← #13016903
Subgroups — #13016903
Order 1280.
420 conjugacy classes of subgroups.
The subgroup lattice has 336 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #10610379 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10605000 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10605000 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10604687 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#170⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10598787 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10598787 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #41×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 9 | #65748 | C22 | 1 | 1 | yes | characteristic |
| 10 | #65748 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C4×#170 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C4×#170 | C22 | 1 | 1 | yes | characteristic |
| 13 | C5×#408 | C4 | 1 | 1 | yes | characteristic |
| 14 | #65523 | C4 | 1 | 1 | yes | characteristic |
| 15 | #65408 | C4 | 1 | 1 | yes | characteristic |
| 16 | M16×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 17 | #65391 | C4 | 1 | 1 | yes | characteristic |
| 18 | C2×C8×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 19 | #65009 | — | 2 | 1 | no | — |
| 20 | #65009 | — | 2 | 1 | no | — |
| 21 | #65007 | — | 2 | 1 | no | — |
| 22 | #65007 | — | 2 | 1 | no | — |
| 23 | Q8⋊#170 | C4 | 1 | 2 | yes | — |
| 24 | Q8⋊#170 | C4 | 1 | 2 | yes | — |
| 25 | D4⋊#170 | C4 | 1 | 2 | yes | — |
| 26 | D4⋊#170 | C4 | 1 | 2 | yes | — |
| 27 | #170⋊C8 | — | 2 | 1 | no | — |
| 28 | #170⋊C8 | — | 2 | 1 | no | — |
| 29 | #64879 | — | 2 | 1 | no | — |
| 30 | C8×#170 | — | 2 | 1 | no | — |
| 31 | #8189 | — | 5 | 1 | no | maximal, Sylow |
| 32 | C2×C5×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C22×#170 | — | 2 | 1 | no | — |
| 34 | C22×#170 | — | 2 | 1 | no | — |
| 35 | C2×C5×M16 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C22×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | Q8×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | Q8×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | D4×Dic5 | — | 2 | 1 | no | — |
| 40 | D4×Dic5 | — | 2 | 1 | no | — |
| 41 | D4×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | D4×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | #4072 | — | 2 | 1 | no | — |
| 44 | #4072 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | C2×C4×Dic5 | — | 2 | 1 | no | — |
| 46 | C2×C4×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 47 | C2×#681 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | C22×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C22⋊#170 | — | 2 | 1 | no | — |
| 50 | C22⋊#170 | — | 2 | 1 | no | — |
| 51 | C4⋊#170 | C2×C4 | 1 | 2 | yes | — |
| 52 | C4⋊#170 | C2×C4 | 1 | 2 | yes | — |
| 53 | C4×#170 | — | 2 | 1 | no | — |
| 54 | C4×#170 | — | 2 | 1 | no | — |
| 55 | C4×#170 | — | 2 | 1 | no | — |
| 56 | C4×#170 | — | 2 | 1 | no | — |
| 57 | C4×#170 | C2×C4 | 1 | 2 | yes | — |
| 58 | C4×#170 | C2×C4 | 1 | 2 | yes | — |
| 59 | C5×C22⋊C8 | — | 2 | 1 | no | — |
| 60 | C5×C22⋊C8 | — | 2 | 1 | no | — |
| 61 | C22⋊#168 | — | 2 | 1 | no | — |
| 62 | C22⋊#168 | — | 2 | 1 | no | — |
| 63 | #3947 | — | 2 | 1 | no | — |
| 64 | #3946 | — | 2 | 1 | no | — |
| 65 | #3946 | — | 2 | 1 | no | — |
| 66 | #3946 | — | 2 | 1 | no | — |
| 67 | #3946 | — | 2 | 1 | no | — |
| 68 | C8×Dic5 | — | 2 | 1 | no | — |
| 69 | C8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 70 | C8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 71 | C8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 72 | C8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 73 | #2922 | — | 5 | 1 | no | — |
| 74 | #1476 | — | 5 | 1 | no | — |
| 75 | #1476 | — | 5 | 1 | no | — |
| 76 | #1450 | — | 5 | 1 | no | — |
| 77 | C2×#322 | — | 5 | 1 | no | — |
| 78 | #1278 | — | 5 | 1 | no | — |
| 79 | #1278 | — | 5 | 1 | no | — |
| 80 | C5×#41 | — | 2 | 2 | no | — |
| 81 | C5×#41 | — | 2 | 2 | no | — |
| 82 | C2×C5×Q8 | C42 | 1 | 1 | yes | characteristic |
| 83 | C2×C5×D4 | — | 2 | 1 | no | — |
| 84 | C2×C5×D4 | C42 | 1 | 1 | yes | characteristic |
| 85 | C22×C4×C5 | — | 2 | 1 | no | — |
| 86 | C22×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 87 | C22×Dic5 | — | 2 | 1 | no | — |
| 88 | C22×Dic5 | — | 2 | 1 | no | — |
| 89 | C22×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 90 | C22×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 91 | C2×#170 | — | 2 | 4 | no | — |
| 92 | C2×#170 | — | 2 | 2 | no | — |
| 93 | C2×#170 | — | 2 | 2 | no | — |
| 94 | C2×#170 | — | 2 | 2 | no | — |
| 95 | C2×#170 | — | 2 | 2 | no | — |
| 96 | C2×#170 | — | 4 | 2 | no | — |
| 97 | C5×M16 | — | 2 | 2 | no | — |
| 98 | C2×C5×C8 | — | 2 | 2 | no | — |
| 99 | C2×C5×C8 | — | 2 | 1 | no | — |
| 100 | C2×C5×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 101 | C2×C5×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 102 | C22⋊Dic5 | — | 2 | 1 | no | — |
| 103 | C22⋊Dic5 | — | 2 | 1 | no | — |
| 104 | C22⋊Dic5 | — | 2 | 1 | no | — |
| 105 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 106 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 107 | C4⋊Dic5 | C42 | 1 | 1 | yes | characteristic |
| 108 | C4⋊Dic5 | C42 | 1 | 1 | yes | characteristic |
| 109 | C4×Dic5 | — | 2 | 1 | no | — |
| 110 | C4×Dic5 | — | 2 | 1 | no | — |
| 111 | C4×Dic5 | — | 2 | 1 | no | — |
| 112 | C4×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 113 | C4×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 114 | C4×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 115 | C4×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | #681 | — | 2 | 2 | no | — |
| 117 | C2×#168 | — | 2 | 2 | no | — |
| 118 | C2×#168 | — | 2 | 1 | no | — |
| 119 | C2×#168 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 120 | C2×#168 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 121 | C4×#41 | — | 5 | 1 | no | — |
| 122 | #433 | — | 10 | 1 | no | — |
| 123 | #433 | — | 10 | 1 | no | — |
| 124 | #431 | — | 5 | 1 | no | — |
| 125 | #431 | — | 5 | 1 | no | — |
| 126 | #431 | — | 10 | 1 | no | — |
| 127 | #431 | — | 10 | 1 | no | — |
| 128 | #424 | — | 5 | 1 | no | — |
| 129 | #424 | — | 5 | 1 | no | — |
| 130 | #408 | — | 5 | 1 | no | — |
| 131 | #408 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 132 | C4×M16 | — | 5 | 1 | no | — |
| 133 | C2×C4×C8 | — | 5 | 1 | no | — |
| 134 | C2×C4×C8 | — | 5 | 1 | no | — |
| 135 | C2×C4×C8 | — | 5 | 1 | no | — |
| 136 | Q8⋊C8 | — | 5 | 2 | no | — |
| 137 | Q8⋊C8 | — | 5 | 2 | no | — |
| 138 | D4⋊C8 | — | 5 | 2 | no | — |
| 139 | D4⋊C8 | — | 5 | 2 | no | — |
| 140 | #322 | — | 10 | 1 | no | — |
| 141 | #322 | — | 10 | 1 | no | — |
| 142 | #322 | — | 10 | 1 | no | — |
| 143 | C82 | — | 10 | 1 | no | — |
| 144 | C23×C5 | — | 2 | 1 | no | — |
| 145 | C23×C5 | #117 | 1 | 1 | yes | characteristic |
| 146 | C5×Q8 | — | 2 | 2 | no | — |
| 147 | C5×D4 | — | 2 | 2 | no | — |
| 148 | C5×D4 | — | 4 | 2 | no | — |
| 149 | C2×C4×C5 | — | 2 | 1 | no | — |
| 150 | C2×C4×C5 | — | 4 | 1 | no | — |
| 151 | C2×C4×C5 | — | 4 | 1 | no | — |
| 152 | C2×C4×C5 | C2×C42 | 1 | 1 | yes | characteristic |
| 153 | C2×C4×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 154 | C2×C4×C5 | #131 | 1 | 2 | yes | — |
| 155 | C2×C4×C5 | #131 | 1 | 2 | yes | — |
| 156 | C2×Dic5 | — | 2 | 2 | no | — |
| 157 | C2×Dic5 | — | 2 | 2 | no | — |
| 158 | C2×Dic5 | — | 2 | 1 | no | — |
| 159 | C2×Dic5 | — | 2 | 1 | no | — |
| 160 | C2×Dic5 | — | 4 | 2 | no | — |
| 161 | C2×Dic5 | — | 4 | 2 | no | — |
| 162 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 163 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 164 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 165 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 166 | #170 | — | 2 | 4 | no | — |
| 167 | #170 | — | 4 | 2 | no | — |
| 168 | #170 | — | 4 | 2 | no | — |
| 169 | #170 | — | 4 | 2 | no | — |
| 170 | C5×C8 | — | 2 | 1 | no | — |
| 171 | C5×C8 | — | 2 | 1 | no | — |
| 172 | C5×C8 | — | 4 | 1 | no | — |
| 173 | #168 | — | 2 | 1 | no | — |
| 174 | #168 | — | 2 | 1 | no | — |
| 175 | #168 | — | 4 | 1 | no | — |
| 176 | C2×#41 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 177 | C2×M16 | — | 5 | 1 | no | — |
| 178 | C2×M16 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 179 | C22×C8 | — | 5 | 1 | no | — |
| 180 | C22×C8 | — | 10 | 1 | no | — |
| 181 | C22×C8 | — | 10 | 1 | no | — |
| 182 | C22×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 183 | C4×Q8 | — | 5 | 1 | no | — |
| 184 | C4×Q8 | — | 5 | 1 | no | — |
| 185 | C4×D4 | — | 5 | 1 | no | — |
| 186 | C4×D4 | — | 5 | 1 | no | — |
| 187 | C4×D4 | — | 10 | 1 | no | — |
| 188 | C4×D4 | — | 10 | 1 | no | — |
| 189 | #117 | — | 5 | 1 | no | — |
| 190 | #117 | — | 10 | 1 | no | — |
| 191 | C2×C42 | — | 5 | 1 | no | — |
| 192 | C2×C42 | — | 10 | 1 | no | — |
| 193 | C4⋊C8 | — | 5 | 2 | no | — |
| 194 | C4⋊C8 | — | 5 | 2 | no | — |
| 195 | C4⋊C8 | — | 10 | 1 | no | — |
| 196 | C4⋊C8 | — | 10 | 1 | no | — |
| 197 | C4⋊C8 | — | 10 | 1 | no | — |
| 198 | C4⋊C8 | — | 10 | 1 | no | — |
| 199 | C22⋊C8 | — | 2 | 1 | no | — |
| 200 | C22⋊C8 | — | 2 | 1 | no | — |
| 201 | C22⋊C8 | — | 10 | 1 | no | — |
| 202 | C22⋊C8 | — | 10 | 1 | no | — |
| 203 | C22⋊C8 | — | 10 | 1 | no | — |
| 204 | C22⋊C8 | — | 10 | 1 | no | — |
| 205 | #97 | — | 10 | 1 | no | — |
| 206 | C4×C8 | — | 5 | 1 | no | — |
| 207 | C4×C8 | — | 5 | 1 | no | — |
| 208 | C4×C8 | — | 5 | 1 | no | — |
| 209 | C4×C8 | — | 5 | 1 | no | — |
| 210 | C4×C8 | — | 5 | 2 | no | — |
| 211 | C4×C8 | — | 5 | 2 | no | — |
| 212 | C4×C8 | — | 10 | 1 | no | — |
| 213 | C4×C8 | — | 10 | 1 | no | — |
| 214 | C4×C8 | — | 10 | 1 | no | — |
| 215 | C4×C8 | — | 10 | 1 | no | — |
| 216 | C4×C8 | — | 10 | 1 | no | — |
| 217 | C22×C5 | — | 2 | 1 | no | — |
| 218 | C22×C5 | — | 2 | 1 | no | — |
| 219 | C22×C5 | — | 4 | 1 | no | — |
| 220 | C22×C5 | — | 4 | 1 | no | — |
| 221 | C22×C5 | — | 4 | 1 | no | — |
| 222 | C22×C5 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 223 | C22×C5 | #433 | 1 | 2 | yes | — |
| 224 | C4×C5 | — | 4 | 1 | no | — |
| 225 | C4×C5 | #405 | 1 | 2 | yes | — |
| 226 | C4×C5 | #408 | 1 | 1 | yes | characteristic |
| 227 | C4×C5 | #408 | 1 | 1 | yes | characteristic |
| 228 | Dic5 | — | 2 | 1 | no | — |
| 229 | Dic5 | — | 2 | 1 | no | — |
| 230 | Dic5 | — | 4 | 1 | no | — |
| 231 | Dic5 | — | 4 | 1 | no | — |
| 232 | Dic5 | #443 | 1 | 2 | yes | — |
| 233 | Dic5 | #444 | 1 | 2 | yes | — |
| 234 | #41 | — | 2 | 2 | no | — |
| 235 | #41 | — | 2 | 2 | no | — |
| 236 | C2×Q8 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 237 | C2×D4 | — | 2 | 1 | no | — |
| 238 | C2×D4 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 239 | C22×C4 | — | 2 | 1 | no | — |
| 240 | C22×C4 | — | 5 | 1 | no | — |
| 241 | C22×C4 | — | 5 | 1 | no | — |
| 242 | C22×C4 | — | 10 | 1 | no | — |
| 243 | C22×C4 | — | 10 | 1 | no | — |
| 244 | C22×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 245 | M16 | — | 2 | 2 | no | — |
| 246 | M16 | — | 10 | 2 | no | — |
| 247 | C2×C8 | — | 2 | 2 | no | — |
| 248 | C2×C8 | — | 2 | 1 | no | — |
| 249 | C2×C8 | — | 5 | 1 | no | — |
| 250 | C2×C8 | — | 5 | 1 | no | — |
| 251 | C2×C8 | — | 10 | 2 | no | — |
| 252 | C2×C8 | — | 10 | 4 | no | — |
| 253 | C2×C8 | — | 10 | 2 | no | — |
| 254 | C2×C8 | — | 10 | 2 | no | — |
| 255 | C2×C8 | — | 10 | 2 | no | — |
| 256 | C2×C8 | — | 10 | 2 | no | — |
| 257 | C2×C8 | — | 10 | 1 | no | — |
| 258 | C2×C8 | — | 20 | 2 | no | — |
| 259 | C2×C8 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 260 | C2×C8 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 261 | C4⋊C4 | — | 5 | 1 | no | — |
| 262 | C4⋊C4 | — | 5 | 1 | no | — |
| 263 | C4⋊C4 | — | 10 | 1 | no | — |
| 264 | C4⋊C4 | — | 10 | 1 | no | — |
| 265 | C22⋊C4 | — | 10 | 1 | no | — |
| 266 | C22⋊C4 | — | 10 | 1 | no | — |
| 267 | C22⋊C4 | — | 10 | 1 | no | — |
| 268 | C42 | — | 5 | 1 | no | — |
| 269 | C42 | — | 5 | 1 | no | — |
| 270 | C42 | — | 5 | 1 | no | — |
| 271 | C42 | — | 5 | 1 | no | — |
| 272 | C42 | — | 10 | 1 | no | — |
| 273 | C42 | — | 10 | 1 | no | — |
| 274 | C42 | — | 10 | 1 | no | — |
| 275 | C2×C5 | — | 2 | 2 | no | — |
| 276 | C2×C5 | — | 8 | 1 | no | — |
| 277 | C2×C5 | #1750 | 1 | 1 | yes | characteristic |
| 278 | C2×C5 | #1764 | 1 | 1 | yes | characteristic |
| 279 | C2×C5 | #1765 | 1 | 1 | yes | characteristic |
| 280 | C23 | — | 2 | 1 | no | — |
| 281 | C23 | #4130 | 1 | 1 | yes | characteristic |
| 282 | Q8 | — | 2 | 2 | no | — |
| 283 | D4 | — | 2 | 2 | no | — |
| 284 | D4 | — | 4 | 2 | no | — |
| 285 | C2×C4 | — | 2 | 1 | no | — |
| 286 | C2×C4 | — | 4 | 1 | no | — |
| 287 | C2×C4 | — | 4 | 1 | no | — |
| 288 | C2×C4 | — | 5 | 1 | no | — |
| 289 | C2×C4 | — | 5 | 1 | no | — |
| 290 | C2×C4 | — | 5 | 1 | no | — |
| 291 | C2×C4 | — | 5 | 1 | no | — |
| 292 | C2×C4 | — | 10 | 2 | no | — |
| 293 | C2×C4 | — | 10 | 2 | no | — |
| 294 | C2×C4 | — | 10 | 1 | no | — |
| 295 | C2×C4 | — | 10 | 1 | no | — |
| 296 | C2×C4 | — | 20 | 2 | no | — |
| 297 | C2×C4 | — | 20 | 2 | no | — |
| 298 | C2×C4 | C2×C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | #4131 | 1 | 2 | yes | — |
| 300 | C2×C4 | #4131 | 1 | 2 | yes | — |
| 301 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 302 | C8 | — | 2 | 1 | no | — |
| 303 | C8 | — | 2 | 1 | no | — |
| 304 | C8 | — | 4 | 1 | no | — |
| 305 | C8 | — | 10 | 4 | no | — |
| 306 | C8 | — | 10 | 1 | no | — |
| 307 | C8 | — | 10 | 1 | no | — |
| 308 | C8 | — | 20 | 2 | no | — |
| 309 | C8 | — | 20 | 2 | no | — |
| 310 | C8 | — | 20 | 2 | no | — |
| 311 | C8 | — | 20 | 1 | no | — |
| 312 | C5 | #8189 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 313 | C22 | — | 2 | 1 | no | — |
| 314 | C22 | — | 2 | 1 | no | — |
| 315 | C22 | — | 4 | 1 | no | — |
| 316 | C22 | — | 4 | 1 | no | — |
| 317 | C22 | — | 4 | 1 | no | — |
| 318 | C22 | #65697 | 1 | 2 | yes | — |
| 319 | C22 | C4×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 320 | C4 | — | 4 | 1 | no | — |
| 321 | C4 | — | 5 | 2 | no | — |
| 322 | C4 | — | 5 | 2 | no | — |
| 323 | C4 | — | 10 | 1 | no | — |
| 324 | C4 | — | 10 | 1 | no | — |
| 325 | C4 | — | 20 | 1 | no | — |
| 326 | C4 | — | 20 | 1 | no | — |
| 327 | C4 | #65691 | 1 | 2 | yes | — |
| 328 | C4 | #65775 | 1 | 1 | yes | characteristic |
| 329 | C4 | #65776 | 1 | 1 | yes | characteristic |
| 330 | C2 | — | 2 | 2 | no | — |
| 331 | C2 | — | 8 | 1 | no | — |
| 332 | C2 | #10604224 | 1 | 1 | yes | characteristic |
| 333 | C2 | #10604777 | 1 | 1 | yes | characteristic |
| 334 | C2 | #10604778 | 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