← #14305186
Subgroups — #14305186
Order 1920.
480 conjugacy classes of subgroups.
The subgroup lattice has 288 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C3×C5×#106 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#91102 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#91099 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11751085 | C2 | 1 | 2 | yes | maximal |
| 5 | #11746350 | C2 | 1 | 2 | yes | maximal |
| 6 | #10601355 | — | 3 | 1 | no | maximal, Hall |
| 7 | C22×C3×C5×C8 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×C3×C5×C4⋊C4 | C22 | 1 | 2 | yes | — |
| 9 | C2×C4⋊Dic15 | — | 2 | 2 | no | — |
| 10 | C2×C4⋊Dic15 | C22 | 1 | 2 | yes | — |
| 11 | C2×C4×Dic15 | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×C3⋊#168 | C22 | 1 | 1 | yes | characteristic |
| 13 | C3×C5×#106 | C4 | 1 | 4 | yes | complemented |
| 14 | #91115 | — | 2 | 2 | no | — |
| 15 | #91102 | — | 2 | 2 | no | — |
| 16 | #91099 | — | 2 | 2 | no | — |
| 17 | #70562 | — | 5 | 1 | no | maximal, Hall |
| 18 | C2×C5×#106 | D3 | 1 | 1 | yes | characteristic |
| 19 | C2×#3950 | — | 3 | 1 | no | — |
| 20 | C2×#3947 | — | 3 | 1 | no | — |
| 21 | #65274 | — | 3 | 2 | no | — |
| 22 | #64749 | — | 3 | 2 | no | — |
| 23 | C22×C3×C4×C5 | — | 2 | 2 | no | — |
| 24 | C22×C3×C4×C5 | C23 | 1 | 1 | yes | characteristic |
| 25 | C22×Dic15 | — | 2 | 2 | no | — |
| 26 | C22×Dic15 | D4 | 1 | 1 | yes | characteristic |
| 27 | C22×Dic15 | D4 | 1 | 1 | yes | characteristic |
| 28 | C2×C3×C5×C8 | C2×C4 | 1 | 2 | yes | — |
| 29 | C2×C3×C5×C8 | D4 | 1 | 2 | yes | — |
| 30 | C2×C3×C5×C8 | Q8 | 1 | 2 | yes | — |
| 31 | C3×C5×C4⋊C4 | — | 2 | 2 | no | — |
| 32 | C3×C5×C4⋊C4 | C2×C4 | 1 | 4 | yes | — |
| 33 | C4⋊Dic15 | — | 2 | 2 | no | — |
| 34 | C4⋊Dic15 | — | 2 | 2 | no | — |
| 35 | C4⋊Dic15 | — | 4 | 4 | no | — |
| 36 | C4×Dic15 | — | 2 | 1 | no | — |
| 37 | C4×Dic15 | — | 2 | 1 | no | — |
| 38 | C2×C3⋊#168 | — | 2 | 1 | no | — |
| 39 | C2×C3⋊#168 | — | 2 | 1 | no | — |
| 40 | C2×C3⋊#168 | Q8 | 1 | 2 | yes | — |
| 41 | C2×C3×#106 | D5 | 1 | 1 | yes | characteristic |
| 42 | C2×#818 | — | 5 | 1 | no | — |
| 43 | C2×#815 | — | 5 | 1 | no | — |
| 44 | #4984 | — | 5 | 2 | no | — |
| 45 | #4529 | — | 5 | 2 | no | — |
| 46 | C22×C5×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 47 | C2×C5×C4⋊C4 | C2×D3 | 1 | 2 | yes | — |
| 48 | C2×C4⋊Dic5 | — | 3 | 2 | no | — |
| 49 | C2×C4⋊Dic5 | — | 6 | 2 | no | — |
| 50 | C2×C4×Dic5 | — | 3 | 1 | no | — |
| 51 | C22×#168 | — | 3 | 1 | no | — |
| 52 | C5×#106 | Dic3 | 1 | 4 | yes | complemented |
| 53 | #3963 | — | 6 | 2 | no | — |
| 54 | #3950 | — | 6 | 2 | no | — |
| 55 | #3947 | — | 6 | 2 | no | — |
| 56 | #1944 | — | 15 | 1 | no | Sylow |
| 57 | C23×C3×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 58 | C2×C3×C4×C5 | — | 2 | 4 | no | — |
| 59 | C2×C3×C4×C5 | — | 4 | 2 | no | — |
| 60 | C2×C3×C4×C5 | — | 4 | 2 | no | — |
| 61 | C2×C3×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 62 | C2×C3×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C2×C3×C4×C5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 64 | C2×C3×C4×C5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 65 | C2×C3×C4×C5 | #41 | 1 | 1 | yes | characteristic |
| 66 | C2×C3×C4×C5 | #41 | 1 | 1 | yes | characteristic |
| 67 | C2×Dic15 | — | 2 | 1 | no | — |
| 68 | C2×Dic15 | — | 2 | 1 | no | — |
| 69 | C2×Dic15 | — | 2 | 2 | no | — |
| 70 | C2×Dic15 | — | 2 | 2 | no | — |
| 71 | C2×Dic15 | — | 4 | 2 | no | — |
| 72 | C2×Dic15 | — | 4 | 2 | no | — |
| 73 | C2×Dic15 | — | 4 | 2 | no | — |
| 74 | C3×C5×C8 | C4⋊C4 | 1 | 4 | yes | complemented |
| 75 | C3⋊#168 | — | 2 | 2 | no | — |
| 76 | C22×C3×C8 | C2×D5 | 1 | 1 | yes | characteristic |
| 77 | C2×C3×C4⋊C4 | C2×D5 | 1 | 2 | yes | — |
| 78 | C2×C4⋊Dic3 | — | 5 | 2 | no | — |
| 79 | C2×C4⋊Dic3 | — | 10 | 2 | no | — |
| 80 | C2×C4×Dic3 | — | 5 | 1 | no | — |
| 81 | C22×C3⋊C8 | — | 5 | 1 | no | — |
| 82 | C3×#106 | Dic5 | 1 | 4 | yes | complemented |
| 83 | C3⋊#95 | — | 10 | 2 | no | — |
| 84 | #818 | — | 10 | 2 | no | — |
| 85 | #815 | — | 10 | 2 | no | — |
| 86 | C22×C4×C5 | — | 2 | 2 | no | — |
| 87 | C22×C4×C5 | C22×D3 | 1 | 1 | yes | characteristic |
| 88 | C22×Dic5 | — | 3 | 1 | no | — |
| 89 | C22×Dic5 | — | 3 | 1 | no | — |
| 90 | C22×Dic5 | — | 6 | 2 | no | — |
| 91 | C2×C5×C8 | Dic6 | 1 | 2 | yes | — |
| 92 | C2×C5×C8 | D12 | 1 | 2 | yes | — |
| 93 | C2×C5×C8 | C2×Dic3 | 1 | 2 | yes | — |
| 94 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 95 | C5×C4⋊C4 | C2×Dic3 | 1 | 4 | yes | — |
| 96 | C4⋊Dic5 | — | 6 | 2 | no | — |
| 97 | C4⋊Dic5 | — | 6 | 2 | no | — |
| 98 | C4⋊Dic5 | — | 12 | 4 | no | — |
| 99 | C4×Dic5 | — | 6 | 1 | no | — |
| 100 | C4×Dic5 | — | 6 | 1 | no | — |
| 101 | C2×#168 | — | 3 | 2 | no | — |
| 102 | C2×#168 | — | 6 | 1 | no | — |
| 103 | C2×#168 | — | 6 | 1 | no | — |
| 104 | C2×#107 | — | 15 | 1 | no | — |
| 105 | C2×#106 | D15 | 1 | 1 | yes | characteristic |
| 106 | C2×#97 | — | 15 | 1 | no | — |
| 107 | #389 | — | 15 | 2 | no | — |
| 108 | #340 | — | 15 | 2 | no | — |
| 109 | C22×C3×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 110 | C22×C3×C5 | #121 | 1 | 1 | yes | characteristic |
| 111 | C22×C3×C5 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 112 | C22×C3×C5 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 113 | C22×C3×C5 | #137 | 1 | 2 | yes | — |
| 114 | C3×C4×C5 | — | 4 | 4 | no | — |
| 115 | C3×C4×C5 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 116 | C3×C4×C5 | C4×Q8 | 1 | 1 | yes | characteristic |
| 117 | C3×C4×C5 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 118 | C3×C4×C5 | #125 | 1 | 1 | yes | characteristic |
| 119 | Dic15 | — | 4 | 1 | no | — |
| 120 | Dic15 | — | 4 | 1 | no | — |
| 121 | Dic15 | — | 8 | 2 | no | — |
| 122 | C22×C3×C4 | — | 2 | 2 | no | — |
| 123 | C22×C3×C4 | C22×D5 | 1 | 1 | yes | characteristic |
| 124 | C22×Dic3 | — | 5 | 1 | no | — |
| 125 | C22×Dic3 | — | 5 | 1 | no | — |
| 126 | C22×Dic3 | — | 10 | 2 | no | — |
| 127 | C2×C3×C8 | Dic10 | 1 | 2 | yes | — |
| 128 | C2×C3×C8 | D20 | 1 | 2 | yes | — |
| 129 | C2×C3×C8 | C2×Dic5 | 1 | 2 | yes | — |
| 130 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 131 | C3×C4⋊C4 | C2×Dic5 | 1 | 4 | yes | — |
| 132 | C4⋊Dic3 | — | 10 | 2 | no | — |
| 133 | C4⋊Dic3 | — | 10 | 2 | no | — |
| 134 | C4⋊Dic3 | — | 20 | 4 | no | — |
| 135 | C4×Dic3 | — | 10 | 1 | no | — |
| 136 | C4×Dic3 | — | 10 | 1 | no | — |
| 137 | C2×C3⋊C8 | — | 5 | 2 | no | — |
| 138 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 139 | C2×C3⋊C8 | — | 10 | 1 | no | — |
| 140 | C23×C5 | D3×D4 | 1 | 1 | yes | characteristic |
| 141 | C2×C4×C5 | — | 2 | 4 | no | — |
| 142 | C2×C4×C5 | — | 4 | 2 | no | — |
| 143 | C2×C4×C5 | — | 4 | 2 | no | — |
| 144 | C2×C4×C5 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 145 | C2×C4×C5 | C2×D12 | 1 | 1 | yes | characteristic |
| 146 | C2×C4×C5 | #237 | 1 | 1 | yes | characteristic |
| 147 | C2×C4×C5 | D3×Q8 | 1 | 1 | yes | characteristic |
| 148 | C2×C4×C5 | #239 | 1 | 1 | yes | characteristic |
| 149 | C2×C4×C5 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 150 | C2×Dic5 | — | 6 | 1 | no | — |
| 151 | C2×Dic5 | — | 6 | 1 | no | — |
| 152 | C2×Dic5 | — | 6 | 2 | no | — |
| 153 | C2×Dic5 | — | 6 | 2 | no | — |
| 154 | C2×Dic5 | — | 12 | 2 | no | — |
| 155 | C2×Dic5 | — | 12 | 2 | no | — |
| 156 | C2×Dic5 | — | 12 | 2 | no | — |
| 157 | C5×C8 | C4⋊Dic3 | 1 | 4 | yes | complemented |
| 158 | #168 | — | 6 | 2 | no | — |
| 159 | C22×C8 | — | 15 | 1 | no | — |
| 160 | C22×C8 | C2×D15 | 1 | 1 | yes | characteristic |
| 161 | C2×C4⋊C4 | — | 15 | 2 | no | — |
| 162 | C2×C4⋊C4 | — | 30 | 2 | no | — |
| 163 | C2×C4⋊C4 | C2×D15 | 1 | 2 | yes | — |
| 164 | C2×C42 | — | 15 | 1 | no | — |
| 165 | #107 | — | 30 | 2 | no | — |
| 166 | #106 | Dic15 | 1 | 4 | yes | complemented |
| 167 | #97 | — | 30 | 2 | no | — |
| 168 | #95 | — | 30 | 2 | no | — |
| 169 | C2×C3×C5 | #389 | 1 | 1 | yes | characteristic |
| 170 | C2×C3×C5 | #440 | 1 | 2 | yes | — |
| 171 | C2×C3×C5 | #469 | 1 | 1 | yes | characteristic |
| 172 | C2×C3×C5 | #470 | 1 | 1 | yes | characteristic |
| 173 | C2×C3×C5 | #501 | 1 | 2 | yes | — |
| 174 | C23×C3 | D4×D5 | 1 | 1 | yes | characteristic |
| 175 | C2×C3×C4 | — | 2 | 4 | no | — |
| 176 | C2×C3×C4 | — | 4 | 2 | no | — |
| 177 | C2×C3×C4 | — | 4 | 2 | no | — |
| 178 | C2×C3×C4 | C2×Dic10 | 1 | 1 | yes | characteristic |
| 179 | C2×C3×C4 | C2×D20 | 1 | 1 | yes | characteristic |
| 180 | C2×C3×C4 | #711 | 1 | 1 | yes | characteristic |
| 181 | C2×C3×C4 | Q8×D5 | 1 | 1 | yes | characteristic |
| 182 | C2×C3×C4 | #713 | 1 | 1 | yes | characteristic |
| 183 | C2×C3×C4 | C22×Dic5 | 1 | 1 | yes | characteristic |
| 184 | C2×Dic3 | — | 10 | 1 | no | — |
| 185 | C2×Dic3 | — | 10 | 1 | no | — |
| 186 | C2×Dic3 | — | 10 | 2 | no | — |
| 187 | C2×Dic3 | — | 10 | 2 | no | — |
| 188 | C2×Dic3 | — | 20 | 2 | no | — |
| 189 | C2×Dic3 | — | 20 | 2 | no | — |
| 190 | C2×Dic3 | — | 20 | 2 | no | — |
| 191 | C3×C8 | C4⋊Dic5 | 1 | 4 | yes | complemented |
| 192 | C3⋊C8 | — | 10 | 2 | no | — |
| 193 | C22×C5 | #888 | 1 | 1 | yes | characteristic |
| 194 | C22×C5 | #895 | 1 | 1 | yes | characteristic |
| 195 | C22×C5 | #914 | 1 | 2 | yes | — |
| 196 | C22×C5 | #915 | 1 | 2 | yes | — |
| 197 | C22×C5 | D4×Dic3 | 1 | 1 | yes | characteristic |
| 198 | C4×C5 | — | 4 | 4 | no | — |
| 199 | C4×C5 | #890 | 1 | 1 | yes | characteristic |
| 200 | C4×C5 | #897 | 1 | 1 | yes | characteristic |
| 201 | C4×C5 | C2×C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 202 | C4×C5 | Q8×Dic3 | 1 | 1 | yes | characteristic |
| 203 | Dic5 | — | 12 | 1 | no | — |
| 204 | Dic5 | — | 12 | 1 | no | — |
| 205 | Dic5 | — | 24 | 2 | no | — |
| 206 | C22×C4 | — | 2 | 2 | no | — |
| 207 | C22×C4 | — | 15 | 1 | no | — |
| 208 | C22×C4 | — | 15 | 1 | no | — |
| 209 | C22×C4 | — | 30 | 2 | no | — |
| 210 | C22×C4 | C22×D15 | 1 | 1 | yes | characteristic |
| 211 | C2×C8 | — | 15 | 2 | no | — |
| 212 | C2×C8 | — | 30 | 1 | no | — |
| 213 | C2×C8 | — | 30 | 1 | no | — |
| 214 | C2×C8 | Dic30 | 1 | 2 | yes | — |
| 215 | C2×C8 | D60 | 1 | 2 | yes | — |
| 216 | C2×C8 | C2×Dic15 | 1 | 2 | yes | — |
| 217 | C4⋊C4 | — | 2 | 2 | no | — |
| 218 | C4⋊C4 | — | 30 | 2 | no | — |
| 219 | C4⋊C4 | — | 30 | 2 | no | — |
| 220 | C4⋊C4 | — | 60 | 4 | no | — |
| 221 | C4⋊C4 | C2×Dic15 | 1 | 4 | yes | — |
| 222 | C42 | — | 30 | 1 | no | — |
| 223 | C42 | — | 30 | 1 | no | — |
| 224 | C3×C5 | #1944 | 1 | 1 | yes | characteristic, complemented, Hall |
| 225 | C22×C3 | #4034 | 1 | 1 | yes | characteristic |
| 226 | C22×C3 | #4041 | 1 | 1 | yes | characteristic |
| 227 | C22×C3 | #4060 | 1 | 2 | yes | — |
| 228 | C22×C3 | #4061 | 1 | 2 | yes | — |
| 229 | C22×C3 | D4×Dic5 | 1 | 1 | yes | characteristic |
| 230 | C3×C4 | — | 4 | 4 | no | — |
| 231 | C3×C4 | #4036 | 1 | 1 | yes | characteristic |
| 232 | C3×C4 | #4043 | 1 | 1 | yes | characteristic |
| 233 | C3×C4 | C2×C4⋊Dic5 | 1 | 1 | yes | characteristic |
| 234 | C3×C4 | Q8×Dic5 | 1 | 1 | yes | characteristic |
| 235 | Dic3 | — | 20 | 1 | no | — |
| 236 | Dic3 | — | 20 | 1 | no | — |
| 237 | Dic3 | — | 40 | 2 | no | — |
| 238 | C2×C5 | #4856 | 1 | 2 | yes | — |
| 239 | C2×C5 | #4865 | 1 | 1 | yes | characteristic |
| 240 | C2×C5 | #4867 | 1 | 1 | yes | characteristic |
| 241 | C2×C5 | #4984 | 1 | 1 | yes | characteristic |
| 242 | C2×C5 | #5164 | 1 | 2 | yes | — |
| 243 | C23 | D4×D15 | 1 | 1 | yes | characteristic |
| 244 | C2×C4 | — | 2 | 4 | no | — |
| 245 | C2×C4 | — | 4 | 2 | no | — |
| 246 | C2×C4 | — | 4 | 2 | no | — |
| 247 | C2×C4 | — | 30 | 1 | no | — |
| 248 | C2×C4 | — | 30 | 1 | no | — |
| 249 | C2×C4 | — | 30 | 2 | no | — |
| 250 | C2×C4 | — | 30 | 2 | no | — |
| 251 | C2×C4 | — | 60 | 2 | no | — |
| 252 | C2×C4 | — | 60 | 2 | no | — |
| 253 | C2×C4 | — | 60 | 2 | no | — |
| 254 | C2×C4 | C2×Dic30 | 1 | 1 | yes | characteristic |
| 255 | C2×C4 | C2×D60 | 1 | 1 | yes | characteristic |
| 256 | C2×C4 | #6819 | 1 | 1 | yes | characteristic |
| 257 | C2×C4 | Q8×D15 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | #6821 | 1 | 1 | yes | characteristic |
| 259 | C2×C4 | C22×Dic15 | 1 | 1 | yes | characteristic |
| 260 | C8 | — | 30 | 2 | no | — |
| 261 | C8 | C4⋊Dic15 | 1 | 4 | yes | complemented |
| 262 | C2×C3 | #65145 | 1 | 2 | yes | — |
| 263 | C2×C3 | #65154 | 1 | 1 | yes | characteristic |
| 264 | C2×C3 | #65157 | 1 | 1 | yes | characteristic |
| 265 | C2×C3 | #65274 | 1 | 1 | yes | characteristic |
| 266 | C2×C3 | #65453 | 1 | 2 | yes | — |
| 267 | C5 | #70562 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 268 | C22 | #91777 | 1 | 1 | yes | characteristic |
| 269 | C22 | #91784 | 1 | 1 | yes | characteristic |
| 270 | C22 | #91803 | 1 | 2 | yes | — |
| 271 | C22 | #91804 | 1 | 2 | yes | — |
| 272 | C22 | D4×Dic15 | 1 | 1 | yes | characteristic |
| 273 | C4 | — | 4 | 4 | no | — |
| 274 | C4 | — | 60 | 1 | no | — |
| 275 | C4 | — | 60 | 1 | no | — |
| 276 | C4 | — | 120 | 2 | no | — |
| 277 | C4 | #91779 | 1 | 1 | yes | characteristic |
| 278 | C4 | #91786 | 1 | 1 | yes | characteristic |
| 279 | C4 | C2×C4⋊Dic15 | 1 | 1 | yes | characteristic |
| 280 | C4 | Q8×Dic15 | 1 | 1 | yes | characteristic |
| 281 | C3 | #10601355 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 282 | C2 | #11750957 | 1 | 2 | yes | — |
| 283 | C2 | #11750966 | 1 | 1 | yes | characteristic |
| 284 | C2 | #11750968 | 1 | 1 | yes | characteristic |
| 285 | C2 | #11751085 | 1 | 1 | yes | characteristic |
| 286 | C2 | #11751265 | 1 | 2 | yes | — |
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