← #14281761
Subgroups — #14281761
Order 1920.
496 conjugacy classes of subgroups.
The subgroup lattice has 316 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#91333 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11752407 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×#65748 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11746495 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #10604271 | — | 3 | 1 | no | maximal, Hall |
| 6 | C22×Dic30 | C4 | 1 | 1 | yes | characteristic |
| 7 | C2×C5×C4⋊Dic3 | C4 | 1 | 1 | yes | characteristic |
| 8 | C2×C3×C4×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#6669 | — | 2 | 1 | no | — |
| 10 | C2×Dic3×Dic5 | — | 2 | 1 | no | — |
| 11 | #91333 | — | 2 | 1 | no | — |
| 12 | #91333 | — | 2 | 1 | no | — |
| 13 | #91333 | C4 | 1 | 2 | yes | — |
| 14 | #91333 | C22 | 1 | 2 | yes | — |
| 15 | #91241 | — | 2 | 1 | no | — |
| 16 | C4⋊(C3⋊#170) | C22 | 1 | 2 | yes | — |
| 17 | C4×C3⋊#170 | — | 2 | 1 | no | — |
| 18 | C3×C22⋊#170 | — | 2 | 1 | no | — |
| 19 | C3×C4⋊#170 | — | 2 | 1 | no | — |
| 20 | C3×C4×#170 | C22 | 1 | 2 | yes | — |
| 21 | #70767 | — | 5 | 1 | no | maximal, Hall |
| 22 | C2×#4033 | — | 3 | 1 | no | — |
| 23 | #65748 | — | 3 | 1 | no | — |
| 24 | #65748 | D3 | 1 | 1 | yes | characteristic |
| 25 | #64872 | — | 3 | 4 | no | — |
| 26 | C22×C3×C4×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | C22×Dic15 | — | 2 | 1 | no | — |
| 28 | C2×Dic30 | — | 2 | 1 | no | — |
| 29 | C2×Dic30 | — | 2 | 2 | no | — |
| 30 | C2×Dic30 | C8 | 1 | 4 | yes | complemented |
| 31 | C2×Dic30 | C2×C4 | 1 | 2 | yes | — |
| 32 | C22×C5×Dic3 | — | 2 | 1 | no | — |
| 33 | C22×C3×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 34 | C22×C3×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 35 | C2×C3⋊#170 | — | 2 | 2 | no | — |
| 36 | C2×C3×#170 | — | 2 | 2 | no | — |
| 37 | C5×C4⋊Dic3 | — | 2 | 1 | no | — |
| 38 | C5×C4⋊Dic3 | C2×C4 | 1 | 2 | yes | — |
| 39 | C3×C4×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C3×C4×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 41 | C3×C4×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 42 | C3×C4×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 43 | #6669 | — | 2 | 2 | no | — |
| 44 | #6669 | — | 2 | 2 | no | — |
| 45 | Dic3×Dic5 | — | 2 | 2 | no | — |
| 46 | Dic3×Dic5 | — | 2 | 2 | no | — |
| 47 | C2×C4×Dic6 | — | 5 | 1 | no | — |
| 48 | C3×#431 | — | 5 | 1 | no | — |
| 49 | #4925 | — | 5 | 1 | no | — |
| 50 | #4456 | — | 5 | 4 | no | — |
| 51 | C22×Dic10 | — | 3 | 1 | no | — |
| 52 | C2×C5×C4⋊C4 | — | 3 | 1 | no | — |
| 53 | C2×#683 | — | 6 | 1 | no | — |
| 54 | C2×C4×Dic5 | — | 6 | 1 | no | — |
| 55 | C2×C4×Dic5 | C2×D3 | 1 | 1 | yes | characteristic |
| 56 | #4033 | — | 3 | 2 | no | — |
| 57 | #4033 | — | 3 | 2 | no | — |
| 58 | #4033 | — | 6 | 1 | no | — |
| 59 | #4033 | — | 6 | 1 | no | — |
| 60 | C22⋊#170 | — | 2 | 1 | no | — |
| 61 | C22⋊#170 | — | 6 | 1 | no | — |
| 62 | C4⋊#170 | — | 2 | 1 | no | — |
| 63 | C4⋊#170 | — | 3 | 2 | no | — |
| 64 | C4×#170 | — | 6 | 1 | no | — |
| 65 | C4×#170 | C2×D3 | 1 | 2 | yes | — |
| 66 | #1479 | — | 15 | 1 | no | Sylow |
| 67 | C23×C3×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 68 | C2×C3×C4×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 69 | C2×C3×C4×C5 | C2×C8 | 1 | 2 | yes | — |
| 70 | C2×C3×C4×C5 | M16 | 1 | 2 | yes | — |
| 71 | C2×C3×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 72 | C2×Dic15 | — | 2 | 4 | no | — |
| 73 | C2×Dic15 | — | 2 | 2 | no | — |
| 74 | Dic30 | — | 2 | 4 | no | — |
| 75 | Dic30 | — | 2 | 2 | no | — |
| 76 | Dic30 | C2×C8 | 1 | 4 | yes | — |
| 77 | C2×C5×Dic3 | — | 2 | 2 | no | — |
| 78 | C2×C5×Dic3 | — | 4 | 2 | no | — |
| 79 | C2×C3×Dic5 | — | 2 | 2 | no | — |
| 80 | C2×C3×Dic5 | — | 4 | 1 | no | — |
| 81 | C2×C3×Dic5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 82 | C2×C3×Dic5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 83 | C2×C3×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C3×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C3⋊#170 | — | 4 | 2 | no | — |
| 86 | C3×#170 | — | 4 | 2 | no | — |
| 87 | C22×Dic6 | — | 5 | 1 | no | — |
| 88 | C2×C3×C42 | — | 5 | 1 | no | — |
| 89 | C2×C4⋊Dic3 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 90 | C2×Dic3⋊C4 | — | 10 | 1 | no | — |
| 91 | C2×C4×Dic3 | — | 10 | 1 | no | — |
| 92 | C4×Dic6 | — | 5 | 2 | no | — |
| 93 | C4×Dic6 | — | 5 | 2 | no | — |
| 94 | C4×Dic6 | — | 10 | 1 | no | — |
| 95 | C4×Dic6 | — | 10 | 1 | no | — |
| 96 | C3×C4⋊C8 | — | 10 | 1 | no | — |
| 97 | C3×C22⋊C8 | — | 10 | 1 | no | — |
| 98 | C3×C4×C8 | — | 5 | 2 | no | — |
| 99 | #830 | — | 10 | 1 | no | — |
| 100 | C4⋊(C3⋊C8) | — | 5 | 2 | no | — |
| 101 | C4×C3⋊C8 | — | 10 | 1 | no | — |
| 102 | C22×C4×C5 | — | 6 | 1 | no | — |
| 103 | C22×C4×C5 | C4×D3 | 1 | 1 | yes | characteristic |
| 104 | C22×Dic5 | — | 6 | 1 | no | — |
| 105 | C22×Dic5 | D12 | 1 | 1 | yes | characteristic |
| 106 | C22×Dic5 | #67 | 1 | 1 | yes | characteristic |
| 107 | C2×Dic10 | — | 3 | 4 | no | — |
| 108 | C2×Dic10 | — | 3 | 2 | no | — |
| 109 | C2×Dic10 | — | 6 | 2 | no | — |
| 110 | C2×Dic10 | — | 6 | 1 | no | — |
| 111 | C2×#170 | — | 2 | 2 | no | — |
| 112 | C2×#170 | — | 6 | 2 | no | — |
| 113 | C5×C4⋊C4 | — | 3 | 2 | no | — |
| 114 | C5×C4⋊C4 | — | 6 | 1 | no | — |
| 115 | #683 | — | 6 | 2 | no | — |
| 116 | #683 | — | 6 | 2 | no | — |
| 117 | C4×Dic5 | — | 6 | 2 | no | — |
| 118 | C4×Dic5 | — | 6 | 2 | no | — |
| 119 | C4×Dic5 | C4×D3 | 1 | 1 | yes | characteristic |
| 120 | C4×Dic5 | D12 | 1 | 1 | yes | characteristic |
| 121 | C4×Dic5 | #67 | 1 | 1 | yes | characteristic |
| 122 | C4×Dic5 | C22×D3 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×Q8 | — | 15 | 1 | no | — |
| 124 | #431 | — | 5 | 1 | no | — |
| 125 | #431 | — | 15 | 1 | no | — |
| 126 | Q8⋊C8 | — | 15 | 4 | no | — |
| 127 | C22×C3×C5 | — | 2 | 1 | no | — |
| 128 | C22×C3×C5 | — | 2 | 1 | no | — |
| 129 | C22×C3×C5 | C22⋊C8 | 1 | 2 | yes | — |
| 130 | C22×C3×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 131 | C3×C4×C5 | C22⋊C8 | 1 | 2 | yes | — |
| 132 | C3×C4×C5 | C22×C8 | 1 | 1 | yes | characteristic |
| 133 | C3×C4×C5 | C2×M16 | 1 | 1 | yes | characteristic |
| 134 | Dic15 | — | 2 | 4 | no | — |
| 135 | C3×Dic5 | — | 2 | 1 | no | — |
| 136 | C3×Dic5 | — | 2 | 1 | no | — |
| 137 | C3×Dic5 | — | 2 | 1 | no | — |
| 138 | C3×Dic5 | #137 | 1 | 2 | yes | — |
| 139 | C5×Dic3 | — | 4 | 2 | no | — |
| 140 | C22×C3×C4 | — | 5 | 1 | no | — |
| 141 | C22×C3×C4 | — | 5 | 1 | no | — |
| 142 | C22×C3×C4 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 143 | C22×Dic3 | — | 2 | 1 | no | — |
| 144 | C22×Dic3 | — | 10 | 1 | no | — |
| 145 | C2×Dic6 | — | 5 | 4 | no | — |
| 146 | C2×Dic6 | — | 5 | 2 | no | — |
| 147 | C2×Dic6 | — | 10 | 2 | no | — |
| 148 | C2×Dic6 | — | 10 | 1 | no | — |
| 149 | C2×C3×C8 | — | 10 | 2 | no | — |
| 150 | C3×C42 | — | 5 | 1 | no | — |
| 151 | C3×C42 | — | 5 | 1 | no | — |
| 152 | C3×C42 | — | 5 | 1 | no | — |
| 153 | C3×C42 | — | 5 | 1 | no | — |
| 154 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 155 | C4⋊Dic3 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 156 | Dic3⋊C4 | — | 10 | 2 | no | — |
| 157 | Dic3⋊C4 | — | 10 | 2 | no | — |
| 158 | C4×Dic3 | — | 10 | 2 | no | — |
| 159 | C4×Dic3 | — | 10 | 2 | no | — |
| 160 | C2×C3⋊C8 | — | 10 | 2 | no | — |
| 161 | C23×C5 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 162 | C2×C4×C5 | — | 6 | 2 | no | — |
| 163 | C2×C4×C5 | — | 12 | 2 | no | — |
| 164 | C2×C4×C5 | D3×C8 | 1 | 2 | yes | — |
| 165 | C2×C4×C5 | ΓL(1,𝔽25) | 1 | 2 | yes | — |
| 166 | C2×C4×C5 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 167 | C2×C4×C5 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 168 | C2×Dic5 | — | 2 | 2 | no | — |
| 169 | C2×Dic5 | — | 4 | 1 | no | — |
| 170 | C2×Dic5 | — | 6 | 4 | no | — |
| 171 | C2×Dic5 | — | 6 | 2 | no | — |
| 172 | C2×Dic5 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 173 | C2×Dic5 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 174 | C2×Dic5 | C2×D12 | 1 | 1 | yes | characteristic |
| 175 | C2×Dic5 | C2×#67 | 1 | 1 | yes | characteristic |
| 176 | Dic10 | — | 3 | 4 | no | — |
| 177 | Dic10 | — | 6 | 4 | no | — |
| 178 | Dic10 | — | 6 | 2 | no | — |
| 179 | #170 | — | 4 | 2 | no | — |
| 180 | #170 | — | 12 | 2 | no | — |
| 181 | C22×Q8 | — | 15 | 1 | no | — |
| 182 | C4×Q8 | — | 15 | 2 | no | — |
| 183 | C4×Q8 | — | 15 | 2 | no | — |
| 184 | C4×Q8 | — | 30 | 1 | no | — |
| 185 | C4×Q8 | — | 30 | 1 | no | — |
| 186 | C2×C4⋊C4 | — | 3 | 1 | no | — |
| 187 | C2×C4⋊C4 | — | 30 | 1 | no | — |
| 188 | C2×C42 | — | 5 | 1 | no | — |
| 189 | C2×C42 | — | 30 | 1 | no | — |
| 190 | C4⋊C8 | — | 10 | 1 | no | — |
| 191 | C4⋊C8 | — | 15 | 2 | no | — |
| 192 | C22⋊C8 | — | 10 | 1 | no | — |
| 193 | C22⋊C8 | — | 30 | 1 | no | — |
| 194 | C4×C8 | — | 5 | 2 | no | — |
| 195 | C4×C8 | — | 30 | 1 | no | — |
| 196 | C2×C3×C5 | — | 2 | 2 | no | — |
| 197 | C2×C3×C5 | C2×C22⋊C8 | 1 | 1 | yes | characteristic |
| 198 | C2×C3×C5 | #419 | 1 | 1 | yes | characteristic |
| 199 | C2×C3×C5 | #421 | 1 | 1 | yes | characteristic |
| 200 | C23×C3 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 201 | C2×C3×C4 | — | 5 | 1 | no | — |
| 202 | C2×C3×C4 | — | 5 | 1 | no | — |
| 203 | C2×C3×C4 | — | 5 | 1 | no | — |
| 204 | C2×C3×C4 | — | 5 | 1 | no | — |
| 205 | C2×C3×C4 | — | 10 | 2 | no | — |
| 206 | C2×C3×C4 | — | 20 | 1 | no | — |
| 207 | C2×C3×C4 | D5⋊C8 | 1 | 2 | yes | — |
| 208 | C2×C3×C4 | #700 | 1 | 2 | yes | — |
| 209 | C2×C3×C4 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 210 | C2×C3×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 211 | C2×Dic3 | — | 2 | 2 | no | — |
| 212 | C2×Dic3 | — | 4 | 2 | no | — |
| 213 | C2×Dic3 | — | 10 | 2 | no | — |
| 214 | C2×Dic3 | — | 10 | 4 | no | — |
| 215 | Dic6 | — | 5 | 4 | no | — |
| 216 | Dic6 | — | 10 | 2 | no | — |
| 217 | Dic6 | — | 10 | 4 | no | — |
| 218 | C3×C8 | — | 20 | 2 | no | — |
| 219 | C3⋊C8 | — | 20 | 2 | no | — |
| 220 | C22×C5 | — | 2 | 1 | no | — |
| 221 | C22×C5 | — | 2 | 1 | no | — |
| 222 | C22×C5 | (C2×D3)⋊C8 | 1 | 2 | yes | — |
| 223 | C22×C5 | C2×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 224 | C4×C5 | — | 12 | 2 | no | — |
| 225 | C4×C5 | (C2×D3)⋊C8 | 1 | 2 | yes | — |
| 226 | C4×C5 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 227 | C4×C5 | C2×ΓL(1,𝔽25) | 1 | 1 | yes | characteristic |
| 228 | Dic5 | — | 2 | 1 | no | — |
| 229 | Dic5 | — | 2 | 1 | no | — |
| 230 | Dic5 | — | 2 | 1 | no | — |
| 231 | Dic5 | — | 6 | 4 | no | — |
| 232 | Dic5 | #909 | 1 | 2 | yes | — |
| 233 | C2×Q8 | — | 15 | 4 | no | — |
| 234 | C2×Q8 | — | 15 | 2 | no | — |
| 235 | C2×Q8 | — | 30 | 1 | no | — |
| 236 | C2×Q8 | — | 30 | 2 | no | — |
| 237 | C22×C4 | — | 5 | 1 | no | — |
| 238 | C22×C4 | — | 5 | 1 | no | — |
| 239 | C22×C4 | — | 6 | 1 | no | — |
| 240 | C22×C4 | — | 30 | 1 | no | — |
| 241 | C22×C4 | D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 242 | C2×C8 | — | 10 | 2 | no | — |
| 243 | C2×C8 | — | 30 | 2 | no | — |
| 244 | C4⋊C4 | — | 3 | 2 | no | — |
| 245 | C4⋊C4 | — | 6 | 1 | no | — |
| 246 | C4⋊C4 | — | 30 | 2 | no | — |
| 247 | C4⋊C4 | — | 30 | 2 | no | — |
| 248 | C42 | — | 5 | 1 | no | — |
| 249 | C42 | — | 5 | 1 | no | — |
| 250 | C42 | — | 5 | 1 | no | — |
| 251 | C42 | — | 5 | 1 | no | — |
| 252 | C42 | — | 30 | 2 | no | — |
| 253 | C42 | — | 30 | 2 | no | — |
| 254 | C3×C5 | #1479 | 1 | 1 | yes | characteristic, complemented, Hall |
| 255 | C22×C3 | — | 2 | 1 | no | — |
| 256 | C22×C3 | — | 2 | 1 | no | — |
| 257 | C22×C3 | #4003 | 1 | 2 | yes | — |
| 258 | C22×C3 | C2×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 259 | C3×C4 | — | 5 | 2 | no | — |
| 260 | C3×C4 | — | 10 | 1 | no | — |
| 261 | C3×C4 | — | 10 | 1 | no | — |
| 262 | C3×C4 | — | 10 | 1 | no | — |
| 263 | C3×C4 | #4003 | 1 | 2 | yes | — |
| 264 | C3×C4 | C2×D5⋊C8 | 1 | 1 | yes | characteristic |
| 265 | C3×C4 | C2×#700 | 1 | 1 | yes | characteristic |
| 266 | Dic3 | — | 4 | 2 | no | — |
| 267 | Dic3 | — | 10 | 4 | no | — |
| 268 | C2×C5 | — | 2 | 2 | no | — |
| 269 | C2×C5 | #5005 | 1 | 1 | yes | characteristic |
| 270 | C2×C5 | C2×(C2×D3)⋊C8 | 1 | 1 | yes | characteristic |
| 271 | C2×C5 | #5120 | 1 | 1 | yes | characteristic |
| 272 | C23 | (C2×D3)⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 273 | Q8 | — | 15 | 4 | no | — |
| 274 | Q8 | — | 30 | 2 | no | — |
| 275 | Q8 | — | 30 | 4 | no | — |
| 276 | C2×C4 | — | 5 | 1 | no | — |
| 277 | C2×C4 | — | 5 | 1 | no | — |
| 278 | C2×C4 | — | 5 | 1 | no | — |
| 279 | C2×C4 | — | 5 | 1 | no | — |
| 280 | C2×C4 | — | 6 | 2 | no | — |
| 281 | C2×C4 | — | 10 | 2 | no | — |
| 282 | C2×C4 | — | 12 | 2 | no | — |
| 283 | C2×C4 | — | 20 | 1 | no | — |
| 284 | C2×C4 | — | 30 | 2 | no | — |
| 285 | C2×C4 | — | 30 | 4 | no | — |
| 286 | C2×C4 | (C2×D3)⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | #6738 | 1 | 2 | yes | — |
| 288 | C2×C4 | #6740 | 1 | 2 | yes | — |
| 289 | C2×C4 | C2×D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 290 | C8 | — | 20 | 2 | no | — |
| 291 | C8 | — | 60 | 2 | no | — |
| 292 | C2×C3 | — | 2 | 2 | no | — |
| 293 | C2×C3 | C2×#4003 | 1 | 1 | yes | characteristic |
| 294 | C2×C3 | #65769 | 1 | 1 | yes | characteristic |
| 295 | C2×C3 | #65782 | 1 | 1 | yes | characteristic |
| 296 | C5 | #70767 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 297 | C22 | — | 2 | 1 | no | — |
| 298 | C22 | — | 2 | 1 | no | — |
| 299 | C22 | #91171 | 1 | 2 | yes | — |
| 300 | C22 | C2×(C2×D3)⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 301 | C4 | — | 5 | 2 | no | — |
| 302 | C4 | — | 10 | 1 | no | — |
| 303 | C4 | — | 10 | 1 | no | — |
| 304 | C4 | — | 10 | 1 | no | — |
| 305 | C4 | — | 12 | 2 | no | — |
| 306 | C4 | — | 30 | 4 | no | — |
| 307 | C4 | #91171 | 1 | 2 | yes | — |
| 308 | C4 | C2×#6738 | 1 | 1 | yes | characteristic |
| 309 | C4 | C2×#6740 | 1 | 1 | yes | characteristic |
| 310 | C3 | #10604271 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 311 | C2 | — | 2 | 2 | no | — |
| 312 | C2 | #11751712 | 1 | 1 | yes | characteristic |
| 313 | C2 | #11751719 | 1 | 1 | yes | characteristic |
| 314 | C2 | C2×#91171 | 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