← #13509258
Subgroups — #13509258
Order 1792.
794 conjugacy classes of subgroups.
The subgroup lattice has 282 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C7×#508 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11741352 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×#89967 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C22×Dic56 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11729253 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #11729236 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #11729219 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #11729153 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×C7×#135 | C22 | 1 | 2 | yes | — |
| 10 | #90516 | — | 2 | 2 | no | — |
| 11 | C2×#6514 | — | 2 | 2 | no | — |
| 12 | C22×Dic28 | C22 | 1 | 2 | yes | — |
| 13 | C7×#508 | — | 2 | 4 | no | — |
| 14 | C2×C7×Q32 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C7×SD32 | C22 | 1 | 2 | yes | — |
| 16 | C2×C7×D16 | C22 | 1 | 1 | yes | characteristic |
| 17 | C22×C7×C16 | C22 | 1 | 1 | yes | characteristic |
| 18 | #89967 | — | 2 | 8 | no | — |
| 19 | #89953 | C22 | 1 | 4 | yes | — |
| 20 | C2×#6403 | C22 | 1 | 4 | yes | — |
| 21 | #89939 | — | 2 | 4 | no | — |
| 22 | #89927 | C22 | 1 | 4 | yes | — |
| 23 | C2×#6401 | C22 | 1 | 4 | yes | — |
| 24 | #89873 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#6386 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×Dic56 | — | 2 | 4 | no | — |
| 27 | C2×Dic56 | C22 | 1 | 2 | yes | — |
| 28 | C2×Dic56 | C22 | 1 | 2 | yes | — |
| 29 | #89297 | C22 | 1 | 2 | yes | — |
| 30 | C16×Dic7 | C22 | 1 | 2 | yes | — |
| 31 | #33721 | — | 7 | 1 | no | maximal, Sylow |
| 32 | C2×C7×#41 | — | 2 | 2 | no | — |
| 33 | C22×Dic14 | — | 2 | 2 | no | — |
| 34 | C7×#135 | — | 2 | 4 | no | — |
| 35 | C7×#135 | — | 2 | 4 | no | — |
| 36 | C2×C7×Q16 | C23 | 1 | 2 | yes | — |
| 37 | C2×C7×SD16 | — | 2 | 2 | no | — |
| 38 | C2×C7×D8 | C23 | 1 | 2 | yes | — |
| 39 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 40 | #6514 | — | 4 | 8 | no | — |
| 41 | #6507 | — | 2 | 4 | no | — |
| 42 | C2×#1144 | — | 2 | 4 | no | — |
| 43 | #6499 | — | 2 | 4 | no | — |
| 44 | #6498 | — | 4 | 4 | no | — |
| 45 | C2×#1142 | — | 2 | 4 | no | — |
| 46 | #6489 | D4 | 1 | 1 | yes | characteristic |
| 47 | #6487 | — | 4 | 4 | no | — |
| 48 | C2×#1136 | D4 | 1 | 1 | yes | characteristic |
| 49 | C2×Dic28 | — | 2 | 4 | no | — |
| 50 | C2×Dic28 | — | 2 | 4 | no | — |
| 51 | C2×Dic28 | — | 2 | 2 | no | — |
| 52 | C2×Dic28 | C23 | 1 | 4 | yes | — |
| 53 | C7×Q32 | — | 2 | 2 | no | — |
| 54 | C7×SD32 | — | 2 | 4 | no | — |
| 55 | C7×D16 | — | 2 | 2 | no | — |
| 56 | C2×C7×C16 | D4 | 1 | 4 | yes | — |
| 57 | C2×C7×C16 | C23 | 1 | 1 | yes | characteristic |
| 58 | C2×C7×C16 | C23 | 1 | 1 | yes | characteristic |
| 59 | #6403 | — | 2 | 8 | no | — |
| 60 | #6401 | — | 2 | 8 | no | — |
| 61 | #6389 | D4 | 1 | 2 | yes | — |
| 62 | C8×Dic7 | C23 | 1 | 2 | yes | — |
| 63 | #6386 | D4 | 1 | 4 | yes | — |
| 64 | C2×C7⋊C16 | C23 | 1 | 2 | yes | — |
| 65 | Dic56 | — | 2 | 4 | no | — |
| 66 | Dic56 | — | 2 | 4 | no | — |
| 67 | C2×#510 | — | 7 | 2 | no | — |
| 68 | C2×#508 | D7 | 1 | 1 | yes | characteristic |
| 69 | C22×Q32 | — | 7 | 1 | no | — |
| 70 | #3156 | — | 7 | 2 | no | — |
| 71 | #2251 | — | 7 | 4 | no | — |
| 72 | #2249 | — | 7 | 2 | no | — |
| 73 | #2247 | — | 7 | 2 | no | — |
| 74 | #2108 | — | 7 | 1 | no | — |
| 75 | C7×#41 | — | 4 | 4 | no | — |
| 76 | C7×#41 | — | 4 | 4 | no | — |
| 77 | C2×C7×Q8 | — | 2 | 2 | no | — |
| 78 | C2×C7×D4 | — | 2 | 2 | no | — |
| 79 | C2×C7×D4 | — | 4 | 2 | no | — |
| 80 | C22×C4×C7 | — | 4 | 2 | no | — |
| 81 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C22×Dic7 | — | 4 | 2 | no | — |
| 83 | C2×Dic14 | — | 2 | 4 | no | — |
| 84 | C2×Dic14 | — | 4 | 4 | no | — |
| 85 | C2×Dic14 | — | 4 | 4 | no | — |
| 86 | C2×Dic14 | — | 4 | 2 | no | — |
| 87 | C7×Q16 | — | 2 | 2 | no | — |
| 88 | C7×Q16 | — | 2 | 2 | no | — |
| 89 | C7×SD16 | — | 4 | 4 | no | — |
| 90 | C7×D8 | — | 2 | 2 | no | — |
| 91 | C7×D8 | — | 2 | 2 | no | — |
| 92 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 94 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 95 | C2×C7×C8 | C24 | 1 | 1 | yes | characteristic |
| 96 | C22⋊Dic7 | — | 4 | 8 | no | — |
| 97 | C22⋊Dic7 | D8 | 1 | 2 | yes | — |
| 98 | #1144 | — | 4 | 8 | no | — |
| 99 | #1142 | — | 4 | 8 | no | — |
| 100 | C4⋊Dic7 | D8 | 1 | 2 | yes | — |
| 101 | Dic7⋊C4 | — | 4 | 8 | no | — |
| 102 | C4×Dic7 | C2×D4 | 1 | 2 | yes | — |
| 103 | #1136 | D8 | 1 | 4 | yes | — |
| 104 | C2×C7⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 105 | Dic28 | — | 2 | 4 | no | — |
| 106 | Dic28 | — | 2 | 4 | no | — |
| 107 | Dic28 | — | 4 | 4 | no | — |
| 108 | C7×C16 | C2×D4 | 1 | 2 | yes | — |
| 109 | C7×C16 | C2×D4 | 1 | 2 | yes | — |
| 110 | C7⋊C16 | C2×D4 | 1 | 4 | yes | — |
| 111 | C2×#137 | — | 14 | 2 | no | — |
| 112 | C2×#135 | C2×D7 | 1 | 2 | yes | — |
| 113 | C22×Q16 | — | 7 | 2 | no | — |
| 114 | #536 | — | 14 | 2 | no | — |
| 115 | #510 | — | 14 | 8 | no | — |
| 116 | #508 | — | 2 | 4 | no | — |
| 117 | C2×Q32 | — | 7 | 4 | no | — |
| 118 | C2×Q32 | — | 7 | 2 | no | — |
| 119 | C2×Q32 | — | 7 | 2 | no | — |
| 120 | C2×Q32 | — | 14 | 4 | no | — |
| 121 | C2×Q32 | C2×D7 | 1 | 1 | yes | characteristic |
| 122 | C2×SD32 | — | 7 | 4 | no | — |
| 123 | C2×SD32 | C2×D7 | 1 | 2 | yes | — |
| 124 | C2×D16 | C2×D7 | 1 | 1 | yes | characteristic |
| 125 | C2×M32 | — | 7 | 1 | no | — |
| 126 | C22×C16 | C2×D7 | 1 | 1 | yes | characteristic |
| 127 | #497 | — | 14 | 4 | no | — |
| 128 | #495 | — | 7 | 4 | no | — |
| 129 | #494 | — | 7 | 4 | no | — |
| 130 | #405 | — | 7 | 1 | no | — |
| 131 | #346 | — | 7 | 2 | no | — |
| 132 | C4×C16 | — | 7 | 2 | no | — |
| 133 | C23×C7 | — | 4 | 2 | no | — |
| 134 | C23×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 135 | C7×Q8 | — | 4 | 2 | no | — |
| 136 | C7×Q8 | — | 4 | 2 | no | — |
| 137 | C7×D4 | — | 4 | 2 | no | — |
| 138 | C7×D4 | — | 4 | 2 | no | — |
| 139 | C7×D4 | — | 8 | 4 | no | — |
| 140 | C2×C4×C7 | — | 4 | 2 | no | — |
| 141 | C2×C4×C7 | — | 8 | 4 | no | — |
| 142 | C2×C4×C7 | #127 | 1 | 2 | yes | — |
| 143 | C2×C4×C7 | C2×D8 | 1 | 2 | yes | — |
| 144 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 145 | C2×C4×C7 | C22×D4 | 1 | 1 | yes | characteristic |
| 146 | C2×Dic7 | — | 4 | 4 | no | — |
| 147 | C2×Dic7 | — | 8 | 4 | no | — |
| 148 | C2×Dic7 | C2×D8 | 1 | 4 | yes | — |
| 149 | Dic14 | — | 4 | 4 | no | — |
| 150 | Dic14 | — | 4 | 4 | no | — |
| 151 | Dic14 | — | 8 | 4 | no | — |
| 152 | C7×C8 | #127 | 1 | 2 | yes | — |
| 153 | C7×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 154 | C7×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 155 | C7⋊C8 | C2×D8 | 1 | 4 | yes | — |
| 156 | C2×#41 | — | 2 | 2 | no | — |
| 157 | C22×Q8 | — | 14 | 2 | no | — |
| 158 | #137 | — | 28 | 8 | no | — |
| 159 | #135 | — | 2 | 4 | no | — |
| 160 | #135 | — | 2 | 4 | no | — |
| 161 | C2×Q16 | — | 7 | 4 | no | — |
| 162 | C2×Q16 | — | 14 | 4 | no | — |
| 163 | C2×Q16 | — | 14 | 4 | no | — |
| 164 | C2×Q16 | — | 14 | 2 | no | — |
| 165 | C2×Q16 | — | 14 | 4 | no | — |
| 166 | C2×Q16 | C22×D7 | 1 | 2 | yes | — |
| 167 | C2×SD16 | — | 2 | 2 | no | — |
| 168 | C2×SD16 | — | 14 | 4 | no | — |
| 169 | C2×D8 | C22×D7 | 1 | 2 | yes | — |
| 170 | C2×M16 | — | 7 | 1 | no | — |
| 171 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 172 | C4⋊Q8 | — | 14 | 4 | no | — |
| 173 | #124 | — | 14 | 4 | no | — |
| 174 | #123 | — | 28 | 4 | no | — |
| 175 | C22⋊Q8 | — | 28 | 4 | no | — |
| 176 | #117 | — | 7 | 1 | no | — |
| 177 | Q32 | — | 2 | 2 | no | — |
| 178 | Q32 | — | 14 | 8 | no | — |
| 179 | Q32 | — | 14 | 4 | no | — |
| 180 | Q32 | — | 14 | 4 | no | — |
| 181 | SD32 | — | 2 | 4 | no | — |
| 182 | SD32 | — | 14 | 8 | no | — |
| 183 | D16 | — | 2 | 2 | no | — |
| 184 | M32 | — | 7 | 4 | no | — |
| 185 | C2×C16 | — | 7 | 2 | no | — |
| 186 | C2×C16 | C22⋊D7 | 1 | 4 | yes | — |
| 187 | C2×C16 | C22×D7 | 1 | 1 | yes | characteristic |
| 188 | C2×C16 | C22×D7 | 1 | 1 | yes | characteristic |
| 189 | #97 | — | 7 | 2 | no | — |
| 190 | C4×C8 | — | 7 | 2 | no | — |
| 191 | C22×C7 | — | 2 | 2 | no | — |
| 192 | C22×C7 | — | 8 | 2 | no | — |
| 193 | C22×C7 | — | 8 | 4 | no | — |
| 194 | C22×C7 | #493 | 1 | 2 | yes | — |
| 195 | C22×C7 | C22×D8 | 1 | 1 | yes | characteristic |
| 196 | C4×C7 | — | 8 | 2 | no | — |
| 197 | C4×C7 | #493 | 1 | 2 | yes | — |
| 198 | C4×C7 | C2×#127 | 1 | 1 | yes | characteristic |
| 199 | C4×C7 | C22×D8 | 1 | 1 | yes | characteristic |
| 200 | Dic7 | — | 2 | 4 | no | — |
| 201 | Dic7 | — | 8 | 4 | no | — |
| 202 | #41 | — | 4 | 4 | no | — |
| 203 | #41 | — | 4 | 4 | no | — |
| 204 | C2×Q8 | — | 2 | 2 | no | — |
| 205 | C2×Q8 | — | 14 | 4 | no | — |
| 206 | C2×Q8 | — | 28 | 2 | no | — |
| 207 | C2×Q8 | — | 28 | 4 | no | — |
| 208 | C2×Q8 | — | 28 | 4 | no | — |
| 209 | C2×D4 | — | 2 | 2 | no | — |
| 210 | C2×D4 | — | 4 | 2 | no | — |
| 211 | C22×C4 | — | 4 | 2 | no | — |
| 212 | C22×C4 | — | 28 | 2 | no | — |
| 213 | C22×C4 | D4×D7 | 1 | 1 | yes | characteristic |
| 214 | Q16 | — | 2 | 2 | no | — |
| 215 | Q16 | — | 2 | 2 | no | — |
| 216 | Q16 | — | 14 | 4 | no | — |
| 217 | Q16 | — | 14 | 4 | no | — |
| 218 | Q16 | — | 28 | 8 | no | — |
| 219 | Q16 | — | 28 | 4 | no | — |
| 220 | SD16 | — | 4 | 4 | no | — |
| 221 | SD16 | — | 28 | 8 | no | — |
| 222 | D8 | — | 2 | 2 | no | — |
| 223 | D8 | — | 2 | 2 | no | — |
| 224 | M16 | — | 7 | 4 | no | — |
| 225 | C2×C8 | — | 7 | 2 | no | — |
| 226 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 227 | C2×C8 | D4×D7 | 1 | 2 | yes | — |
| 228 | C2×C8 | C2×C22⋊D7 | 1 | 2 | yes | — |
| 229 | C2×C8 | C23×D7 | 1 | 1 | yes | characteristic |
| 230 | C4⋊C4 | — | 7 | 2 | no | — |
| 231 | C4⋊C4 | — | 28 | 8 | no | — |
| 232 | C22⋊C4 | — | 7 | 2 | no | — |
| 233 | C22⋊C4 | — | 28 | 8 | no | — |
| 234 | C42 | — | 7 | 2 | no | — |
| 235 | C16 | — | 7 | 4 | no | — |
| 236 | C16 | C2×C22⋊D7 | 1 | 2 | yes | — |
| 237 | C16 | C2×C22⋊D7 | 1 | 2 | yes | — |
| 238 | C2×C7 | — | 2 | 2 | no | — |
| 239 | C2×C7 | — | 16 | 2 | no | — |
| 240 | C2×C7 | C2×#493 | 1 | 1 | yes | characteristic |
| 241 | C2×C7 | #3417 | 1 | 2 | yes | — |
| 242 | C23 | — | 4 | 2 | no | — |
| 243 | C23 | D7×D8 | 1 | 1 | yes | characteristic |
| 244 | Q8 | — | 4 | 2 | no | — |
| 245 | Q8 | — | 4 | 2 | no | — |
| 246 | Q8 | — | 28 | 4 | no | — |
| 247 | Q8 | — | 28 | 4 | no | — |
| 248 | Q8 | — | 56 | 4 | no | — |
| 249 | D4 | — | 4 | 2 | no | — |
| 250 | D4 | — | 4 | 2 | no | — |
| 251 | D4 | — | 8 | 4 | no | — |
| 252 | C2×C4 | — | 4 | 2 | no | — |
| 253 | C2×C4 | — | 7 | 4 | no | — |
| 254 | C2×C4 | — | 8 | 4 | no | — |
| 255 | C2×C4 | — | 28 | 4 | no | — |
| 256 | C2×C4 | — | 56 | 4 | no | — |
| 257 | C2×C4 | D7×D8 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | D7×D8 | 1 | 2 | yes | — |
| 259 | C2×C4 | #6503 | 1 | 2 | yes | — |
| 260 | C2×C4 | C2×D4×D7 | 1 | 1 | yes | characteristic |
| 261 | C8 | — | 7 | 4 | no | — |
| 262 | C8 | #6503 | 1 | 2 | yes | — |
| 263 | C8 | C2×D4×D7 | 1 | 1 | yes | characteristic |
| 264 | C8 | C22×C22⋊D7 | 1 | 1 | yes | characteristic |
| 265 | C7 | #33721 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 266 | C22 | — | 2 | 2 | no | — |
| 267 | C22 | — | 8 | 4 | no | — |
| 268 | C22 | — | 8 | 2 | no | — |
| 269 | C22 | #89923 | 1 | 2 | yes | — |
| 270 | C22 | C2×D7×D8 | 1 | 1 | yes | characteristic |
| 271 | C4 | — | 8 | 2 | no | — |
| 272 | C4 | — | 14 | 4 | no | — |
| 273 | C4 | — | 56 | 4 | no | — |
| 274 | C4 | #89923 | 1 | 2 | yes | — |
| 275 | C4 | C2×D7×D8 | 1 | 1 | yes | characteristic |
| 276 | C4 | C2×#6503 | 1 | 1 | yes | characteristic |
| 277 | C2 | — | 2 | 2 | no | — |
| 278 | C2 | — | 16 | 2 | no | — |
| 279 | C2 | #11739478 | 1 | 2 | yes | — |
| 280 | C2 | C2×#89923 | 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