← #13017284
Subgroups — #13017284
Order 1280.
342 conjugacy classes of subgroups.
The subgroup lattice has 286 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #10604695 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10604694 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10604689 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10604687 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #10599389 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #10598696 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #10598696 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #65748 | C22 | 1 | 1 | yes | characteristic |
| 9 | #65748 | C22 | 1 | 1 | yes | characteristic |
| 10 | #65748 | C22 | 1 | 1 | yes | characteristic |
| 11 | #65748 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C8×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C8×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 14 | C42×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 15 | #64884 | C4 | 1 | 2 | yes | complemented |
| 16 | #64883 | C4 | 1 | 2 | yes | complemented |
| 17 | #64882 | C4 | 1 | 2 | yes | complemented |
| 18 | #64881 | C4 | 1 | 2 | yes | complemented |
| 19 | #170⋊C8 | — | 2 | 1 | no | — |
| 20 | #170⋊C8 | — | 2 | 1 | no | — |
| 21 | #64879 | — | 2 | 1 | no | — |
| 22 | C8×#170 | — | 2 | 1 | no | — |
| 23 | C5×#336 | C4 | 1 | 1 | yes | characteristic |
| 24 | #64770 | C4 | 1 | 1 | yes | characteristic |
| 25 | #64770 | C4 | 1 | 1 | yes | characteristic |
| 26 | #64744 | C4 | 1 | 1 | yes | characteristic |
| 27 | #8270 | — | 5 | 1 | no | maximal, Sylow |
| 28 | C22×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C22×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C2×C42×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | C2×C4×Dic5 | — | 2 | 1 | no | — |
| 32 | C2×C4×Dic5 | — | 2 | 1 | no | — |
| 33 | C2×C4×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C2×C4×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 35 | C22×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C22×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C22⋊#170 | — | 2 | 1 | no | — |
| 38 | C22⋊#170 | — | 2 | 1 | no | — |
| 39 | C22⋊#170 | — | 2 | 1 | no | — |
| 40 | C22⋊#170 | — | 2 | 1 | no | — |
| 41 | C4⋊#170 | C2×C4 | 1 | 2 | yes | — |
| 42 | C4⋊#170 | C2×C4 | 1 | 2 | yes | — |
| 43 | C4⋊#170 | C2×C4 | 1 | 2 | yes | — |
| 44 | C4⋊#170 | C2×C4 | 1 | 2 | yes | — |
| 45 | C4×#170 | — | 2 | 1 | no | — |
| 46 | C4×#170 | — | 2 | 1 | no | — |
| 47 | C4×#170 | — | 2 | 1 | no | — |
| 48 | C4×#170 | — | 2 | 1 | no | — |
| 49 | C8×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | C8×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C8×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C8×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 54 | C8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 55 | C8×Dic5 | Q8 | 1 | 1 | yes | characteristic |
| 56 | C8×Dic5 | Q8 | 1 | 1 | yes | characteristic |
| 57 | #1744 | — | 5 | 1 | no | — |
| 58 | #1564 | — | 5 | 1 | no | — |
| 59 | #1564 | — | 5 | 1 | no | — |
| 60 | #1452 | — | 5 | 1 | no | — |
| 61 | #1450 | — | 5 | 1 | no | — |
| 62 | #1276 | — | 5 | 1 | no | — |
| 63 | #1276 | — | 5 | 1 | no | — |
| 64 | C22×C4×C5 | — | 2 | 1 | no | — |
| 65 | C22×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 66 | C22×Dic5 | — | 2 | 1 | no | — |
| 67 | C22×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 68 | C22×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 69 | C2×#170 | — | 2 | 2 | no | — |
| 70 | C2×#170 | — | 2 | 2 | no | — |
| 71 | C2×#170 | — | 2 | 2 | no | — |
| 72 | C2×#170 | — | 2 | 2 | no | — |
| 73 | C2×C5×C8 | — | 2 | 1 | no | — |
| 74 | C2×C5×C8 | — | 2 | 1 | no | — |
| 75 | C2×C5×C8 | — | 2 | 1 | no | — |
| 76 | C2×C5×C8 | — | 2 | 1 | no | — |
| 77 | C2×C5×C8 | C42 | 1 | 1 | yes | characteristic |
| 78 | C2×C5×C8 | C42 | 1 | 1 | yes | characteristic |
| 79 | C2×C5×C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 80 | C2×C5×C8 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 81 | C42×C5 | — | 2 | 1 | no | — |
| 82 | C42×C5 | — | 2 | 1 | no | — |
| 83 | C4×Dic5 | — | 2 | 4 | no | — |
| 84 | C4×Dic5 | — | 2 | 1 | no | — |
| 85 | C4×Dic5 | — | 2 | 1 | no | — |
| 86 | C4×Dic5 | — | 4 | 2 | no | — |
| 87 | C4×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C4×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C4×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C4×Dic5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 91 | C2×#168 | — | 2 | 1 | no | — |
| 92 | C2×#168 | — | 2 | 1 | no | — |
| 93 | C2×#168 | — | 2 | 1 | no | — |
| 94 | C2×#168 | — | 2 | 1 | no | — |
| 95 | C2×#168 | C42 | 1 | 1 | yes | characteristic |
| 96 | C2×#168 | C42 | 1 | 1 | yes | characteristic |
| 97 | C2×#168 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 98 | C2×#168 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 99 | #431 | — | 5 | 1 | no | — |
| 100 | #431 | — | 5 | 1 | no | — |
| 101 | #431 | — | 5 | 1 | no | — |
| 102 | #431 | — | 5 | 1 | no | — |
| 103 | C2×C4×C8 | — | 5 | 1 | no | — |
| 104 | C2×C4×C8 | — | 5 | 1 | no | — |
| 105 | C43 | — | 5 | 1 | no | — |
| 106 | #336 | — | 5 | 1 | no | — |
| 107 | #336 | — | 5 | 1 | no | — |
| 108 | #336 | — | 5 | 1 | no | — |
| 109 | #336 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 110 | #335 | — | 5 | 2 | no | — |
| 111 | #335 | — | 5 | 2 | no | — |
| 112 | #334 | — | 5 | 2 | no | — |
| 113 | #334 | — | 5 | 2 | no | — |
| 114 | #322 | — | 10 | 1 | no | — |
| 115 | #322 | — | 10 | 1 | no | — |
| 116 | #322 | — | 10 | 1 | no | — |
| 117 | C82 | — | 10 | 1 | no | — |
| 118 | C23×C5 | #117 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C5 | — | 2 | 2 | no | — |
| 120 | C2×C4×C5 | — | 4 | 1 | no | — |
| 121 | C2×C4×C5 | — | 4 | 1 | no | — |
| 122 | C2×C4×C5 | C2×C42 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C5 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C5 | #131 | 1 | 1 | yes | characteristic |
| 125 | C2×C4×C5 | #131 | 1 | 1 | yes | characteristic |
| 126 | C2×C4×C5 | #131 | 1 | 1 | yes | characteristic |
| 127 | C2×C4×C5 | #131 | 1 | 1 | yes | characteristic |
| 128 | C2×Dic5 | — | 2 | 2 | no | — |
| 129 | C2×Dic5 | — | 2 | 2 | no | — |
| 130 | C2×Dic5 | — | 2 | 2 | no | — |
| 131 | C2×Dic5 | — | 4 | 1 | no | — |
| 132 | C2×Dic5 | — | 4 | 1 | no | — |
| 133 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 134 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 135 | C2×Dic5 | C4×Q8 | 1 | 1 | yes | characteristic |
| 136 | C2×Dic5 | C4×Q8 | 1 | 1 | yes | characteristic |
| 137 | #170 | — | 4 | 2 | no | — |
| 138 | #170 | — | 4 | 2 | no | — |
| 139 | #170 | — | 4 | 2 | no | — |
| 140 | #170 | — | 4 | 2 | no | — |
| 141 | C5×C8 | — | 2 | 1 | no | — |
| 142 | C5×C8 | — | 2 | 1 | no | — |
| 143 | C5×C8 | — | 2 | 1 | no | — |
| 144 | C5×C8 | — | 2 | 1 | no | — |
| 145 | #168 | — | 2 | 1 | no | — |
| 146 | #168 | — | 2 | 1 | no | — |
| 147 | #168 | — | 2 | 1 | no | — |
| 148 | #168 | — | 2 | 1 | no | — |
| 149 | C22×C8 | — | 5 | 1 | no | — |
| 150 | C22×C8 | — | 5 | 1 | no | — |
| 151 | C22×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 152 | C22×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 153 | C2×C42 | — | 5 | 1 | no | — |
| 154 | C2×C42 | — | 5 | 1 | no | — |
| 155 | C2×C42 | — | 10 | 1 | no | — |
| 156 | C2×C42 | — | 10 | 1 | no | — |
| 157 | C2×C42 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 158 | C4⋊C8 | — | 5 | 2 | no | — |
| 159 | C4⋊C8 | — | 5 | 2 | no | — |
| 160 | C4⋊C8 | — | 5 | 2 | no | — |
| 161 | C4⋊C8 | — | 5 | 2 | no | — |
| 162 | C22⋊C8 | — | 10 | 1 | no | — |
| 163 | C22⋊C8 | — | 10 | 1 | no | — |
| 164 | C22⋊C8 | — | 10 | 1 | no | — |
| 165 | C22⋊C8 | — | 10 | 1 | no | — |
| 166 | C4×C8 | — | 5 | 1 | no | — |
| 167 | C4×C8 | — | 5 | 1 | no | — |
| 168 | C4×C8 | — | 5 | 1 | no | — |
| 169 | C4×C8 | — | 5 | 1 | no | — |
| 170 | C4×C8 | — | 5 | 1 | no | — |
| 171 | C4×C8 | — | 5 | 1 | no | — |
| 172 | C4×C8 | — | 5 | 1 | no | — |
| 173 | C4×C8 | — | 5 | 1 | no | — |
| 174 | C4×C8 | — | 10 | 1 | no | — |
| 175 | C4×C8 | — | 10 | 1 | no | — |
| 176 | C4×C8 | — | 10 | 1 | no | — |
| 177 | C4×C8 | — | 10 | 1 | no | — |
| 178 | C22×C5 | — | 2 | 1 | no | — |
| 179 | C22×C5 | — | 2 | 1 | no | — |
| 180 | C22×C5 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 181 | C22×C5 | #433 | 1 | 1 | yes | characteristic |
| 182 | C22×C5 | #433 | 1 | 1 | yes | characteristic |
| 183 | C4×C5 | — | 4 | 2 | no | — |
| 184 | C4×C5 | #405 | 1 | 1 | yes | characteristic |
| 185 | C4×C5 | #405 | 1 | 1 | yes | characteristic |
| 186 | C4×C5 | #424 | 1 | 1 | yes | characteristic |
| 187 | C4×C5 | #424 | 1 | 1 | yes | characteristic |
| 188 | Dic5 | — | 2 | 1 | no | — |
| 189 | Dic5 | — | 2 | 1 | no | — |
| 190 | Dic5 | — | 4 | 2 | no | — |
| 191 | Dic5 | #443 | 1 | 2 | yes | — |
| 192 | Dic5 | #444 | 1 | 2 | yes | — |
| 193 | C22×C4 | — | 2 | 1 | no | — |
| 194 | C22×C4 | — | 5 | 1 | no | — |
| 195 | C22×C4 | — | 5 | 1 | no | — |
| 196 | C22×C4 | — | 10 | 1 | no | — |
| 197 | C22×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 198 | C2×C8 | — | 2 | 1 | no | — |
| 199 | C2×C8 | — | 2 | 1 | no | — |
| 200 | C2×C8 | — | 2 | 1 | no | — |
| 201 | C2×C8 | — | 2 | 1 | no | — |
| 202 | C2×C8 | — | 5 | 1 | no | — |
| 203 | C2×C8 | — | 5 | 1 | no | — |
| 204 | C2×C8 | — | 5 | 1 | no | — |
| 205 | C2×C8 | — | 5 | 1 | no | — |
| 206 | C2×C8 | — | 10 | 1 | no | — |
| 207 | C2×C8 | — | 10 | 1 | no | — |
| 208 | C2×C8 | — | 10 | 1 | no | — |
| 209 | C2×C8 | — | 10 | 1 | no | — |
| 210 | C2×C8 | — | 10 | 2 | no | — |
| 211 | C2×C8 | — | 10 | 2 | no | — |
| 212 | C2×C8 | — | 10 | 2 | no | — |
| 213 | C2×C8 | — | 10 | 2 | no | — |
| 214 | C2×C8 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 215 | C2×C8 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 216 | C2×C8 | C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 217 | C2×C8 | C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 218 | C42 | — | 2 | 1 | no | — |
| 219 | C42 | — | 2 | 1 | no | — |
| 220 | C42 | — | 5 | 1 | no | — |
| 221 | C42 | — | 5 | 1 | no | — |
| 222 | C42 | — | 5 | 1 | no | — |
| 223 | C42 | — | 5 | 1 | no | — |
| 224 | C42 | — | 10 | 4 | no | — |
| 225 | C42 | — | 10 | 1 | no | — |
| 226 | C42 | — | 10 | 1 | no | — |
| 227 | C42 | — | 20 | 2 | no | — |
| 228 | C2×C5 | — | 2 | 1 | no | — |
| 229 | C2×C5 | — | 2 | 1 | no | — |
| 230 | C2×C5 | #1768 | 1 | 1 | yes | characteristic |
| 231 | C2×C5 | #1772 | 1 | 1 | yes | characteristic |
| 232 | C2×C5 | #1773 | 1 | 1 | yes | characteristic |
| 233 | C23 | #4130 | 1 | 1 | yes | characteristic |
| 234 | C2×C4 | — | 2 | 2 | no | — |
| 235 | C2×C4 | — | 4 | 1 | no | — |
| 236 | C2×C4 | — | 4 | 1 | no | — |
| 237 | C2×C4 | — | 5 | 1 | no | — |
| 238 | C2×C4 | — | 5 | 1 | no | — |
| 239 | C2×C4 | — | 5 | 1 | no | — |
| 240 | C2×C4 | — | 5 | 1 | no | — |
| 241 | C2×C4 | — | 10 | 2 | no | — |
| 242 | C2×C4 | — | 10 | 2 | no | — |
| 243 | C2×C4 | — | 10 | 2 | no | — |
| 244 | C2×C4 | — | 20 | 1 | no | — |
| 245 | C2×C4 | — | 20 | 1 | no | — |
| 246 | C2×C4 | C2×C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 247 | C2×C4 | C2×C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 248 | C2×C4 | #4131 | 1 | 1 | yes | characteristic |
| 249 | C2×C4 | #4131 | 1 | 1 | yes | characteristic |
| 250 | C2×C4 | #4131 | 1 | 1 | yes | characteristic |
| 251 | C2×C4 | #4133 | 1 | 1 | yes | characteristic |
| 252 | C8 | — | 2 | 1 | no | — |
| 253 | C8 | — | 2 | 1 | no | — |
| 254 | C8 | — | 2 | 1 | no | — |
| 255 | C8 | — | 2 | 1 | no | — |
| 256 | C8 | — | 10 | 1 | no | — |
| 257 | C8 | — | 10 | 1 | no | — |
| 258 | C8 | — | 10 | 1 | no | — |
| 259 | C8 | — | 10 | 1 | no | — |
| 260 | C8 | — | 20 | 2 | no | — |
| 261 | C8 | — | 20 | 2 | no | — |
| 262 | C8 | — | 20 | 2 | no | — |
| 263 | C8 | — | 20 | 2 | no | — |
| 264 | C5 | #8270 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 265 | C22 | — | 2 | 1 | no | — |
| 266 | C22 | — | 2 | 1 | no | — |
| 267 | C22 | C4×C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 268 | C22 | #65697 | 1 | 1 | yes | characteristic |
| 269 | C22 | #65706 | 1 | 1 | yes | characteristic |
| 270 | C4 | — | 4 | 2 | no | — |
| 271 | C4 | — | 5 | 2 | no | — |
| 272 | C4 | — | 5 | 2 | no | — |
| 273 | C4 | — | 10 | 1 | no | — |
| 274 | C4 | — | 10 | 1 | no | — |
| 275 | C4 | — | 20 | 2 | no | — |
| 276 | C4 | #65691 | 1 | 1 | yes | characteristic |
| 277 | C4 | #65696 | 1 | 1 | yes | characteristic |
| 278 | C4 | #65701 | 1 | 1 | yes | characteristic |
| 279 | C4 | #65708 | 1 | 1 | yes | characteristic |
| 280 | C2 | — | 2 | 1 | no | — |
| 281 | C2 | — | 2 | 1 | no | — |
| 282 | C2 | #10603778 | 1 | 1 | yes | characteristic |
| 283 | C2 | #10603851 | 1 | 1 | yes | characteristic |
| 284 | C2 | #10603885 | 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