← #14298762
Subgroups — #14298762
Order 1920.
344 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×#91106 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11751223 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11751099 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C3×C5×#344 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #11746378 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #11746377 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #11746352 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #10599806 | — | 3 | 1 | no | maximal, Hall |
| 9 | C2×C3×C5×M16 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×C5×M16 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×C5×#117 | C22 | 1 | 1 | yes | characteristic |
| 12 | #91815 | C22 | 1 | 1 | yes | characteristic |
| 13 | #91815 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#6710 | C22 | 1 | 1 | yes | characteristic |
| 15 | C22×C3⋊#168 | C22 | 1 | 1 | yes | characteristic |
| 16 | #91106 | — | 2 | 1 | no | — |
| 17 | #91106 | — | 2 | 1 | no | — |
| 18 | #91099 | C4 | 1 | 2 | yes | complemented |
| 19 | C8×Dic15 | C4 | 1 | 2 | yes | complemented |
| 20 | #91092 | — | 2 | 1 | no | — |
| 21 | #91091 | — | 2 | 1 | no | — |
| 22 | #69013 | — | 5 | 1 | no | maximal, Hall |
| 23 | C2×#3954 | — | 3 | 1 | no | — |
| 24 | #65411 | — | 3 | 1 | no | — |
| 25 | #65287 | — | 3 | 1 | no | — |
| 26 | C5×#344 | D3 | 1 | 1 | yes | characteristic |
| 27 | #64777 | — | 3 | 1 | no | — |
| 28 | #64776 | — | 3 | 1 | no | — |
| 29 | #64751 | — | 3 | 1 | no | — |
| 30 | C22×C3×C4×C5 | C23 | 1 | 1 | yes | characteristic |
| 31 | C3×C5×M16 | — | 2 | 1 | no | — |
| 32 | C3×C5×M16 | — | 2 | 1 | no | — |
| 33 | C3×C5×M16 | — | 2 | 1 | no | — |
| 34 | C3×C5×M16 | — | 2 | 1 | no | — |
| 35 | C2×C3×C5×C8 | — | 2 | 1 | no | — |
| 36 | C2×C3×C5×C8 | C2×C4 | 1 | 2 | yes | — |
| 37 | C3×C5×C4⋊C4 | — | 2 | 1 | no | — |
| 38 | C3×C5×C22⋊C4 | — | 2 | 1 | no | — |
| 39 | C3×C42×C5 | — | 2 | 1 | no | — |
| 40 | #6719 | — | 2 | 1 | no | — |
| 41 | #6719 | — | 2 | 1 | no | — |
| 42 | C4⋊Dic15 | — | 2 | 1 | no | — |
| 43 | C4⋊Dic15 | — | 2 | 1 | no | — |
| 44 | C4×Dic15 | — | 2 | 1 | no | — |
| 45 | C4×Dic15 | C2×C4 | 1 | 2 | yes | — |
| 46 | #6710 | — | 2 | 1 | no | — |
| 47 | #6710 | — | 2 | 1 | no | — |
| 48 | C2×C3⋊#168 | — | 2 | 1 | no | — |
| 49 | C2×C3⋊#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | C2×C3⋊#168 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C2×C3⋊#168 | D4 | 1 | 1 | yes | characteristic |
| 52 | C2×C3⋊#168 | D4 | 1 | 1 | yes | characteristic |
| 53 | C2×C3⋊#168 | Q8 | 1 | 1 | yes | characteristic |
| 54 | C2×C3⋊#168 | Q8 | 1 | 1 | yes | characteristic |
| 55 | C2×#822 | — | 5 | 1 | no | — |
| 56 | #5122 | — | 5 | 1 | no | — |
| 57 | #4998 | — | 5 | 1 | no | — |
| 58 | C3×#344 | D5 | 1 | 1 | yes | characteristic |
| 59 | #4557 | — | 5 | 1 | no | — |
| 60 | #4556 | — | 5 | 1 | no | — |
| 61 | #4531 | — | 5 | 1 | no | — |
| 62 | C2×C5×M16 | C2×D3 | 1 | 1 | yes | characteristic |
| 63 | C2×C5×M16 | C2×D3 | 1 | 1 | yes | characteristic |
| 64 | C5×#117 | C2×D3 | 1 | 1 | yes | characteristic |
| 65 | #4072 | — | 3 | 1 | no | — |
| 66 | #4072 | — | 3 | 1 | no | — |
| 67 | C2×#681 | — | 3 | 1 | no | — |
| 68 | C22×#168 | — | 3 | 1 | no | — |
| 69 | #3954 | — | 6 | 1 | no | — |
| 70 | #3954 | — | 6 | 1 | no | — |
| 71 | #3947 | — | 3 | 2 | no | — |
| 72 | C8×Dic5 | — | 3 | 2 | no | — |
| 73 | #3940 | — | 6 | 1 | no | — |
| 74 | #3939 | — | 6 | 1 | no | — |
| 75 | #1833 | — | 15 | 1 | no | Sylow |
| 76 | C23×C3×C5 | #41 | 1 | 1 | yes | characteristic |
| 77 | C2×C3×C4×C5 | — | 4 | 1 | no | — |
| 78 | C2×C3×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | C2×C3×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C2×C3×C4×C5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 81 | C2×C3×C4×C5 | #41 | 1 | 1 | yes | characteristic |
| 82 | C2×C3×C4×C5 | #41 | 1 | 1 | yes | characteristic |
| 83 | C2×C3×C4×C5 | #41 | 1 | 1 | yes | characteristic |
| 84 | C2×Dic15 | — | 2 | 2 | no | — |
| 85 | C2×Dic15 | — | 4 | 1 | no | — |
| 86 | C3×C5×C8 | — | 2 | 2 | no | — |
| 87 | C3×C5×C8 | — | 4 | 1 | no | — |
| 88 | C3⋊#168 | — | 4 | 1 | no | — |
| 89 | C3⋊#168 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 90 | C3⋊#168 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 91 | C3⋊#168 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 92 | C3⋊#168 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 93 | C2×C3×M16 | C2×D5 | 1 | 1 | yes | characteristic |
| 94 | C2×C3×M16 | C2×D5 | 1 | 1 | yes | characteristic |
| 95 | C3×#117 | C2×D5 | 1 | 1 | yes | characteristic |
| 96 | #926 | — | 5 | 1 | no | — |
| 97 | #926 | — | 5 | 1 | no | — |
| 98 | C2×#208 | — | 5 | 1 | no | — |
| 99 | C22×C3⋊C8 | — | 5 | 1 | no | — |
| 100 | #822 | — | 10 | 1 | no | — |
| 101 | #822 | — | 10 | 1 | no | — |
| 102 | #815 | — | 5 | 2 | no | — |
| 103 | C8×Dic3 | — | 5 | 2 | no | — |
| 104 | #808 | — | 10 | 1 | no | — |
| 105 | #807 | — | 10 | 1 | no | — |
| 106 | C22×C4×C5 | C22×D3 | 1 | 1 | yes | characteristic |
| 107 | C5×M16 | — | 2 | 1 | no | — |
| 108 | C5×M16 | — | 2 | 1 | no | — |
| 109 | C5×M16 | — | 2 | 1 | no | — |
| 110 | C5×M16 | — | 2 | 1 | no | — |
| 111 | C2×C5×C8 | — | 2 | 1 | no | — |
| 112 | C2×C5×C8 | C4×D3 | 1 | 2 | yes | — |
| 113 | C5×C4⋊C4 | — | 2 | 1 | no | — |
| 114 | C5×C22⋊C4 | — | 2 | 1 | no | — |
| 115 | C42×C5 | — | 2 | 1 | no | — |
| 116 | C22⋊Dic5 | — | 6 | 1 | no | — |
| 117 | C22⋊Dic5 | — | 6 | 1 | no | — |
| 118 | C4⋊Dic5 | — | 6 | 1 | no | — |
| 119 | C4⋊Dic5 | — | 6 | 1 | no | — |
| 120 | C4×Dic5 | — | 3 | 2 | no | — |
| 121 | C4×Dic5 | — | 6 | 1 | no | — |
| 122 | #681 | — | 6 | 1 | no | — |
| 123 | #681 | — | 6 | 1 | no | — |
| 124 | C2×#168 | — | 3 | 1 | no | — |
| 125 | C2×#168 | — | 3 | 1 | no | — |
| 126 | C2×#168 | — | 3 | 1 | no | — |
| 127 | C2×#168 | — | 3 | 1 | no | — |
| 128 | C2×#168 | — | 3 | 1 | no | — |
| 129 | C2×#168 | — | 3 | 1 | no | — |
| 130 | C2×#168 | — | 6 | 1 | no | — |
| 131 | C2×#108 | — | 15 | 1 | no | — |
| 132 | #427 | — | 15 | 1 | no | — |
| 133 | #405 | — | 15 | 1 | no | — |
| 134 | #344 | — | 15 | 1 | no | — |
| 135 | #344 | D15 | 1 | 1 | yes | characteristic |
| 136 | H(3,ℤ/4) | — | 15 | 1 | no | — |
| 137 | H(3,ℤ/4) | — | 15 | 1 | no | — |
| 138 | C22×C3×C5 | — | 4 | 1 | no | — |
| 139 | C22×C3×C5 | #117 | 1 | 1 | yes | characteristic |
| 140 | C22×C3×C5 | #124 | 1 | 1 | yes | characteristic |
| 141 | C22×C3×C5 | #125 | 1 | 1 | yes | characteristic |
| 142 | C3×C4×C5 | — | 8 | 1 | no | — |
| 143 | C3×C4×C5 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 144 | C3×C4×C5 | #117 | 1 | 1 | yes | characteristic |
| 145 | C3×C4×C5 | #124 | 1 | 1 | yes | characteristic |
| 146 | C3×C4×C5 | #125 | 1 | 1 | yes | characteristic |
| 147 | Dic15 | — | 4 | 2 | no | — |
| 148 | Dic15 | — | 8 | 1 | no | — |
| 149 | C22×C3×C4 | C22×D5 | 1 | 1 | yes | characteristic |
| 150 | C3×M16 | — | 2 | 1 | no | — |
| 151 | C3×M16 | — | 2 | 1 | no | — |
| 152 | C3×M16 | — | 2 | 1 | no | — |
| 153 | C3×M16 | — | 2 | 1 | no | — |
| 154 | C2×C3×C8 | — | 2 | 1 | no | — |
| 155 | C2×C3×C8 | C4×D5 | 1 | 2 | yes | — |
| 156 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 157 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 158 | C3×C42 | — | 2 | 1 | no | — |
| 159 | #217 | — | 10 | 1 | no | — |
| 160 | #217 | — | 10 | 1 | no | — |
| 161 | C4⋊Dic3 | — | 10 | 1 | no | — |
| 162 | C4⋊Dic3 | — | 10 | 1 | no | — |
| 163 | C4×Dic3 | — | 5 | 2 | no | — |
| 164 | C4×Dic3 | — | 10 | 1 | no | — |
| 165 | #208 | — | 10 | 1 | no | — |
| 166 | #208 | — | 10 | 1 | no | — |
| 167 | C2×C3⋊C8 | — | 5 | 1 | no | — |
| 168 | C2×C3⋊C8 | — | 5 | 1 | no | — |
| 169 | C2×C3⋊C8 | — | 5 | 1 | no | — |
| 170 | C2×C3⋊C8 | — | 5 | 1 | no | — |
| 171 | C2×C3⋊C8 | — | 5 | 1 | no | — |
| 172 | C2×C3⋊C8 | — | 5 | 1 | no | — |
| 173 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 174 | C23×C5 | #235 | 1 | 1 | yes | characteristic |
| 175 | C2×C4×C5 | — | 4 | 1 | no | — |
| 176 | C2×C4×C5 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 177 | C2×C4×C5 | #235 | 1 | 1 | yes | characteristic |
| 178 | C2×C4×C5 | D3×D4 | 1 | 1 | yes | characteristic |
| 179 | C2×C4×C5 | #237 | 1 | 1 | yes | characteristic |
| 180 | C2×C4×C5 | #237 | 1 | 1 | yes | characteristic |
| 181 | C2×C4×C5 | D3×Q8 | 1 | 1 | yes | characteristic |
| 182 | C2×Dic5 | — | 6 | 2 | no | — |
| 183 | C2×Dic5 | — | 12 | 1 | no | — |
| 184 | C5×C8 | — | 2 | 2 | no | — |
| 185 | C5×C8 | — | 4 | 1 | no | — |
| 186 | #168 | — | 3 | 1 | no | — |
| 187 | #168 | — | 3 | 1 | no | — |
| 188 | #168 | — | 3 | 1 | no | — |
| 189 | #168 | — | 3 | 1 | no | — |
| 190 | #168 | — | 12 | 1 | no | — |
| 191 | C2×M16 | — | 15 | 1 | no | — |
| 192 | C2×M16 | C2×D15 | 1 | 1 | yes | characteristic |
| 193 | C2×M16 | C2×D15 | 1 | 1 | yes | characteristic |
| 194 | C22×C8 | — | 15 | 1 | no | — |
| 195 | #117 | — | 15 | 1 | no | — |
| 196 | #117 | — | 15 | 1 | no | — |
| 197 | #117 | C2×D15 | 1 | 1 | yes | characteristic |
| 198 | #108 | — | 30 | 1 | no | — |
| 199 | #108 | — | 30 | 1 | no | — |
| 200 | #107 | — | 30 | 1 | no | — |
| 201 | #106 | — | 30 | 1 | no | — |
| 202 | #97 | — | 15 | 2 | no | — |
| 203 | C4×C8 | — | 15 | 2 | no | — |
| 204 | C2×C3×C5 | — | 2 | 1 | no | — |
| 205 | C2×C3×C5 | — | 2 | 1 | no | — |
| 206 | C2×C3×C5 | — | 2 | 1 | no | — |
| 207 | C2×C3×C5 | #382 | 1 | 1 | yes | characteristic |
| 208 | C23×C3 | #709 | 1 | 1 | yes | characteristic |
| 209 | C2×C3×C4 | — | 4 | 1 | no | — |
| 210 | C2×C3×C4 | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 211 | C2×C3×C4 | #709 | 1 | 1 | yes | characteristic |
| 212 | C2×C3×C4 | D4×D5 | 1 | 1 | yes | characteristic |
| 213 | C2×C3×C4 | #711 | 1 | 1 | yes | characteristic |
| 214 | C2×C3×C4 | #711 | 1 | 1 | yes | characteristic |
| 215 | C2×C3×C4 | Q8×D5 | 1 | 1 | yes | characteristic |
| 216 | C2×Dic3 | — | 10 | 2 | no | — |
| 217 | C2×Dic3 | — | 20 | 1 | no | — |
| 218 | C3×C8 | — | 2 | 2 | no | — |
| 219 | C3×C8 | — | 4 | 1 | no | — |
| 220 | C3⋊C8 | — | 5 | 1 | no | — |
| 221 | C3⋊C8 | — | 5 | 1 | no | — |
| 222 | C3⋊C8 | — | 5 | 1 | no | — |
| 223 | C3⋊C8 | — | 5 | 1 | no | — |
| 224 | C3⋊C8 | — | 20 | 1 | no | — |
| 225 | C22×C5 | — | 4 | 1 | no | — |
| 226 | C22×C5 | #872 | 1 | 1 | yes | characteristic |
| 227 | C22×C5 | #885 | 1 | 1 | yes | characteristic |
| 228 | C22×C5 | #889 | 1 | 1 | yes | characteristic |
| 229 | C4×C5 | — | 8 | 1 | no | — |
| 230 | C4×C5 | #877 | 1 | 1 | yes | characteristic |
| 231 | C4×C5 | #885 | 1 | 1 | yes | characteristic |
| 232 | C4×C5 | #889 | 1 | 1 | yes | characteristic |
| 233 | C4×C5 | D3×C4⋊C4 | 1 | 1 | yes | characteristic |
| 234 | Dic5 | — | 12 | 2 | no | — |
| 235 | Dic5 | — | 24 | 1 | no | — |
| 236 | C22×C4 | C22×D15 | 1 | 1 | yes | characteristic |
| 237 | M16 | — | 2 | 1 | no | — |
| 238 | M16 | — | 2 | 1 | no | — |
| 239 | M16 | — | 2 | 1 | no | — |
| 240 | M16 | — | 2 | 1 | no | — |
| 241 | M16 | — | 30 | 1 | no | — |
| 242 | M16 | — | 30 | 1 | no | — |
| 243 | C2×C8 | — | 2 | 1 | no | — |
| 244 | C2×C8 | — | 15 | 1 | no | — |
| 245 | C2×C8 | — | 15 | 1 | no | — |
| 246 | C2×C8 | — | 15 | 1 | no | — |
| 247 | C2×C8 | — | 15 | 1 | no | — |
| 248 | C2×C8 | — | 15 | 1 | no | — |
| 249 | C2×C8 | — | 15 | 1 | no | — |
| 250 | C2×C8 | — | 30 | 1 | no | — |
| 251 | C2×C8 | C4×D15 | 1 | 2 | yes | — |
| 252 | C4⋊C4 | — | 2 | 1 | no | — |
| 253 | C4⋊C4 | — | 30 | 1 | no | — |
| 254 | C4⋊C4 | — | 30 | 1 | no | — |
| 255 | C22⋊C4 | — | 2 | 1 | no | — |
| 256 | C22⋊C4 | — | 30 | 1 | no | — |
| 257 | C22⋊C4 | — | 30 | 1 | no | — |
| 258 | C42 | — | 2 | 1 | no | — |
| 259 | C42 | — | 15 | 2 | no | — |
| 260 | C42 | — | 30 | 1 | no | — |
| 261 | C3×C5 | #1833 | 1 | 1 | yes | characteristic, complemented, Hall |
| 262 | C22×C3 | — | 4 | 1 | no | — |
| 263 | C22×C3 | #4018 | 1 | 1 | yes | characteristic |
| 264 | C22×C3 | #4031 | 1 | 1 | yes | characteristic |
| 265 | C22×C3 | #4035 | 1 | 1 | yes | characteristic |
| 266 | C3×C4 | — | 8 | 1 | no | — |
| 267 | C3×C4 | #4023 | 1 | 1 | yes | characteristic |
| 268 | C3×C4 | #4031 | 1 | 1 | yes | characteristic |
| 269 | C3×C4 | #4035 | 1 | 1 | yes | characteristic |
| 270 | C3×C4 | D5×C4⋊C4 | 1 | 1 | yes | characteristic |
| 271 | Dic3 | — | 20 | 2 | no | — |
| 272 | Dic3 | — | 40 | 1 | no | — |
| 273 | C2×C5 | — | 2 | 1 | no | — |
| 274 | C2×C5 | — | 2 | 1 | no | — |
| 275 | C2×C5 | — | 2 | 1 | no | — |
| 276 | C2×C5 | #4650 | 1 | 1 | yes | characteristic |
| 277 | C23 | #6817 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | — | 4 | 1 | no | — |
| 279 | C2×C4 | — | 30 | 2 | no | — |
| 280 | C2×C4 | — | 60 | 1 | no | — |
| 281 | C2×C4 | C2×C4×D15 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | #6817 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | D4×D15 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #6819 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | #6819 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | Q8×D15 | 1 | 1 | yes | characteristic |
| 287 | C8 | — | 2 | 2 | no | — |
| 288 | C8 | — | 4 | 1 | no | — |
| 289 | C8 | — | 15 | 1 | no | — |
| 290 | C8 | — | 15 | 1 | no | — |
| 291 | C8 | — | 15 | 1 | no | — |
| 292 | C8 | — | 15 | 1 | no | — |
| 293 | C8 | — | 60 | 1 | no | — |
| 294 | C2×C3 | — | 2 | 1 | no | — |
| 295 | C2×C3 | — | 2 | 1 | no | — |
| 296 | C2×C3 | — | 2 | 1 | no | — |
| 297 | C2×C3 | #64939 | 1 | 1 | yes | characteristic |
| 298 | C5 | #69013 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 299 | C22 | — | 4 | 1 | no | — |
| 300 | C22 | #91761 | 1 | 1 | yes | characteristic |
| 301 | C22 | #91774 | 1 | 1 | yes | characteristic |
| 302 | C22 | #91778 | 1 | 1 | yes | characteristic |
| 303 | C4 | — | 8 | 1 | no | — |
| 304 | C4 | — | 60 | 2 | no | — |
| 305 | C4 | — | 120 | 1 | no | — |
| 306 | C4 | #91766 | 1 | 1 | yes | characteristic |
| 307 | C4 | #91774 | 1 | 1 | yes | characteristic |
| 308 | C4 | #91778 | 1 | 1 | yes | characteristic |
| 309 | C4 | C4⋊C4×D15 | 1 | 1 | yes | characteristic |
| 310 | C3 | #10599806 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 311 | C2 | — | 2 | 1 | no | — |
| 312 | C2 | — | 2 | 1 | no | — |
| 313 | C2 | — | 2 | 1 | no | — |
| 314 | C2 | #11750751 | 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