← #14305025
Subgroups — #14305025
Order 1920.
496 conjugacy classes of subgroups.
The subgroup lattice has 300 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C4⋊(C3⋊#170) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C3×#4002 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#91177 | C2 | 1 | 2 | yes | maximal |
| 4 | #11748638 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #11746570 | C2 | 1 | 2 | yes | maximal |
| 6 | #10603797 | — | 3 | 1 | no | maximal, Hall |
| 7 | C22×C3⋊#170 | C22 | 1 | 2 | yes | — |
| 8 | C22×C3×#170 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×C3×#170 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×C4×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#6668 | C4 | 1 | 2 | yes | — |
| 12 | C2×Dic3×Dic5 | C22 | 1 | 2 | yes | — |
| 13 | C4⋊(C3⋊#170) | — | 2 | 2 | no | — |
| 14 | C3×#4002 | — | 2 | 2 | no | — |
| 15 | #91177 | — | 2 | 4 | no | — |
| 16 | #91115 | C4 | 1 | 1 | yes | characteristic |
| 17 | C5×C3⋊#95 | C4 | 1 | 1 | yes | characteristic |
| 18 | #69836 | — | 5 | 1 | no | maximal, Hall |
| 19 | C2×#4004 | — | 3 | 2 | no | — |
| 20 | C2×#4002 | D3 | 1 | 1 | yes | characteristic |
| 21 | C2×C4⋊#170 | — | 3 | 1 | no | — |
| 22 | #64938 | — | 3 | 1 | no | — |
| 23 | #64918 | — | 3 | 2 | no | — |
| 24 | C22×C3×C4×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C22×Dic15 | C2×C4 | 1 | 2 | yes | — |
| 26 | C22×C5×Dic3 | C2×C4 | 1 | 2 | yes | — |
| 27 | C22×C3×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C22×C3×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 29 | C2×C3⋊#170 | — | 2 | 2 | no | — |
| 30 | C2×C3⋊#170 | — | 2 | 2 | no | — |
| 31 | C2×C3⋊#170 | — | 2 | 2 | no | — |
| 32 | C2×C3×#170 | — | 2 | 1 | no | — |
| 33 | C2×C3×#170 | — | 2 | 1 | no | — |
| 34 | C2×C3×#170 | — | 2 | 1 | no | — |
| 35 | C2×C3×#170 | — | 2 | 1 | no | — |
| 36 | C2×C3×#170 | D4 | 1 | 2 | yes | — |
| 37 | C2×C3×#170 | Q8 | 1 | 2 | yes | — |
| 38 | C3×C4×Dic5 | D4 | 1 | 2 | yes | — |
| 39 | C3×C4×Dic5 | Q8 | 1 | 2 | yes | — |
| 40 | #6668 | — | 2 | 4 | no | — |
| 41 | Dic3×Dic5 | — | 2 | 2 | no | — |
| 42 | Dic3×Dic5 | — | 2 | 2 | no | — |
| 43 | C2×C3×#97 | — | 5 | 1 | no | — |
| 44 | C2×Dic3⋊C8 | — | 5 | 2 | no | — |
| 45 | C4×C4⋊Dic3 | — | 5 | 1 | no | — |
| 46 | C2×C4⋊(C3⋊C8) | — | 5 | 1 | no | — |
| 47 | #4550 | — | 5 | 2 | no | — |
| 48 | C22×#170 | — | 3 | 2 | no | — |
| 49 | C22×#170 | C2×D3 | 1 | 1 | yes | characteristic |
| 50 | C22×#170 | C2×D3 | 1 | 1 | yes | characteristic |
| 51 | C2×#683 | — | 3 | 2 | no | — |
| 52 | C2×C4×Dic5 | — | 3 | 2 | no | — |
| 53 | C2×C4×Dic5 | C2×D3 | 1 | 1 | yes | characteristic |
| 54 | #4004 | — | 6 | 4 | no | — |
| 55 | #4002 | — | 2 | 2 | no | — |
| 56 | C4⋊#170 | — | 6 | 2 | no | — |
| 57 | C5×#95 | — | 3 | 1 | no | — |
| 58 | #3963 | — | 3 | 1 | no | — |
| 59 | #1940 | — | 15 | 1 | no | Sylow |
| 60 | C23×C3×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 61 | C2×C3×C4×C5 | — | 2 | 2 | no | — |
| 62 | C2×C3×C4×C5 | C4⋊C4 | 1 | 2 | yes | — |
| 63 | C2×Dic15 | — | 2 | 2 | no | — |
| 64 | C2×Dic15 | — | 2 | 2 | no | — |
| 65 | C2×Dic15 | M16 | 1 | 4 | yes | — |
| 66 | C2×C5×Dic3 | — | 2 | 2 | no | — |
| 67 | C2×C5×Dic3 | — | 2 | 2 | no | — |
| 68 | C2×C5×Dic3 | — | 2 | 2 | no | — |
| 69 | C2×C3×Dic5 | — | 2 | 2 | no | — |
| 70 | C2×C3×Dic5 | C4⋊C4 | 1 | 2 | yes | — |
| 71 | C2×C3×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C2×C3×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | C2×C3×Dic5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 74 | C2×C3×Dic5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 75 | C2×C3×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 76 | C2×C3×Dic5 | #41 | 1 | 1 | yes | characteristic |
| 77 | C3⋊#170 | — | 4 | 2 | no | — |
| 78 | C3×#170 | — | 2 | 2 | no | — |
| 79 | C3×#170 | — | 2 | 2 | no | — |
| 80 | C22×C3×C8 | — | 5 | 1 | no | — |
| 81 | C22×C3×C8 | — | 5 | 1 | no | — |
| 82 | C2×C3×C42 | — | 5 | 1 | no | — |
| 83 | C2×C4⋊Dic3 | — | 5 | 2 | no | — |
| 84 | C2×C4×Dic3 | — | 5 | 2 | no | — |
| 85 | C22×C3⋊C8 | — | 5 | 2 | no | — |
| 86 | C3×#97 | — | 10 | 2 | no | — |
| 87 | C3⋊#95 | — | 5 | 1 | no | — |
| 88 | C3⋊#95 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 89 | Dic3⋊C8 | — | 10 | 4 | no | — |
| 90 | C4⋊(C3⋊C8) | — | 10 | 2 | no | — |
| 91 | C22×C4×C5 | — | 3 | 2 | no | — |
| 92 | C22×C4×C5 | C4×D3 | 1 | 1 | yes | characteristic |
| 93 | C22×Dic5 | — | 3 | 2 | no | — |
| 94 | C22×Dic5 | C4×D3 | 1 | 1 | yes | characteristic |
| 95 | C22×Dic5 | C22×D3 | 1 | 1 | yes | characteristic |
| 96 | C2×#170 | — | 2 | 1 | no | — |
| 97 | C2×#170 | — | 2 | 1 | no | — |
| 98 | C2×#170 | — | 2 | 1 | no | — |
| 99 | C2×#170 | — | 2 | 1 | no | — |
| 100 | C2×#170 | — | 6 | 2 | no | — |
| 101 | C2×#170 | — | 6 | 2 | no | — |
| 102 | C2×#170 | — | 6 | 2 | no | — |
| 103 | C2×#170 | Dic6 | 1 | 2 | yes | — |
| 104 | C2×#170 | D12 | 1 | 2 | yes | — |
| 105 | #683 | — | 6 | 4 | no | — |
| 106 | C4×Dic5 | — | 6 | 2 | no | — |
| 107 | C4×Dic5 | — | 6 | 2 | no | — |
| 108 | C4×Dic5 | Dic6 | 1 | 2 | yes | — |
| 109 | C4×Dic5 | #67 | 1 | 2 | yes | — |
| 110 | C2×C4⋊C8 | — | 15 | 1 | no | — |
| 111 | C2×C4⋊C8 | — | 15 | 2 | no | — |
| 112 | C2×#97 | — | 5 | 1 | no | — |
| 113 | C4×C4⋊C4 | — | 15 | 1 | no | — |
| 114 | #336 | — | 15 | 2 | no | — |
| 115 | C22×C3×C5 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 116 | C22×C3×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 117 | C22×C3×C5 | C4×Q8 | 1 | 1 | yes | characteristic |
| 118 | C22×C3×C5 | C2×M16 | 1 | 2 | yes | — |
| 119 | C22×C3×C5 | #131 | 1 | 2 | yes | — |
| 120 | C3×C4×C5 | — | 2 | 2 | no | — |
| 121 | Dic15 | — | 2 | 4 | no | — |
| 122 | C3×Dic5 | — | 2 | 2 | no | — |
| 123 | C3×Dic5 | #121 | 1 | 1 | yes | characteristic |
| 124 | C3×Dic5 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 125 | C3×Dic5 | #125 | 1 | 1 | yes | characteristic |
| 126 | C3×Dic5 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 127 | C5×Dic3 | — | 4 | 2 | no | — |
| 128 | C22×C3×C4 | — | 5 | 1 | no | — |
| 129 | C22×C3×C4 | — | 5 | 1 | no | — |
| 130 | C22×C3×C4 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 131 | C22×Dic3 | — | 5 | 2 | no | — |
| 132 | C22×Dic3 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 133 | C2×C3×C8 | — | 5 | 2 | no | — |
| 134 | C2×C3×C8 | — | 5 | 2 | no | — |
| 135 | C2×C3×C8 | — | 10 | 1 | no | — |
| 136 | C2×C3×C8 | — | 10 | 1 | no | — |
| 137 | C2×C3×C8 | — | 10 | 1 | no | — |
| 138 | C2×C3×C8 | — | 10 | 1 | no | — |
| 139 | C3×C42 | — | 5 | 2 | no | — |
| 140 | C3×C42 | — | 5 | 2 | no | — |
| 141 | C4⋊Dic3 | — | 10 | 4 | no | — |
| 142 | C4×Dic3 | — | 10 | 2 | no | — |
| 143 | C4×Dic3 | — | 10 | 2 | no | — |
| 144 | C2×C3⋊C8 | — | 10 | 2 | no | — |
| 145 | C2×C3⋊C8 | — | 10 | 2 | no | — |
| 146 | C2×C3⋊C8 | — | 10 | 2 | no | — |
| 147 | C23×C5 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 148 | C2×C4×C5 | — | 2 | 2 | no | — |
| 149 | C2×C4×C5 | — | 6 | 2 | no | — |
| 150 | C2×C4×C5 | — | 6 | 2 | no | — |
| 151 | C2×C4×C5 | — | 6 | 2 | no | — |
| 152 | C2×C4×C5 | Dic3⋊C4 | 1 | 2 | yes | — |
| 153 | C2×Dic5 | — | 2 | 2 | no | — |
| 154 | C2×Dic5 | — | 3 | 4 | no | — |
| 155 | C2×Dic5 | — | 6 | 2 | no | — |
| 156 | C2×Dic5 | — | 6 | 2 | no | — |
| 157 | C2×Dic5 | Dic3⋊C4 | 1 | 2 | yes | — |
| 158 | C2×Dic5 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 159 | C2×Dic5 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 160 | C2×Dic5 | C2×D12 | 1 | 1 | yes | characteristic |
| 161 | C2×Dic5 | #235 | 1 | 1 | yes | characteristic |
| 162 | C2×Dic5 | #235 | 1 | 1 | yes | characteristic |
| 163 | C2×Dic5 | C2×#67 | 1 | 1 | yes | characteristic |
| 164 | #170 | — | 2 | 2 | no | — |
| 165 | #170 | — | 2 | 2 | no | — |
| 166 | #170 | — | 12 | 2 | no | — |
| 167 | C22×C8 | — | 5 | 1 | no | — |
| 168 | C22×C8 | — | 5 | 1 | no | — |
| 169 | C22×C8 | — | 15 | 2 | no | — |
| 170 | C2×C4⋊C4 | — | 15 | 2 | no | — |
| 171 | C2×C42 | — | 5 | 1 | no | — |
| 172 | C2×C42 | — | 15 | 2 | no | — |
| 173 | C4⋊C8 | — | 30 | 2 | no | — |
| 174 | C4⋊C8 | — | 30 | 4 | no | — |
| 175 | #97 | — | 10 | 2 | no | — |
| 176 | #95 | — | 3 | 1 | no | — |
| 177 | #95 | — | 15 | 1 | no | — |
| 178 | C2×C3×C5 | #389 | 1 | 1 | yes | characteristic |
| 179 | C2×C3×C5 | #423 | 1 | 1 | yes | characteristic |
| 180 | C2×C3×C5 | #424 | 1 | 1 | yes | characteristic |
| 181 | C2×C3×C5 | #435 | 1 | 2 | yes | — |
| 182 | C2×C3×C5 | #446 | 1 | 2 | yes | — |
| 183 | C23×C3 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 184 | C2×C3×C4 | — | 2 | 2 | no | — |
| 185 | C2×C3×C4 | — | 5 | 1 | no | — |
| 186 | C2×C3×C4 | — | 5 | 1 | no | — |
| 187 | C2×C3×C4 | — | 5 | 1 | no | — |
| 188 | C2×C3×C4 | — | 5 | 1 | no | — |
| 189 | C2×C3×C4 | — | 5 | 1 | no | — |
| 190 | C2×C3×C4 | — | 5 | 1 | no | — |
| 191 | C2×C3×C4 | — | 5 | 2 | no | — |
| 192 | C2×C3×C4 | — | 10 | 2 | no | — |
| 193 | C2×C3×C4 | C4⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 194 | C2×Dic3 | — | 2 | 2 | no | — |
| 195 | C2×Dic3 | — | 2 | 2 | no | — |
| 196 | C2×Dic3 | — | 2 | 2 | no | — |
| 197 | C2×Dic3 | — | 5 | 4 | no | — |
| 198 | C2×Dic3 | — | 10 | 2 | no | — |
| 199 | C2×Dic3 | — | 10 | 2 | no | — |
| 200 | C3×C8 | — | 10 | 2 | no | — |
| 201 | C3×C8 | — | 10 | 2 | no | — |
| 202 | C3⋊C8 | — | 20 | 2 | no | — |
| 203 | C22×C5 | C4×Dic6 | 1 | 1 | yes | characteristic |
| 204 | C22×C5 | C4×D12 | 1 | 1 | yes | characteristic |
| 205 | C22×C5 | D3×M16 | 1 | 2 | yes | — |
| 206 | C22×C5 | #907 | 1 | 2 | yes | — |
| 207 | C22×C5 | C2×Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 208 | C4×C5 | — | 2 | 2 | no | — |
| 209 | C4×C5 | — | 12 | 2 | no | — |
| 210 | Dic5 | — | 2 | 2 | no | — |
| 211 | Dic5 | — | 6 | 4 | no | — |
| 212 | Dic5 | #869 | 1 | 1 | yes | characteristic |
| 213 | Dic5 | #870 | 1 | 1 | yes | characteristic |
| 214 | Dic5 | #924 | 1 | 1 | yes | characteristic |
| 215 | Dic5 | #930 | 1 | 1 | yes | characteristic |
| 216 | C22×C4 | — | 3 | 2 | no | — |
| 217 | C22×C4 | — | 5 | 1 | no | — |
| 218 | C22×C4 | — | 5 | 1 | no | — |
| 219 | C22×C4 | — | 15 | 2 | no | — |
| 220 | C22×C4 | D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 221 | C2×C8 | — | 5 | 2 | no | — |
| 222 | C2×C8 | — | 5 | 2 | no | — |
| 223 | C2×C8 | — | 10 | 1 | no | — |
| 224 | C2×C8 | — | 10 | 1 | no | — |
| 225 | C2×C8 | — | 10 | 1 | no | — |
| 226 | C2×C8 | — | 10 | 1 | no | — |
| 227 | C2×C8 | — | 30 | 2 | no | — |
| 228 | C2×C8 | — | 30 | 2 | no | — |
| 229 | C2×C8 | — | 30 | 2 | no | — |
| 230 | C4⋊C4 | — | 30 | 4 | no | — |
| 231 | C42 | — | 5 | 2 | no | — |
| 232 | C42 | — | 5 | 2 | no | — |
| 233 | C42 | — | 30 | 2 | no | — |
| 234 | C42 | — | 30 | 2 | no | — |
| 235 | C3×C5 | #1940 | 1 | 1 | yes | characteristic, complemented, Hall |
| 236 | C22×C3 | D5⋊M16 | 1 | 2 | yes | — |
| 237 | C22×C3 | C2×C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 238 | C22×C3 | #4131 | 1 | 2 | yes | — |
| 239 | C22×C3 | D4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 240 | C22×C3 | Q8×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 241 | C3×C4 | — | 2 | 2 | no | — |
| 242 | C3×C4 | — | 5 | 1 | no | — |
| 243 | C3×C4 | — | 5 | 1 | no | — |
| 244 | C3×C4 | — | 5 | 1 | no | — |
| 245 | C3×C4 | — | 5 | 1 | no | — |
| 246 | C3×C4 | — | 10 | 2 | no | — |
| 247 | Dic3 | — | 4 | 2 | no | — |
| 248 | Dic3 | — | 10 | 4 | no | — |
| 249 | C2×C5 | #4701 | 1 | 2 | yes | — |
| 250 | C2×C5 | #4706 | 1 | 2 | yes | — |
| 251 | C2×C5 | #4928 | 1 | 1 | yes | characteristic |
| 252 | C2×C5 | #5118 | 1 | 1 | yes | characteristic |
| 253 | C2×C5 | #5119 | 1 | 1 | yes | characteristic |
| 254 | C23 | C2×D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 255 | C2×C4 | — | 2 | 2 | no | — |
| 256 | C2×C4 | — | 5 | 1 | no | — |
| 257 | C2×C4 | — | 5 | 1 | no | — |
| 258 | C2×C4 | — | 5 | 1 | no | — |
| 259 | C2×C4 | — | 5 | 1 | no | — |
| 260 | C2×C4 | — | 5 | 1 | no | — |
| 261 | C2×C4 | — | 5 | 1 | no | — |
| 262 | C2×C4 | — | 5 | 2 | no | — |
| 263 | C2×C4 | — | 6 | 2 | no | — |
| 264 | C2×C4 | — | 6 | 2 | no | — |
| 265 | C2×C4 | — | 6 | 2 | no | — |
| 266 | C2×C4 | — | 10 | 2 | no | — |
| 267 | C2×C4 | — | 15 | 4 | no | — |
| 268 | C2×C4 | — | 30 | 2 | no | — |
| 269 | C2×C4 | — | 30 | 2 | no | — |
| 270 | C2×C4 | Dic3⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 271 | C8 | — | 10 | 2 | no | — |
| 272 | C8 | — | 10 | 2 | no | — |
| 273 | C8 | — | 60 | 2 | no | — |
| 274 | C2×C3 | #65682 | 1 | 1 | yes | characteristic |
| 275 | C2×C3 | #65694 | 1 | 2 | yes | — |
| 276 | C2×C3 | #65696 | 1 | 1 | yes | characteristic |
| 277 | C2×C3 | #65707 | 1 | 2 | yes | — |
| 278 | C2×C3 | #65712 | 1 | 1 | yes | characteristic |
| 279 | C5 | #69836 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 280 | C22 | AGL(1,𝔽5)×Dic6 | 1 | 1 | yes | characteristic |
| 281 | C22 | #91915 | 1 | 2 | yes | — |
| 282 | C22 | AGL(1,𝔽5)×D12 | 1 | 1 | yes | characteristic |
| 283 | C22 | C2×Dic3⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 284 | C22 | #91929 | 1 | 2 | yes | — |
| 285 | C4 | — | 2 | 2 | no | — |
| 286 | C4 | — | 5 | 1 | no | — |
| 287 | C4 | — | 5 | 1 | no | — |
| 288 | C4 | — | 5 | 1 | no | — |
| 289 | C4 | — | 5 | 1 | no | — |
| 290 | C4 | — | 10 | 2 | no | — |
| 291 | C4 | — | 12 | 2 | no | — |
| 292 | C4 | — | 30 | 4 | no | — |
| 293 | C3 | #10603797 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 294 | C2 | #11751780 | 1 | 2 | yes | — |
| 295 | C2 | #11751829 | 1 | 1 | yes | characteristic |
| 296 | C2 | #11751859 | 1 | 1 | yes | characteristic |
| 297 | C2 | #11751895 | 1 | 2 | yes | — |
| 298 | C2 | #11752014 | 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