← #13031863
Subgroups — #13031863
Order 1280.
614 conjugacy classes of subgroups.
The subgroup lattice has 328 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #10617200 | C2 | 1 | 2 | yes | maximal |
| 2 | C4×D5×Q16 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10604841 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10604438 | C2 | 1 | 2 | yes | maximal |
| 5 | #10604068 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10603982 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #10603815 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #10603809 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C2×D5×Q16 | C4 | 1 | 1 | yes | characteristic |
| 10 | C4×Q8×D5 | C22 | 1 | 2 | yes | — |
| 11 | #65788 | C22 | 1 | 4 | yes | — |
| 12 | #65709 | C22 | 1 | 4 | yes | — |
| 13 | #65707 | — | 2 | 4 | no | — |
| 14 | #65684 | — | 2 | 2 | no | — |
| 15 | #65683 | — | 2 | 2 | no | — |
| 16 | #65680 | C22 | 1 | 2 | yes | — |
| 17 | #65678 | C22 | 1 | 1 | yes | characteristic |
| 18 | #65678 | C22 | 1 | 1 | yes | characteristic |
| 19 | C4×C5×Q16 | C4 | 1 | 1 | yes | characteristic |
| 20 | Q16×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 21 | C4×#689 | C4 | 1 | 2 | yes | — |
| 22 | D5×#107 | C4 | 1 | 1 | yes | characteristic |
| 23 | #65162 | C22 | 1 | 1 | yes | characteristic |
| 24 | D5×#103 | C4 | 1 | 2 | yes | — |
| 25 | #65079 | C22 | 1 | 2 | yes | — |
| 26 | C4×Dic20 | C4 | 1 | 1 | yes | characteristic |
| 27 | C4×C8×D5 | C22 | 1 | 1 | yes | characteristic |
| 28 | Q8⋊#170 | C22 | 1 | 4 | yes | — |
| 29 | #64883 | C22 | 1 | 2 | yes | — |
| 30 | #64882 | C22 | 1 | 2 | yes | — |
| 31 | #64879 | C22 | 1 | 2 | yes | — |
| 32 | C8×#170 | C22 | 1 | 2 | yes | — |
| 33 | #64872 | C22 | 1 | 4 | yes | — |
| 34 | #17897 | — | 5 | 1 | no | maximal, Sylow |
| 35 | C2×Q8×D5 | C2×C4 | 1 | 2 | yes | — |
| 36 | C2×C5×Q16 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C4×C5×Q8 | C2×C4 | 1 | 2 | yes | — |
| 38 | Q8×Dic5 | C23 | 1 | 2 | yes | — |
| 39 | C2×#689 | C2×C4 | 1 | 2 | yes | — |
| 40 | D5×Q16 | — | 2 | 4 | no | — |
| 41 | C2×Dic20 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C2×C8×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C2×C8×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | D5×C4⋊C4 | — | 2 | 2 | no | — |
| 45 | D5×C4⋊C4 | C2×C4 | 1 | 2 | yes | — |
| 46 | #4033 | — | 2 | 2 | no | — |
| 47 | #4033 | C23 | 1 | 2 | yes | — |
| 48 | C42×D5 | — | 2 | 2 | no | — |
| 49 | C42×D5 | C23 | 1 | 1 | yes | characteristic |
| 50 | C4×Dic10 | — | 2 | 2 | no | — |
| 51 | C4×Dic10 | C2×C4 | 1 | 2 | yes | — |
| 52 | #4004 | — | 2 | 4 | no | — |
| 53 | #4004 | — | 2 | 1 | no | — |
| 54 | #4004 | D4 | 1 | 2 | yes | — |
| 55 | #4003 | — | 2 | 2 | no | — |
| 56 | #4003 | — | 2 | 1 | no | — |
| 57 | #4003 | D4 | 1 | 1 | yes | characteristic |
| 58 | #4003 | D4 | 1 | 1 | yes | characteristic |
| 59 | #4002 | — | 2 | 4 | no | — |
| 60 | #4002 | — | 2 | 4 | no | — |
| 61 | C4⋊#170 | — | 2 | 4 | no | — |
| 62 | C4⋊#170 | C23 | 1 | 4 | yes | — |
| 63 | C4×#170 | — | 2 | 2 | no | — |
| 64 | C4×#170 | C23 | 1 | 2 | yes | — |
| 65 | C4×#170 | C23 | 1 | 2 | yes | — |
| 66 | C5×#107 | C2×C4 | 1 | 1 | yes | characteristic |
| 67 | C5×#103 | C2×C4 | 1 | 2 | yes | — |
| 68 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 69 | #3967 | C2×C4 | 1 | 2 | yes | — |
| 70 | #3950 | C2×C4 | 1 | 1 | yes | characteristic |
| 71 | #3948 | C2×C4 | 1 | 2 | yes | — |
| 72 | C8×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 73 | C8×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 74 | #3942 | C2×C4 | 1 | 2 | yes | — |
| 75 | #3939 | C2×C4 | 1 | 2 | yes | — |
| 76 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 77 | #2983 | — | 5 | 2 | no | — |
| 78 | C2×C4×Q16 | — | 5 | 1 | no | — |
| 79 | #1593 | — | 5 | 4 | no | — |
| 80 | #1591 | — | 5 | 2 | no | — |
| 81 | #1591 | — | 5 | 2 | no | — |
| 82 | #1564 | — | 5 | 1 | no | — |
| 83 | #1479 | — | 5 | 2 | no | — |
| 84 | #1450 | — | 5 | 1 | no | — |
| 85 | C2×C5×Q8 | C22×C4 | 1 | 2 | yes | — |
| 86 | Q8×D5 | — | 2 | 4 | no | — |
| 87 | Q8×D5 | — | 2 | 4 | no | — |
| 88 | C2×C4×D5 | — | 2 | 2 | no | — |
| 89 | C2×C4×D5 | — | 2 | 2 | no | — |
| 90 | C2×C4×D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×D5 | #41 | 1 | 1 | yes | characteristic |
| 93 | C2×Dic10 | — | 2 | 2 | no | — |
| 94 | C2×Dic10 | C22×C4 | 1 | 2 | yes | — |
| 95 | C2×#170 | — | 2 | 4 | no | — |
| 96 | C2×#170 | — | 2 | 2 | no | — |
| 97 | C2×#170 | C2×D4 | 1 | 2 | yes | — |
| 98 | C2×#170 | C2×D4 | 1 | 2 | yes | — |
| 99 | C5×Q16 | — | 2 | 2 | no | — |
| 100 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 101 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 102 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 103 | C5×C4⋊C4 | C22×C4 | 1 | 2 | yes | — |
| 104 | C42×C5 | — | 2 | 2 | no | — |
| 105 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 106 | #689 | — | 2 | 4 | no | — |
| 107 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 108 | C4⋊Dic5 | C22×C4 | 1 | 2 | yes | — |
| 109 | #683 | — | 2 | 2 | no | — |
| 110 | #683 | — | 2 | 2 | no | — |
| 111 | #683 | — | 2 | 2 | no | — |
| 112 | C4×Dic5 | — | 2 | 2 | no | — |
| 113 | C4×Dic5 | — | 2 | 2 | no | — |
| 114 | C4×Dic5 | — | 2 | 2 | no | — |
| 115 | C4×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C4×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 117 | C4×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 118 | C2×#168 | C22×C4 | 1 | 1 | yes | characteristic |
| 119 | C2×#168 | C22×C4 | 1 | 1 | yes | characteristic |
| 120 | Dic20 | — | 2 | 2 | no | — |
| 121 | C8×D5 | — | 2 | 2 | no | — |
| 122 | C8×D5 | M16 | 1 | 4 | yes | — |
| 123 | C22×Q16 | — | 5 | 1 | no | — |
| 124 | C2×C4×Q8 | — | 5 | 2 | no | — |
| 125 | #446 | — | 5 | 4 | no | — |
| 126 | #446 | — | 5 | 4 | no | — |
| 127 | #446 | — | 10 | 4 | no | — |
| 128 | C4×Q16 | — | 5 | 1 | no | — |
| 129 | C4×Q16 | — | 5 | 2 | no | — |
| 130 | C4×Q16 | — | 5 | 1 | no | — |
| 131 | C4×Q16 | — | 5 | 2 | no | — |
| 132 | C4×Q16 | — | 5 | 1 | no | — |
| 133 | C4×Q16 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 134 | #432 | — | 10 | 2 | no | — |
| 135 | #432 | — | 10 | 2 | no | — |
| 136 | #431 | — | 5 | 1 | no | — |
| 137 | #431 | — | 5 | 1 | no | — |
| 138 | #431 | — | 5 | 2 | no | — |
| 139 | C2×#107 | — | 5 | 1 | no | — |
| 140 | C2×#103 | — | 5 | 2 | no | — |
| 141 | C2×C4×C8 | — | 5 | 1 | no | — |
| 142 | #335 | — | 5 | 2 | no | — |
| 143 | #334 | — | 5 | 2 | no | — |
| 144 | Q8⋊C8 | — | 5 | 4 | no | — |
| 145 | Q8⋊C8 | — | 5 | 4 | no | — |
| 146 | #322 | — | 5 | 2 | no | — |
| 147 | C82 | — | 5 | 2 | no | — |
| 148 | C22×D5 | C4×D4 | 1 | 1 | yes | characteristic |
| 149 | C5×Q8 | — | 2 | 2 | no | — |
| 150 | C5×Q8 | — | 2 | 2 | no | — |
| 151 | C2×C4×C5 | — | 2 | 2 | no | — |
| 152 | C2×C4×C5 | — | 2 | 2 | no | — |
| 153 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 154 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 155 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 156 | C2×Dic5 | — | 2 | 2 | no | — |
| 157 | C2×Dic5 | — | 2 | 2 | no | — |
| 158 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 159 | C2×Dic5 | C22×D4 | 1 | 1 | yes | characteristic |
| 160 | C2×Dic5 | C2×#41 | 1 | 1 | yes | characteristic |
| 161 | C4×D5 | — | 2 | 2 | no | — |
| 162 | C4×D5 | — | 4 | 1 | no | — |
| 163 | C4×D5 | — | 4 | 4 | no | — |
| 164 | C4×D5 | — | 8 | 2 | no | — |
| 165 | C4×D5 | C2×M16 | 1 | 2 | yes | — |
| 166 | C4×D5 | #131 | 1 | 2 | yes | — |
| 167 | Dic10 | — | 2 | 2 | no | — |
| 168 | Dic10 | — | 2 | 2 | no | — |
| 169 | Dic10 | — | 4 | 4 | no | — |
| 170 | #170 | — | 2 | 2 | no | — |
| 171 | #170 | — | 2 | 2 | no | — |
| 172 | #170 | — | 4 | 2 | no | — |
| 173 | #170 | — | 4 | 4 | no | — |
| 174 | C5×C8 | — | 2 | 1 | no | — |
| 175 | C5×C8 | C2×M16 | 1 | 2 | yes | — |
| 176 | #168 | — | 2 | 1 | no | — |
| 177 | #168 | C2×M16 | 1 | 2 | yes | — |
| 178 | C22×Q8 | — | 5 | 2 | no | — |
| 179 | C2×Q16 | — | 5 | 2 | no | — |
| 180 | C2×Q16 | — | 5 | 1 | no | — |
| 181 | C2×Q16 | — | 10 | 4 | no | — |
| 182 | C2×Q16 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 183 | C22×C8 | — | 5 | 1 | no | — |
| 184 | C22×C8 | — | 5 | 1 | no | — |
| 185 | C4×Q8 | — | 5 | 2 | no | — |
| 186 | C4×Q8 | — | 5 | 2 | no | — |
| 187 | C4×Q8 | — | 5 | 2 | no | — |
| 188 | C4×Q8 | — | 10 | 2 | no | — |
| 189 | C4×Q8 | — | 10 | 2 | no | — |
| 190 | C4×Q8 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 191 | C2×C4⋊C4 | — | 5 | 2 | no | — |
| 192 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 193 | C2×C42 | — | 5 | 1 | no | — |
| 194 | C2×C42 | — | 10 | 2 | no | — |
| 195 | #107 | — | 5 | 1 | no | — |
| 196 | #107 | — | 5 | 2 | no | — |
| 197 | #107 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 198 | C4⋊C8 | — | 5 | 2 | no | — |
| 199 | C4⋊C8 | — | 5 | 4 | no | — |
| 200 | C4⋊C8 | — | 10 | 1 | no | — |
| 201 | C4⋊C8 | — | 10 | 4 | no | — |
| 202 | C4⋊C8 | — | 10 | 4 | no | — |
| 203 | #103 | — | 5 | 2 | no | — |
| 204 | #103 | — | 5 | 2 | no | — |
| 205 | #103 | — | 5 | 2 | no | — |
| 206 | #103 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 207 | C22⋊C8 | — | 5 | 1 | no | — |
| 208 | C22⋊C8 | — | 5 | 1 | no | — |
| 209 | C22⋊C8 | — | 10 | 1 | no | — |
| 210 | C22⋊C8 | — | 10 | 2 | no | — |
| 211 | #97 | — | 10 | 4 | no | — |
| 212 | #97 | — | 10 | 4 | no | — |
| 213 | C4×C8 | — | 5 | 2 | no | — |
| 214 | C4×C8 | — | 5 | 1 | no | — |
| 215 | C4×C8 | — | 5 | 1 | no | — |
| 216 | C4×C8 | — | 5 | 2 | no | — |
| 217 | C4×C8 | — | 5 | 1 | no | — |
| 218 | C4×C8 | — | 10 | 2 | no | — |
| 219 | C4×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 220 | C22×C5 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 221 | C2×D5 | — | 2 | 1 | no | — |
| 222 | C2×D5 | — | 2 | 1 | no | — |
| 223 | C2×D5 | #436 | 1 | 2 | yes | — |
| 224 | C4×C5 | — | 2 | 1 | no | — |
| 225 | C4×C5 | — | 2 | 1 | no | — |
| 226 | C4×C5 | — | 4 | 2 | no | — |
| 227 | C4×C5 | — | 4 | 2 | no | — |
| 228 | C4×C5 | C22×M16 | 1 | 1 | yes | characteristic |
| 229 | C4×C5 | C2×#131 | 1 | 1 | yes | characteristic |
| 230 | Dic5 | — | 2 | 1 | no | — |
| 231 | Dic5 | — | 4 | 2 | no | — |
| 232 | Dic5 | — | 4 | 2 | no | — |
| 233 | Dic5 | #436 | 1 | 2 | yes | — |
| 234 | Dic5 | #578 | 1 | 2 | yes | — |
| 235 | C2×Q8 | — | 5 | 2 | no | — |
| 236 | C2×Q8 | — | 10 | 2 | no | — |
| 237 | C2×Q8 | — | 10 | 4 | no | — |
| 238 | C2×Q8 | — | 10 | 4 | no | — |
| 239 | C2×Q8 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 240 | C22×C4 | — | 5 | 1 | no | — |
| 241 | C22×C4 | — | 5 | 1 | no | — |
| 242 | C22×C4 | — | 5 | 1 | no | — |
| 243 | C22×C4 | — | 10 | 2 | no | — |
| 244 | C22×C4 | — | 10 | 2 | no | — |
| 245 | Q16 | — | 2 | 2 | no | — |
| 246 | Q16 | — | 10 | 4 | no | — |
| 247 | Q16 | — | 10 | 2 | no | — |
| 248 | C2×C8 | — | 5 | 1 | no | — |
| 249 | C2×C8 | — | 5 | 4 | no | — |
| 250 | C2×C8 | — | 5 | 2 | no | — |
| 251 | C2×C8 | — | 5 | 2 | no | — |
| 252 | C2×C8 | — | 5 | 1 | no | — |
| 253 | C2×C8 | — | 10 | 2 | no | — |
| 254 | C2×C8 | — | 10 | 2 | no | — |
| 255 | C2×C8 | — | 10 | 4 | no | — |
| 256 | C2×C8 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 258 | C4⋊C4 | — | 2 | 2 | no | — |
| 259 | C4⋊C4 | — | 5 | 2 | no | — |
| 260 | C4⋊C4 | — | 10 | 2 | no | — |
| 261 | C4⋊C4 | — | 10 | 2 | no | — |
| 262 | C4⋊C4 | — | 10 | 2 | no | — |
| 263 | C4⋊C4 | — | 10 | 2 | no | — |
| 264 | C4⋊C4 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 265 | C42 | — | 2 | 2 | no | — |
| 266 | C42 | — | 5 | 1 | no | — |
| 267 | C42 | — | 5 | 1 | no | — |
| 268 | C42 | — | 5 | 1 | no | — |
| 269 | C42 | — | 10 | 2 | no | — |
| 270 | C42 | — | 10 | 2 | no | — |
| 271 | C42 | — | 10 | 2 | no | — |
| 272 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 273 | C2×C5 | C2×#436 | 1 | 1 | yes | characteristic |
| 274 | C2×C5 | #2950 | 1 | 1 | yes | characteristic |
| 275 | C2×C5 | #2957 | 1 | 1 | yes | characteristic |
| 276 | D5 | — | 2 | 2 | no | — |
| 277 | C23 | — | 5 | 1 | no | — |
| 278 | Q8 | — | 2 | 2 | no | — |
| 279 | Q8 | — | 2 | 2 | no | — |
| 280 | Q8 | — | 10 | 2 | no | — |
| 281 | Q8 | — | 10 | 2 | no | — |
| 282 | Q8 | — | 20 | 4 | no | — |
| 283 | C2×C4 | — | 2 | 2 | no | — |
| 284 | C2×C4 | — | 2 | 2 | no | — |
| 285 | C2×C4 | — | 5 | 1 | no | — |
| 286 | C2×C4 | — | 5 | 1 | no | — |
| 287 | C2×C4 | — | 5 | 2 | no | — |
| 288 | C2×C4 | — | 5 | 2 | no | — |
| 289 | C2×C4 | — | 5 | 1 | no | — |
| 290 | C2×C4 | — | 10 | 2 | no | — |
| 291 | C2×C4 | — | 10 | 2 | no | — |
| 292 | C2×C4 | — | 10 | 2 | no | — |
| 293 | C2×C4 | — | 20 | 1 | no | — |
| 294 | C2×C4 | — | 20 | 4 | no | — |
| 295 | C2×C4 | — | 40 | 2 | no | — |
| 296 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 299 | C8 | — | 2 | 1 | no | — |
| 300 | C8 | — | 5 | 2 | no | — |
| 301 | C8 | — | 10 | 2 | no | — |
| 302 | C8 | — | 10 | 2 | no | — |
| 303 | C8 | — | 10 | 1 | no | — |
| 304 | C8 | — | 20 | 4 | no | — |
| 305 | C8 | — | 20 | 2 | no | — |
| 306 | C8 | D5⋊M16 | 1 | 2 | yes | — |
| 307 | C5 | #17897 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 308 | C22 | — | 5 | 2 | no | — |
| 309 | C22 | — | 10 | 1 | no | — |
| 310 | C22 | — | 10 | 1 | no | — |
| 311 | C22 | C2×D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 312 | C4 | — | 2 | 1 | no | — |
| 313 | C4 | — | 2 | 1 | no | — |
| 314 | C4 | — | 4 | 2 | no | — |
| 315 | C4 | — | 4 | 2 | no | — |
| 316 | C4 | — | 5 | 2 | no | — |
| 317 | C4 | — | 5 | 2 | no | — |
| 318 | C4 | — | 10 | 1 | no | — |
| 319 | C4 | — | 20 | 2 | no | — |
| 320 | C4 | — | 20 | 2 | no | — |
| 321 | C4 | C2×D5⋊M16 | 1 | 1 | yes | characteristic |
| 322 | C4 | D5⋊#131 | 1 | 1 | yes | characteristic |
| 323 | C2 | — | 10 | 2 | no | — |
| 324 | C2 | D5⋊#436 | 1 | 1 | yes | characteristic |
| 325 | C2 | #10617434 | 1 | 1 | yes | characteristic |
| 326 | C2 | #10617445 | 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