← #13506705
Subgroups — #13506705
Order 1792.
1176 conjugacy classes of subgroups.
The subgroup lattice has 280 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#90443 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#2913 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#89912 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11728406 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #11728405 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #11728352 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×#89286 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C2×#6553 | — | 2 | 1 | no | — |
| 9 | C2×C7×#131 | C22 | 1 | 1 | yes | characteristic |
| 10 | C7×#520 | — | 2 | 1 | no | — |
| 11 | #90443 | — | 2 | 4 | no | — |
| 12 | #90443 | C4 | 1 | 8 | yes | complemented |
| 13 | C2×#6471 | — | 2 | 3 | no | — |
| 14 | C2×#6467 | C22 | 1 | 3 | yes | — |
| 15 | C22×D56 | C22 | 1 | 3 | yes | — |
| 16 | C7×#428 | — | 2 | 3 | no | — |
| 17 | C7×#427 | C22 | 1 | 3 | yes | complemented |
| 18 | C2×C7×#107 | C22 | 1 | 3 | yes | complemented |
| 19 | #89912 | — | 2 | 4 | no | — |
| 20 | C2×#6386 | C22 | 1 | 3 | yes | complemented |
| 21 | C22×C7⋊C16 | C22 | 1 | 3 | yes | complemented |
| 22 | #89287 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | #89287 | C22 | 1 | 3 | yes | complemented |
| 24 | #89286 | C22 | 1 | 3 | yes | complemented |
| 25 | #89286 | C22 | 1 | 3 | yes | complemented |
| 26 | #89286 | C22 | 1 | 6 | yes | complemented |
| 27 | #33381 | — | 7 | 1 | no | maximal, Sylow |
| 28 | C2×C7×#41 | D4 | 1 | 1 | yes | characteristic |
| 29 | #6553 | — | 2 | 8 | no | — |
| 30 | #6553 | — | 4 | 4 | no | — |
| 31 | C2×#1161 | — | 4 | 1 | no | — |
| 32 | C2×D4×D7 | — | 4 | 3 | no | — |
| 33 | C2×#1157 | — | 2 | 3 | no | — |
| 34 | C22×D28 | — | 2 | 3 | no | — |
| 35 | C7×#131 | C2×C4 | 1 | 4 | yes | — |
| 36 | C7×#131 | D4 | 1 | 4 | yes | — |
| 37 | C2×C7×M16 | D4 | 1 | 3 | yes | — |
| 38 | C22×C7×C8 | C23 | 1 | 3 | yes | — |
| 39 | C4×C7×Q8 | — | 4 | 1 | no | — |
| 40 | C4×C7×D4 | — | 4 | 3 | no | — |
| 41 | C7×#117 | — | 2 | 3 | no | — |
| 42 | C2×C7×C4⋊C4 | — | 2 | 3 | no | — |
| 43 | #6471 | — | 2 | 24 | no | — |
| 44 | C2×Dic28 | C23 | 1 | 1 | yes | characteristic |
| 45 | #6467 | — | 2 | 6 | no | — |
| 46 | #6467 | C2×C4 | 1 | 12 | yes | complemented |
| 47 | C2×D56 | — | 2 | 3 | no | — |
| 48 | C2×D56 | — | 2 | 6 | no | — |
| 49 | C2×D56 | C2×C4 | 1 | 12 | yes | complemented |
| 50 | C2×D56 | C23 | 1 | 3 | yes | — |
| 51 | C2×#1132 | — | 2 | 3 | no | — |
| 52 | C7×#107 | — | 2 | 3 | no | — |
| 53 | C7×#107 | C23 | 1 | 3 | yes | — |
| 54 | C7×#107 | C23 | 1 | 1 | yes | characteristic |
| 55 | C7×#106 | — | 2 | 3 | no | — |
| 56 | #6386 | — | 2 | 6 | no | — |
| 57 | C2×C7⋊C16 | — | 2 | 6 | no | — |
| 58 | C2×C7⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 59 | C2×C7⋊C16 | C23 | 1 | 3 | yes | — |
| 60 | C2×#576 | — | 7 | 1 | no | — |
| 61 | C2×#504 | — | 7 | 1 | no | — |
| 62 | #2913 | D7 | 1 | 1 | yes | characteristic, complemented |
| 63 | #2140 | — | 7 | 3 | no | — |
| 64 | #2139 | — | 7 | 3 | no | — |
| 65 | #2138 | — | 7 | 3 | no | — |
| 66 | C2×#357 | — | 7 | 3 | no | — |
| 67 | C23×D7 | — | 4 | 3 | no | — |
| 68 | C7×#41 | C22⋊C4 | 1 | 4 | yes | — |
| 69 | C7×#41 | SD16 | 1 | 4 | yes | — |
| 70 | C2×C7×Q8 | D8 | 1 | 1 | yes | characteristic |
| 71 | C2×C7×D4 | D8 | 1 | 3 | yes | — |
| 72 | C22×C4×C7 | — | 4 | 3 | no | — |
| 73 | C22×C4×C7 | C2×D4 | 1 | 3 | yes | — |
| 74 | C2×C22⋊D7 | — | 4 | 3 | no | — |
| 75 | #1161 | — | 4 | 8 | no | — |
| 76 | D4×D7 | — | 4 | 24 | no | — |
| 77 | #1157 | — | 2 | 12 | no | — |
| 78 | #1157 | — | 4 | 6 | no | — |
| 79 | C2×D28 | — | 2 | 12 | no | — |
| 80 | C2×D28 | — | 2 | 3 | no | — |
| 81 | C2×D28 | — | 4 | 3 | no | — |
| 82 | C2×D28 | — | 4 | 6 | no | — |
| 83 | C2×C4×D7 | — | 4 | 3 | no | — |
| 84 | C2×Dic14 | — | 2 | 1 | no | — |
| 85 | C7×M16 | C22⋊C4 | 1 | 12 | yes | — |
| 86 | C2×C7×C8 | C22×C4 | 1 | 6 | yes | — |
| 87 | C2×C7×C8 | C2×D4 | 1 | 3 | yes | — |
| 88 | C2×C7×C8 | C2×D4 | 1 | 6 | yes | — |
| 89 | C2×C7×C8 | C24 | 1 | 1 | yes | characteristic |
| 90 | C7×C4⋊C4 | — | 2 | 1 | no | — |
| 91 | C7×C4⋊C4 | — | 2 | 3 | no | — |
| 92 | C7×C4⋊C4 | — | 4 | 3 | no | — |
| 93 | C7×C22⋊C4 | — | 4 | 3 | no | — |
| 94 | C42×C7 | — | 4 | 3 | no | — |
| 95 | Dic28 | — | 2 | 1 | no | — |
| 96 | Dic28 | C22×C4 | 1 | 2 | yes | — |
| 97 | D56 | — | 2 | 12 | no | — |
| 98 | D56 | — | 2 | 3 | no | — |
| 99 | D56 | C22×C4 | 1 | 6 | yes | — |
| 100 | #1132 | — | 2 | 12 | no | — |
| 101 | C7⋊C16 | — | 2 | 1 | no | — |
| 102 | C7⋊C16 | — | 2 | 3 | no | — |
| 103 | C2×H(5,𝔽2) | — | 14 | 1 | no | — |
| 104 | #576 | — | 7 | 8 | no | — |
| 105 | #576 | — | 14 | 4 | no | — |
| 106 | C2×AGL(1,ℤ/8) | — | 14 | 3 | no | — |
| 107 | C2×#135 | — | 7 | 3 | no | — |
| 108 | C22×D8 | — | 7 | 3 | no | — |
| 109 | C2×#131 | C2×D7 | 1 | 1 | yes | characteristic |
| 110 | #520 | — | 2 | 1 | no | — |
| 111 | #504 | — | 14 | 4 | no | — |
| 112 | C2×M32 | — | 7 | 3 | no | — |
| 113 | C22×C16 | — | 7 | 3 | no | — |
| 114 | #428 | — | 2 | 3 | no | — |
| 115 | #427 | C2×D7 | 1 | 3 | yes | complemented |
| 116 | C2×#107 | C2×D7 | 1 | 3 | yes | complemented |
| 117 | #358 | — | 7 | 1 | no | — |
| 118 | #358 | — | 7 | 3 | no | — |
| 119 | #357 | — | 7 | 3 | no | — |
| 120 | #357 | — | 7 | 6 | no | — |
| 121 | #357 | — | 7 | 3 | no | — |
| 122 | C23×C7 | C2×D8 | 1 | 3 | yes | — |
| 123 | C22×D7 | — | 4 | 3 | no | — |
| 124 | C22×D7 | — | 4 | 12 | no | — |
| 125 | C22×D7 | — | 16 | 3 | no | — |
| 126 | C22×D7 | — | 16 | 3 | no | — |
| 127 | C7×Q8 | #102 | 1 | 4 | yes | — |
| 128 | C7×D4 | #102 | 1 | 12 | yes | — |
| 129 | C2×C4×C7 | — | 4 | 1 | no | — |
| 130 | C2×C4×C7 | — | 4 | 3 | no | — |
| 131 | C2×C4×C7 | — | 16 | 3 | no | — |
| 132 | C2×C4×C7 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 133 | C2×C4×C7 | C2×D8 | 1 | 3 | yes | — |
| 134 | C2×C4×C7 | C2×SD16 | 1 | 6 | yes | — |
| 135 | C2×C4×C7 | C22×D4 | 1 | 1 | yes | characteristic |
| 136 | C22⋊D7 | — | 4 | 12 | no | — |
| 137 | C2×Dic7 | — | 4 | 1 | no | — |
| 138 | D28 | — | 2 | 6 | no | — |
| 139 | D28 | — | 4 | 3 | no | — |
| 140 | D28 | — | 4 | 12 | no | — |
| 141 | C4×D7 | — | 4 | 12 | no | — |
| 142 | Dic14 | — | 2 | 2 | no | — |
| 143 | Dic14 | — | 4 | 1 | no | — |
| 144 | C7×C8 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 145 | C7×C8 | C23×C4 | 1 | 1 | yes | characteristic |
| 146 | C7×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 147 | H(5,𝔽2) | — | 14 | 8 | no | — |
| 148 | H(5,𝔽2) | — | 28 | 4 | no | — |
| 149 | C2×#41 | — | 14 | 3 | no | — |
| 150 | C2×#41 | — | 28 | 1 | no | — |
| 151 | C2×#41 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 152 | C22×D4 | — | 14 | 3 | no | — |
| 153 | C22×D4 | — | 28 | 3 | no | — |
| 154 | AGL(1,ℤ/8) | — | 14 | 24 | no | — |
| 155 | #135 | — | 7 | 12 | no | — |
| 156 | #135 | — | 14 | 6 | no | — |
| 157 | C2×Q16 | — | 7 | 1 | no | — |
| 158 | C2×SD16 | — | 14 | 3 | no | — |
| 159 | C2×D8 | — | 7 | 12 | no | — |
| 160 | C2×D8 | — | 7 | 3 | no | — |
| 161 | C2×D8 | — | 14 | 6 | no | — |
| 162 | C2×D8 | — | 14 | 3 | no | — |
| 163 | #131 | C4×D7 | 1 | 4 | yes | — |
| 164 | #131 | D28 | 1 | 4 | yes | — |
| 165 | C2×M16 | C22⋊D7 | 1 | 3 | yes | — |
| 166 | C22×C8 | C22×D7 | 1 | 3 | yes | — |
| 167 | C4×Q8 | — | 4 | 1 | no | — |
| 168 | C4×D4 | — | 4 | 3 | no | — |
| 169 | #117 | — | 2 | 3 | no | — |
| 170 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 171 | M32 | — | 14 | 6 | no | — |
| 172 | C2×C16 | — | 7 | 1 | no | — |
| 173 | C2×C16 | — | 7 | 3 | no | — |
| 174 | C2×C16 | — | 14 | 6 | no | — |
| 175 | #107 | — | 2 | 3 | no | — |
| 176 | #107 | C22×D7 | 1 | 3 | yes | — |
| 177 | #107 | C22×D7 | 1 | 1 | yes | characteristic |
| 178 | #106 | — | 2 | 3 | no | — |
| 179 | C22×C7 | — | 2 | 3 | no | — |
| 180 | C22×C7 | — | 2 | 3 | no | — |
| 181 | C22×C7 | C2×#102 | 1 | 6 | yes | — |
| 182 | C22×C7 | C22×D8 | 1 | 1 | yes | characteristic |
| 183 | C2×D7 | — | 4 | 6 | no | — |
| 184 | C2×D7 | — | 8 | 3 | no | — |
| 185 | C2×D7 | — | 8 | 3 | no | — |
| 186 | C2×D7 | — | 16 | 12 | no | — |
| 187 | C4×C7 | — | 8 | 3 | no | — |
| 188 | C4×C7 | — | 8 | 1 | no | — |
| 189 | C4×C7 | C2×#102 | 1 | 6 | yes | — |
| 190 | C4×C7 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 191 | C4×C7 | C22×SD16 | 1 | 1 | yes | characteristic |
| 192 | Dic7 | — | 4 | 2 | no | — |
| 193 | C24 | — | 28 | 3 | no | — |
| 194 | #41 | — | 14 | 12 | no | — |
| 195 | #41 | — | 28 | 8 | no | — |
| 196 | #41 | — | 28 | 6 | no | — |
| 197 | #41 | (C2×D7)⋊C4 | 1 | 4 | yes | — |
| 198 | #41 | #1143 | 1 | 4 | yes | — |
| 199 | C2×Q8 | — | 14 | 1 | no | — |
| 200 | C2×Q8 | #1141 | 1 | 1 | yes | characteristic |
| 201 | C2×D4 | — | 14 | 12 | no | — |
| 202 | C2×D4 | — | 14 | 3 | no | — |
| 203 | C2×D4 | — | 28 | 3 | no | — |
| 204 | C2×D4 | — | 28 | 24 | no | — |
| 205 | C2×D4 | — | 28 | 3 | no | — |
| 206 | C2×D4 | — | 28 | 6 | no | — |
| 207 | C2×D4 | #1141 | 1 | 3 | yes | — |
| 208 | C22×C4 | — | 4 | 3 | no | — |
| 209 | C22×C4 | — | 28 | 3 | no | — |
| 210 | C22×C4 | C2×C22⋊D7 | 1 | 3 | yes | — |
| 211 | Q16 | — | 7 | 2 | no | — |
| 212 | Q16 | — | 14 | 1 | no | — |
| 213 | SD16 | — | 14 | 12 | no | — |
| 214 | D8 | — | 7 | 6 | no | — |
| 215 | D8 | — | 14 | 12 | no | — |
| 216 | D8 | — | 14 | 3 | no | — |
| 217 | M16 | (C2×D7)⋊C4 | 1 | 12 | yes | — |
| 218 | C2×C8 | C2×C4×D7 | 1 | 6 | yes | — |
| 219 | C2×C8 | C2×D28 | 1 | 6 | yes | — |
| 220 | C2×C8 | C2×C22⋊D7 | 1 | 3 | yes | — |
| 221 | C2×C8 | C23×D7 | 1 | 1 | yes | characteristic |
| 222 | C4⋊C4 | — | 2 | 1 | no | — |
| 223 | C4⋊C4 | — | 2 | 3 | no | — |
| 224 | C4⋊C4 | — | 4 | 3 | no | — |
| 225 | C22⋊C4 | — | 4 | 3 | no | — |
| 226 | C42 | — | 4 | 3 | no | — |
| 227 | C16 | — | 14 | 1 | no | — |
| 228 | C16 | — | 14 | 3 | no | — |
| 229 | C2×C7 | — | 2 | 6 | no | — |
| 230 | C2×C7 | C22×#102 | 1 | 1 | yes | characteristic |
| 231 | C2×C7 | #3415 | 1 | 1 | yes | characteristic |
| 232 | C2×C7 | #3416 | 1 | 1 | yes | characteristic |
| 233 | D7 | — | 8 | 6 | no | — |
| 234 | C23 | — | 28 | 3 | no | — |
| 235 | C23 | — | 28 | 12 | no | — |
| 236 | C23 | — | 112 | 3 | no | — |
| 237 | C23 | — | 112 | 3 | no | — |
| 238 | C23 | C2×#1141 | 1 | 3 | yes | — |
| 239 | Q8 | — | 14 | 2 | no | — |
| 240 | Q8 | — | 28 | 1 | no | — |
| 241 | Q8 | #6383 | 1 | 4 | yes | — |
| 242 | D4 | — | 14 | 6 | no | — |
| 243 | D4 | — | 28 | 12 | no | — |
| 244 | D4 | — | 28 | 3 | no | — |
| 245 | D4 | — | 28 | 12 | no | — |
| 246 | D4 | #6383 | 1 | 12 | yes | — |
| 247 | C2×C4 | — | 4 | 1 | no | — |
| 248 | C2×C4 | — | 4 | 3 | no | — |
| 249 | C2×C4 | — | 16 | 3 | no | — |
| 250 | C2×C4 | — | 28 | 1 | no | — |
| 251 | C2×C4 | — | 28 | 12 | no | — |
| 252 | C2×C4 | C2×(C2×D7)⋊C4 | 1 | 6 | yes | — |
| 253 | C2×C4 | C2×#1141 | 1 | 3 | yes | — |
| 254 | C2×C4 | C2×#1143 | 1 | 6 | yes | — |
| 255 | C2×C4 | C22×C22⋊D7 | 1 | 1 | yes | characteristic |
| 256 | C8 | C2×(C2×D7)⋊C4 | 1 | 6 | yes | — |
| 257 | C8 | C22×C4×D7 | 1 | 1 | yes | characteristic |
| 258 | C8 | C22×D28 | 1 | 1 | yes | characteristic |
| 259 | C7 | #33381 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 260 | C22 | — | 2 | 3 | no | — |
| 261 | C22 | — | 2 | 3 | no | — |
| 262 | C22 | — | 28 | 6 | no | — |
| 263 | C22 | — | 56 | 3 | no | — |
| 264 | C22 | — | 56 | 3 | no | — |
| 265 | C22 | — | 112 | 12 | no | — |
| 266 | C22 | C2×#6383 | 1 | 6 | yes | — |
| 267 | C22 | C22×#1141 | 1 | 1 | yes | characteristic |
| 268 | C4 | — | 8 | 3 | no | — |
| 269 | C4 | — | 8 | 1 | no | — |
| 270 | C4 | — | 28 | 2 | no | — |
| 271 | C4 | C2×#6383 | 1 | 6 | yes | — |
| 272 | C4 | C22×(C2×D7)⋊C4 | 1 | 1 | yes | characteristic |
| 273 | C4 | C22×#1143 | 1 | 1 | yes | characteristic |
| 274 | C2 | — | 2 | 6 | no | — |
| 275 | C2 | — | 56 | 6 | no | — |
| 276 | C2 | C22×#6383 | 1 | 1 | yes | characteristic |
| 277 | C2 | #11741452 | 1 | 1 | yes | characteristic |
| 278 | C2 | #11741461 | 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