← #13031802
Subgroups — #13031802
Order 1280.
682 conjugacy classes of subgroups.
The subgroup lattice has 342 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #10617196 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C4×#4065 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10604841 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10604436 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10604067 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10603979 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #10603815 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C8×D5⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#4065 | C4 | 1 | 1 | yes | characteristic |
| 10 | C4×#713 | C22 | 1 | 2 | yes | — |
| 11 | #65788 | C22 | 1 | 4 | yes | — |
| 12 | #65705 | C22 | 1 | 4 | yes | — |
| 13 | #65704 | — | 2 | 4 | no | — |
| 14 | #65680 | C22 | 1 | 2 | yes | — |
| 15 | #65678 | — | 2 | 2 | no | — |
| 16 | C4×#700 | — | 2 | 2 | no | — |
| 17 | C4×D5⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 18 | C4×D5⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 19 | C4×C5×Q16 | C4 | 1 | 1 | yes | characteristic |
| 20 | Q16×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 21 | C4×#688 | C4 | 1 | 2 | yes | — |
| 22 | #65169 | C4 | 1 | 1 | yes | characteristic |
| 23 | #65161 | C22 | 1 | 1 | yes | characteristic |
| 24 | #65093 | C4 | 1 | 2 | yes | — |
| 25 | #65077 | C22 | 1 | 2 | yes | — |
| 26 | C4×D40 | 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 | C8×#170 | C22 | 1 | 2 | yes | — |
| 32 | C8×#170 | C22 | 1 | 2 | yes | — |
| 33 | #64871 | C22 | 1 | 4 | yes | — |
| 34 | #17886 | — | 5 | 1 | no | maximal, Sylow |
| 35 | C2×#713 | C2×C4 | 1 | 2 | yes | — |
| 36 | C2×#700 | — | 2 | 4 | no | — |
| 37 | C2×D5⋊C8 | — | 2 | 1 | no | — |
| 38 | C2×D5⋊C8 | D4 | 1 | 2 | yes | — |
| 39 | C2×C5×Q16 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C4×C5×Q8 | C2×C4 | 1 | 2 | yes | — |
| 41 | Q8×Dic5 | C23 | 1 | 2 | yes | — |
| 42 | C2×#688 | C2×C4 | 1 | 2 | yes | — |
| 43 | #4065 | — | 2 | 4 | no | — |
| 44 | C2×D40 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | C2×C8×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | C2×C8×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | #4039 | — | 2 | 2 | no | — |
| 48 | #4039 | C23 | 1 | 2 | yes | — |
| 49 | #4038 | — | 2 | 2 | no | — |
| 50 | #4038 | C2×C4 | 1 | 2 | yes | — |
| 51 | C4×D20 | — | 2 | 2 | no | — |
| 52 | C4×D20 | C2×C4 | 1 | 2 | yes | — |
| 53 | C42×D5 | — | 2 | 2 | no | — |
| 54 | C42×D5 | C23 | 1 | 1 | yes | characteristic |
| 55 | #4004 | — | 2 | 4 | no | — |
| 56 | #4003 | — | 2 | 2 | no | — |
| 57 | #4003 | — | 2 | 4 | no | — |
| 58 | #4002 | — | 2 | 4 | no | — |
| 59 | C4⋊#170 | C23 | 1 | 4 | yes | — |
| 60 | C4×#170 | — | 2 | 2 | no | — |
| 61 | C4×#170 | — | 2 | 1 | no | — |
| 62 | C4×#170 | D4 | 1 | 1 | yes | characteristic |
| 63 | C4×#170 | D4 | 1 | 1 | yes | characteristic |
| 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 | #3953 | C2×C4 | 1 | 2 | yes | — |
| 71 | #3950 | C2×C4 | 1 | 1 | yes | characteristic |
| 72 | C8×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 73 | C8×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 74 | #3941 | C2×C4 | 1 | 2 | yes | — |
| 75 | #3940 | C2×C4 | 1 | 2 | yes | — |
| 76 | C4×#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 77 | #2965 | — | 5 | 2 | no | — |
| 78 | C4×#135 | — | 5 | 1 | no | — |
| 79 | #1591 | — | 5 | 2 | no | — |
| 80 | #1590 | — | 5 | 4 | no | — |
| 81 | #1589 | — | 5 | 2 | no | — |
| 82 | #1564 | — | 5 | 1 | no | — |
| 83 | #1476 | — | 5 | 2 | no | — |
| 84 | C2×C82 | — | 5 | 1 | no | — |
| 85 | C2×C5×Q8 | C22×C4 | 1 | 2 | yes | — |
| 86 | #713 | — | 2 | 4 | no | — |
| 87 | #713 | — | 2 | 4 | no | — |
| 88 | C2×D20 | — | 2 | 2 | no | — |
| 89 | C2×D20 | C22×C4 | 1 | 2 | yes | — |
| 90 | C2×C4×D5 | — | 2 | 2 | no | — |
| 91 | C2×C4×D5 | — | 2 | 2 | no | — |
| 92 | C2×C4×D5 | — | 2 | 2 | no | — |
| 93 | C2×C4×D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 94 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×D5 | #41 | 1 | 1 | yes | characteristic |
| 96 | C2×#170 | — | 2 | 2 | no | — |
| 97 | C2×#170 | — | 2 | 4 | no | — |
| 98 | C2×#170 | C2×D4 | 1 | 2 | yes | — |
| 99 | C2×#170 | C2×D4 | 1 | 2 | yes | — |
| 100 | #700 | — | 4 | 8 | no | — |
| 101 | D5⋊C8 | — | 2 | 4 | no | — |
| 102 | D5⋊C8 | — | 2 | 4 | no | — |
| 103 | C5×Q16 | — | 2 | 2 | no | — |
| 104 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 105 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 106 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 107 | C5×C4⋊C4 | C22×C4 | 1 | 2 | yes | — |
| 108 | C42×C5 | — | 2 | 2 | no | — |
| 109 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 110 | #688 | — | 2 | 4 | no | — |
| 111 | #685 | — | 2 | 2 | no | — |
| 112 | #685 | — | 2 | 2 | no | — |
| 113 | #685 | — | 2 | 2 | no | — |
| 114 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 115 | C4⋊Dic5 | C22×C4 | 1 | 2 | yes | — |
| 116 | C4×Dic5 | — | 2 | 2 | no | — |
| 117 | C4×Dic5 | — | 2 | 2 | no | — |
| 118 | C4×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 119 | C4×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 120 | C4×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 121 | C2×#168 | C22×C4 | 1 | 1 | yes | characteristic |
| 122 | C2×#168 | C22×C4 | 1 | 1 | yes | characteristic |
| 123 | D40 | — | 2 | 2 | no | — |
| 124 | C8×D5 | — | 2 | 2 | no | — |
| 125 | C8×D5 | M16 | 1 | 4 | yes | complemented |
| 126 | C2×#135 | — | 5 | 1 | no | — |
| 127 | C4×#41 | — | 5 | 2 | no | — |
| 128 | #446 | — | 5 | 4 | no | — |
| 129 | C4×Q16 | — | 5 | 1 | no | — |
| 130 | C4×Q16 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 131 | C4×SD16 | — | 5 | 2 | no | — |
| 132 | C4×SD16 | — | 5 | 2 | no | — |
| 133 | C4×D8 | — | 5 | 1 | no | — |
| 134 | C4×D8 | — | 5 | 1 | no | — |
| 135 | #436 | — | 5 | 4 | no | — |
| 136 | #435 | — | 10 | 4 | no | — |
| 137 | #431 | — | 5 | 2 | no | — |
| 138 | #431 | — | 10 | 2 | no | — |
| 139 | #427 | — | 5 | 1 | no | — |
| 140 | #416 | — | 5 | 2 | no | — |
| 141 | C4×M16 | — | 10 | 2 | no | — |
| 142 | C2×C4×C8 | — | 5 | 1 | no | — |
| 143 | C2×C4×C8 | — | 5 | 1 | no | — |
| 144 | C2×C4×C8 | — | 5 | 1 | no | — |
| 145 | #335 | — | 5 | 2 | no | — |
| 146 | #334 | — | 5 | 2 | no | — |
| 147 | Q8⋊C8 | — | 5 | 4 | no | — |
| 148 | D4⋊C8 | — | 5 | 4 | no | — |
| 149 | C82 | — | 5 | 2 | no | — |
| 150 | C82 | — | 5 | 2 | no | — |
| 151 | C22×D5 | — | 2 | 2 | no | — |
| 152 | C22×D5 | C4×D4 | 1 | 1 | yes | characteristic |
| 153 | C5×Q8 | — | 2 | 2 | no | — |
| 154 | C5×Q8 | — | 2 | 2 | no | — |
| 155 | C2×C4×C5 | — | 2 | 2 | no | — |
| 156 | C2×C4×C5 | — | 2 | 2 | no | — |
| 157 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 158 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 159 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 160 | C2×Dic5 | — | 2 | 2 | no | — |
| 161 | C2×Dic5 | C4×D4 | 1 | 1 | yes | characteristic |
| 162 | C2×Dic5 | C22×D4 | 1 | 1 | yes | characteristic |
| 163 | C2×Dic5 | C2×#41 | 1 | 1 | yes | characteristic |
| 164 | D20 | — | 2 | 2 | no | — |
| 165 | D20 | — | 2 | 2 | no | — |
| 166 | D20 | — | 4 | 4 | no | — |
| 167 | C4×D5 | — | 2 | 2 | no | — |
| 168 | C4×D5 | — | 2 | 2 | no | — |
| 169 | C4×D5 | — | 4 | 4 | no | — |
| 170 | C4×D5 | — | 4 | 4 | no | — |
| 171 | C4×D5 | — | 4 | 4 | no | — |
| 172 | C4×D5 | C2×M16 | 1 | 2 | yes | — |
| 173 | C4×D5 | #131 | 1 | 2 | yes | — |
| 174 | #170 | — | 2 | 4 | no | — |
| 175 | #170 | — | 2 | 2 | no | — |
| 176 | #170 | — | 4 | 4 | no | — |
| 177 | #170 | #135 | 1 | 4 | yes | complemented |
| 178 | C5×C8 | — | 2 | 1 | no | — |
| 179 | C5×C8 | C2×M16 | 1 | 2 | yes | — |
| 180 | #168 | — | 2 | 1 | no | — |
| 181 | #168 | C2×M16 | 1 | 2 | yes | — |
| 182 | C2×#41 | — | 5 | 2 | no | — |
| 183 | #135 | — | 10 | 4 | no | — |
| 184 | C2×Q16 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 185 | C2×SD16 | — | 5 | 2 | no | — |
| 186 | C2×D8 | — | 5 | 1 | no | — |
| 187 | C2×M16 | — | 10 | 4 | no | — |
| 188 | C22×C8 | — | 5 | 1 | no | — |
| 189 | C22×C8 | — | 5 | 2 | no | — |
| 190 | C22×C8 | — | 5 | 1 | no | — |
| 191 | C22×C8 | — | 10 | 1 | no | — |
| 192 | C4×Q8 | — | 5 | 2 | no | — |
| 193 | C4×Q8 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 194 | C4×D4 | — | 5 | 2 | no | — |
| 195 | C4×D4 | — | 5 | 2 | no | — |
| 196 | C4×D4 | — | 10 | 2 | no | — |
| 197 | C4×D4 | — | 10 | 2 | no | — |
| 198 | #117 | — | 5 | 2 | no | — |
| 199 | #117 | — | 10 | 2 | no | — |
| 200 | C2×C42 | — | 5 | 1 | no | — |
| 201 | C2×C42 | — | 10 | 2 | no | — |
| 202 | #107 | — | 5 | 1 | no | — |
| 203 | #107 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 204 | #106 | — | 5 | 2 | no | — |
| 205 | C4⋊C8 | — | 5 | 4 | no | — |
| 206 | C4⋊C8 | — | 10 | 4 | no | — |
| 207 | #103 | — | 5 | 2 | no | — |
| 208 | #103 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 209 | #102 | — | 5 | 2 | no | — |
| 210 | #102 | — | 5 | 2 | no | — |
| 211 | C22⋊C8 | — | 10 | 4 | no | — |
| 212 | C22⋊C8 | — | 10 | 2 | no | — |
| 213 | #97 | — | 10 | 4 | no | — |
| 214 | C4×C8 | — | 5 | 2 | no | — |
| 215 | C4×C8 | — | 5 | 1 | no | — |
| 216 | C4×C8 | — | 5 | 1 | no | — |
| 217 | C4×C8 | — | 5 | 1 | no | — |
| 218 | C4×C8 | — | 5 | 1 | no | — |
| 219 | C4×C8 | — | 5 | 2 | no | — |
| 220 | C4×C8 | — | 5 | 1 | no | — |
| 221 | C4×C8 | — | 10 | 1 | no | — |
| 222 | C4×C8 | — | 10 | 2 | no | — |
| 223 | C4×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 224 | C22×C5 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 225 | C2×D5 | — | 2 | 1 | no | — |
| 226 | C2×D5 | — | 2 | 1 | no | — |
| 227 | C2×D5 | — | 4 | 2 | no | — |
| 228 | C2×D5 | — | 4 | 2 | no | — |
| 229 | C2×D5 | — | 4 | 2 | no | — |
| 230 | C2×D5 | #436 | 1 | 2 | yes | — |
| 231 | C4×C5 | — | 2 | 1 | no | — |
| 232 | C4×C5 | — | 4 | 2 | no | — |
| 233 | C4×C5 | — | 4 | 2 | no | — |
| 234 | C4×C5 | #443 | 1 | 2 | yes | — |
| 235 | C4×C5 | C22×M16 | 1 | 1 | yes | characteristic |
| 236 | C4×C5 | C2×#131 | 1 | 1 | yes | characteristic |
| 237 | Dic5 | — | 2 | 1 | no | — |
| 238 | Dic5 | — | 4 | 2 | no | — |
| 239 | Dic5 | #436 | 1 | 2 | yes | — |
| 240 | Dic5 | C2×#135 | 1 | 2 | yes | — |
| 241 | #41 | — | 10 | 4 | no | — |
| 242 | #41 | — | 10 | 4 | no | — |
| 243 | C2×Q8 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 244 | C2×D4 | — | 5 | 2 | no | — |
| 245 | C2×D4 | — | 10 | 2 | no | — |
| 246 | C22×C4 | — | 5 | 1 | no | — |
| 247 | C22×C4 | — | 5 | 1 | no | — |
| 248 | C22×C4 | — | 5 | 1 | no | — |
| 249 | C22×C4 | — | 10 | 2 | no | — |
| 250 | C22×C4 | — | 10 | 2 | no | — |
| 251 | C22×C4 | — | 10 | 2 | no | — |
| 252 | Q16 | — | 2 | 2 | no | — |
| 253 | SD16 | — | 10 | 4 | no | — |
| 254 | D8 | — | 10 | 2 | no | — |
| 255 | M16 | — | 20 | 8 | no | — |
| 256 | C2×C8 | — | 5 | 2 | no | — |
| 257 | C2×C8 | — | 5 | 1 | no | — |
| 258 | C2×C8 | — | 5 | 4 | no | — |
| 259 | C2×C8 | — | 5 | 2 | no | — |
| 260 | C2×C8 | — | 5 | 1 | no | — |
| 261 | C2×C8 | — | 10 | 2 | no | — |
| 262 | C2×C8 | — | 10 | 4 | no | — |
| 263 | C2×C8 | — | 10 | 2 | no | — |
| 264 | C2×C8 | — | 10 | 4 | no | — |
| 265 | C2×C8 | — | 10 | 4 | no | — |
| 266 | C2×C8 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 267 | C2×C8 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 268 | C4⋊C4 | — | 2 | 2 | no | — |
| 269 | C4⋊C4 | — | 5 | 2 | no | — |
| 270 | C4⋊C4 | — | 10 | 2 | no | — |
| 271 | C4⋊C4 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 272 | C22⋊C4 | — | 10 | 2 | no | — |
| 273 | C22⋊C4 | — | 10 | 2 | no | — |
| 274 | C22⋊C4 | — | 10 | 2 | no | — |
| 275 | C42 | — | 2 | 2 | no | — |
| 276 | C42 | — | 5 | 1 | no | — |
| 277 | C42 | — | 5 | 1 | no | — |
| 278 | C42 | — | 5 | 1 | no | — |
| 279 | C42 | — | 10 | 2 | no | — |
| 280 | C42 | — | 10 | 2 | no | — |
| 281 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 282 | C2×C5 | C2×#436 | 1 | 1 | yes | characteristic |
| 283 | C2×C5 | C4×#135 | 1 | 1 | yes | characteristic |
| 284 | C2×C5 | C2×#443 | 1 | 1 | yes | characteristic |
| 285 | D5 | — | 2 | 2 | no | — |
| 286 | D5 | — | 8 | 2 | no | — |
| 287 | C23 | — | 5 | 1 | no | — |
| 288 | C23 | — | 10 | 2 | no | — |
| 289 | Q8 | — | 2 | 2 | no | — |
| 290 | Q8 | — | 2 | 2 | no | — |
| 291 | D4 | — | 10 | 2 | no | — |
| 292 | D4 | — | 10 | 2 | no | — |
| 293 | D4 | — | 20 | 4 | no | — |
| 294 | C2×C4 | — | 2 | 2 | no | — |
| 295 | C2×C4 | — | 2 | 2 | no | — |
| 296 | C2×C4 | — | 5 | 1 | no | — |
| 297 | C2×C4 | — | 5 | 1 | no | — |
| 298 | C2×C4 | — | 5 | 2 | no | — |
| 299 | C2×C4 | — | 5 | 2 | no | — |
| 300 | C2×C4 | — | 5 | 1 | no | — |
| 301 | C2×C4 | — | 10 | 2 | no | — |
| 302 | C2×C4 | — | 10 | 2 | no | — |
| 303 | C2×C4 | — | 10 | 2 | no | — |
| 304 | C2×C4 | — | 20 | 4 | no | — |
| 305 | C2×C4 | — | 20 | 4 | no | — |
| 306 | C2×C4 | — | 20 | 4 | no | — |
| 307 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 308 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 309 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 310 | C8 | — | 2 | 1 | no | — |
| 311 | C8 | — | 5 | 4 | no | — |
| 312 | C8 | — | 5 | 2 | no | — |
| 313 | C8 | — | 10 | 2 | no | — |
| 314 | C8 | — | 10 | 4 | no | — |
| 315 | C8 | — | 10 | 1 | no | — |
| 316 | C8 | — | 20 | 4 | no | — |
| 317 | C8 | D5⋊M16 | 1 | 2 | yes | — |
| 318 | C5 | #17886 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 319 | C22 | — | 5 | 2 | no | — |
| 320 | C22 | — | 10 | 1 | no | — |
| 321 | C22 | — | 10 | 1 | no | — |
| 322 | C22 | — | 20 | 2 | no | — |
| 323 | C22 | — | 20 | 2 | no | — |
| 324 | C22 | — | 20 | 2 | no | — |
| 325 | C22 | C2×D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 326 | C4 | — | 2 | 1 | no | — |
| 327 | C4 | — | 4 | 2 | no | — |
| 328 | C4 | — | 4 | 2 | no | — |
| 329 | C4 | — | 5 | 2 | no | — |
| 330 | C4 | — | 5 | 2 | no | — |
| 331 | C4 | — | 10 | 1 | no | — |
| 332 | C4 | — | 20 | 2 | no | — |
| 333 | C4 | #65740 | 1 | 2 | yes | — |
| 334 | C4 | C2×D5⋊M16 | 1 | 1 | yes | characteristic |
| 335 | C4 | D5⋊#131 | 1 | 1 | yes | characteristic |
| 336 | C2 | — | 10 | 2 | no | — |
| 337 | C2 | — | 40 | 2 | no | — |
| 338 | C2 | D5⋊#436 | 1 | 1 | yes | characteristic |
| 339 | C2 | C2×#65740 | 1 | 1 | yes | characteristic |
| 340 | C2 | AGL(1,𝔽5)×#135 | 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