← #14313896
Subgroups — #14313896
Order 1920.
464 conjugacy classes of subgroups.
The subgroup lattice has 296 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C3×C5×#135 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#91102 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#91096 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11746385 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11746383 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10602871 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×C3×C5×#41 | — | 2 | 1 | no | — |
| 8 | C3×C5×#135 | — | 2 | 2 | no | — |
| 9 | C3×C5×#135 | C4 | 1 | 4 | yes | complemented |
| 10 | C2×C3×C5×Q16 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×C5×SD16 | — | 2 | 1 | no | — |
| 12 | C2×C3×C5×D8 | C22 | 1 | 1 | yes | characteristic |
| 13 | C22×C3×C5×C8 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×C4⋊Dic15 | — | 2 | 1 | no | — |
| 15 | #91102 | — | 2 | 1 | no | — |
| 16 | #91102 | C22 | 1 | 2 | yes | — |
| 17 | #91096 | — | 2 | 2 | no | — |
| 18 | C2×C3⋊#672 | C22 | 1 | 2 | yes | — |
| 19 | #72078 | — | 5 | 1 | no | maximal, Hall |
| 20 | C2×C5×#135 | D3 | 1 | 1 | yes | characteristic |
| 21 | C2×#3950 | — | 3 | 1 | no | — |
| 22 | C2×#3944 | — | 3 | 1 | no | — |
| 23 | #64784 | — | 3 | 2 | no | — |
| 24 | #64782 | — | 3 | 2 | no | — |
| 25 | C3×C5×#41 | — | 2 | 4 | no | — |
| 26 | C3×C5×#41 | — | 4 | 2 | no | — |
| 27 | C2×C3×C5×Q8 | — | 2 | 1 | no | — |
| 28 | C2×C3×C5×D4 | — | 2 | 1 | no | — |
| 29 | C2×C3×C5×D4 | — | 4 | 1 | no | — |
| 30 | C22×C3×C4×C5 | — | 4 | 1 | no | — |
| 31 | C22×C3×C4×C5 | D4 | 1 | 1 | yes | characteristic |
| 32 | C22×Dic15 | — | 4 | 1 | no | — |
| 33 | C3×C5×Q16 | — | 2 | 1 | no | — |
| 34 | C3×C5×Q16 | C2×C4 | 1 | 2 | yes | complemented |
| 35 | C3×C5×SD16 | — | 2 | 4 | no | — |
| 36 | C3×C5×D8 | — | 2 | 1 | no | — |
| 37 | C3×C5×D8 | C2×C4 | 1 | 2 | yes | complemented |
| 38 | C2×C3×C5×C8 | C2×C4 | 1 | 2 | yes | — |
| 39 | C2×C3×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 40 | C2×C3×C5×C8 | D4 | 1 | 2 | yes | — |
| 41 | C2×C3×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 42 | C4⋊Dic15 | — | 2 | 2 | no | — |
| 43 | C4⋊Dic15 | — | 4 | 1 | no | — |
| 44 | C3⋊#672 | — | 2 | 2 | no | — |
| 45 | C2×C3×#135 | D5 | 1 | 1 | yes | characteristic |
| 46 | C2×#818 | — | 5 | 1 | no | — |
| 47 | C2×#812 | — | 5 | 1 | no | — |
| 48 | #4564 | — | 5 | 2 | no | — |
| 49 | #4562 | — | 5 | 2 | no | — |
| 50 | C2×C5×#41 | — | 2 | 1 | no | — |
| 51 | C5×#135 | — | 2 | 2 | no | — |
| 52 | C5×#135 | Dic3 | 1 | 4 | yes | complemented |
| 53 | C2×C5×Q16 | C2×D3 | 1 | 1 | yes | characteristic |
| 54 | C2×C5×SD16 | — | 2 | 1 | no | — |
| 55 | C2×C5×D8 | C2×D3 | 1 | 1 | yes | characteristic |
| 56 | C22×C5×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 57 | C2×C4⋊Dic5 | — | 6 | 1 | no | — |
| 58 | #3950 | — | 3 | 2 | no | — |
| 59 | #3950 | — | 6 | 1 | no | — |
| 60 | #3944 | — | 6 | 2 | no | — |
| 61 | C2×#672 | — | 3 | 2 | no | — |
| 62 | #2139 | — | 15 | 1 | no | Sylow |
| 63 | C23×C3×C5 | — | 4 | 1 | no | — |
| 64 | C23×C3×C5 | D8 | 1 | 1 | yes | characteristic |
| 65 | C3×C5×Q8 | — | 2 | 2 | no | — |
| 66 | C3×C5×Q8 | — | 4 | 1 | no | — |
| 67 | C3×C5×D4 | — | 2 | 2 | no | — |
| 68 | C3×C5×D4 | — | 4 | 4 | no | — |
| 69 | C3×C5×D4 | — | 4 | 1 | no | — |
| 70 | C2×C3×C4×C5 | — | 4 | 1 | no | — |
| 71 | C2×C3×C4×C5 | — | 4 | 4 | no | — |
| 72 | C2×C3×C4×C5 | C22⋊C4 | 1 | 2 | yes | — |
| 73 | C2×C3×C4×C5 | D8 | 1 | 1 | yes | characteristic |
| 74 | C2×C3×C4×C5 | SD16 | 1 | 2 | yes | — |
| 75 | C2×C3×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C2×Dic15 | — | 4 | 2 | no | — |
| 77 | C2×Dic15 | — | 8 | 2 | no | — |
| 78 | C3×C5×C8 | C22⋊C4 | 1 | 2 | yes | — |
| 79 | C3×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 80 | C3×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | C2×C3×#41 | — | 2 | 1 | no | — |
| 82 | C3×#135 | — | 2 | 2 | no | — |
| 83 | C3×#135 | Dic5 | 1 | 4 | yes | complemented |
| 84 | C2×C3×Q16 | C2×D5 | 1 | 1 | yes | characteristic |
| 85 | C2×C3×SD16 | — | 2 | 1 | no | — |
| 86 | C2×C3×D8 | C2×D5 | 1 | 1 | yes | characteristic |
| 87 | C22×C3×C8 | C2×D5 | 1 | 1 | yes | characteristic |
| 88 | C2×C4⋊Dic3 | — | 10 | 1 | no | — |
| 89 | #818 | — | 5 | 2 | no | — |
| 90 | #818 | — | 10 | 1 | no | — |
| 91 | #812 | — | 10 | 2 | no | — |
| 92 | C2×C3⋊C16 | — | 5 | 2 | no | — |
| 93 | C5×#41 | — | 2 | 4 | no | — |
| 94 | C5×#41 | — | 4 | 2 | no | — |
| 95 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 96 | C2×C5×D4 | — | 2 | 1 | no | — |
| 97 | C2×C5×D4 | — | 4 | 1 | no | — |
| 98 | C22×C4×C5 | — | 4 | 1 | no | — |
| 99 | C22×C4×C5 | #67 | 1 | 1 | yes | characteristic |
| 100 | C22×Dic5 | — | 12 | 1 | no | — |
| 101 | C5×Q16 | — | 2 | 1 | no | — |
| 102 | C5×Q16 | C2×Dic3 | 1 | 2 | yes | complemented |
| 103 | C5×SD16 | — | 2 | 4 | no | — |
| 104 | C5×D8 | — | 2 | 1 | no | — |
| 105 | C5×D8 | C2×Dic3 | 1 | 2 | yes | complemented |
| 106 | C2×C5×C8 | C2×Dic3 | 1 | 2 | yes | — |
| 107 | C2×C5×C8 | #67 | 1 | 1 | yes | characteristic |
| 108 | C2×C5×C8 | #67 | 1 | 2 | yes | — |
| 109 | C2×C5×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 110 | C4⋊Dic5 | — | 6 | 2 | no | — |
| 111 | C4⋊Dic5 | — | 12 | 1 | no | — |
| 112 | #672 | — | 6 | 2 | no | — |
| 113 | C2×#135 | D15 | 1 | 1 | yes | characteristic |
| 114 | C2×M32 | — | 15 | 1 | no | — |
| 115 | C2×#107 | — | 15 | 1 | no | — |
| 116 | #358 | — | 15 | 2 | no | — |
| 117 | #357 | — | 15 | 2 | no | — |
| 118 | C22×C3×C5 | — | 2 | 1 | no | — |
| 119 | C22×C3×C5 | — | 2 | 1 | no | — |
| 120 | C22×C3×C5 | — | 4 | 2 | no | — |
| 121 | C22×C3×C5 | — | 8 | 1 | no | — |
| 122 | C22×C3×C5 | — | 8 | 1 | no | — |
| 123 | C22×C3×C5 | #102 | 1 | 2 | yes | — |
| 124 | C22×C3×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 125 | C3×C4×C5 | — | 4 | 2 | no | — |
| 126 | C3×C4×C5 | #102 | 1 | 2 | yes | — |
| 127 | C3×C4×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 128 | C3×C4×C5 | C2×SD16 | 1 | 1 | yes | characteristic |
| 129 | Dic15 | — | 8 | 2 | no | — |
| 130 | C3×#41 | — | 2 | 4 | no | — |
| 131 | C3×#41 | — | 4 | 2 | no | — |
| 132 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 133 | C2×C3×D4 | — | 2 | 1 | no | — |
| 134 | C2×C3×D4 | — | 4 | 1 | no | — |
| 135 | C22×C3×C4 | — | 4 | 1 | no | — |
| 136 | C22×C3×C4 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 137 | C22×Dic3 | — | 20 | 1 | no | — |
| 138 | C3×Q16 | — | 2 | 1 | no | — |
| 139 | C3×Q16 | C2×Dic5 | 1 | 2 | yes | complemented |
| 140 | C3×SD16 | — | 2 | 4 | no | — |
| 141 | C3×D8 | — | 2 | 1 | no | — |
| 142 | C3×D8 | C2×Dic5 | 1 | 2 | yes | complemented |
| 143 | C2×C3×C8 | C2×Dic5 | 1 | 2 | yes | — |
| 144 | C2×C3×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 145 | C2×C3×C8 | C22⋊D5 | 1 | 2 | yes | — |
| 146 | C2×C3×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 147 | C4⋊Dic3 | — | 10 | 2 | no | — |
| 148 | C4⋊Dic3 | — | 20 | 1 | no | — |
| 149 | C3⋊C16 | — | 10 | 2 | no | — |
| 150 | C23×C5 | — | 4 | 1 | no | — |
| 151 | C23×C5 | #213 | 1 | 1 | yes | characteristic |
| 152 | C5×Q8 | — | 2 | 2 | no | — |
| 153 | C5×Q8 | — | 4 | 1 | no | — |
| 154 | C5×D4 | — | 2 | 2 | no | — |
| 155 | C5×D4 | — | 4 | 4 | no | — |
| 156 | C5×D4 | — | 4 | 1 | no | — |
| 157 | C2×C4×C5 | — | 4 | 1 | no | — |
| 158 | C2×C4×C5 | — | 4 | 4 | no | — |
| 159 | C2×C4×C5 | #213 | 1 | 1 | yes | characteristic |
| 160 | C2×C4×C5 | #214 | 1 | 2 | yes | — |
| 161 | C2×C4×C5 | #217 | 1 | 2 | yes | — |
| 162 | C2×C4×C5 | C2×#67 | 1 | 1 | yes | characteristic |
| 163 | C2×Dic5 | — | 12 | 2 | no | — |
| 164 | C2×Dic5 | — | 24 | 2 | no | — |
| 165 | C5×C8 | #217 | 1 | 2 | yes | — |
| 166 | C5×C8 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 167 | C5×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 168 | C2×#41 | — | 2 | 1 | no | — |
| 169 | #135 | — | 2 | 2 | no | — |
| 170 | #135 | Dic15 | 1 | 4 | yes | complemented |
| 171 | C2×Q16 | C2×D15 | 1 | 1 | yes | characteristic |
| 172 | C2×SD16 | — | 2 | 1 | no | — |
| 173 | C2×D8 | C2×D15 | 1 | 1 | yes | characteristic |
| 174 | C22×C8 | C2×D15 | 1 | 1 | yes | characteristic |
| 175 | C2×C4⋊C4 | — | 30 | 1 | no | — |
| 176 | M32 | — | 30 | 2 | no | — |
| 177 | C2×C16 | — | 15 | 2 | no | — |
| 178 | #107 | — | 15 | 2 | no | — |
| 179 | #107 | — | 30 | 1 | no | — |
| 180 | C2×C3×C5 | — | 2 | 2 | no | — |
| 181 | C2×C3×C5 | — | 8 | 2 | no | — |
| 182 | C2×C3×C5 | C2×#102 | 1 | 1 | yes | characteristic |
| 183 | C2×C3×C5 | #509 | 1 | 1 | yes | characteristic |
| 184 | C2×C3×C5 | #510 | 1 | 1 | yes | characteristic |
| 185 | C23×C3 | — | 4 | 1 | no | — |
| 186 | C23×C3 | #686 | 1 | 1 | yes | characteristic |
| 187 | C3×Q8 | — | 2 | 2 | no | — |
| 188 | C3×Q8 | — | 4 | 1 | no | — |
| 189 | C3×D4 | — | 2 | 2 | no | — |
| 190 | C3×D4 | — | 4 | 4 | no | — |
| 191 | C3×D4 | — | 4 | 1 | no | — |
| 192 | C2×C3×C4 | — | 4 | 1 | no | — |
| 193 | C2×C3×C4 | — | 4 | 4 | no | — |
| 194 | C2×C3×C4 | #686 | 1 | 1 | yes | characteristic |
| 195 | C2×C3×C4 | #687 | 1 | 2 | yes | — |
| 196 | C2×C3×C4 | C22⋊Dic5 | 1 | 2 | yes | — |
| 197 | C2×C3×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 198 | C2×Dic3 | — | 20 | 2 | no | — |
| 199 | C2×Dic3 | — | 40 | 2 | no | — |
| 200 | C3×C8 | C22⋊Dic5 | 1 | 2 | yes | — |
| 201 | C3×C8 | C22×Dic5 | 1 | 1 | yes | characteristic |
| 202 | C3×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 203 | C22×C5 | — | 2 | 1 | no | — |
| 204 | C22×C5 | — | 2 | 1 | no | — |
| 205 | C22×C5 | — | 4 | 2 | no | — |
| 206 | C22×C5 | — | 8 | 1 | no | — |
| 207 | C22×C5 | — | 8 | 1 | no | — |
| 208 | C22×C5 | #832 | 1 | 2 | yes | — |
| 209 | C22×C5 | C2×#213 | 1 | 1 | yes | characteristic |
| 210 | C4×C5 | — | 4 | 2 | no | — |
| 211 | C4×C5 | #832 | 1 | 2 | yes | — |
| 212 | C4×C5 | C2×#214 | 1 | 1 | yes | characteristic |
| 213 | C4×C5 | C2×#217 | 1 | 1 | yes | characteristic |
| 214 | Dic5 | — | 24 | 2 | no | — |
| 215 | #41 | — | 2 | 4 | no | — |
| 216 | #41 | — | 4 | 2 | no | — |
| 217 | C2×Q8 | — | 2 | 1 | no | — |
| 218 | C2×D4 | — | 2 | 1 | no | — |
| 219 | C2×D4 | — | 4 | 1 | no | — |
| 220 | C22×C4 | — | 4 | 1 | no | — |
| 221 | C22×C4 | — | 60 | 1 | no | — |
| 222 | C22×C4 | #1220 | 1 | 1 | yes | characteristic |
| 223 | Q16 | — | 2 | 1 | no | — |
| 224 | Q16 | C2×Dic15 | 1 | 2 | yes | complemented |
| 225 | SD16 | — | 2 | 4 | no | — |
| 226 | D8 | — | 2 | 1 | no | — |
| 227 | D8 | C2×Dic15 | 1 | 2 | yes | complemented |
| 228 | C2×C8 | C2×Dic15 | 1 | 2 | yes | — |
| 229 | C2×C8 | #1220 | 1 | 1 | yes | characteristic |
| 230 | C2×C8 | #1220 | 1 | 2 | yes | — |
| 231 | C2×C8 | C22×D15 | 1 | 1 | yes | characteristic |
| 232 | C4⋊C4 | — | 30 | 2 | no | — |
| 233 | C4⋊C4 | — | 60 | 1 | no | — |
| 234 | C16 | — | 30 | 2 | no | — |
| 235 | C3×C5 | #2139 | 1 | 1 | yes | characteristic, complemented, Hall |
| 236 | C22×C3 | — | 2 | 1 | no | — |
| 237 | C22×C3 | — | 2 | 1 | no | — |
| 238 | C22×C3 | — | 4 | 2 | no | — |
| 239 | C22×C3 | — | 8 | 1 | no | — |
| 240 | C22×C3 | — | 8 | 1 | no | — |
| 241 | C22×C3 | #3964 | 1 | 2 | yes | — |
| 242 | C22×C3 | C2×#686 | 1 | 1 | yes | characteristic |
| 243 | C3×C4 | — | 4 | 2 | no | — |
| 244 | C3×C4 | #3964 | 1 | 2 | yes | — |
| 245 | C3×C4 | C2×#687 | 1 | 1 | yes | characteristic |
| 246 | C3×C4 | C2×C22⋊Dic5 | 1 | 1 | yes | characteristic |
| 247 | Dic3 | — | 40 | 2 | no | — |
| 248 | C2×C5 | — | 2 | 2 | no | — |
| 249 | C2×C5 | — | 8 | 2 | no | — |
| 250 | C2×C5 | #5193 | 1 | 1 | yes | characteristic |
| 251 | C2×C5 | #5195 | 1 | 1 | yes | characteristic |
| 252 | C2×C5 | C2×#832 | 1 | 1 | yes | characteristic |
| 253 | C23 | — | 4 | 1 | no | — |
| 254 | C23 | #6715 | 1 | 1 | yes | characteristic |
| 255 | Q8 | — | 2 | 2 | no | — |
| 256 | Q8 | — | 4 | 1 | no | — |
| 257 | D4 | — | 2 | 2 | no | — |
| 258 | D4 | — | 4 | 4 | no | — |
| 259 | D4 | — | 4 | 1 | no | — |
| 260 | C2×C4 | — | 4 | 1 | no | — |
| 261 | C2×C4 | — | 4 | 4 | no | — |
| 262 | C2×C4 | — | 60 | 2 | no | — |
| 263 | C2×C4 | — | 120 | 2 | no | — |
| 264 | C2×C4 | #6715 | 1 | 1 | yes | characteristic |
| 265 | C2×C4 | #6716 | 1 | 2 | yes | — |
| 266 | C2×C4 | #6719 | 1 | 2 | yes | — |
| 267 | C2×C4 | C2×#1220 | 1 | 1 | yes | characteristic |
| 268 | C8 | #6719 | 1 | 2 | yes | — |
| 269 | C8 | C22×Dic15 | 1 | 1 | yes | characteristic |
| 270 | C8 | C2×#1220 | 1 | 1 | yes | characteristic |
| 271 | C2×C3 | — | 2 | 2 | no | — |
| 272 | C2×C3 | — | 8 | 2 | no | — |
| 273 | C2×C3 | #65482 | 1 | 1 | yes | characteristic |
| 274 | C2×C3 | #65484 | 1 | 1 | yes | characteristic |
| 275 | C2×C3 | C2×#3964 | 1 | 1 | yes | characteristic |
| 276 | C5 | #72078 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 277 | C22 | — | 2 | 1 | no | — |
| 278 | C22 | — | 2 | 1 | no | — |
| 279 | C22 | — | 4 | 2 | no | — |
| 280 | C22 | — | 8 | 1 | no | — |
| 281 | C22 | — | 8 | 1 | no | — |
| 282 | C22 | #91116 | 1 | 2 | yes | — |
| 283 | C22 | C2×#6715 | 1 | 1 | yes | characteristic |
| 284 | C4 | — | 4 | 2 | no | — |
| 285 | C4 | — | 120 | 2 | no | — |
| 286 | C4 | #91116 | 1 | 2 | yes | — |
| 287 | C4 | C2×#6716 | 1 | 1 | yes | characteristic |
| 288 | C4 | C2×#6719 | 1 | 1 | yes | characteristic |
| 289 | C3 | #10602871 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 290 | C2 | — | 2 | 2 | no | — |
| 291 | C2 | — | 8 | 2 | no | — |
| 292 | C2 | #11751294 | 1 | 1 | yes | characteristic |
| 293 | C2 | #11751296 | 1 | 1 | yes | characteristic |
| 294 | C2 | C2×#91116 | 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