← C7×#10555 #13187771
Subgroups — C7×#10555 #13187771
Order 1792.
364 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C7×#3165 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#1950 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C7×#1550 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C7×#1549 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×C7×#333 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C7×#1481 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C7×#1283 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C7×#533 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C7×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 10 | C7×#501 | — | 2 | 2 | no | — |
| 11 | C7×#500 | — | 2 | 1 | no | — |
| 12 | C7×#498 | — | 2 | 1 | no | — |
| 13 | C7×#431 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×C7×#108 | — | 2 | 1 | no | — |
| 15 | C7×#428 | — | 2 | 1 | no | — |
| 16 | C7×#428 | — | 2 | 1 | no | — |
| 17 | C7×#423 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×C7×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×C7×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 20 | C4×C7×M16 | C22 | 1 | 1 | yes | characteristic |
| 21 | C7×#341 | — | 2 | 1 | no | — |
| 22 | C7×#333 | — | 2 | 1 | no | — |
| 23 | C7×#333 | — | 2 | 2 | no | — |
| 24 | C7×#332 | — | 2 | 2 | no | — |
| 25 | C7×#332 | — | 2 | 1 | no | — |
| 26 | C7×Q8⋊C8 | — | 2 | 2 | no | — |
| 27 | #10555 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 28 | C22×C7×Q8 | — | 2 | 1 | no | — |
| 29 | C2×C7×M16 | — | 2 | 1 | no | — |
| 30 | C2×C7×M16 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×C7×M16 | D4 | 1 | 1 | yes | characteristic |
| 32 | C22×C7×C8 | — | 2 | 1 | no | — |
| 33 | C22×C7×C8 | — | 2 | 1 | no | — |
| 34 | C7×C4⋊Q8 | — | 2 | 1 | no | — |
| 35 | C7×C4⋊Q8 | — | 2 | 1 | no | — |
| 36 | C7×C4⋊Q8 | — | 2 | 1 | no | — |
| 37 | C7×C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 38 | C7×C4⋊Q8 | D4 | 1 | 2 | yes | — |
| 39 | C7×C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 40 | C7×#125 | — | 2 | 1 | no | — |
| 41 | C7×C22⋊Q8 | — | 2 | 1 | no | — |
| 42 | C7×C22⋊Q8 | — | 2 | 1 | no | — |
| 43 | C4×C7×Q8 | — | 2 | 1 | no | — |
| 44 | C4×C7×Q8 | — | 2 | 1 | no | — |
| 45 | C7×#117 | — | 2 | 1 | no | — |
| 46 | C2×C7×C4⋊C4 | — | 2 | 1 | no | — |
| 47 | C2×C7×C4⋊C4 | — | 2 | 1 | no | — |
| 48 | C2×C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 49 | C7×#108 | — | 4 | 2 | no | — |
| 50 | C7×#107 | — | 2 | 2 | no | — |
| 51 | C7×#107 | — | 2 | 2 | no | — |
| 52 | C7×#106 | — | 2 | 2 | no | — |
| 53 | C7×#106 | — | 2 | 2 | no | — |
| 54 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 55 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 56 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 57 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 58 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 59 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 60 | C7×C4⋊C8 | D4 | 1 | 2 | yes | — |
| 61 | C7×C22⋊C8 | — | 2 | 1 | no | — |
| 62 | C7×#97 | — | 2 | 1 | no | — |
| 63 | C4×C7×C8 | — | 2 | 1 | no | — |
| 64 | C4×C7×C8 | — | 2 | 1 | no | — |
| 65 | #3165 | C2×C7 | 1 | 1 | yes | characteristic |
| 66 | #1950 | C2×C7 | 1 | 1 | yes | characteristic |
| 67 | #1550 | C2×C7 | 1 | 1 | yes | characteristic |
| 68 | #1549 | C2×C7 | 1 | 1 | yes | characteristic |
| 69 | C2×#333 | C2×C7 | 1 | 1 | yes | characteristic |
| 70 | #1481 | C2×C7 | 1 | 1 | yes | characteristic |
| 71 | #1283 | C2×C7 | 1 | 1 | yes | characteristic |
| 72 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 73 | C2×C7×Q8 | — | 2 | 2 | no | — |
| 74 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 75 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 76 | C2×C7×Q8 | — | 8 | 2 | no | — |
| 77 | C22×C4×C7 | — | 4 | 1 | no | — |
| 78 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | C22×C4×C7 | #41 | 1 | 1 | yes | characteristic |
| 80 | C22×C4×C7 | #41 | 1 | 1 | yes | characteristic |
| 81 | C7×M16 | — | 2 | 1 | no | — |
| 82 | C7×M16 | — | 2 | 1 | no | — |
| 83 | C7×M16 | — | 4 | 1 | no | — |
| 84 | C7×M16 | — | 4 | 1 | no | — |
| 85 | C7×M16 | SD16 | 1 | 2 | yes | — |
| 86 | C7×M16 | Q16 | 1 | 2 | yes | — |
| 87 | C2×C7×C8 | — | 2 | 1 | no | — |
| 88 | C2×C7×C8 | — | 2 | 1 | no | — |
| 89 | C2×C7×C8 | — | 2 | 2 | no | — |
| 90 | C2×C7×C8 | — | 4 | 1 | no | — |
| 91 | C2×C7×C8 | — | 4 | 1 | no | — |
| 92 | C2×C7×C8 | — | 4 | 1 | no | — |
| 93 | C2×C7×C8 | — | 4 | 1 | no | — |
| 94 | C2×C7×C8 | — | 4 | 1 | no | — |
| 95 | C2×C7×C8 | — | 8 | 1 | no | — |
| 96 | C7×C4⋊C4 | — | 2 | 1 | no | — |
| 97 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 98 | C7×C4⋊C4 | — | 2 | 1 | no | — |
| 99 | C7×C4⋊C4 | — | 2 | 1 | no | — |
| 100 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 101 | C7×C4⋊C4 | — | 2 | 1 | no | — |
| 102 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 103 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 104 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 105 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 106 | C7×C22⋊C4 | — | 4 | 1 | no | — |
| 107 | C42×C7 | — | 4 | 1 | no | — |
| 108 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C42×C7 | #41 | 1 | 1 | yes | characteristic |
| 112 | #533 | C22×C7 | 1 | 1 | yes | characteristic |
| 113 | C2×C4⋊Q8 | C22×C7 | 1 | 1 | yes | characteristic |
| 114 | #501 | — | 2 | 2 | no | — |
| 115 | #500 | — | 2 | 1 | no | — |
| 116 | #498 | — | 2 | 1 | no | — |
| 117 | #431 | C22×C7 | 1 | 1 | yes | characteristic |
| 118 | C2×#108 | — | 2 | 1 | no | — |
| 119 | #428 | — | 2 | 1 | no | — |
| 120 | #428 | — | 2 | 1 | no | — |
| 121 | #423 | C22×C7 | 1 | 1 | yes | characteristic |
| 122 | C2×C4⋊C8 | C22×C7 | 1 | 1 | yes | characteristic |
| 123 | C2×C4⋊C8 | C22×C7 | 1 | 1 | yes | characteristic |
| 124 | C4×M16 | C22×C7 | 1 | 1 | yes | characteristic |
| 125 | #341 | — | 2 | 1 | no | — |
| 126 | #333 | — | 2 | 1 | no | — |
| 127 | #333 | — | 2 | 2 | no | — |
| 128 | #332 | — | 2 | 2 | no | — |
| 129 | #332 | — | 2 | 1 | no | — |
| 130 | Q8⋊C8 | — | 2 | 2 | no | — |
| 131 | C23×C7 | #124 | 1 | 1 | yes | characteristic |
| 132 | C7×Q8 | — | 4 | 1 | no | — |
| 133 | C7×Q8 | — | 4 | 1 | no | — |
| 134 | C7×Q8 | — | 8 | 1 | no | — |
| 135 | C7×Q8 | — | 8 | 2 | no | — |
| 136 | C7×Q8 | — | 8 | 2 | no | — |
| 137 | C2×C4×C7 | — | 2 | 1 | no | — |
| 138 | C2×C4×C7 | — | 2 | 1 | no | — |
| 139 | C2×C4×C7 | — | 4 | 1 | no | — |
| 140 | C2×C4×C7 | — | 4 | 1 | no | — |
| 141 | C2×C4×C7 | — | 4 | 1 | no | — |
| 142 | C2×C4×C7 | — | 4 | 2 | no | — |
| 143 | C2×C4×C7 | — | 8 | 2 | no | — |
| 144 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 145 | C2×C4×C7 | #121 | 1 | 1 | yes | characteristic |
| 146 | C2×C4×C7 | #121 | 1 | 1 | yes | characteristic |
| 147 | C2×C4×C7 | #121 | 1 | 1 | yes | characteristic |
| 148 | C2×C4×C7 | #123 | 1 | 1 | yes | characteristic |
| 149 | C2×C4×C7 | #123 | 1 | 1 | yes | characteristic |
| 150 | C2×C4×C7 | C2×SD16 | 1 | 1 | yes | characteristic |
| 151 | C2×C4×C7 | C2×Q16 | 1 | 1 | yes | characteristic |
| 152 | C2×C4×C7 | #137 | 1 | 1 | yes | characteristic |
| 153 | C2×C4×C7 | #137 | 1 | 1 | yes | characteristic |
| 154 | C7×C8 | — | 2 | 2 | no | — |
| 155 | C7×C8 | — | 2 | 2 | no | — |
| 156 | C7×C8 | — | 4 | 2 | no | — |
| 157 | C7×C8 | — | 8 | 1 | no | — |
| 158 | C7×C8 | — | 8 | 1 | no | — |
| 159 | C7×C8 | — | 8 | 1 | no | — |
| 160 | C22×Q8 | — | 2 | 1 | no | — |
| 161 | C2×M16 | — | 2 | 1 | no | — |
| 162 | C2×M16 | C7×D4 | 1 | 1 | yes | characteristic |
| 163 | C2×M16 | C7×D4 | 1 | 1 | yes | characteristic |
| 164 | C22×C8 | — | 2 | 1 | no | — |
| 165 | C22×C8 | — | 2 | 1 | no | — |
| 166 | C4⋊Q8 | — | 2 | 1 | no | — |
| 167 | C4⋊Q8 | — | 2 | 1 | no | — |
| 168 | C4⋊Q8 | — | 2 | 1 | no | — |
| 169 | C4⋊Q8 | C7×D4 | 1 | 2 | yes | — |
| 170 | C4⋊Q8 | C7×D4 | 1 | 1 | yes | characteristic |
| 171 | C4⋊Q8 | C7×D4 | 1 | 1 | yes | characteristic |
| 172 | #125 | — | 2 | 1 | no | — |
| 173 | C22⋊Q8 | — | 2 | 1 | no | — |
| 174 | C22⋊Q8 | — | 2 | 1 | no | — |
| 175 | C4×Q8 | — | 2 | 1 | no | — |
| 176 | C4×Q8 | — | 2 | 1 | no | — |
| 177 | #117 | — | 2 | 1 | no | — |
| 178 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 179 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 180 | C2×C42 | C23×C7 | 1 | 1 | yes | characteristic |
| 181 | #108 | — | 4 | 2 | no | — |
| 182 | #107 | — | 2 | 2 | no | — |
| 183 | #107 | — | 2 | 2 | no | — |
| 184 | #106 | — | 2 | 2 | no | — |
| 185 | #106 | — | 2 | 2 | no | — |
| 186 | C4⋊C8 | — | 2 | 1 | no | — |
| 187 | C4⋊C8 | — | 2 | 1 | no | — |
| 188 | C4⋊C8 | — | 2 | 1 | no | — |
| 189 | C4⋊C8 | — | 2 | 1 | no | — |
| 190 | C4⋊C8 | — | 2 | 1 | no | — |
| 191 | C4⋊C8 | — | 2 | 1 | no | — |
| 192 | C4⋊C8 | C7×D4 | 1 | 2 | yes | — |
| 193 | C22⋊C8 | — | 2 | 1 | no | — |
| 194 | #97 | — | 2 | 1 | no | — |
| 195 | C4×C8 | — | 2 | 1 | no | — |
| 196 | C4×C8 | — | 2 | 1 | no | — |
| 197 | C22×C7 | — | 2 | 1 | no | — |
| 198 | C22×C7 | — | 2 | 1 | no | — |
| 199 | C22×C7 | #394 | 1 | 1 | yes | characteristic |
| 200 | C22×C7 | #487 | 1 | 1 | yes | characteristic |
| 201 | C22×C7 | #491 | 1 | 1 | yes | characteristic |
| 202 | C4×C7 | — | 2 | 1 | no | — |
| 203 | C4×C7 | — | 2 | 1 | no | — |
| 204 | C4×C7 | — | 2 | 1 | no | — |
| 205 | C4×C7 | — | 8 | 2 | no | — |
| 206 | C4×C7 | — | 8 | 1 | no | — |
| 207 | C4×C7 | — | 8 | 1 | no | — |
| 208 | C4×C7 | #448 | 1 | 1 | yes | characteristic |
| 209 | C4×C7 | #451 | 1 | 1 | yes | characteristic |
| 210 | C4×C7 | #461 | 1 | 1 | yes | characteristic |
| 211 | C4×C7 | #462 | 1 | 1 | yes | characteristic |
| 212 | C4×C7 | #471 | 1 | 1 | yes | characteristic |
| 213 | C4×C7 | #473 | 1 | 1 | yes | characteristic |
| 214 | C2×Q8 | — | 2 | 1 | no | — |
| 215 | C2×Q8 | — | 2 | 2 | no | — |
| 216 | C2×Q8 | — | 2 | 1 | no | — |
| 217 | C2×Q8 | — | 4 | 1 | no | — |
| 218 | C2×Q8 | — | 8 | 2 | no | — |
| 219 | C22×C4 | — | 4 | 1 | no | — |
| 220 | C22×C4 | C2×C7×D4 | 1 | 1 | yes | characteristic |
| 221 | C22×C4 | C7×#41 | 1 | 1 | yes | characteristic |
| 222 | C22×C4 | C7×#41 | 1 | 1 | yes | characteristic |
| 223 | M16 | — | 2 | 1 | no | — |
| 224 | M16 | — | 2 | 1 | no | — |
| 225 | M16 | — | 4 | 1 | no | — |
| 226 | M16 | — | 4 | 1 | no | — |
| 227 | M16 | C7×SD16 | 1 | 2 | yes | — |
| 228 | M16 | C7×Q16 | 1 | 2 | yes | — |
| 229 | C2×C8 | — | 2 | 1 | no | — |
| 230 | C2×C8 | — | 2 | 1 | no | — |
| 231 | C2×C8 | — | 2 | 2 | no | — |
| 232 | C2×C8 | — | 4 | 1 | no | — |
| 233 | C2×C8 | — | 4 | 1 | no | — |
| 234 | C2×C8 | — | 4 | 1 | no | — |
| 235 | C2×C8 | — | 4 | 1 | no | — |
| 236 | C2×C8 | — | 4 | 1 | no | — |
| 237 | C2×C8 | — | 8 | 1 | no | — |
| 238 | C4⋊C4 | — | 2 | 1 | no | — |
| 239 | C4⋊C4 | — | 2 | 2 | no | — |
| 240 | C4⋊C4 | — | 2 | 1 | no | — |
| 241 | C4⋊C4 | — | 2 | 1 | no | — |
| 242 | C4⋊C4 | — | 2 | 2 | no | — |
| 243 | C4⋊C4 | — | 2 | 1 | no | — |
| 244 | C4⋊C4 | — | 4 | 1 | no | — |
| 245 | C4⋊C4 | — | 4 | 1 | no | — |
| 246 | C4⋊C4 | — | 4 | 1 | no | — |
| 247 | C4⋊C4 | — | 4 | 1 | no | — |
| 248 | C22⋊C4 | — | 4 | 1 | no | — |
| 249 | C42 | — | 4 | 1 | no | — |
| 250 | C42 | C2×C7×D4 | 1 | 1 | yes | characteristic |
| 251 | C42 | C2×C7×D4 | 1 | 1 | yes | characteristic |
| 252 | C42 | C2×C7×D4 | 1 | 1 | yes | characteristic |
| 253 | C42 | C7×#41 | 1 | 1 | yes | characteristic |
| 254 | C2×C7 | — | 2 | 1 | no | — |
| 255 | C2×C7 | — | 2 | 1 | no | — |
| 256 | C2×C7 | #2043 | 1 | 1 | yes | characteristic |
| 257 | C2×C7 | #2048 | 1 | 1 | yes | characteristic |
| 258 | C2×C7 | #2052 | 1 | 1 | yes | characteristic |
| 259 | C23 | C7×#124 | 1 | 1 | yes | characteristic |
| 260 | Q8 | — | 4 | 1 | no | — |
| 261 | Q8 | — | 4 | 1 | no | — |
| 262 | Q8 | — | 8 | 1 | no | — |
| 263 | Q8 | — | 8 | 2 | no | — |
| 264 | Q8 | — | 8 | 2 | no | — |
| 265 | C2×C4 | — | 2 | 1 | no | — |
| 266 | C2×C4 | — | 2 | 1 | no | — |
| 267 | C2×C4 | — | 4 | 1 | no | — |
| 268 | C2×C4 | — | 4 | 1 | no | — |
| 269 | C2×C4 | — | 4 | 1 | no | — |
| 270 | C2×C4 | — | 4 | 2 | no | — |
| 271 | C2×C4 | — | 8 | 2 | no | — |
| 272 | C2×C4 | C7×#120 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | C7×#121 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | C7×#121 | 1 | 1 | yes | characteristic |
| 275 | C2×C4 | C7×#121 | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | C7×#123 | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | C7×#123 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | C2×C7×SD16 | 1 | 1 | yes | characteristic |
| 279 | C2×C4 | C2×C7×Q16 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | C7×#137 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | C7×#137 | 1 | 1 | yes | characteristic |
| 282 | C8 | — | 2 | 2 | no | — |
| 283 | C8 | — | 2 | 2 | no | — |
| 284 | C8 | — | 4 | 2 | no | — |
| 285 | C8 | — | 8 | 1 | no | — |
| 286 | C8 | — | 8 | 1 | no | — |
| 287 | C8 | — | 8 | 1 | no | — |
| 288 | C7 | #10555 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 289 | C22 | — | 2 | 1 | no | — |
| 290 | C22 | — | 2 | 1 | no | — |
| 291 | C22 | C7×#394 | 1 | 1 | yes | characteristic |
| 292 | C22 | C7×#487 | 1 | 1 | yes | characteristic |
| 293 | C22 | C7×#491 | 1 | 1 | yes | characteristic |
| 294 | C4 | — | 2 | 1 | no | — |
| 295 | C4 | — | 2 | 1 | no | — |
| 296 | C4 | — | 2 | 1 | no | — |
| 297 | C4 | — | 8 | 2 | no | — |
| 298 | C4 | — | 8 | 1 | no | — |
| 299 | C4 | — | 8 | 1 | no | — |
| 300 | C4 | C7×#448 | 1 | 1 | yes | characteristic |
| 301 | C4 | C7×#451 | 1 | 1 | yes | characteristic |
| 302 | C4 | C7×#461 | 1 | 1 | yes | characteristic |
| 303 | C4 | C7×#462 | 1 | 1 | yes | characteristic |
| 304 | C4 | C7×#471 | 1 | 1 | yes | characteristic |
| 305 | C4 | C7×#473 | 1 | 1 | yes | characteristic |
| 306 | C2 | — | 2 | 1 | no | — |
| 307 | C2 | — | 2 | 1 | no | — |
| 308 | C2 | C7×#2043 | 1 | 1 | yes | characteristic |
| 309 | C2 | C7×#2048 | 1 | 1 | yes | characteristic |
| 310 | C2 | C7×#2052 | 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