C2×D4 | — | 5 | 1 | no | — |
| 215 | C22×C4 | — | 2 | 1 | no | — |
| 216 | C22×C4 | — | 5 | 2 | no | — |
| 217 | C22×C4 | — | 10 | 2 | no | — |
| 218 | C22×C4 | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 219 | M16 | — | 5 | 2 | no | — |
| 220 | M16 | — | 5 | 2 | no | — |
| 221 | M16 | — | 10 | 4 | no | — |
| 222 | M16 | — | 10 | 1 | no | — |
| 223 | M16 | — | 10 | 1 | no | — |
| 224 | M16 | — | 10 | 4 | no | — |
| 225 | C2×C8 | — | 2 | 1 | no | — |
| 226 | C2×C8 | — | 2 | 1 | no | — |
| 227 | C2×C8 | — | 5 | 2 | no | — |
| 228 | C2×C8 | — | 5 | 2 | no | — |
| 229 | C2×C8 | — | 10 | 4 | no | — |
| 230 | C2×C8 | — | 10 | 4 | no | — |
| 231 | C2×C8 | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 232 | C2×C8 | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 233 | C2×C8 | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 234 | C2×C8 | #709 | 1 | 2 | yes | — |
| 235 | C2×C8 | #709 | 1 | 2 | yes | — |
| 236 | C2×C8 | C23×D5 | 1 | 1 | yes | characteristic |
| 237 | C4⋊C4 | — | 10 | 2 | no | — |
| 238 | C4⋊C4 | — | 10 | 1 | no | — |
| 239 | C22⋊C4 | — | 10 | 2 | no | — |
| 240 | C22⋊C4 | — | 10 | 1 | no | — |
| 241 | C42 | — | 2 | 1 | no | — |
| 242 | C42 | — | 5 | 2 | no | — |
| 243 | C42 | — | 10 | 2 | no | — |
| 244 | C42 | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 245 | C42 | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 246 | C16 | — | 2 | 4 | no | — |
| 247 | C16 | — | 10 | 4 | no | — |
| 248 | C2×C5 | — | 2 | 1 | no | — |
| 249 | C2×C5 | — | 2 | 1 | no | — |
| 250 | C2×C5 | — | 2 | 1 | no | — |
| 251 | C2×C5 | C2×#432 | 1 | 1 | yes | characteristic |
| 252 | D5 | — | 4 | 2 | no | — |
| 253 | C23 | — | 5 | 2 | no | — |
| 254 | C23 | D5×M16 | 1 | 1 | yes | characteristic |
| 255 | Q8 | — | 10 | 1 | no | — |
| 256 | Q8 | — | 10 | 1 | no | — |
| 257 | D4 | — | 10 | 2 | no | — |
| 258 | D4 | — | 10 | 2 | no | — |
| 259 | D4 | — | 10 | 1 | no | — |
| 260 | D4 | — | 10 | 1 | no | — |
| 261 | C2×C4 | — | 2 | 1 | no | — |
| 262 | C2×C4 | — | 2 | 1 | no | — |
| 263 | C2×C4 | — | 5 | 2 | no | — |
| 264 | C2×C4 | — | 8 | 1 | no | — |
| 265 | C2×C4 | — | 10 | 2 | no | — |
| 266 | C2×C4 | — | 10 | 2 | no | — |
| 267 | C2×C4 | — | 10 | 2 | no | — |
| 268 | C2×C4 | — | 40 | 2 | no | — |
| 269 | C2×C4 | #4018 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | #4018 | 1 | 1 | yes | characteristic |
| 271 | C2×C4 | D5×M16 | 1 | 1 | yes | characteristic |
| 272 | C2×C4 | D5×M16 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | D5×M16 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | C22×C4×D5 | 1 | 1 | yes | characteristic |
| 275 | C8 | — | 2 | 1 | no | — |
| 276 | C8 | — | 2 | 1 | no | — |
| 277 | C8 | — | 5 | 4 | no | — |
| 278 | C8 | — | 10 | 2 | no | — |
| 279 | C8 | #4018 | 1 | 2 | yes | — |
| 280 | C8 | C2×#709 | 1 | 2 | yes | — |
| 281 | C5 | #33426 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 282 | C22 | — | 4 | 1 | no | — |
| 283 | C22 | — | 10 | 2 | no | — |
| 284 | C22 | — | 20 | 2 | no | — |
| 285 | C22 | #64994 | 1 | 1 | yes | characteristic |
| 286 | C22 | #64994 | 1 | 1 | yes | characteristic |
| 287 | C22 | C2×D5×M16 | 1 | 1 | yes | characteristic |
| 288 | C4 | — | 4 | 1 | no | — |
| 289 | C4 | — | 4 | 1 | no | — |
| 290 | C4 | — | 10 | 2 | no | — |
| 291 | C4 | — | 20 | 2 | no | — |
| 292 | C4 | #64994 | 1 | 1 | yes | characteristic |
| 293 | C4 | #64994 | 1 | 1 | yes | characteristic |
| 294 | C4 | C2×#4018 | 1 | 1 | yes | characteristic |
| 295 | C4 | C2×D5×M16 | 1 | 1 | yes | characteristic |
| 296 | C2 | — | 2 | 1 | no | — |
| 297 | C2 | — | 2 | 1 | no | — |
| 298 | C2 | — | 2 | 1 | no | — |
| 299 | C2 | — | 20 | 2 | no | — |
| 300 | C2 | C2×#64994 | 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