← #14285452
Subgroups — #14285452
Order 1920.
420 conjugacy classes of subgroups.
The subgroup lattice has 324 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #11756229 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×C5×#423 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11751025 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11746307 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11746303 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10601304 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×C3×C5×M16 | — | 2 | 1 | no | — |
| 8 | C2×C3×C42×C5 | C22 | 1 | 1 | yes | characteristic |
| 9 | #91815 | C4 | 1 | 2 | yes | — |
| 10 | #91813 | — | 2 | 1 | no | — |
| 11 | #91813 | — | 2 | 1 | no | — |
| 12 | C2×#6710 | — | 2 | 1 | no | — |
| 13 | #91759 | C4 | 1 | 2 | yes | — |
| 14 | #91758 | C22 | 1 | 1 | yes | characteristic |
| 15 | #91758 | C22 | 1 | 1 | yes | characteristic |
| 16 | C4×Dic30 | — | 2 | 1 | no | — |
| 17 | C4×Dic30 | — | 2 | 1 | no | — |
| 18 | C3×C5×C4⋊C8 | — | 2 | 1 | no | — |
| 19 | C3×C5×C4⋊C8 | C22 | 1 | 2 | yes | — |
| 20 | C4⋊(C3⋊#168) | — | 2 | 1 | no | — |
| 21 | C4⋊(C3⋊#168) | C22 | 1 | 2 | yes | — |
| 22 | #70511 | — | 5 | 1 | no | maximal, Hall |
| 23 | #65804 | — | 3 | 1 | no | — |
| 24 | C5×#423 | D3 | 1 | 1 | yes | characteristic |
| 25 | #65213 | — | 3 | 1 | no | — |
| 26 | #64706 | — | 3 | 2 | no | — |
| 27 | #64702 | — | 3 | 2 | no | — |
| 28 | C22×C3×C4×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C22×C3×C4×C5 | D4 | 1 | 1 | yes | characteristic |
| 30 | C22×C3×C4×C5 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×Dic30 | — | 2 | 1 | no | — |
| 32 | C2×Dic30 | — | 2 | 1 | no | — |
| 33 | C3×C5×M16 | — | 4 | 2 | no | — |
| 34 | C2×C3×C5×C8 | — | 2 | 2 | no | — |
| 35 | C3×C42×C5 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C3×C42×C5 | D4 | 1 | 1 | yes | characteristic |
| 37 | C3×C42×C5 | D4 | 1 | 1 | yes | characteristic |
| 38 | C3×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 39 | #6719 | — | 2 | 2 | no | — |
| 40 | C4⋊Dic15 | — | 2 | 1 | no | — |
| 41 | C4⋊Dic15 | — | 2 | 1 | no | — |
| 42 | C4⋊Dic15 | C2×C4 | 1 | 2 | yes | — |
| 43 | C4⋊Dic15 | C2×C4 | 1 | 2 | yes | — |
| 44 | #6712 | — | 2 | 2 | no | — |
| 45 | #6712 | — | 2 | 2 | no | — |
| 46 | C4×Dic15 | — | 2 | 2 | no | — |
| 47 | #6710 | — | 4 | 2 | no | — |
| 48 | C2×C3⋊#168 | — | 2 | 2 | no | — |
| 49 | #5470 | — | 5 | 1 | no | — |
| 50 | C3×#423 | D5 | 1 | 1 | yes | characteristic |
| 51 | #4924 | — | 5 | 1 | no | — |
| 52 | #4486 | — | 5 | 2 | no | — |
| 53 | #4482 | — | 5 | 2 | no | — |
| 54 | C2×C5×M16 | — | 2 | 1 | no | — |
| 55 | C2×C42×C5 | C2×D3 | 1 | 1 | yes | characteristic |
| 56 | #4072 | — | 3 | 2 | no | — |
| 57 | #4070 | — | 6 | 1 | no | — |
| 58 | #4070 | — | 6 | 1 | no | — |
| 59 | C2×#681 | — | 6 | 1 | no | — |
| 60 | #4016 | — | 3 | 2 | no | — |
| 61 | #4015 | — | 3 | 1 | no | — |
| 62 | #4015 | — | 3 | 1 | no | — |
| 63 | C4×Dic10 | — | 6 | 1 | no | — |
| 64 | C4×Dic10 | — | 6 | 1 | no | — |
| 65 | C5×C4⋊C8 | — | 2 | 1 | no | — |
| 66 | C5×C4⋊C8 | C2×D3 | 1 | 2 | yes | — |
| 67 | C4⋊#168 | — | 3 | 2 | no | — |
| 68 | C4⋊#168 | — | 6 | 1 | no | — |
| 69 | #1561 | — | 15 | 1 | no | Sylow |
| 70 | C23×C3×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C2×C3×C4×C5 | — | 2 | 2 | no | — |
| 72 | C2×C3×C4×C5 | — | 2 | 1 | no | — |
| 73 | C2×C3×C4×C5 | — | 2 | 1 | no | — |
| 74 | C2×C3×C4×C5 | — | 2 | 1 | no | — |
| 75 | C2×C3×C4×C5 | — | 2 | 1 | no | — |
| 76 | C2×C3×C4×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 77 | C2×C3×C4×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 78 | C2×C3×C4×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 79 | C2×C3×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 80 | C2×C3×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | C2×C3×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C2×Dic15 | — | 2 | 2 | no | — |
| 83 | C2×Dic15 | — | 2 | 2 | no | — |
| 84 | Dic30 | — | 4 | 1 | no | — |
| 85 | Dic30 | — | 4 | 1 | no | — |
| 86 | Dic30 | — | 4 | 1 | no | — |
| 87 | Dic30 | — | 4 | 1 | no | — |
| 88 | C3×C5×C8 | — | 4 | 2 | no | — |
| 89 | C3⋊#168 | — | 4 | 2 | no | — |
| 90 | C2×C3×M16 | — | 2 | 1 | no | — |
| 91 | C2×C3×C42 | C2×D5 | 1 | 1 | yes | characteristic |
| 92 | #926 | — | 5 | 2 | no | — |
| 93 | #924 | — | 10 | 1 | no | — |
| 94 | #924 | — | 10 | 1 | no | — |
| 95 | C2×#208 | — | 10 | 1 | no | — |
| 96 | #870 | — | 5 | 2 | no | — |
| 97 | #869 | — | 5 | 1 | no | — |
| 98 | #869 | — | 5 | 1 | no | — |
| 99 | C4×Dic6 | — | 10 | 1 | no | — |
| 100 | C4×Dic6 | — | 10 | 1 | no | — |
| 101 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 102 | C3×C4⋊C8 | C2×D5 | 1 | 2 | yes | — |
| 103 | C4⋊(C3⋊C8) | — | 5 | 2 | no | — |
| 104 | C4⋊(C3⋊C8) | — | 10 | 1 | no | — |
| 105 | C22×C4×C5 | C4×D3 | 1 | 1 | yes | characteristic |
| 106 | C22×C4×C5 | D12 | 1 | 1 | yes | characteristic |
| 107 | C22×C4×C5 | #67 | 1 | 1 | yes | characteristic |
| 108 | C2×Dic10 | — | 6 | 1 | no | — |
| 109 | C2×Dic10 | — | 6 | 1 | no | — |
| 110 | C5×M16 | — | 4 | 2 | no | — |
| 111 | C2×C5×C8 | — | 2 | 2 | no | — |
| 112 | C42×C5 | C4×D3 | 1 | 1 | yes | characteristic |
| 113 | C42×C5 | D12 | 1 | 1 | yes | characteristic |
| 114 | C42×C5 | #67 | 1 | 1 | yes | characteristic |
| 115 | C42×C5 | C22×D3 | 1 | 1 | yes | characteristic |
| 116 | C22⋊Dic5 | — | 6 | 2 | no | — |
| 117 | C4⋊Dic5 | — | 3 | 2 | no | — |
| 118 | C4⋊Dic5 | — | 3 | 2 | no | — |
| 119 | C4⋊Dic5 | — | 6 | 1 | no | — |
| 120 | C4⋊Dic5 | — | 6 | 1 | no | — |
| 121 | #683 | — | 6 | 2 | no | — |
| 122 | #683 | — | 6 | 2 | no | — |
| 123 | C4×Dic5 | — | 6 | 2 | no | — |
| 124 | #681 | — | 12 | 2 | no | — |
| 125 | C2×#168 | — | 6 | 2 | no | — |
| 126 | #533 | — | 15 | 1 | no | — |
| 127 | #423 | — | 15 | 1 | no | — |
| 128 | #423 | D15 | 1 | 1 | yes | characteristic |
| 129 | #332 | — | 15 | 2 | no | — |
| 130 | #332 | — | 15 | 2 | no | — |
| 131 | C22×C3×C5 | — | 2 | 1 | no | — |
| 132 | C22×C3×C5 | — | 2 | 1 | no | — |
| 133 | C22×C3×C5 | — | 2 | 1 | no | — |
| 134 | C22×C3×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 135 | C3×C4×C5 | — | 2 | 1 | no | — |
| 136 | C3×C4×C5 | — | 2 | 1 | no | — |
| 137 | C3×C4×C5 | — | 2 | 1 | no | — |
| 138 | C3×C4×C5 | #101 | 1 | 2 | yes | — |
| 139 | C3×C4×C5 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 140 | C3×C4×C5 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 141 | C3×C4×C5 | #137 | 1 | 1 | yes | characteristic |
| 142 | C3×C4×C5 | #137 | 1 | 1 | yes | characteristic |
| 143 | Dic15 | — | 4 | 2 | no | — |
| 144 | Dic15 | — | 4 | 2 | no | — |
| 145 | C22×C3×C4 | C4×D5 | 1 | 1 | yes | characteristic |
| 146 | C22×C3×C4 | D20 | 1 | 1 | yes | characteristic |
| 147 | C22×C3×C4 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 148 | C2×Dic6 | — | 10 | 1 | no | — |
| 149 | C2×Dic6 | — | 10 | 1 | no | — |
| 150 | C3×M16 | — | 4 | 2 | no | — |
| 151 | C2×C3×C8 | — | 2 | 2 | no | — |
| 152 | C3×C42 | C4×D5 | 1 | 1 | yes | characteristic |
| 153 | C3×C42 | D20 | 1 | 1 | yes | characteristic |
| 154 | C3×C42 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 155 | C3×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 156 | #217 | — | 10 | 2 | no | — |
| 157 | C4⋊Dic3 | — | 5 | 2 | no | — |
| 158 | C4⋊Dic3 | — | 5 | 2 | no | — |
| 159 | C4⋊Dic3 | — | 10 | 1 | no | — |
| 160 | C4⋊Dic3 | — | 10 | 1 | no | — |
| 161 | Dic3⋊C4 | — | 10 | 2 | no | — |
| 162 | Dic3⋊C4 | — | 10 | 2 | no | — |
| 163 | C4×Dic3 | — | 10 | 2 | no | — |
| 164 | #208 | — | 20 | 2 | no | — |
| 165 | C2×C3⋊C8 | — | 10 | 2 | no | — |
| 166 | C23×C5 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 167 | C2×C4×C5 | — | 2 | 2 | no | — |
| 168 | C2×C4×C5 | — | 2 | 1 | no | — |
| 169 | C2×C4×C5 | — | 2 | 1 | no | — |
| 170 | C2×C4×C5 | — | 2 | 1 | no | — |
| 171 | C2×C4×C5 | — | 2 | 1 | no | — |
| 172 | C2×C4×C5 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 173 | C2×C4×C5 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 174 | C2×C4×C5 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 175 | C2×C4×C5 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 176 | C2×C4×C5 | C2×D12 | 1 | 1 | yes | characteristic |
| 177 | C2×C4×C5 | C2×#67 | 1 | 1 | yes | characteristic |
| 178 | C2×Dic5 | — | 6 | 2 | no | — |
| 179 | C2×Dic5 | — | 6 | 2 | no | — |
| 180 | Dic10 | — | 12 | 1 | no | — |
| 181 | Dic10 | — | 12 | 1 | no | — |
| 182 | Dic10 | — | 12 | 1 | no | — |
| 183 | Dic10 | — | 12 | 1 | no | — |
| 184 | C5×C8 | — | 4 | 2 | no | — |
| 185 | #168 | — | 12 | 2 | no | — |
| 186 | C2×M16 | — | 2 | 1 | no | — |
| 187 | C2×M16 | — | 30 | 1 | no | — |
| 188 | C4⋊Q8 | — | 15 | 1 | no | — |
| 189 | C4⋊Q8 | — | 15 | 1 | no | — |
| 190 | #125 | — | 15 | 2 | no | — |
| 191 | C22⋊Q8 | — | 30 | 1 | no | — |
| 192 | C22⋊Q8 | — | 30 | 1 | no | — |
| 193 | C4×Q8 | — | 30 | 1 | no | — |
| 194 | C4×Q8 | — | 30 | 1 | no | — |
| 195 | #117 | — | 15 | 2 | no | — |
| 196 | C2×C42 | C2×D15 | 1 | 1 | yes | characteristic |
| 197 | C4⋊C8 | — | 2 | 1 | no | — |
| 198 | C4⋊C8 | — | 15 | 2 | no | — |
| 199 | C4⋊C8 | — | 30 | 1 | no | — |
| 200 | C4⋊C8 | C2×D15 | 1 | 2 | yes | — |
| 201 | C2×C3×C5 | — | 4 | 1 | no | — |
| 202 | C2×C3×C5 | C2×#101 | 1 | 1 | yes | characteristic |
| 203 | C2×C3×C5 | #417 | 1 | 1 | yes | characteristic |
| 204 | C2×C3×C5 | #417 | 1 | 1 | yes | characteristic |
| 205 | C23×C3 | #685 | 1 | 1 | yes | characteristic |
| 206 | C2×C3×C4 | — | 2 | 2 | no | — |
| 207 | C2×C3×C4 | — | 2 | 1 | no | — |
| 208 | C2×C3×C4 | — | 2 | 1 | no | — |
| 209 | C2×C3×C4 | — | 2 | 1 | no | — |
| 210 | C2×C3×C4 | — | 2 | 1 | no | — |
| 211 | C2×C3×C4 | #685 | 1 | 1 | yes | characteristic |
| 212 | C2×C3×C4 | #685 | 1 | 1 | yes | characteristic |
| 213 | C2×C3×C4 | #685 | 1 | 1 | yes | characteristic |
| 214 | C2×C3×C4 | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 215 | C2×C3×C4 | C2×D20 | 1 | 1 | yes | characteristic |
| 216 | C2×C3×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 217 | C2×Dic3 | — | 10 | 2 | no | — |
| 218 | C2×Dic3 | — | 10 | 2 | no | — |
| 219 | Dic6 | — | 20 | 1 | no | — |
| 220 | Dic6 | — | 20 | 1 | no | — |
| 221 | Dic6 | — | 20 | 1 | no | — |
| 222 | Dic6 | — | 20 | 1 | no | — |
| 223 | C3×C8 | — | 4 | 2 | no | — |
| 224 | C3⋊C8 | — | 20 | 2 | no | — |
| 225 | C22×C5 | — | 2 | 1 | no | — |
| 226 | C22×C5 | — | 2 | 1 | no | — |
| 227 | C22×C5 | — | 2 | 1 | no | — |
| 228 | C22×C5 | C2×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 229 | C4×C5 | — | 2 | 1 | no | — |
| 230 | C4×C5 | — | 2 | 1 | no | — |
| 231 | C4×C5 | — | 2 | 1 | no | — |
| 232 | C4×C5 | #824 | 1 | 2 | yes | — |
| 233 | C4×C5 | #908 | 1 | 1 | yes | characteristic |
| 234 | C4×C5 | #909 | 1 | 1 | yes | characteristic |
| 235 | C4×C5 | #932 | 1 | 1 | yes | characteristic |
| 236 | C4×C5 | #942 | 1 | 1 | yes | characteristic |
| 237 | Dic5 | — | 12 | 2 | no | — |
| 238 | Dic5 | — | 12 | 2 | no | — |
| 239 | C2×Q8 | — | 30 | 1 | no | — |
| 240 | C2×Q8 | — | 30 | 1 | no | — |
| 241 | C22×C4 | C4×D15 | 1 | 1 | yes | characteristic |
| 242 | C22×C4 | D60 | 1 | 1 | yes | characteristic |
| 243 | C22×C4 | #1220 | 1 | 1 | yes | characteristic |
| 244 | M16 | — | 4 | 2 | no | — |
| 245 | M16 | — | 60 | 2 | no | — |
| 246 | C2×C8 | — | 2 | 2 | no | — |
| 247 | C2×C8 | — | 30 | 2 | no | — |
| 248 | C4⋊C4 | — | 15 | 2 | no | — |
| 249 | C4⋊C4 | — | 15 | 2 | no | — |
| 250 | C4⋊C4 | — | 30 | 2 | no | — |
| 251 | C4⋊C4 | — | 30 | 2 | no | — |
| 252 | C4⋊C4 | — | 30 | 1 | no | — |
| 253 | C4⋊C4 | — | 30 | 1 | no | — |
| 254 | C22⋊C4 | — | 30 | 2 | no | — |
| 255 | C42 | — | 30 | 2 | no | — |
| 256 | C42 | C4×D15 | 1 | 1 | yes | characteristic |
| 257 | C42 | D60 | 1 | 1 | yes | characteristic |
| 258 | C42 | #1220 | 1 | 1 | yes | characteristic |
| 259 | C42 | C22×D15 | 1 | 1 | yes | characteristic |
| 260 | C3×C5 | #1561 | 1 | 1 | yes | characteristic, complemented, Hall |
| 261 | C22×C3 | — | 2 | 1 | no | — |
| 262 | C22×C3 | — | 2 | 1 | no | — |
| 263 | C22×C3 | — | 2 | 1 | no | — |
| 264 | C22×C3 | C2×#685 | 1 | 1 | yes | characteristic |
| 265 | C3×C4 | — | 2 | 1 | no | — |
| 266 | C3×C4 | — | 2 | 1 | no | — |
| 267 | C3×C4 | — | 2 | 1 | no | — |
| 268 | C3×C4 | #3956 | 1 | 2 | yes | — |
| 269 | C3×C4 | #4054 | 1 | 1 | yes | characteristic |
| 270 | C3×C4 | #4055 | 1 | 1 | yes | characteristic |
| 271 | C3×C4 | #4078 | 1 | 1 | yes | characteristic |
| 272 | C3×C4 | #4088 | 1 | 1 | yes | characteristic |
| 273 | Dic3 | — | 20 | 2 | no | — |
| 274 | Dic3 | — | 20 | 2 | no | — |
| 275 | C2×C5 | — | 4 | 1 | no | — |
| 276 | C2×C5 | #4966 | 1 | 1 | yes | characteristic |
| 277 | C2×C5 | #5133 | 1 | 1 | yes | characteristic |
| 278 | C2×C5 | C2×#824 | 1 | 1 | yes | characteristic |
| 279 | C23 | #6714 | 1 | 1 | yes | characteristic |
| 280 | Q8 | — | 60 | 1 | no | — |
| 281 | Q8 | — | 60 | 1 | no | — |
| 282 | Q8 | — | 60 | 1 | no | — |
| 283 | Q8 | — | 60 | 1 | no | — |
| 284 | C2×C4 | — | 2 | 1 | no | — |
| 285 | C2×C4 | — | 2 | 1 | no | — |
| 286 | C2×C4 | — | 2 | 1 | no | — |
| 287 | C2×C4 | — | 2 | 1 | no | — |
| 288 | C2×C4 | — | 2 | 2 | no | — |
| 289 | C2×C4 | — | 30 | 2 | no | — |
| 290 | C2×C4 | — | 30 | 2 | no | — |
| 291 | C2×C4 | #6714 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | #6714 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | #6714 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | C2×C4×D15 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | C2×D60 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | C2×#1220 | 1 | 1 | yes | characteristic |
| 297 | C8 | — | 4 | 2 | no | — |
| 298 | C8 | — | 60 | 2 | no | — |
| 299 | C2×C3 | — | 4 | 1 | no | — |
| 300 | C2×C3 | #65255 | 1 | 1 | yes | characteristic |
| 301 | C2×C3 | #65422 | 1 | 1 | yes | characteristic |
| 302 | C2×C3 | C2×#3956 | 1 | 1 | yes | characteristic |
| 303 | C5 | #70511 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 304 | C22 | — | 2 | 1 | no | — |
| 305 | C22 | — | 2 | 1 | no | — |
| 306 | C22 | — | 2 | 1 | no | — |
| 307 | C22 | C2×#6714 | 1 | 1 | yes | characteristic |
| 308 | C4 | — | 2 | 1 | no | — |
| 309 | C4 | — | 2 | 1 | no | — |
| 310 | C4 | — | 2 | 1 | no | — |
| 311 | C4 | — | 60 | 2 | no | — |
| 312 | C4 | — | 60 | 2 | no | — |
| 313 | C4 | #91108 | 1 | 2 | yes | — |
| 314 | C4 | #91797 | 1 | 1 | yes | characteristic |
| 315 | C4 | #91798 | 1 | 1 | yes | characteristic |
| 316 | C4 | #91821 | 1 | 1 | yes | characteristic |
| 317 | C4 | #91831 | 1 | 1 | yes | characteristic |
| 318 | C3 | #10601304 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 319 | C2 | — | 4 | 1 | no | — |
| 320 | C2 | #11751067 | 1 | 1 | yes | characteristic |
| 321 | C2 | #11751234 | 1 | 1 | yes | characteristic |
| 322 | C2 | C2×#91108 | 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