← #13032286
Subgroups — #13032286
Order 1280.
560 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 | #10617381 | C2 | 1 | 2 | yes | maximal |
| 2 | #10613523 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10604842 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10604841 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10604838 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | Q16×#170 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #10604654 | C2 | 1 | 2 | yes | maximal |
| 8 | #10603820 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #10603809 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #66062 | C22 | 1 | 2 | yes | — |
| 11 | #65788 | C22 | 1 | 4 | yes | — |
| 12 | Q8×#170 | C22 | 1 | 4 | yes | — |
| 13 | #65707 | — | 2 | 4 | no | — |
| 14 | #65706 | — | 2 | 4 | no | — |
| 15 | #65683 | C22 | 1 | 2 | yes | — |
| 16 | #65678 | C22 | 1 | 1 | yes | characteristic |
| 17 | #65678 | C22 | 1 | 1 | yes | characteristic |
| 18 | C5×#494 | C4 | 1 | 1 | yes | characteristic |
| 19 | #65477 | C4 | 1 | 2 | yes | — |
| 20 | #65477 | C4 | 1 | 2 | yes | — |
| 21 | Q16×Dic5 | C22 | 1 | 2 | yes | — |
| 22 | Q16×Dic5 | C22 | 1 | 2 | yes | — |
| 23 | #65370 | C4 | 1 | 2 | yes | — |
| 24 | C4×C8×D5 | C22 | 1 | 1 | yes | characteristic |
| 25 | #64971 | C4 | 1 | 1 | yes | characteristic |
| 26 | Q8⋊#170 | C22 | 1 | 4 | yes | — |
| 27 | Q8⋊#170 | C22 | 1 | 4 | yes | — |
| 28 | #64882 | C22 | 1 | 2 | yes | — |
| 29 | #64881 | C22 | 1 | 2 | yes | — |
| 30 | #64879 | C22 | 1 | 1 | yes | characteristic |
| 31 | #64879 | C22 | 1 | 1 | yes | characteristic |
| 32 | C8×#170 | C22 | 1 | 1 | yes | characteristic |
| 33 | C8×#170 | C22 | 1 | 1 | yes | characteristic |
| 34 | #17972 | — | 5 | 1 | no | maximal, Sylow |
| 35 | C2×C5×Q16 | C2×C4 | 1 | 2 | yes | — |
| 36 | C2×C5×Q16 | C2×C4 | 1 | 2 | yes | — |
| 37 | C5×C4⋊Q8 | C2×C4 | 1 | 2 | yes | — |
| 38 | #4092 | C2×C4 | 1 | 4 | yes | — |
| 39 | Q8×Dic5 | C23 | 1 | 4 | yes | — |
| 40 | C2×C8×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C2×C8×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | #4043 | — | 2 | 2 | no | — |
| 43 | #4038 | — | 2 | 2 | no | — |
| 44 | #4036 | — | 2 | 2 | no | — |
| 45 | #4033 | — | 2 | 2 | no | — |
| 46 | C42×D5 | C23 | 1 | 1 | yes | characteristic |
| 47 | #4015 | C2×C4 | 1 | 2 | yes | — |
| 48 | #4004 | — | 2 | 4 | no | — |
| 49 | #4004 | — | 2 | 1 | no | — |
| 50 | #4004 | D4 | 1 | 2 | yes | — |
| 51 | #4003 | — | 2 | 2 | no | — |
| 52 | #4003 | — | 2 | 1 | no | — |
| 53 | #4003 | D4 | 1 | 2 | yes | — |
| 54 | #4002 | — | 2 | 4 | no | — |
| 55 | #4002 | — | 2 | 2 | no | — |
| 56 | C4⋊#170 | — | 2 | 4 | no | — |
| 57 | C4⋊#170 | C23 | 1 | 4 | yes | — |
| 58 | C4×#170 | — | 2 | 4 | no | — |
| 59 | C4×#170 | C23 | 1 | 1 | yes | characteristic |
| 60 | C4×#170 | C23 | 1 | 1 | yes | characteristic |
| 61 | C4×#170 | C23 | 1 | 1 | yes | characteristic |
| 62 | C4×#170 | C23 | 1 | 1 | yes | characteristic |
| 63 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 64 | #3967 | C2×C4 | 1 | 4 | yes | — |
| 65 | #3967 | C2×C4 | 1 | 4 | yes | — |
| 66 | #3950 | C2×C4 | 1 | 2 | yes | — |
| 67 | #3950 | C2×C4 | 1 | 2 | yes | — |
| 68 | C8×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 69 | C8×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 70 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 71 | #3170 | — | 5 | 1 | no | — |
| 72 | #2985 | — | 5 | 2 | no | — |
| 73 | #1593 | — | 5 | 2 | no | — |
| 74 | #1591 | — | 5 | 2 | no | — |
| 75 | #1579 | — | 5 | 2 | no | — |
| 76 | C8×Q16 | — | 5 | 2 | no | — |
| 77 | #1568 | — | 5 | 1 | no | — |
| 78 | #1497 | — | 5 | 2 | no | — |
| 79 | #1450 | — | 5 | 1 | no | — |
| 80 | C2×C5×Q8 | C22×C4 | 1 | 4 | yes | — |
| 81 | C2×C4×D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C2×C4×D5 | #41 | 1 | 1 | yes | characteristic |
| 84 | C2×Dic10 | — | 2 | 2 | no | — |
| 85 | C2×#170 | — | 2 | 4 | no | — |
| 86 | C2×#170 | — | 2 | 1 | no | — |
| 87 | C2×#170 | — | 2 | 1 | no | — |
| 88 | C2×#170 | C2×D4 | 1 | 2 | yes | — |
| 89 | C2×#170 | C2×D4 | 1 | 2 | yes | — |
| 90 | C5×Q16 | — | 2 | 4 | no | — |
| 91 | C5×Q16 | — | 2 | 4 | no | — |
| 92 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 93 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 94 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 95 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 96 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 97 | #685 | — | 2 | 4 | no | — |
| 98 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 99 | C4⋊Dic5 | — | 2 | 4 | no | — |
| 100 | C4⋊Dic5 | C22×C4 | 1 | 4 | yes | — |
| 101 | #683 | — | 2 | 4 | no | — |
| 102 | C4×Dic5 | — | 2 | 4 | no | — |
| 103 | C4×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C4×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 105 | C4×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 106 | C2×#168 | C22×C4 | 1 | 1 | yes | characteristic |
| 107 | C2×#168 | C22×C4 | 1 | 1 | yes | characteristic |
| 108 | C8×D5 | — | 2 | 2 | no | — |
| 109 | C8×D5 | — | 2 | 2 | no | — |
| 110 | #533 | — | 5 | 2 | no | — |
| 111 | #500 | — | 5 | 1 | no | — |
| 112 | #494 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 113 | #487 | — | 5 | 2 | no | — |
| 114 | #467 | — | 5 | 2 | no | — |
| 115 | #467 | — | 5 | 2 | no | — |
| 116 | #446 | — | 5 | 4 | no | — |
| 117 | #446 | — | 10 | 4 | no | — |
| 118 | C8×Q8 | — | 5 | 4 | no | — |
| 119 | C4×Q16 | — | 5 | 2 | no | — |
| 120 | C4×Q16 | — | 5 | 2 | no | — |
| 121 | #433 | — | 10 | 4 | no | — |
| 122 | #432 | — | 5 | 2 | no | — |
| 123 | #431 | — | 5 | 1 | no | — |
| 124 | #431 | — | 5 | 1 | no | — |
| 125 | C2×C4×C8 | — | 5 | 1 | no | — |
| 126 | #335 | — | 5 | 2 | no | — |
| 127 | #334 | — | 5 | 2 | no | — |
| 128 | Q8⋊C8 | — | 5 | 4 | no | — |
| 129 | Q8⋊C8 | — | 5 | 4 | no | — |
| 130 | #322 | — | 5 | 1 | no | — |
| 131 | #322 | — | 5 | 1 | no | — |
| 132 | C82 | — | 5 | 1 | no | — |
| 133 | C82 | — | 5 | 1 | no | — |
| 134 | C22×D5 | C4×D4 | 1 | 1 | yes | characteristic |
| 135 | C5×Q8 | — | 2 | 4 | no | — |
| 136 | C5×Q8 | — | 2 | 4 | no | — |
| 137 | C2×C4×C5 | — | 2 | 4 | no | — |
| 138 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 139 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 140 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 141 | C2×Dic5 | — | 2 | 4 | no | — |
| 142 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 143 | C2×Dic5 | C22×D4 | 1 | 1 | yes | characteristic |
| 144 | C2×Dic5 | C2×#41 | 1 | 1 | yes | characteristic |
| 145 | C4×D5 | — | 2 | 1 | no | — |
| 146 | C4×D5 | — | 2 | 1 | no | — |
| 147 | C4×D5 | — | 2 | 2 | no | — |
| 148 | C4×D5 | — | 4 | 1 | no | — |
| 149 | Dic10 | — | 8 | 2 | no | — |
| 150 | #170 | — | 2 | 2 | no | — |
| 151 | #170 | — | 2 | 2 | no | — |
| 152 | #170 | — | 4 | 1 | no | — |
| 153 | #170 | — | 4 | 1 | no | — |
| 154 | #170 | — | 4 | 4 | no | — |
| 155 | C5×C8 | #131 | 1 | 2 | yes | — |
| 156 | C5×C8 | #131 | 1 | 2 | yes | — |
| 157 | #168 | — | 2 | 1 | no | — |
| 158 | #168 | — | 2 | 1 | no | — |
| 159 | C2×Q16 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 160 | C2×Q16 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 161 | C22×C8 | — | 5 | 1 | no | — |
| 162 | C22×C8 | — | 5 | 1 | no | — |
| 163 | C4⋊Q8 | — | 5 | 2 | no | — |
| 164 | C4⋊Q8 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 165 | #125 | — | 10 | 2 | no | — |
| 166 | C22⋊Q8 | — | 5 | 4 | no | — |
| 167 | C22⋊Q8 | — | 10 | 2 | no | — |
| 168 | C4×Q8 | — | 5 | 4 | no | — |
| 169 | C4×Q8 | — | 10 | 2 | no | — |
| 170 | #117 | — | 10 | 2 | no | — |
| 171 | C2×C42 | — | 5 | 1 | no | — |
| 172 | #107 | — | 5 | 2 | no | — |
| 173 | #107 | — | 5 | 2 | no | — |
| 174 | C4⋊C8 | — | 5 | 2 | no | — |
| 175 | C4⋊C8 | — | 5 | 4 | no | — |
| 176 | C4⋊C8 | — | 10 | 1 | no | — |
| 177 | C4⋊C8 | — | 10 | 4 | no | — |
| 178 | C4⋊C8 | — | 10 | 4 | no | — |
| 179 | #103 | — | 5 | 4 | no | — |
| 180 | #103 | — | 5 | 4 | no | — |
| 181 | C22⋊C8 | — | 5 | 2 | no | — |
| 182 | C22⋊C8 | — | 10 | 1 | no | — |
| 183 | C22⋊C8 | — | 10 | 2 | no | — |
| 184 | #97 | — | 10 | 2 | no | — |
| 185 | #97 | — | 10 | 4 | no | — |
| 186 | C4×C8 | — | 5 | 1 | no | — |
| 187 | C4×C8 | — | 5 | 1 | no | — |
| 188 | C4×C8 | — | 5 | 1 | no | — |
| 189 | C4×C8 | — | 5 | 1 | no | — |
| 190 | C4×C8 | — | 5 | 1 | no | — |
| 191 | C4×C8 | — | 5 | 1 | no | — |
| 192 | C4×C8 | — | 5 | 1 | no | — |
| 193 | C4×C8 | — | 10 | 4 | no | — |
| 194 | C4×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 195 | C22×C5 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 196 | C2×D5 | — | 2 | 1 | no | — |
| 197 | C2×D5 | — | 2 | 1 | no | — |
| 198 | C2×D5 | — | 2 | 1 | no | — |
| 199 | C4×C5 | — | 2 | 1 | no | — |
| 200 | C4×C5 | — | 2 | 1 | no | — |
| 201 | C4×C5 | — | 4 | 4 | no | — |
| 202 | C4×C5 | C2×#131 | 1 | 1 | yes | characteristic |
| 203 | C4×C5 | C2×#131 | 1 | 1 | yes | characteristic |
| 204 | Dic5 | — | 2 | 1 | no | — |
| 205 | Dic5 | — | 2 | 1 | no | — |
| 206 | Dic5 | — | 4 | 4 | no | — |
| 207 | Dic5 | #578 | 1 | 2 | yes | — |
| 208 | C2×Q8 | — | 10 | 2 | no | — |
| 209 | C2×Q8 | C22×AGL(1,𝔽5) | 1 | 4 | yes | — |
| 210 | C22×C4 | — | 5 | 1 | no | — |
| 211 | C22×C4 | — | 5 | 1 | no | — |
| 212 | C22×C4 | — | 5 | 1 | no | — |
| 213 | Q16 | — | 2 | 4 | no | — |
| 214 | Q16 | — | 2 | 4 | no | — |
| 215 | C2×C8 | — | 5 | 2 | no | — |
| 216 | C2×C8 | — | 5 | 2 | no | — |
| 217 | C2×C8 | — | 5 | 1 | no | — |
| 218 | C2×C8 | — | 5 | 1 | no | — |
| 219 | C2×C8 | — | 10 | 2 | no | — |
| 220 | C2×C8 | — | 10 | 2 | no | — |
| 221 | C2×C8 | — | 10 | 1 | no | — |
| 222 | C2×C8 | — | 10 | 1 | no | — |
| 223 | C2×C8 | — | 10 | 4 | no | — |
| 224 | C2×C8 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 225 | C2×C8 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 226 | C4⋊C4 | — | 2 | 2 | no | — |
| 227 | C4⋊C4 | — | 2 | 2 | no | — |
| 228 | C4⋊C4 | — | 5 | 4 | no | — |
| 229 | C4⋊C4 | — | 10 | 4 | no | — |
| 230 | C4⋊C4 | — | 10 | 4 | no | — |
| 231 | C4⋊C4 | — | 10 | 2 | no | — |
| 232 | C22⋊C4 | — | 10 | 4 | no | — |
| 233 | C42 | — | 5 | 1 | no | — |
| 234 | C42 | — | 5 | 1 | no | — |
| 235 | C42 | — | 5 | 1 | no | — |
| 236 | C42 | — | 10 | 4 | no | — |
| 237 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 238 | C2×C5 | #2931 | 1 | 1 | yes | characteristic |
| 239 | C2×C5 | #2950 | 1 | 1 | yes | characteristic |
| 240 | C2×C5 | #2957 | 1 | 1 | yes | characteristic |
| 241 | D5 | — | 4 | 1 | no | — |
| 242 | C23 | — | 5 | 1 | no | — |
| 243 | Q8 | — | 2 | 4 | no | — |
| 244 | Q8 | — | 2 | 4 | no | — |
| 245 | Q8 | — | 40 | 2 | no | — |
| 246 | C2×C4 | — | 2 | 4 | no | — |
| 247 | C2×C4 | — | 5 | 1 | no | — |
| 248 | C2×C4 | — | 5 | 1 | no | — |
| 249 | C2×C4 | — | 5 | 1 | no | — |
| 250 | C2×C4 | — | 10 | 2 | no | — |
| 251 | C2×C4 | — | 10 | 1 | no | — |
| 252 | C2×C4 | — | 10 | 1 | no | — |
| 253 | C2×C4 | — | 10 | 4 | no | — |
| 254 | C2×C4 | — | 20 | 1 | no | — |
| 255 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 256 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 257 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 258 | C8 | — | 10 | 2 | no | — |
| 259 | C8 | — | 10 | 2 | no | — |
| 260 | C8 | — | 10 | 1 | no | — |
| 261 | C8 | — | 10 | 1 | no | — |
| 262 | C8 | — | 20 | 4 | no | — |
| 263 | C8 | — | 20 | 1 | no | — |
| 264 | C8 | — | 20 | 1 | no | — |
| 265 | C8 | #4131 | 1 | 2 | yes | — |
| 266 | C8 | #4131 | 1 | 2 | yes | — |
| 267 | C5 | #17972 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 268 | C22 | — | 10 | 1 | no | — |
| 269 | C22 | — | 10 | 1 | no | — |
| 270 | C22 | — | 10 | 1 | no | — |
| 271 | C22 | C2×D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 272 | C4 | — | 2 | 1 | no | — |
| 273 | C4 | — | 2 | 1 | no | — |
| 274 | C4 | — | 4 | 4 | no | — |
| 275 | C4 | — | 5 | 2 | no | — |
| 276 | C4 | — | 10 | 1 | no | — |
| 277 | C4 | — | 10 | 1 | no | — |
| 278 | C4 | — | 20 | 4 | no | — |
| 279 | C4 | C2×#4131 | 1 | 1 | yes | characteristic |
| 280 | C4 | C2×#4131 | 1 | 1 | yes | characteristic |
| 281 | C2 | — | 20 | 1 | no | — |
| 282 | C2 | #10617361 | 1 | 1 | yes | characteristic |
| 283 | C2 | #10617434 | 1 | 1 | yes | characteristic |
| 284 | C2 | #10617437 | 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