← #13272050
Subgroups — #13272050
Order 1792.
470 conjugacy classes of subgroups.
The subgroup lattice has 340 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#89953 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C7×#362 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#89360 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#89312 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | Dic7×M32 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #11728447 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C2×#89291 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C22×C7×Q16 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#6507 | — | 2 | 1 | no | — |
| 10 | C22×#1144 | — | 2 | 1 | no | — |
| 11 | C22×Dic28 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C7×M32 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C7×#108 | C22 | 1 | 1 | yes | characteristic |
| 14 | #89953 | — | 2 | 1 | no | — |
| 15 | #89953 | — | 2 | 1 | no | — |
| 16 | #89953 | C4 | 1 | 2 | yes | — |
| 17 | #89953 | C4 | 1 | 2 | yes | — |
| 18 | C2×#6396 | — | 2 | 1 | no | — |
| 19 | #89890 | — | 2 | 1 | no | — |
| 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 | 2 | no | — |
| 24 | #89360 | — | 2 | 2 | no | — |
| 25 | #89312 | — | 2 | 2 | no | — |
| 26 | #89297 | — | 2 | 1 | no | — |
| 27 | C16×Dic7 | — | 2 | 1 | no | — |
| 28 | #89291 | — | 2 | 2 | no | — |
| 29 | #13212 | — | 7 | 1 | no | maximal, Sylow |
| 30 | C22×C7×Q8 | — | 2 | 1 | no | — |
| 31 | C22×Dic14 | — | 2 | 1 | no | — |
| 32 | C2×C7×Q16 | — | 2 | 1 | no | — |
| 33 | C2×C7×Q16 | — | 2 | 1 | no | — |
| 34 | C2×C7×Q16 | — | 2 | 1 | no | — |
| 35 | C2×C7×Q16 | — | 2 | 1 | no | — |
| 36 | C2×C7×Q16 | — | 2 | 1 | no | — |
| 37 | C2×C7×Q16 | C2×C4 | 1 | 2 | yes | — |
| 38 | C2×C7×M16 | — | 2 | 1 | no | — |
| 39 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 40 | #6507 | — | 2 | 2 | no | — |
| 41 | #6507 | — | 2 | 2 | no | — |
| 42 | #6507 | — | 4 | 2 | no | — |
| 43 | C2×#1144 | — | 2 | 2 | no | — |
| 44 | C2×#1144 | — | 2 | 2 | no | — |
| 45 | C2×#1144 | — | 4 | 2 | no | — |
| 46 | C2×#1144 | — | 4 | 2 | no | — |
| 47 | C2×Dic7⋊C4 | — | 4 | 2 | no | — |
| 48 | C2×C4×Dic7 | D4 | 1 | 1 | yes | characteristic |
| 49 | C2×#1136 | — | 2 | 1 | no | — |
| 50 | C22×C7⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 51 | C2×Dic28 | — | 2 | 1 | no | — |
| 52 | C2×Dic28 | — | 2 | 1 | no | — |
| 53 | C2×Dic28 | — | 2 | 1 | no | — |
| 54 | C2×Dic28 | — | 2 | 1 | no | — |
| 55 | C2×Dic28 | — | 2 | 1 | no | — |
| 56 | C2×Dic28 | C2×C4 | 1 | 2 | yes | — |
| 57 | C7×M32 | — | 2 | 1 | no | — |
| 58 | C7×M32 | D4 | 1 | 2 | yes | — |
| 59 | C2×C7×C16 | — | 2 | 1 | no | — |
| 60 | C7×#108 | — | 2 | 1 | no | — |
| 61 | C7×#108 | — | 2 | 1 | no | — |
| 62 | #6396 | — | 4 | 2 | no | — |
| 63 | #6393 | — | 2 | 1 | no | — |
| 64 | #6393 | — | 2 | 1 | no | — |
| 65 | Dic7⋊C8 | — | 4 | 2 | no | — |
| 66 | C8×Dic7 | C2×C4 | 1 | 1 | yes | characteristic |
| 67 | C8×Dic7 | C2×C4 | 1 | 1 | yes | characteristic |
| 68 | C8×Dic7 | D4 | 1 | 1 | yes | characteristic |
| 69 | C8×Dic7 | D4 | 1 | 1 | yes | characteristic |
| 70 | #6386 | — | 2 | 1 | no | — |
| 71 | #6386 | D4 | 1 | 2 | yes | — |
| 72 | C2×C7⋊C16 | — | 2 | 1 | no | — |
| 73 | C2×#494 | — | 7 | 1 | no | — |
| 74 | C2×#362 | — | 7 | 1 | no | — |
| 75 | C2×#362 | — | 7 | 1 | no | — |
| 76 | C2×#362 | — | 7 | 1 | no | — |
| 77 | C2×#362 | D7 | 1 | 1 | yes | characteristic |
| 78 | C4×M32 | — | 7 | 1 | no | — |
| 79 | #1848 | — | 7 | 1 | no | — |
| 80 | C2×C7×Q8 | — | 2 | 2 | no | — |
| 81 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 82 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 83 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 84 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 85 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 86 | C22×C4×C7 | — | 4 | 1 | no | — |
| 87 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C22×Dic7 | — | 4 | 1 | no | — |
| 89 | C22×Dic7 | SD16 | 1 | 2 | yes | — |
| 90 | C2×Dic14 | — | 2 | 2 | no | — |
| 91 | C2×Dic14 | — | 4 | 1 | no | — |
| 92 | C2×Dic14 | — | 4 | 1 | no | — |
| 93 | C2×Dic14 | — | 4 | 1 | no | — |
| 94 | C2×Dic14 | — | 4 | 1 | no | — |
| 95 | C2×Dic14 | — | 4 | 1 | no | — |
| 96 | C7×Q16 | — | 2 | 2 | no | — |
| 97 | C7×Q16 | — | 2 | 2 | no | — |
| 98 | C7×Q16 | — | 4 | 1 | no | — |
| 99 | C7×Q16 | — | 4 | 1 | no | — |
| 100 | C7×M16 | — | 4 | 1 | no | — |
| 101 | C7×M16 | — | 4 | 1 | no | — |
| 102 | C2×C7×C8 | — | 4 | 1 | no | — |
| 103 | C2×C7×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 104 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 105 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 106 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 107 | C2×C7×C8 | #41 | 1 | 1 | yes | characteristic |
| 108 | C2×C7×C8 | #41 | 1 | 1 | yes | characteristic |
| 109 | #1144 | — | 4 | 4 | no | — |
| 110 | #1144 | — | 4 | 4 | no | — |
| 111 | Dic7⋊C4 | — | 4 | 4 | no | — |
| 112 | Dic7⋊C4 | — | 4 | 4 | no | — |
| 113 | C4×Dic7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 114 | C4×Dic7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 115 | C4×Dic7 | D8 | 1 | 2 | yes | — |
| 116 | #1136 | — | 4 | 1 | no | — |
| 117 | #1136 | — | 4 | 1 | no | — |
| 118 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 119 | C2×C7⋊C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 120 | C2×C7⋊C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 121 | C2×C7⋊C8 | D8 | 1 | 2 | yes | — |
| 122 | C2×C7⋊C8 | SD16 | 1 | 2 | yes | — |
| 123 | Dic28 | — | 2 | 2 | no | — |
| 124 | Dic28 | — | 2 | 2 | no | — |
| 125 | Dic28 | — | 4 | 1 | no | — |
| 126 | Dic28 | — | 4 | 1 | no | — |
| 127 | C7×C16 | — | 2 | 2 | no | — |
| 128 | C7⋊C16 | — | 2 | 2 | no | — |
| 129 | C22×Q16 | — | 7 | 1 | no | — |
| 130 | C22×Q16 | — | 14 | 1 | no | — |
| 131 | C22×Q16 | C2×D7 | 1 | 1 | yes | characteristic |
| 132 | C2×C4⋊Q8 | — | 14 | 1 | no | — |
| 133 | C2×M32 | — | 7 | 1 | no | — |
| 134 | C2×M32 | C2×D7 | 1 | 1 | yes | characteristic |
| 135 | #494 | — | 7 | 2 | no | — |
| 136 | #494 | — | 7 | 2 | no | — |
| 137 | #494 | — | 14 | 1 | no | — |
| 138 | #494 | — | 14 | 1 | no | — |
| 139 | C2×#108 | — | 7 | 1 | no | — |
| 140 | C2×#108 | — | 14 | 1 | no | — |
| 141 | C2×#108 | C2×D7 | 1 | 1 | yes | characteristic |
| 142 | #423 | — | 14 | 1 | no | — |
| 143 | C2×C4×C8 | — | 7 | 1 | no | — |
| 144 | #362 | — | 2 | 2 | no | — |
| 145 | #362 | — | 14 | 2 | no | — |
| 146 | #362 | — | 14 | 2 | no | — |
| 147 | #362 | — | 14 | 2 | no | — |
| 148 | #346 | — | 14 | 1 | no | — |
| 149 | C4×C16 | — | 14 | 1 | no | — |
| 150 | C23×C7 | C2×SD16 | 1 | 1 | yes | characteristic |
| 151 | C7×Q8 | — | 4 | 2 | no | — |
| 152 | C7×Q8 | — | 4 | 2 | no | — |
| 153 | C7×Q8 | — | 8 | 1 | no | — |
| 154 | C7×Q8 | — | 8 | 1 | no | — |
| 155 | C2×C4×C7 | — | 4 | 2 | no | — |
| 156 | C2×C4×C7 | — | 8 | 1 | no | — |
| 157 | C2×C4×C7 | — | 8 | 1 | no | — |
| 158 | C2×C4×C7 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 159 | C2×C4×C7 | #124 | 1 | 1 | yes | characteristic |
| 160 | C2×C4×C7 | #127 | 1 | 1 | yes | characteristic |
| 161 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 162 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 163 | C2×C4×C7 | C2×SD16 | 1 | 1 | yes | characteristic |
| 164 | C2×Dic7 | — | 2 | 4 | no | — |
| 165 | C2×Dic7 | — | 4 | 2 | no | — |
| 166 | C2×Dic7 | — | 8 | 1 | no | — |
| 167 | C2×Dic7 | — | 8 | 1 | no | — |
| 168 | C2×Dic7 | #102 | 1 | 2 | yes | — |
| 169 | C2×Dic7 | #102 | 1 | 2 | yes | — |
| 170 | Dic14 | — | 4 | 2 | no | — |
| 171 | Dic14 | — | 4 | 2 | no | — |
| 172 | Dic14 | — | 8 | 1 | no | — |
| 173 | Dic14 | — | 8 | 1 | no | — |
| 174 | C7×C8 | — | 8 | 1 | no | — |
| 175 | C7×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 176 | C7×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 177 | C7×C8 | #121 | 1 | 1 | yes | characteristic |
| 178 | C7×C8 | #121 | 1 | 1 | yes | characteristic |
| 179 | C7⋊C8 | — | 8 | 1 | no | — |
| 180 | C7⋊C8 | #102 | 1 | 2 | yes | — |
| 181 | C7⋊C8 | #102 | 1 | 2 | yes | — |
| 182 | C22×Q8 | — | 2 | 1 | no | — |
| 183 | C22×Q8 | — | 14 | 1 | no | — |
| 184 | C2×Q16 | — | 2 | 1 | no | — |
| 185 | C2×Q16 | — | 2 | 1 | no | — |
| 186 | C2×Q16 | — | 2 | 1 | no | — |
| 187 | C2×Q16 | — | 2 | 1 | no | — |
| 188 | C2×Q16 | — | 2 | 1 | no | — |
| 189 | C2×Q16 | — | 7 | 2 | no | — |
| 190 | C2×Q16 | — | 14 | 2 | no | — |
| 191 | C2×Q16 | — | 14 | 2 | no | — |
| 192 | C2×Q16 | — | 14 | 1 | no | — |
| 193 | C2×Q16 | — | 14 | 1 | no | — |
| 194 | C2×Q16 | — | 14 | 1 | no | — |
| 195 | C2×Q16 | — | 14 | 1 | no | — |
| 196 | C2×Q16 | — | 14 | 1 | no | — |
| 197 | C2×Q16 | — | 28 | 2 | no | — |
| 198 | C2×Q16 | — | 28 | 2 | no | — |
| 199 | C2×Q16 | C4×D7 | 1 | 2 | yes | — |
| 200 | C2×M16 | — | 2 | 1 | no | — |
| 201 | C2×M16 | — | 14 | 1 | no | — |
| 202 | C22×C8 | — | 7 | 1 | no | — |
| 203 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 204 | C4⋊Q8 | — | 14 | 2 | no | — |
| 205 | C4⋊Q8 | — | 14 | 2 | no | — |
| 206 | C4⋊Q8 | — | 28 | 2 | no | — |
| 207 | C2×C4⋊C4 | — | 28 | 2 | no | — |
| 208 | C2×C42 | — | 7 | 1 | no | — |
| 209 | M32 | — | 2 | 1 | no | — |
| 210 | M32 | — | 7 | 2 | no | — |
| 211 | M32 | — | 14 | 1 | no | — |
| 212 | M32 | C22⋊D7 | 1 | 2 | yes | — |
| 213 | C2×C16 | — | 2 | 1 | no | — |
| 214 | C2×C16 | — | 14 | 1 | no | — |
| 215 | #108 | — | 2 | 1 | no | — |
| 216 | #108 | — | 2 | 1 | no | — |
| 217 | #108 | — | 14 | 1 | no | — |
| 218 | #108 | — | 14 | 1 | no | — |
| 219 | #108 | — | 28 | 2 | no | — |
| 220 | C4⋊C8 | — | 28 | 2 | no | — |
| 221 | C4×C8 | — | 7 | 1 | no | — |
| 222 | C4×C8 | — | 7 | 1 | no | — |
| 223 | C4×C8 | — | 7 | 1 | no | — |
| 224 | C4×C8 | — | 7 | 1 | no | — |
| 225 | C22×C7 | — | 2 | 2 | no | — |
| 226 | C22×C7 | C2×#102 | 1 | 1 | yes | characteristic |
| 227 | C22×C7 | #486 | 1 | 1 | yes | characteristic |
| 228 | C22×C7 | #492 | 1 | 1 | yes | characteristic |
| 229 | C4×C7 | — | 8 | 2 | no | — |
| 230 | C4×C7 | #390 | 1 | 1 | yes | characteristic |
| 231 | C4×C7 | C2×#102 | 1 | 1 | yes | characteristic |
| 232 | C4×C7 | #486 | 1 | 1 | yes | characteristic |
| 233 | C4×C7 | #493 | 1 | 1 | yes | characteristic |
| 234 | Dic7 | — | 2 | 2 | no | — |
| 235 | Dic7 | — | 8 | 2 | no | — |
| 236 | Dic7 | #362 | 1 | 4 | yes | complemented |
| 237 | C2×Q8 | — | 2 | 2 | no | — |
| 238 | C2×Q8 | — | 4 | 1 | no | — |
| 239 | C2×Q8 | — | 4 | 1 | no | — |
| 240 | C2×Q8 | — | 4 | 1 | no | — |
| 241 | C2×Q8 | — | 4 | 1 | no | — |
| 242 | C2×Q8 | — | 4 | 1 | no | — |
| 243 | C2×Q8 | — | 14 | 2 | no | — |
| 244 | C2×Q8 | — | 28 | 1 | no | — |
| 245 | C2×Q8 | — | 28 | 1 | no | — |
| 246 | C2×Q8 | — | 28 | 1 | no | — |
| 247 | C2×Q8 | — | 28 | 1 | no | — |
| 248 | C2×Q8 | — | 28 | 1 | no | — |
| 249 | C22×C4 | — | 4 | 1 | no | — |
| 250 | C22×C4 | — | 7 | 2 | no | — |
| 251 | C22×C4 | — | 28 | 1 | no | — |
| 252 | C22×C4 | D4×D7 | 1 | 1 | yes | characteristic |
| 253 | Q16 | — | 2 | 2 | no | — |
| 254 | Q16 | — | 2 | 2 | no | — |
| 255 | Q16 | — | 4 | 1 | no | — |
| 256 | Q16 | — | 4 | 1 | no | — |
| 257 | Q16 | — | 14 | 2 | no | — |
| 258 | Q16 | — | 14 | 2 | no | — |
| 259 | Q16 | — | 28 | 4 | no | — |
| 260 | Q16 | — | 28 | 4 | no | — |
| 261 | Q16 | — | 28 | 1 | no | — |
| 262 | Q16 | — | 28 | 1 | no | — |
| 263 | M16 | — | 4 | 1 | no | — |
| 264 | M16 | — | 4 | 1 | no | — |
| 265 | M16 | — | 28 | 1 | no | — |
| 266 | M16 | — | 28 | 1 | no | — |
| 267 | C2×C8 | — | 4 | 1 | no | — |
| 268 | C2×C8 | — | 7 | 1 | no | — |
| 269 | C2×C8 | — | 7 | 1 | no | — |
| 270 | C2×C8 | — | 7 | 2 | no | — |
| 271 | C2×C8 | — | 7 | 2 | no | — |
| 272 | C2×C8 | — | 28 | 1 | no | — |
| 273 | C2×C8 | C2×C4×D7 | 1 | 1 | yes | characteristic |
| 274 | C2×C8 | #1157 | 1 | 1 | yes | characteristic |
| 275 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 276 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 277 | C2×C8 | #1159 | 1 | 1 | yes | characteristic |
| 278 | C2×C8 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 279 | C4⋊C4 | — | 28 | 4 | no | — |
| 280 | C4⋊C4 | — | 28 | 4 | no | — |
| 281 | C42 | — | 7 | 1 | no | — |
| 282 | C42 | — | 7 | 1 | no | — |
| 283 | C42 | — | 7 | 2 | no | — |
| 284 | C16 | — | 2 | 2 | no | — |
| 285 | C16 | — | 14 | 2 | no | — |
| 286 | C2×C7 | — | 2 | 1 | no | — |
| 287 | C2×C7 | — | 2 | 1 | no | — |
| 288 | C2×C7 | #1957 | 1 | 1 | yes | characteristic |
| 289 | C2×C7 | C2×#362 | 1 | 2 | yes | — |
| 290 | C23 | D7×SD16 | 1 | 1 | yes | characteristic |
| 291 | Q8 | — | 4 | 2 | no | — |
| 292 | Q8 | — | 4 | 2 | no | — |
| 293 | Q8 | — | 8 | 1 | no | — |
| 294 | Q8 | — | 8 | 1 | no | — |
| 295 | Q8 | — | 28 | 2 | no | — |
| 296 | Q8 | — | 28 | 2 | no | — |
| 297 | Q8 | — | 56 | 1 | no | — |
| 298 | Q8 | — | 56 | 1 | no | — |
| 299 | C2×C4 | — | 4 | 2 | no | — |
| 300 | C2×C4 | — | 7 | 2 | no | — |
| 301 | C2×C4 | — | 7 | 2 | no | — |
| 302 | C2×C4 | — | 8 | 1 | no | — |
| 303 | C2×C4 | — | 8 | 1 | no | — |
| 304 | C2×C4 | — | 14 | 4 | no | — |
| 305 | C2×C4 | — | 28 | 2 | no | — |
| 306 | C2×C4 | — | 56 | 1 | no | — |
| 307 | C2×C4 | — | 56 | 1 | no | — |
| 308 | C2×C4 | D7×C22⋊C4 | 1 | 1 | yes | characteristic |
| 309 | C2×C4 | #6448 | 1 | 1 | yes | characteristic |
| 310 | C2×C4 | D7×D8 | 1 | 1 | yes | characteristic |
| 311 | C2×C4 | D7×D8 | 1 | 1 | yes | characteristic |
| 312 | C2×C4 | D7×SD16 | 1 | 1 | yes | characteristic |
| 313 | C2×C4 | #6503 | 1 | 1 | yes | characteristic |
| 314 | C8 | — | 7 | 2 | no | — |
| 315 | C8 | — | 7 | 2 | no | — |
| 316 | C8 | — | 8 | 1 | no | — |
| 317 | C8 | — | 56 | 1 | no | — |
| 318 | C8 | (C22⋊D7)⋊C4 | 1 | 1 | yes | characteristic |
| 319 | C8 | #6447 | 1 | 1 | yes | characteristic |
| 320 | C8 | C4×C22⋊D7 | 1 | 1 | yes | characteristic |
| 321 | C8 | #6502 | 1 | 1 | yes | characteristic |
| 322 | C7 | #13212 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 323 | C22 | — | 2 | 2 | no | — |
| 324 | C22 | #89532 | 1 | 1 | yes | characteristic |
| 325 | C22 | D7×#102 | 1 | 1 | yes | characteristic |
| 326 | C22 | #89947 | 1 | 1 | yes | characteristic |
| 327 | C4 | — | 7 | 4 | no | — |
| 328 | C4 | — | 8 | 2 | no | — |
| 329 | C4 | — | 14 | 2 | no | — |
| 330 | C4 | — | 56 | 2 | no | — |
| 331 | C4 | #89532 | 1 | 1 | yes | characteristic |
| 332 | C4 | D7×#102 | 1 | 1 | yes | characteristic |
| 333 | C4 | #89729 | 1 | 1 | yes | characteristic |
| 334 | C4 | #89923 | 1 | 1 | yes | characteristic |
| 335 | C2 | — | 2 | 1 | no | — |
| 336 | C2 | — | 2 | 1 | no | — |
| 337 | C2 | D7×#362 | 1 | 2 | yes | — |
| 338 | C2 | #11727911 | 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