← C7×#12816 #13190032
Subgroups — C7×#12816 #13190032
Order 1792.
446 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 | C4×C7×#135 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#2150 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×C7×#360 | C2 | 1 | 2 | yes | maximal |
| 4 | C7×#1844 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C7×#1386 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×C7×#135 | C22 | 1 | 1 | yes | characteristic |
| 7 | C4×C7×#41 | — | 2 | 1 | no | — |
| 8 | C2×C7×M32 | C22 | 1 | 2 | yes | — |
| 9 | C4×C7×Q16 | C4 | 1 | 2 | yes | complemented |
| 10 | C4×C7×SD16 | — | 2 | 2 | no | — |
| 11 | C4×C7×D8 | C4 | 1 | 2 | yes | complemented |
| 12 | C7×#427 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C7×#106 | — | 2 | 1 | no | — |
| 14 | C2×C7×#106 | C22 | 1 | 2 | yes | — |
| 15 | C7×#416 | — | 2 | 1 | no | — |
| 16 | C2×C4×C7×C8 | C22 | 1 | 1 | yes | characteristic |
| 17 | C7×#384 | — | 2 | 1 | no | — |
| 18 | C7×C4⋊C16 | — | 2 | 2 | no | — |
| 19 | C7×#360 | — | 2 | 4 | no | — |
| 20 | #12816 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 21 | C2×C7×#41 | — | 2 | 1 | no | — |
| 22 | C7×#135 | — | 2 | 1 | no | — |
| 23 | C7×#135 | — | 2 | 1 | no | — |
| 24 | C7×#135 | D4 | 1 | 2 | yes | — |
| 25 | C7×#135 | Q8 | 1 | 2 | yes | — |
| 26 | C2×C7×Q16 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | C2×C7×SD16 | — | 2 | 1 | no | — |
| 28 | C2×C7×D8 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C22×C7×C8 | D4 | 1 | 1 | yes | characteristic |
| 30 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 31 | C4×C7×Q8 | — | 2 | 2 | no | — |
| 32 | C4×C7×D4 | — | 2 | 2 | no | — |
| 33 | C4×C7×D4 | — | 4 | 2 | no | — |
| 34 | C7×#117 | — | 2 | 1 | no | — |
| 35 | C7×#117 | — | 4 | 1 | no | — |
| 36 | C2×C7×C4⋊C4 | — | 2 | 2 | no | — |
| 37 | C2×C7×C4⋊C4 | — | 4 | 2 | no | — |
| 38 | C2×C42×C7 | — | 4 | 1 | no | — |
| 39 | C2×C42×C7 | D4 | 1 | 1 | yes | characteristic |
| 40 | C7×M32 | — | 2 | 2 | no | — |
| 41 | C7×M32 | — | 2 | 2 | no | — |
| 42 | C2×C7×C16 | — | 2 | 2 | no | — |
| 43 | C7×#107 | C2×C4 | 1 | 2 | yes | — |
| 44 | C7×#106 | — | 2 | 1 | no | — |
| 45 | C7×#106 | — | 2 | 2 | no | — |
| 46 | C7×#106 | — | 2 | 2 | no | — |
| 47 | C7×#106 | — | 4 | 2 | no | — |
| 48 | C7×#103 | — | 2 | 2 | no | — |
| 49 | C7×#102 | — | 2 | 2 | no | — |
| 50 | C4×C7×C8 | C2×C4 | 1 | 2 | yes | — |
| 51 | C4×C7×C8 | D4 | 1 | 2 | yes | — |
| 52 | C4×#135 | C2×C7 | 1 | 1 | yes | characteristic |
| 53 | #2150 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 54 | C2×#360 | C2×C7 | 1 | 2 | yes | — |
| 55 | #1844 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 56 | #1386 | C2×C7 | 1 | 2 | yes | complemented |
| 57 | C7×#41 | — | 2 | 2 | no | — |
| 58 | C7×#41 | — | 2 | 2 | no | — |
| 59 | C7×#41 | — | 4 | 1 | no | — |
| 60 | C7×#41 | — | 4 | 1 | no | — |
| 61 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 62 | C2×C7×D4 | — | 2 | 1 | no | — |
| 63 | C2×C7×D4 | — | 4 | 1 | no | — |
| 64 | C22×C4×C7 | — | 4 | 1 | no | — |
| 65 | C22×C4×C7 | — | 4 | 2 | no | — |
| 66 | C22×C4×C7 | — | 4 | 2 | no | — |
| 67 | C22×C4×C7 | D8 | 1 | 2 | yes | — |
| 68 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | C7×Q16 | — | 2 | 1 | no | — |
| 70 | C7×Q16 | C4⋊C4 | 1 | 2 | yes | complemented |
| 71 | C7×SD16 | — | 2 | 2 | no | — |
| 72 | C7×SD16 | — | 2 | 2 | no | — |
| 73 | C7×D8 | — | 2 | 1 | no | — |
| 74 | C7×D8 | C4⋊C4 | 1 | 2 | yes | complemented |
| 75 | C2×C7×C8 | — | 2 | 1 | no | — |
| 76 | C2×C7×C8 | — | 2 | 1 | no | — |
| 77 | C2×C7×C8 | C22⋊C4 | 1 | 2 | yes | — |
| 78 | C2×C7×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | C2×C7×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 82 | C2×C7×C8 | #41 | 1 | 1 | yes | characteristic |
| 83 | C2×C7×C8 | #41 | 1 | 1 | yes | characteristic |
| 84 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 85 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 86 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 87 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 88 | C7×C4⋊C4 | — | 8 | 4 | no | — |
| 89 | C7×C22⋊C4 | — | 4 | 1 | no | — |
| 90 | C7×C22⋊C4 | — | 4 | 2 | no | — |
| 91 | C42×C7 | — | 4 | 2 | no | — |
| 92 | C42×C7 | — | 4 | 1 | no | — |
| 93 | C42×C7 | C22⋊C4 | 1 | 2 | yes | — |
| 94 | C42×C7 | SD16 | 1 | 1 | yes | characteristic |
| 95 | C42×C7 | SD16 | 1 | 1 | yes | characteristic |
| 96 | C7×C16 | — | 4 | 2 | no | — |
| 97 | C2×#135 | C22×C7 | 1 | 1 | yes | characteristic |
| 98 | C4×#41 | — | 2 | 1 | no | — |
| 99 | C2×M32 | C22×C7 | 1 | 2 | yes | — |
| 100 | C4×Q16 | C4×C7 | 1 | 2 | yes | complemented |
| 101 | C4×SD16 | — | 2 | 2 | no | — |
| 102 | C4×D8 | C4×C7 | 1 | 2 | yes | complemented |
| 103 | #427 | C22×C7 | 1 | 1 | yes | characteristic |
| 104 | C2×#106 | — | 2 | 1 | no | — |
| 105 | C2×#106 | C22×C7 | 1 | 2 | yes | — |
| 106 | #416 | — | 2 | 1 | no | — |
| 107 | C2×C4×C8 | C22×C7 | 1 | 1 | yes | characteristic |
| 108 | #384 | — | 2 | 1 | no | — |
| 109 | C4⋊C16 | — | 2 | 2 | no | — |
| 110 | #360 | — | 2 | 4 | no | — |
| 111 | C23×C7 | — | 4 | 1 | no | — |
| 112 | C23×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 113 | C7×Q8 | — | 2 | 2 | no | — |
| 114 | C7×Q8 | — | 4 | 1 | no | — |
| 115 | C7×D4 | — | 2 | 2 | no | — |
| 116 | C7×D4 | — | 4 | 2 | no | — |
| 117 | C7×D4 | — | 4 | 2 | no | — |
| 118 | C7×D4 | — | 4 | 1 | no | — |
| 119 | C2×C4×C7 | — | 2 | 4 | no | — |
| 120 | C2×C4×C7 | — | 4 | 1 | no | — |
| 121 | C2×C4×C7 | — | 4 | 2 | no | — |
| 122 | C2×C4×C7 | — | 4 | 2 | no | — |
| 123 | C2×C4×C7 | — | 4 | 2 | no | — |
| 124 | C2×C4×C7 | — | 8 | 4 | no | — |
| 125 | C2×C4×C7 | — | 8 | 4 | no | — |
| 126 | C2×C4×C7 | — | 8 | 2 | no | — |
| 127 | C2×C4×C7 | #102 | 1 | 2 | yes | — |
| 128 | C2×C4×C7 | #102 | 1 | 2 | yes | — |
| 129 | C2×C4×C7 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C7 | #121 | 1 | 1 | yes | characteristic |
| 131 | C2×C4×C7 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C7 | C2×SD16 | 1 | 1 | yes | characteristic |
| 133 | C2×C4×C7 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 134 | C2×C4×C7 | #137 | 1 | 1 | yes | characteristic |
| 135 | C7×C8 | — | 2 | 2 | no | — |
| 136 | C7×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 137 | C7×C8 | #117 | 1 | 1 | yes | characteristic |
| 138 | C7×C8 | #121 | 1 | 1 | yes | characteristic |
| 139 | C7×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 140 | C2×#41 | — | 2 | 1 | no | — |
| 141 | #135 | — | 2 | 1 | no | — |
| 142 | #135 | — | 2 | 1 | no | — |
| 143 | #135 | C7×D4 | 1 | 2 | yes | — |
| 144 | #135 | C7×Q8 | 1 | 2 | yes | — |
| 145 | C2×Q16 | C2×C4×C7 | 1 | 1 | yes | characteristic |
| 146 | C2×SD16 | — | 2 | 1 | no | — |
| 147 | C2×D8 | C2×C4×C7 | 1 | 1 | yes | characteristic |
| 148 | C22×C8 | C7×D4 | 1 | 1 | yes | characteristic |
| 149 | C22×C8 | C23×C7 | 1 | 1 | yes | characteristic |
| 150 | C4×Q8 | — | 2 | 2 | no | — |
| 151 | C4×D4 | — | 2 | 2 | no | — |
| 152 | C4×D4 | — | 4 | 2 | no | — |
| 153 | #117 | — | 2 | 1 | no | — |
| 154 | #117 | — | 4 | 1 | no | — |
| 155 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 156 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 157 | C2×C42 | — | 4 | 1 | no | — |
| 158 | C2×C42 | C7×D4 | 1 | 1 | yes | characteristic |
| 159 | M32 | — | 2 | 2 | no | — |
| 160 | M32 | — | 2 | 2 | no | — |
| 161 | C2×C16 | — | 2 | 2 | no | — |
| 162 | #107 | C2×C4×C7 | 1 | 2 | yes | — |
| 163 | #106 | — | 2 | 1 | no | — |
| 164 | #106 | — | 2 | 2 | no | — |
| 165 | #106 | — | 2 | 2 | no | — |
| 166 | #106 | — | 4 | 2 | no | — |
| 167 | #103 | — | 2 | 2 | no | — |
| 168 | #102 | — | 2 | 2 | no | — |
| 169 | C4×C8 | C2×C4×C7 | 1 | 2 | yes | — |
| 170 | C4×C8 | C7×D4 | 1 | 2 | yes | — |
| 171 | C22×C7 | — | 2 | 2 | no | — |
| 172 | C22×C7 | — | 4 | 2 | no | — |
| 173 | C22×C7 | — | 8 | 2 | no | — |
| 174 | C22×C7 | C2×#102 | 1 | 1 | yes | characteristic |
| 175 | C22×C7 | #459 | 1 | 1 | yes | characteristic |
| 176 | C22×C7 | #474 | 1 | 1 | yes | characteristic |
| 177 | C4×C7 | — | 2 | 2 | no | — |
| 178 | C4×C7 | — | 4 | 2 | no | — |
| 179 | C4×C7 | — | 8 | 2 | no | — |
| 180 | C4×C7 | — | 16 | 2 | no | — |
| 181 | C4×C7 | #360 | 1 | 4 | yes | complemented |
| 182 | C4×C7 | #380 | 1 | 1 | yes | characteristic |
| 183 | C4×C7 | #417 | 1 | 1 | yes | characteristic |
| 184 | C4×C7 | #461 | 1 | 1 | yes | characteristic |
| 185 | C4×C7 | #476 | 1 | 1 | yes | characteristic |
| 186 | #41 | — | 2 | 2 | no | — |
| 187 | #41 | — | 2 | 2 | no | — |
| 188 | #41 | — | 4 | 1 | no | — |
| 189 | #41 | — | 4 | 1 | no | — |
| 190 | C2×Q8 | — | 2 | 1 | no | — |
| 191 | C2×D4 | — | 2 | 1 | no | — |
| 192 | C2×D4 | — | 4 | 1 | no | — |
| 193 | C22×C4 | — | 4 | 1 | no | — |
| 194 | C22×C4 | — | 4 | 2 | no | — |
| 195 | C22×C4 | — | 4 | 2 | no | — |
| 196 | C22×C4 | C7×D8 | 1 | 2 | yes | — |
| 197 | C22×C4 | C2×C7×D4 | 1 | 1 | yes | characteristic |
| 198 | Q16 | — | 2 | 1 | no | — |
| 199 | Q16 | C7×C4⋊C4 | 1 | 2 | yes | complemented |
| 200 | SD16 | — | 2 | 2 | no | — |
| 201 | SD16 | — | 2 | 2 | no | — |
| 202 | D8 | — | 2 | 1 | no | — |
| 203 | D8 | C7×C4⋊C4 | 1 | 2 | yes | complemented |
| 204 | C2×C8 | — | 2 | 1 | no | — |
| 205 | C2×C8 | — | 2 | 1 | no | — |
| 206 | C2×C8 | C7×C22⋊C4 | 1 | 2 | yes | — |
| 207 | C2×C8 | C22×C4×C7 | 1 | 1 | yes | characteristic |
| 208 | C2×C8 | C2×C7×D4 | 1 | 1 | yes | characteristic |
| 209 | C2×C8 | C2×C7×D4 | 1 | 1 | yes | characteristic |
| 210 | C2×C8 | C2×C7×Q8 | 1 | 1 | yes | characteristic |
| 211 | C2×C8 | C7×#41 | 1 | 1 | yes | characteristic |
| 212 | C2×C8 | C7×#41 | 1 | 1 | yes | characteristic |
| 213 | C4⋊C4 | — | 2 | 2 | no | — |
| 214 | C4⋊C4 | — | 4 | 2 | no | — |
| 215 | C4⋊C4 | — | 4 | 2 | no | — |
| 216 | C4⋊C4 | — | 4 | 2 | no | — |
| 217 | C4⋊C4 | — | 8 | 4 | no | — |
| 218 | C22⋊C4 | — | 4 | 1 | no | — |
| 219 | C22⋊C4 | — | 4 | 2 | no | — |
| 220 | C42 | — | 4 | 2 | no | — |
| 221 | C42 | — | 4 | 1 | no | — |
| 222 | C42 | C7×C22⋊C4 | 1 | 2 | yes | — |
| 223 | C42 | C7×SD16 | 1 | 1 | yes | characteristic |
| 224 | C42 | C7×SD16 | 1 | 1 | yes | characteristic |
| 225 | C16 | — | 4 | 2 | no | — |
| 226 | C2×C7 | — | 2 | 1 | no | — |
| 227 | C2×C7 | — | 2 | 1 | no | — |
| 228 | C2×C7 | — | 8 | 2 | no | — |
| 229 | C2×C7 | #1802 | 1 | 1 | yes | characteristic |
| 230 | C2×C7 | C2×#360 | 1 | 2 | yes | — |
| 231 | C23 | — | 4 | 1 | no | — |
| 232 | C23 | C2×C7×D8 | 1 | 1 | yes | characteristic |
| 233 | Q8 | — | 2 | 2 | no | — |
| 234 | Q8 | — | 4 | 1 | no | — |
| 235 | D4 | — | 2 | 2 | no | — |
| 236 | D4 | — | 4 | 2 | no | — |
| 237 | D4 | — | 4 | 2 | no | — |
| 238 | D4 | — | 4 | 1 | no | — |
| 239 | C2×C4 | — | 2 | 4 | no | — |
| 240 | C2×C4 | — | 4 | 1 | no | — |
| 241 | C2×C4 | — | 4 | 2 | no | — |
| 242 | C2×C4 | — | 4 | 2 | no | — |
| 243 | C2×C4 | — | 4 | 2 | no | — |
| 244 | C2×C4 | — | 8 | 4 | no | — |
| 245 | C2×C4 | — | 8 | 4 | no | — |
| 246 | C2×C4 | — | 8 | 2 | no | — |
| 247 | C2×C4 | C7×#102 | 1 | 2 | yes | — |
| 248 | C2×C4 | C7×#102 | 1 | 2 | yes | — |
| 249 | C2×C4 | C2×C7×C22⋊C4 | 1 | 1 | yes | characteristic |
| 250 | C2×C4 | C7×#121 | 1 | 1 | yes | characteristic |
| 251 | C2×C4 | C7×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 252 | C2×C4 | C2×C7×SD16 | 1 | 1 | yes | characteristic |
| 253 | C2×C4 | C7×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 254 | C2×C4 | C7×#137 | 1 | 1 | yes | characteristic |
| 255 | C8 | — | 2 | 2 | no | — |
| 256 | C8 | C2×C7×C4⋊C4 | 1 | 1 | yes | characteristic |
| 257 | C8 | C7×#117 | 1 | 1 | yes | characteristic |
| 258 | C8 | C7×#121 | 1 | 1 | yes | characteristic |
| 259 | C8 | C7×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 260 | C7 | #12816 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 261 | C22 | — | 2 | 2 | no | — |
| 262 | C22 | — | 4 | 2 | no | — |
| 263 | C22 | — | 8 | 2 | no | — |
| 264 | C22 | C2×C7×#102 | 1 | 1 | yes | characteristic |
| 265 | C22 | C7×#459 | 1 | 1 | yes | characteristic |
| 266 | C22 | C7×#474 | 1 | 1 | yes | characteristic |
| 267 | C4 | — | 2 | 2 | no | — |
| 268 | C4 | — | 4 | 2 | no | — |
| 269 | C4 | — | 8 | 2 | no | — |
| 270 | C4 | — | 16 | 2 | no | — |
| 271 | C4 | C7×#360 | 1 | 4 | yes | complemented |
| 272 | C4 | C7×#380 | 1 | 1 | yes | characteristic |
| 273 | C4 | C7×#417 | 1 | 1 | yes | characteristic |
| 274 | C4 | C7×#461 | 1 | 1 | yes | characteristic |
| 275 | C4 | C7×#476 | 1 | 1 | yes | characteristic |
| 276 | C2 | — | 2 | 1 | no | — |
| 277 | C2 | — | 2 | 1 | no | — |
| 278 | C2 | — | 8 | 2 | no | — |
| 279 | C2 | C7×#1802 | 1 | 1 | yes | characteristic |
| 280 | C2 | C2×C7×#360 | 1 | 2 | yes | — |
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