← #13324456
Subgroups — #13324456
Order 1792.
640 conjugacy classes of subgroups.
The subgroup lattice has 308 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C7×#3169 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11740725 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #11729557 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11728636 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11728635 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #11728632 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | D8×C7⋊C8 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #11727486 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #11727478 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C7×#532 | C22 | 1 | 2 | yes | — |
| 11 | C7×#500 | C4 | 1 | 1 | yes | characteristic |
| 12 | C7×#493 | C4 | 1 | 1 | yes | characteristic |
| 13 | C7×#486 | C4 | 1 | 2 | yes | — |
| 14 | C7×#466 | C4 | 1 | 2 | yes | — |
| 15 | C7×#466 | C4 | 1 | 2 | yes | — |
| 16 | C4×C7×D8 | C22 | 1 | 2 | yes | — |
| 17 | C4×C7×D8 | C22 | 1 | 2 | yes | — |
| 18 | C2×C4×C7×C8 | C22 | 1 | 1 | yes | characteristic |
| 19 | #90006 | — | 2 | 4 | no | — |
| 20 | #89784 | C22 | 1 | 4 | yes | — |
| 21 | #89783 | — | 2 | 4 | no | — |
| 22 | D4×C7⋊C8 | C22 | 1 | 4 | yes | — |
| 23 | #89698 | C22 | 1 | 2 | yes | — |
| 24 | #89697 | C22 | 1 | 1 | yes | characteristic |
| 25 | #89697 | C22 | 1 | 1 | yes | characteristic |
| 26 | D4⋊(C7⋊C8) | C22 | 1 | 4 | yes | — |
| 27 | D4⋊(C7⋊C8) | C22 | 1 | 4 | yes | — |
| 28 | #89253 | C22 | 1 | 2 | yes | — |
| 29 | #89252 | C22 | 1 | 2 | yes | — |
| 30 | #89250 | C22 | 1 | 1 | yes | characteristic |
| 31 | #89250 | C22 | 1 | 1 | yes | characteristic |
| 32 | C8×C7⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 33 | C8×C7⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 34 | #17970 | — | 7 | 1 | no | maximal, Sylow |
| 35 | C2×C7×#41 | — | 2 | 2 | no | — |
| 36 | C2×C7×D8 | C2×C4 | 1 | 2 | yes | — |
| 37 | C2×C7×D8 | C2×C4 | 1 | 2 | yes | — |
| 38 | C22×C7×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C22×C7×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C7×C4⋊Q8 | C2×C4 | 1 | 2 | yes | — |
| 41 | C7×#127 | C2×C4 | 1 | 2 | yes | — |
| 42 | C7×#124 | — | 2 | 2 | no | — |
| 43 | C7×#121 | — | 2 | 2 | no | — |
| 44 | C7×#121 | C2×C4 | 1 | 4 | yes | — |
| 45 | C4×C7×D4 | — | 2 | 2 | no | — |
| 46 | C4×C7×D4 | C23 | 1 | 4 | yes | — |
| 47 | C2×C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 48 | C2×#1136 | — | 2 | 4 | no | — |
| 49 | C22×C7⋊C8 | — | 2 | 4 | no | — |
| 50 | C7×#107 | C2×C4 | 1 | 2 | yes | — |
| 51 | C7×#107 | C2×C4 | 1 | 2 | yes | — |
| 52 | C7×#102 | C2×C4 | 1 | 4 | yes | — |
| 53 | C7×#102 | C2×C4 | 1 | 4 | yes | — |
| 54 | C4×C7×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | C4×C7×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 57 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 58 | C22⋊(C7⋊C8) | — | 2 | 2 | no | — |
| 59 | C22⋊(C7⋊C8) | — | 2 | 1 | no | — |
| 60 | C22⋊(C7⋊C8) | — | 2 | 4 | no | — |
| 61 | C22⋊(C7⋊C8) | — | 2 | 4 | no | — |
| 62 | C22⋊(C7⋊C8) | D4 | 1 | 2 | yes | — |
| 63 | C4⋊(C7⋊C8) | — | 2 | 1 | no | — |
| 64 | C4⋊(C7⋊C8) | D4 | 1 | 2 | yes | — |
| 65 | C4⋊(C7⋊C8) | C23 | 1 | 4 | yes | — |
| 66 | #6377 | — | 2 | 2 | no | — |
| 67 | C4×C7⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 68 | C4×C7⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 69 | C4×C7⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 70 | C4×C7⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 71 | #3169 | D7 | 1 | 1 | yes | characteristic |
| 72 | #2981 | — | 7 | 2 | no | — |
| 73 | #1594 | — | 7 | 2 | no | — |
| 74 | #1589 | — | 7 | 2 | no | — |
| 75 | #1580 | — | 7 | 2 | no | — |
| 76 | C8×D8 | — | 7 | 2 | no | — |
| 77 | #1568 | — | 7 | 1 | no | — |
| 78 | #1496 | — | 7 | 2 | no | — |
| 79 | #1450 | — | 7 | 1 | no | — |
| 80 | C7×#41 | — | 8 | 4 | no | — |
| 81 | C2×C7×Q8 | — | 2 | 2 | no | — |
| 82 | C2×C7×D4 | — | 2 | 4 | no | — |
| 83 | C2×C7×D4 | — | 2 | 2 | no | — |
| 84 | C2×C7×D4 | — | 2 | 2 | no | — |
| 85 | C2×C7×D4 | C22×C4 | 1 | 4 | yes | — |
| 86 | C22×C4×C7 | — | 2 | 4 | no | — |
| 87 | C22×C4×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C22×C4×C7 | #41 | 1 | 1 | yes | characteristic |
| 90 | C7×D8 | — | 2 | 4 | no | — |
| 91 | C7×D8 | — | 2 | 4 | no | — |
| 92 | C2×C7×C8 | — | 2 | 2 | no | — |
| 93 | C2×C7×C8 | — | 2 | 2 | no | — |
| 94 | C2×C7×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 95 | C2×C7×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 96 | C2×C7×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 97 | C2×C7×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 98 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 99 | C7×C4⋊C4 | C22×C4 | 1 | 4 | yes | — |
| 100 | C7×C22⋊C4 | — | 2 | 4 | no | — |
| 101 | C7×C22⋊C4 | — | 2 | 4 | no | — |
| 102 | C42×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 103 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C42×C7 | #41 | 1 | 1 | yes | characteristic |
| 105 | C42×C7 | C24 | 1 | 1 | yes | characteristic |
| 106 | #1136 | — | 8 | 4 | no | — |
| 107 | C2×C7⋊C8 | — | 2 | 4 | no | — |
| 108 | C2×C7⋊C8 | — | 2 | 1 | no | — |
| 109 | C2×C7⋊C8 | — | 2 | 1 | no | — |
| 110 | C2×C7⋊C8 | — | 8 | 4 | no | — |
| 111 | C2×C7⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 112 | C2×C7⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 113 | #532 | C2×D7 | 1 | 2 | yes | — |
| 114 | #500 | Dic7 | 1 | 1 | yes | characteristic |
| 115 | #493 | Dic7 | 1 | 1 | yes | characteristic |
| 116 | #486 | Dic7 | 1 | 2 | yes | — |
| 117 | #466 | Dic7 | 1 | 2 | yes | — |
| 118 | #466 | Dic7 | 1 | 2 | yes | — |
| 119 | C4×D8 | C2×D7 | 1 | 2 | yes | — |
| 120 | C4×D8 | C2×D7 | 1 | 2 | yes | — |
| 121 | #436 | — | 7 | 4 | no | — |
| 122 | #435 | — | 14 | 4 | no | — |
| 123 | C8×D4 | — | 7 | 4 | no | — |
| 124 | #432 | — | 7 | 2 | no | — |
| 125 | #431 | — | 7 | 1 | no | — |
| 126 | #431 | — | 7 | 1 | no | — |
| 127 | #408 | — | 14 | 4 | no | — |
| 128 | C2×C4×C8 | C2×D7 | 1 | 1 | yes | characteristic |
| 129 | #335 | — | 7 | 2 | no | — |
| 130 | #334 | — | 7 | 2 | no | — |
| 131 | D4⋊C8 | — | 7 | 4 | no | — |
| 132 | D4⋊C8 | — | 7 | 4 | no | — |
| 133 | #322 | — | 7 | 1 | no | — |
| 134 | #322 | — | 7 | 1 | no | — |
| 135 | C82 | — | 7 | 1 | no | — |
| 136 | C82 | — | 7 | 1 | no | — |
| 137 | C23×C7 | — | 2 | 4 | no | — |
| 138 | C23×C7 | C4×D4 | 1 | 1 | yes | characteristic |
| 139 | C7×Q8 | — | 8 | 2 | no | — |
| 140 | C7×D4 | — | 2 | 4 | no | — |
| 141 | C7×D4 | — | 2 | 4 | no | — |
| 142 | C7×D4 | — | 8 | 2 | no | — |
| 143 | C7×D4 | — | 8 | 2 | no | — |
| 144 | C7×D4 | — | 8 | 4 | no | — |
| 145 | C2×C4×C7 | — | 2 | 1 | no | — |
| 146 | C2×C4×C7 | — | 2 | 1 | no | — |
| 147 | C2×C4×C7 | — | 2 | 2 | no | — |
| 148 | C2×C4×C7 | — | 2 | 4 | no | — |
| 149 | C2×C4×C7 | — | 4 | 1 | no | — |
| 150 | C2×C4×C7 | — | 4 | 4 | no | — |
| 151 | C2×C4×C7 | — | 4 | 4 | no | — |
| 152 | C2×C4×C7 | C4×D4 | 1 | 1 | yes | characteristic |
| 153 | C2×C4×C7 | C4×D4 | 1 | 1 | yes | characteristic |
| 154 | C2×C4×C7 | C4×D4 | 1 | 1 | yes | characteristic |
| 155 | C2×C4×C7 | C23×C4 | 1 | 1 | yes | characteristic |
| 156 | C2×C4×C7 | C22×D4 | 1 | 1 | yes | characteristic |
| 157 | C2×C4×C7 | C2×#41 | 1 | 1 | yes | characteristic |
| 158 | C7×C8 | — | 2 | 1 | no | — |
| 159 | C7×C8 | — | 2 | 1 | no | — |
| 160 | C7×C8 | #131 | 1 | 2 | yes | — |
| 161 | C7×C8 | #131 | 1 | 2 | yes | — |
| 162 | C7⋊C8 | — | 2 | 2 | no | — |
| 163 | C7⋊C8 | — | 2 | 2 | no | — |
| 164 | C7⋊C8 | — | 4 | 1 | no | — |
| 165 | C7⋊C8 | — | 4 | 1 | no | — |
| 166 | C7⋊C8 | — | 4 | 4 | no | — |
| 167 | C2×#41 | — | 2 | 2 | no | — |
| 168 | C2×D8 | C2×Dic7 | 1 | 2 | yes | — |
| 169 | C2×D8 | C2×Dic7 | 1 | 2 | yes | — |
| 170 | C2×M16 | — | 14 | 4 | no | — |
| 171 | C22×C8 | — | 14 | 4 | no | — |
| 172 | C22×C8 | C2×Dic7 | 1 | 1 | yes | characteristic |
| 173 | C22×C8 | C2×Dic7 | 1 | 1 | yes | characteristic |
| 174 | C4⋊Q8 | C2×Dic7 | 1 | 2 | yes | — |
| 175 | #127 | C2×Dic7 | 1 | 2 | yes | — |
| 176 | #124 | — | 2 | 2 | no | — |
| 177 | #121 | — | 2 | 2 | no | — |
| 178 | #121 | C2×Dic7 | 1 | 4 | yes | — |
| 179 | C4×D4 | — | 2 | 2 | no | — |
| 180 | C4×D4 | C22×D7 | 1 | 4 | yes | — |
| 181 | C2×C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 182 | #107 | C2×Dic7 | 1 | 2 | yes | — |
| 183 | #107 | C2×Dic7 | 1 | 2 | yes | — |
| 184 | C4⋊C8 | — | 7 | 2 | no | — |
| 185 | C4⋊C8 | — | 7 | 4 | no | — |
| 186 | C4⋊C8 | — | 14 | 1 | no | — |
| 187 | #102 | C2×Dic7 | 1 | 4 | yes | — |
| 188 | #102 | C2×Dic7 | 1 | 4 | yes | — |
| 189 | C22⋊C8 | — | 7 | 2 | no | — |
| 190 | C22⋊C8 | — | 14 | 4 | no | — |
| 191 | C22⋊C8 | — | 14 | 4 | no | — |
| 192 | C22⋊C8 | — | 14 | 2 | no | — |
| 193 | C22⋊C8 | — | 14 | 1 | no | — |
| 194 | #97 | — | 14 | 2 | no | — |
| 195 | C4×C8 | — | 7 | 1 | no | — |
| 196 | C4×C8 | — | 7 | 1 | no | — |
| 197 | C4×C8 | — | 7 | 1 | no | — |
| 198 | C4×C8 | — | 7 | 1 | no | — |
| 199 | C4×C8 | C2×Dic7 | 1 | 1 | yes | characteristic |
| 200 | C4×C8 | C2×Dic7 | 1 | 1 | yes | characteristic |
| 201 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 202 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 203 | C22×C7 | — | 2 | 1 | no | — |
| 204 | C22×C7 | — | 2 | 1 | no | — |
| 205 | C22×C7 | — | 2 | 1 | no | — |
| 206 | C22×C7 | — | 4 | 4 | no | — |
| 207 | C22×C7 | — | 4 | 4 | no | — |
| 208 | C22×C7 | — | 4 | 4 | no | — |
| 209 | C22×C7 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 210 | C4×C7 | — | 2 | 1 | no | — |
| 211 | C4×C7 | — | 2 | 1 | no | — |
| 212 | C4×C7 | — | 2 | 1 | no | — |
| 213 | C4×C7 | — | 2 | 1 | no | — |
| 214 | C4×C7 | — | 4 | 4 | no | — |
| 215 | C4×C7 | C2×#131 | 1 | 1 | yes | characteristic |
| 216 | C4×C7 | C2×#131 | 1 | 1 | yes | characteristic |
| 217 | C4×C7 | #576 | 1 | 2 | yes | — |
| 218 | #41 | — | 8 | 4 | no | — |
| 219 | C2×Q8 | — | 2 | 2 | no | — |
| 220 | C2×D4 | — | 2 | 4 | no | — |
| 221 | C2×D4 | — | 2 | 2 | no | — |
| 222 | C2×D4 | — | 2 | 2 | no | — |
| 223 | C2×D4 | C22×Dic7 | 1 | 4 | yes | — |
| 224 | C22×C4 | — | 2 | 4 | no | — |
| 225 | C22×C4 | D4×D7 | 1 | 1 | yes | characteristic |
| 226 | C22×C4 | #1159 | 1 | 1 | yes | characteristic |
| 227 | C22×C4 | C22×Dic7 | 1 | 1 | yes | characteristic |
| 228 | D8 | — | 2 | 4 | no | — |
| 229 | D8 | — | 2 | 4 | no | — |
| 230 | M16 | — | 56 | 4 | no | — |
| 231 | C2×C8 | — | 2 | 2 | no | — |
| 232 | C2×C8 | — | 2 | 2 | no | — |
| 233 | C2×C8 | — | 7 | 2 | no | — |
| 234 | C2×C8 | — | 7 | 2 | no | — |
| 235 | C2×C8 | — | 14 | 4 | no | — |
| 236 | C2×C8 | — | 14 | 1 | no | — |
| 237 | C2×C8 | — | 14 | 1 | no | — |
| 238 | C2×C8 | — | 56 | 4 | no | — |
| 239 | C2×C8 | C22×Dic7 | 1 | 1 | yes | characteristic |
| 240 | C2×C8 | C22×Dic7 | 1 | 1 | yes | characteristic |
| 241 | C2×C8 | C22×Dic7 | 1 | 1 | yes | characteristic |
| 242 | C2×C8 | C22×Dic7 | 1 | 1 | yes | characteristic |
| 243 | C4⋊C4 | — | 2 | 2 | no | — |
| 244 | C4⋊C4 | C22×Dic7 | 1 | 4 | yes | — |
| 245 | C22⋊C4 | — | 2 | 4 | no | — |
| 246 | C22⋊C4 | — | 2 | 4 | no | — |
| 247 | C42 | D4×D7 | 1 | 1 | yes | characteristic |
| 248 | C42 | #1159 | 1 | 1 | yes | characteristic |
| 249 | C42 | C22×Dic7 | 1 | 1 | yes | characteristic |
| 250 | C42 | C23×D7 | 1 | 1 | yes | characteristic |
| 251 | C2×C7 | — | 4 | 1 | no | — |
| 252 | C2×C7 | — | 8 | 4 | no | — |
| 253 | C2×C7 | #2931 | 1 | 1 | yes | characteristic |
| 254 | C2×C7 | #2948 | 1 | 1 | yes | characteristic |
| 255 | C2×C7 | #2957 | 1 | 1 | yes | characteristic |
| 256 | C23 | — | 2 | 4 | no | — |
| 257 | C23 | D4×Dic7 | 1 | 1 | yes | characteristic |
| 258 | Q8 | — | 8 | 2 | no | — |
| 259 | D4 | — | 2 | 4 | no | — |
| 260 | D4 | — | 2 | 4 | no | — |
| 261 | D4 | — | 8 | 2 | no | — |
| 262 | D4 | — | 8 | 2 | no | — |
| 263 | D4 | — | 8 | 4 | no | — |
| 264 | C2×C4 | — | 2 | 2 | no | — |
| 265 | C2×C4 | — | 2 | 1 | no | — |
| 266 | C2×C4 | — | 2 | 1 | no | — |
| 267 | C2×C4 | — | 2 | 4 | no | — |
| 268 | C2×C4 | — | 4 | 1 | no | — |
| 269 | C2×C4 | — | 4 | 4 | no | — |
| 270 | C2×C4 | — | 4 | 4 | no | — |
| 271 | C2×C4 | D4×Dic7 | 1 | 1 | yes | characteristic |
| 272 | C2×C4 | D4×Dic7 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | D4×Dic7 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | C2×D4×D7 | 1 | 1 | yes | characteristic |
| 275 | C2×C4 | C2×#1159 | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | C23×Dic7 | 1 | 1 | yes | characteristic |
| 277 | C8 | — | 2 | 1 | no | — |
| 278 | C8 | — | 2 | 1 | no | — |
| 279 | C8 | — | 14 | 2 | no | — |
| 280 | C8 | — | 14 | 2 | no | — |
| 281 | C8 | — | 28 | 1 | no | — |
| 282 | C8 | — | 28 | 1 | no | — |
| 283 | C8 | — | 28 | 4 | no | — |
| 284 | C8 | #6511 | 1 | 2 | yes | — |
| 285 | C8 | #6511 | 1 | 2 | yes | — |
| 286 | C7 | #17970 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 287 | C22 | — | 2 | 1 | no | — |
| 288 | C22 | — | 2 | 1 | no | — |
| 289 | C22 | — | 2 | 1 | no | — |
| 290 | C22 | — | 4 | 4 | no | — |
| 291 | C22 | — | 4 | 4 | no | — |
| 292 | C22 | — | 4 | 4 | no | — |
| 293 | C22 | C2×D4×Dic7 | 1 | 1 | yes | characteristic |
| 294 | C4 | — | 2 | 1 | no | — |
| 295 | C4 | — | 2 | 1 | no | — |
| 296 | C4 | — | 2 | 1 | no | — |
| 297 | C4 | — | 2 | 1 | no | — |
| 298 | C4 | — | 4 | 4 | no | — |
| 299 | C4 | #90448 | 1 | 2 | yes | — |
| 300 | C4 | C2×#6511 | 1 | 1 | yes | characteristic |
| 301 | C4 | C2×#6511 | 1 | 1 | yes | characteristic |
| 302 | C2 | — | 4 | 1 | no | — |
| 303 | C2 | — | 8 | 4 | no | — |
| 304 | C2 | #11740720 | 1 | 1 | yes | characteristic |
| 305 | C2 | #11741268 | 1 | 1 | yes | characteristic |
| 306 | C2 | #11741355 | 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