← #13269905
Subgroups — #13269905
Order 1792.
400 conjugacy classes of subgroups.
The subgroup lattice has 332 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#89968 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#2148 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11729835 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#89306 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11728447 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11725626 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #11725537 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×C7×#135 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#6511 | — | 2 | 1 | no | — |
| 10 | C2×C7×M32 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C7×#108 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#6411 | — | 2 | 1 | no | — |
| 13 | #89968 | — | 2 | 1 | no | — |
| 14 | #89968 | — | 2 | 1 | no | — |
| 15 | #89968 | C4 | 1 | 2 | yes | — |
| 16 | #89968 | C4 | 1 | 2 | yes | — |
| 17 | C2×#6396 | — | 2 | 1 | no | — |
| 18 | #89890 | — | 2 | 1 | no | — |
| 19 | C2×#6393 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#6393 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×C8×Dic7 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#6386 | C22 | 1 | 1 | yes | characteristic |
| 23 | C7×#362 | — | 2 | 1 | no | — |
| 24 | C7×#361 | — | 2 | 1 | no | — |
| 25 | #89360 | — | 2 | 1 | no | — |
| 26 | #89358 | — | 2 | 1 | no | — |
| 27 | #89306 | — | 2 | 1 | no | — |
| 28 | #89306 | — | 2 | 1 | no | — |
| 29 | #12825 | — | 7 | 1 | no | maximal, Sylow |
| 30 | C2×C7×#41 | — | 2 | 1 | no | — |
| 31 | C7×#135 | — | 2 | 1 | no | — |
| 32 | C7×#135 | — | 2 | 1 | no | — |
| 33 | C7×#135 | C2×C4 | 1 | 2 | yes | — |
| 34 | C7×#135 | Q8 | 1 | 2 | yes | — |
| 35 | C2×C7×Q16 | D4 | 1 | 1 | yes | characteristic |
| 36 | C2×C7×SD16 | — | 2 | 1 | no | — |
| 37 | C2×C7×D8 | D4 | 1 | 1 | yes | characteristic |
| 38 | C2×C7×M16 | — | 2 | 1 | no | — |
| 39 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 40 | #6511 | — | 2 | 2 | no | — |
| 41 | #6511 | — | 2 | 2 | no | — |
| 42 | #6511 | — | 4 | 1 | no | — |
| 43 | #6511 | — | 4 | 1 | no | — |
| 44 | C2×C4×Dic7 | D4 | 1 | 1 | yes | characteristic |
| 45 | C2×#1136 | — | 2 | 1 | no | — |
| 46 | C2×#1136 | — | 2 | 1 | no | — |
| 47 | C2×#1136 | — | 4 | 1 | no | — |
| 48 | C22×C7⋊C8 | — | 4 | 1 | no | — |
| 49 | C22×C7⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 50 | C7×M32 | — | 2 | 1 | no | — |
| 51 | C7×M32 | — | 2 | 1 | no | — |
| 52 | C2×C7×C16 | — | 2 | 1 | no | — |
| 53 | C7×#108 | — | 2 | 1 | no | — |
| 54 | C7×#108 | — | 2 | 1 | no | — |
| 55 | #6411 | — | 2 | 2 | no | — |
| 56 | #6411 | — | 2 | 2 | no | — |
| 57 | #6396 | — | 4 | 1 | no | — |
| 58 | #6396 | — | 4 | 1 | no | — |
| 59 | #6393 | — | 2 | 1 | no | — |
| 60 | #6393 | — | 2 | 1 | no | — |
| 61 | #6393 | — | 2 | 1 | no | — |
| 62 | #6393 | C2×C4 | 1 | 2 | yes | — |
| 63 | Dic7⋊C8 | — | 4 | 2 | no | — |
| 64 | C8×Dic7 | C2×C4 | 1 | 1 | yes | characteristic |
| 65 | C8×Dic7 | C2×C4 | 1 | 1 | yes | characteristic |
| 66 | C8×Dic7 | D4 | 1 | 1 | yes | characteristic |
| 67 | C8×Dic7 | D4 | 1 | 1 | yes | characteristic |
| 68 | #6386 | — | 2 | 1 | no | — |
| 69 | #6386 | — | 2 | 1 | no | — |
| 70 | C2×C7⋊C16 | — | 2 | 1 | no | — |
| 71 | C2×#443 | — | 7 | 1 | no | — |
| 72 | C2×#364 | — | 7 | 1 | no | — |
| 73 | #2148 | — | 7 | 1 | no | — |
| 74 | #2148 | D7 | 1 | 1 | yes | characteristic |
| 75 | #1848 | — | 7 | 1 | no | — |
| 76 | #1385 | — | 7 | 1 | no | — |
| 77 | #1385 | — | 7 | 1 | no | — |
| 78 | C7×#41 | — | 2 | 2 | no | — |
| 79 | C7×#41 | — | 2 | 2 | no | — |
| 80 | C7×#41 | — | 4 | 1 | no | — |
| 81 | C7×#41 | — | 4 | 1 | no | — |
| 82 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 83 | C2×C7×D4 | — | 2 | 1 | no | — |
| 84 | C2×C7×D4 | — | 4 | 1 | no | — |
| 85 | C22×C4×C7 | — | 4 | 1 | no | — |
| 86 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C22×Dic7 | — | 2 | 1 | no | — |
| 88 | C7×Q16 | — | 2 | 1 | no | — |
| 89 | C7×Q16 | C4⋊C4 | 1 | 2 | yes | — |
| 90 | C7×SD16 | — | 2 | 2 | no | — |
| 91 | C7×SD16 | — | 2 | 2 | no | — |
| 92 | C7×D8 | — | 2 | 1 | no | — |
| 93 | C7×D8 | C4⋊C4 | 1 | 2 | yes | — |
| 94 | C7×M16 | — | 4 | 1 | no | — |
| 95 | C7×M16 | — | 4 | 1 | no | — |
| 96 | C2×C7×C8 | — | 4 | 1 | no | — |
| 97 | C2×C7×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 98 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C2×C7×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 101 | C2×C7×C8 | #41 | 1 | 1 | yes | characteristic |
| 102 | C2×C7×C8 | #41 | 1 | 1 | yes | characteristic |
| 103 | C4×Dic7 | — | 2 | 1 | no | — |
| 104 | C4×Dic7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 105 | C4×Dic7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 106 | #1136 | — | 2 | 2 | no | — |
| 107 | #1136 | — | 4 | 1 | no | — |
| 108 | #1136 | — | 4 | 2 | no | — |
| 109 | #1136 | — | 4 | 2 | no | — |
| 110 | #1136 | — | 4 | 1 | no | — |
| 111 | #1136 | — | 4 | 1 | no | — |
| 112 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 113 | C2×C7⋊C8 | — | 4 | 2 | no | — |
| 114 | C2×C7⋊C8 | — | 4 | 2 | no | — |
| 115 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 116 | C2×C7⋊C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 117 | C2×C7⋊C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 118 | C2×C7⋊C8 | D8 | 1 | 1 | yes | characteristic |
| 119 | C2×C7⋊C8 | D8 | 1 | 1 | yes | characteristic |
| 120 | C2×C7⋊C8 | SD16 | 1 | 1 | yes | characteristic |
| 121 | C2×C7⋊C8 | SD16 | 1 | 1 | yes | characteristic |
| 122 | C7×C16 | — | 4 | 1 | no | — |
| 123 | C7⋊C16 | — | 4 | 1 | no | — |
| 124 | C2×#135 | C2×D7 | 1 | 1 | yes | characteristic |
| 125 | C2×#131 | — | 14 | 1 | no | — |
| 126 | C2×M32 | — | 7 | 1 | no | — |
| 127 | C2×M32 | C2×D7 | 1 | 1 | yes | characteristic |
| 128 | #443 | — | 7 | 2 | no | — |
| 129 | #443 | — | 7 | 2 | no | — |
| 130 | #443 | — | 14 | 1 | no | — |
| 131 | #443 | — | 14 | 1 | no | — |
| 132 | C2×#108 | — | 7 | 1 | no | — |
| 133 | C2×#108 | — | 7 | 1 | no | — |
| 134 | C2×#108 | — | 14 | 1 | no | — |
| 135 | C2×#108 | C2×D7 | 1 | 1 | yes | characteristic |
| 136 | #423 | — | 14 | 1 | no | — |
| 137 | C2×#104 | — | 14 | 1 | no | — |
| 138 | C2×C4×C8 | — | 7 | 1 | no | — |
| 139 | #364 | — | 14 | 1 | no | — |
| 140 | #364 | — | 14 | 1 | no | — |
| 141 | #362 | — | 2 | 1 | no | — |
| 142 | #362 | — | 14 | 1 | no | — |
| 143 | #361 | — | 2 | 1 | no | — |
| 144 | #361 | — | 14 | 1 | no | — |
| 145 | C23×C7 | — | 4 | 1 | no | — |
| 146 | C23×C7 | #137 | 1 | 1 | yes | characteristic |
| 147 | C7×Q8 | — | 2 | 2 | no | — |
| 148 | C7×Q8 | — | 4 | 1 | no | — |
| 149 | C7×D4 | — | 2 | 2 | no | — |
| 150 | C7×D4 | — | 4 | 2 | no | — |
| 151 | C7×D4 | — | 4 | 2 | no | — |
| 152 | C7×D4 | — | 4 | 1 | no | — |
| 153 | C2×C4×C7 | — | 4 | 1 | no | — |
| 154 | C2×C4×C7 | — | 4 | 2 | no | — |
| 155 | C2×C4×C7 | — | 4 | 2 | no | — |
| 156 | C2×C4×C7 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 157 | C2×C4×C7 | #121 | 1 | 1 | yes | characteristic |
| 158 | C2×C4×C7 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 159 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 160 | C2×C4×C7 | C2×SD16 | 1 | 1 | yes | characteristic |
| 161 | C2×C4×C7 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 162 | C2×Dic7 | — | 2 | 1 | no | — |
| 163 | C2×Dic7 | — | 2 | 1 | no | — |
| 164 | C2×Dic7 | — | 4 | 2 | no | — |
| 165 | C7×C8 | — | 8 | 1 | no | — |
| 166 | C7×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 167 | C7×C8 | #117 | 1 | 1 | yes | characteristic |
| 168 | C7×C8 | #121 | 1 | 1 | yes | characteristic |
| 169 | C7×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 170 | C7⋊C8 | — | 4 | 2 | no | — |
| 171 | C7⋊C8 | — | 8 | 1 | no | — |
| 172 | C7⋊C8 | #102 | 1 | 1 | yes | characteristic |
| 173 | C7⋊C8 | #102 | 1 | 1 | yes | characteristic |
| 174 | C7⋊C8 | #102 | 1 | 1 | yes | characteristic |
| 175 | C7⋊C8 | #102 | 1 | 1 | yes | characteristic |
| 176 | C2×#41 | — | 2 | 1 | no | — |
| 177 | #135 | — | 2 | 1 | no | — |
| 178 | #135 | — | 2 | 1 | no | — |
| 179 | #135 | Dic14 | 1 | 2 | yes | — |
| 180 | #135 | C4×D7 | 1 | 2 | yes | — |
| 181 | C2×Q16 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 182 | C2×SD16 | — | 2 | 1 | no | — |
| 183 | C2×D8 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 184 | #131 | — | 14 | 2 | no | — |
| 185 | #131 | — | 14 | 2 | no | — |
| 186 | #131 | — | 28 | 1 | no | — |
| 187 | #131 | — | 28 | 1 | no | — |
| 188 | C2×M16 | — | 2 | 1 | no | — |
| 189 | C2×M16 | — | 14 | 1 | no | — |
| 190 | C2×M16 | — | 14 | 1 | no | — |
| 191 | C2×M16 | — | 28 | 1 | no | — |
| 192 | C22×C8 | — | 7 | 1 | no | — |
| 193 | C22×C8 | — | 28 | 1 | no | — |
| 194 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 195 | C2×C42 | — | 7 | 1 | no | — |
| 196 | M32 | — | 2 | 1 | no | — |
| 197 | M32 | — | 2 | 1 | no | — |
| 198 | M32 | — | 14 | 1 | no | — |
| 199 | M32 | — | 14 | 1 | no | — |
| 200 | C2×C16 | — | 2 | 1 | no | — |
| 201 | C2×C16 | — | 14 | 1 | no | — |
| 202 | #108 | — | 2 | 1 | no | — |
| 203 | #108 | — | 2 | 1 | no | — |
| 204 | #108 | — | 7 | 2 | no | — |
| 205 | #108 | — | 14 | 1 | no | — |
| 206 | #108 | — | 14 | 1 | no | — |
| 207 | #108 | — | 14 | 1 | no | — |
| 208 | #108 | — | 28 | 1 | no | — |
| 209 | #108 | — | 28 | 1 | no | — |
| 210 | C4⋊C8 | — | 28 | 2 | no | — |
| 211 | #104 | — | 14 | 2 | no | — |
| 212 | #104 | — | 14 | 2 | no | — |
| 213 | C4×C8 | — | 7 | 1 | no | — |
| 214 | C4×C8 | — | 7 | 1 | no | — |
| 215 | C4×C8 | — | 7 | 1 | no | — |
| 216 | C4×C8 | — | 7 | 1 | no | — |
| 217 | C22×C7 | — | 4 | 1 | no | — |
| 218 | C22×C7 | — | 4 | 2 | no | — |
| 219 | C22×C7 | — | 8 | 2 | no | — |
| 220 | C22×C7 | #417 | 1 | 1 | yes | characteristic |
| 221 | C22×C7 | #461 | 1 | 1 | yes | characteristic |
| 222 | C22×C7 | #474 | 1 | 1 | yes | characteristic |
| 223 | C4×C7 | — | 4 | 2 | no | — |
| 224 | C4×C7 | #380 | 1 | 1 | yes | characteristic |
| 225 | C4×C7 | C2×#102 | 1 | 1 | yes | characteristic |
| 226 | C4×C7 | #459 | 1 | 1 | yes | characteristic |
| 227 | C4×C7 | #476 | 1 | 1 | yes | characteristic |
| 228 | Dic7 | — | 2 | 2 | no | — |
| 229 | Dic7 | — | 4 | 1 | no | — |
| 230 | #41 | — | 2 | 2 | no | — |
| 231 | #41 | — | 2 | 2 | no | — |
| 232 | #41 | — | 4 | 1 | no | — |
| 233 | #41 | — | 4 | 1 | no | — |
| 234 | C2×Q8 | — | 2 | 1 | no | — |
| 235 | C2×D4 | — | 2 | 1 | no | — |
| 236 | C2×D4 | — | 4 | 1 | no | — |
| 237 | C22×C4 | — | 4 | 1 | no | — |
| 238 | C22×C4 | — | 14 | 1 | no | — |
| 239 | C22×C4 | D4×D7 | 1 | 1 | yes | characteristic |
| 240 | Q16 | — | 2 | 1 | no | — |
| 241 | Q16 | Dic7⋊C4 | 1 | 2 | yes | — |
| 242 | SD16 | — | 2 | 2 | no | — |
| 243 | SD16 | — | 2 | 2 | no | — |
| 244 | D8 | — | 2 | 1 | no | — |
| 245 | D8 | Dic7⋊C4 | 1 | 2 | yes | — |
| 246 | M16 | — | 4 | 1 | no | — |
| 247 | M16 | — | 4 | 1 | no | — |
| 248 | M16 | — | 14 | 2 | no | — |
| 249 | M16 | — | 28 | 2 | no | — |
| 250 | M16 | — | 28 | 2 | no | — |
| 251 | M16 | — | 28 | 1 | no | — |
| 252 | M16 | — | 28 | 1 | no | — |
| 253 | M16 | — | 28 | 1 | no | — |
| 254 | C2×C8 | — | 4 | 1 | no | — |
| 255 | C2×C8 | — | 7 | 1 | no | — |
| 256 | C2×C8 | — | 7 | 1 | no | — |
| 257 | C2×C8 | — | 7 | 1 | no | — |
| 258 | C2×C8 | — | 7 | 1 | no | — |
| 259 | C2×C8 | — | 7 | 1 | no | — |
| 260 | C2×C8 | — | 7 | 1 | no | — |
| 261 | C2×C8 | — | 28 | 2 | no | — |
| 262 | C2×C8 | — | 28 | 2 | no | — |
| 263 | C2×C8 | — | 28 | 1 | no | — |
| 264 | C2×C8 | — | 28 | 1 | no | — |
| 265 | C2×C8 | C2×Dic14 | 1 | 1 | yes | characteristic |
| 266 | C2×C8 | C2×C4×D7 | 1 | 1 | yes | characteristic |
| 267 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 268 | C2×C8 | #1159 | 1 | 1 | yes | characteristic |
| 269 | C2×C8 | #1159 | 1 | 1 | yes | characteristic |
| 270 | C2×C8 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 271 | C42 | — | 7 | 1 | no | — |
| 272 | C42 | — | 7 | 1 | no | — |
| 273 | C42 | — | 14 | 1 | no | — |
| 274 | C16 | — | 4 | 1 | no | — |
| 275 | C16 | — | 28 | 1 | no | — |
| 276 | C2×C7 | — | 2 | 1 | no | — |
| 277 | C2×C7 | — | 2 | 1 | no | — |
| 278 | C2×C7 | — | 2 | 1 | no | — |
| 279 | C2×C7 | — | 8 | 2 | no | — |
| 280 | C2×C7 | #1802 | 1 | 1 | yes | characteristic |
| 281 | C23 | — | 4 | 1 | no | — |
| 282 | C23 | #6478 | 1 | 1 | yes | characteristic |
| 283 | Q8 | — | 2 | 2 | no | — |
| 284 | Q8 | — | 4 | 1 | no | — |
| 285 | D4 | — | 2 | 2 | no | — |
| 286 | D4 | — | 4 | 2 | no | — |
| 287 | D4 | — | 4 | 2 | no | — |
| 288 | D4 | — | 4 | 1 | no | — |
| 289 | C2×C4 | — | 4 | 1 | no | — |
| 290 | C2×C4 | — | 4 | 2 | no | — |
| 291 | C2×C4 | — | 4 | 2 | no | — |
| 292 | C2×C4 | — | 14 | 1 | no | — |
| 293 | C2×C4 | — | 14 | 1 | no | — |
| 294 | C2×C4 | — | 28 | 2 | no | — |
| 295 | C2×C4 | #6441 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | D7×C22⋊C4 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | D7×D8 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #6474 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | D7×SD16 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | #6502 | 1 | 1 | yes | characteristic |
| 301 | C8 | — | 7 | 1 | no | — |
| 302 | C8 | — | 7 | 1 | no | — |
| 303 | C8 | — | 7 | 1 | no | — |
| 304 | C8 | — | 7 | 1 | no | — |
| 305 | C8 | — | 8 | 1 | no | — |
| 306 | C8 | — | 28 | 2 | no | — |
| 307 | C8 | — | 56 | 1 | no | — |
| 308 | C8 | #6440 | 1 | 1 | yes | characteristic |
| 309 | C8 | #6441 | 1 | 1 | yes | characteristic |
| 310 | C8 | C2×Dic7⋊C4 | 1 | 1 | yes | characteristic |
| 311 | C8 | #6502 | 1 | 1 | yes | characteristic |
| 312 | C7 | #12825 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 313 | C22 | — | 4 | 1 | no | — |
| 314 | C22 | — | 4 | 2 | no | — |
| 315 | C22 | — | 8 | 2 | no | — |
| 316 | C22 | #89531 | 1 | 1 | yes | characteristic |
| 317 | C22 | #89543 | 1 | 1 | yes | characteristic |
| 318 | C22 | #89934 | 1 | 1 | yes | characteristic |
| 319 | C4 | — | 4 | 2 | no | — |
| 320 | C4 | — | 14 | 2 | no | — |
| 321 | C4 | — | 28 | 1 | no | — |
| 322 | C4 | #89533 | 1 | 1 | yes | characteristic |
| 323 | C4 | D7×#102 | 1 | 1 | yes | characteristic |
| 324 | C4 | #89714 | 1 | 1 | yes | characteristic |
| 325 | C4 | #89922 | 1 | 1 | yes | characteristic |
| 326 | C2 | — | 2 | 1 | no | — |
| 327 | C2 | — | 2 | 1 | no | — |
| 328 | C2 | — | 2 | 1 | no | — |
| 329 | C2 | — | 8 | 2 | no | — |
| 330 | C2 | #11727907 | 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