Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
442 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D7×#492 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | D7×#364 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C7×#1359 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11725593 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11725376 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×D7×SD16 | — | 2 | 2 | no | — |
| 7 | D7×C4⋊Q8 | C4 | 1 | 1 | yes | characteristic |
| 8 | D7×#127 | C4 | 1 | 1 | yes | characteristic |
| 9 | C7×#492 | C22 | 1 | 1 | yes | characteristic |
| 10 | #89947 | — | 2 | 2 | no | — |
| 11 | #89857 | C4 | 1 | 1 | yes | characteristic |
| 12 | #89848 | C4 | 1 | 1 | yes | characteristic |
| 13 | D7×M32 | — | 2 | 2 | no | — |
| 14 | #89465 | C22 | 1 | 1 | yes | characteristic |
| 15 | C4×C8×D7 | C22 | 1 | 1 | yes | characteristic |
| 16 | C7×#364 | C22 | 1 | 2 | yes | complemented |
| 17 | #89306 | — | 2 | 2 | no | — |
| 18 | #89258 | C22 | 1 | 2 | yes | complemented |
| 19 | C2×#1359 | — | 7 | 1 | no | maximal, Sylow |
| 20 | C2×Q8×D7 | — | 2 | 1 | no | — |
| 21 | C2×D4×D7 | — | 2 | 1 | no | — |
| 22 | C2×D4×D7 | — | 4 | 1 | no | — |
| 23 | C2×C7×SD16 | — | 2 | 2 | no | — |
| 24 | C7×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C7×#127 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | #6507 | — | 2 | 1 | no | — |
| 27 | C2×#1143 | — | 2 | 2 | no | — |
| 28 | #6503 | — | 2 | 1 | no | — |
| 29 | #6503 | — | 4 | 1 | no | — |
| 30 | C2×#1142 | — | 2 | 2 | no | — |
| 31 | D7×SD16 | — | 4 | 2 | no | — |
| 32 | D7×SD16 | — | 4 | 2 | no | — |
| 33 | D7×SD16 | — | 4 | 2 | no | — |
| 34 | D7×SD16 | — | 4 | 2 | no | — |
| 35 | C2×#1132 | — | 2 | 2 | no | — |
| 36 | C2×C8×D7 | D4 | 1 | 2 | yes | — |
| 37 | D7×C4⋊C4 | — | 4 | 1 | no | — |
| 38 | #6451 | — | 4 | 1 | no | — |
| 39 | #6437 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C42×D7 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | #6432 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C7×M32 | — | 2 | 2 | no | — |
| 43 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 44 | C8×Dic7 | D4 | 1 | 2 | yes | — |
| 45 | #6386 | — | 2 | 2 | no | — |
| 46 | C4×C7⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | #6372 | — | 4 | 2 | no | — |
| 48 | D7×C16 | — | 4 | 2 | no | — |
| 49 | C2×#492 | — | 7 | 1 | no | — |
| 50 | C2×#364 | — | 7 | 2 | no | — |
| 51 | #1359 | — | 7 | 1 | no | — |
| 52 | #1359 | — | 7 | 2 | no | — |
| 53 | #1359 | D7 | 1 | 1 | yes | characteristic, direct product factor |
| 54 | C23×D7 | — | 4 | 1 | no | — |
| 55 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 56 | C2×C7×D4 | — | 2 | 1 | no | — |
| 57 | C2×C7×D4 | — | 4 | 1 | no | — |
| 58 | C2×C22⋊D7 | — | 4 | 2 | no | — |
| 59 | C2×C22⋊D7 | — | 4 | 1 | no | — |
| 60 | Q8×D7 | — | 4 | 2 | no | — |
| 61 | Q8×D7 | — | 4 | 2 | no | — |
| 62 | D4×D7 | — | 4 | 2 | no | — |
| 63 | D4×D7 | — | 4 | 2 | no | — |
| 64 | D4×D7 | — | 8 | 4 | no | — |
| 65 | C2×D28 | — | 2 | 1 | no | — |
| 66 | C2×D28 | — | 4 | 1 | no | — |
| 67 | C2×C4×D7 | — | 2 | 1 | no | — |
| 68 | C2×C4×D7 | — | 4 | 1 | no | — |
| 69 | C2×C4×D7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 70 | C2×Dic14 | — | 2 | 1 | no | — |
| 71 | C2×Dic14 | — | 4 | 1 | no | — |
| 72 | C7×SD16 | — | 4 | 2 | no | — |
| 73 | C7×SD16 | — | 4 | 2 | no | — |
| 74 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 75 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 76 | C42×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | #1143 | — | 4 | 2 | no | — |
| 78 | #1143 | — | 4 | 2 | no | — |
| 79 | #1142 | — | 4 | 2 | no | — |
| 80 | #1142 | — | 4 | 2 | no | — |
| 81 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 82 | Dic7⋊C4 | — | 4 | 2 | no | — |
| 83 | C4×Dic7 | — | 2 | 1 | no | — |
| 84 | C4×Dic7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 85 | C2×C7⋊C8 | C22⋊C4 | 1 | 2 | yes | — |
| 86 | #1132 | — | 4 | 2 | no | — |
| 87 | #1132 | — | 4 | 2 | no | — |
| 88 | C8×D7 | D8 | 1 | 2 | yes | — |
| 89 | C8×D7 | D8 | 1 | 2 | yes | — |
| 90 | C8×D7 | SD16 | 1 | 2 | yes | — |
| 91 | C8×D7 | SD16 | 1 | 2 | yes | — |
| 92 | C7×C16 | — | 4 | 2 | no | — |
| 93 | C7⋊C16 | — | 4 | 2 | no | — |
| 94 | C22×SD16 | — | 14 | 2 | no | — |
| 95 | C2×C4⋊Q8 | — | 7 | 1 | no | — |
| 96 | C2×#127 | — | 7 | 1 | no | — |
| 97 | C2×M32 | — | 14 | 2 | no | — |
| 98 | #492 | — | 7 | 1 | no | — |
| 99 | #492 | — | 7 | 1 | no | — |
| 100 | #492 | — | 7 | 1 | no | — |
| 101 | #492 | — | 14 | 2 | no | — |
| 102 | #492 | C2×D7 | 1 | 1 | yes | characteristic |
| 103 | C2×C4×C8 | — | 7 | 1 | no | — |
| 104 | #364 | — | 7 | 2 | no | — |
| 105 | #364 | — | 14 | 2 | no | — |
| 106 | #364 | C2×D7 | 1 | 2 | yes | complemented |
| 107 | C23×C7 | — | 4 | 1 | no | — |
| 108 | C22×D7 | — | 4 | 1 | no | — |
| 109 | C22×D7 | — | 8 | 4 | no | — |
| 110 | C22×D7 | — | 8 | 1 | no | — |
| 111 | C22×D7 | — | 8 | 1 | no | — |
| 112 | C22×D7 | #100 | 1 | 1 | yes | characteristic |
| 113 | C7×Q8 | — | 4 | 2 | no | — |
| 114 | C7×D4 | — | 4 | 2 | no | — |
| 115 | C7×D4 | — | 8 | 2 | no | — |
| 116 | C2×C4×C7 | — | 2 | 1 | no | — |
| 117 | C2×C4×C7 | — | 4 | 1 | no | — |
| 118 | C2×C4×C7 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 119 | C22⋊D7 | — | 8 | 4 | no | — |
| 120 | C22⋊D7 | — | 8 | 2 | no | — |
| 121 | C2×Dic7 | — | 2 | 1 | no | — |
| 122 | C2×Dic7 | — | 4 | 1 | no | — |
| 123 | C2×Dic7 | #100 | 1 | 1 | yes | characteristic |
| 124 | D28 | — | 4 | 2 | no | — |
| 125 | D28 | — | 8 | 2 | no | — |
| 126 | C4×D7 | — | 2 | 4 | no | — |
| 127 | C4×D7 | — | 8 | 1 | no | — |
| 128 | C4×D7 | — | 8 | 1 | no | — |
| 129 | C4×D7 | #102 | 1 | 2 | yes | — |
| 130 | C4×D7 | #102 | 1 | 2 | yes | — |
| 131 | Dic14 | — | 4 | 2 | no | — |
| 132 | Dic14 | — | 8 | 2 | no | — |
| 133 | C7×C8 | C2×D8 | 1 | 2 | yes | — |
| 134 | C7×C8 | C2×SD16 | 1 | 2 | yes | — |
| 135 | C7⋊C8 | #102 | 1 | 2 | yes | — |
| 136 | C7⋊C8 | #102 | 1 | 2 | yes | — |
| 137 | C22×Q8 | — | 14 | 1 | no | — |
| 138 | C22×D4 | — | 14 | 1 | no | — |
| 139 | C22×D4 | — | 28 | 1 | no | — |
| 140 | C2×SD16 | — | 2 | 2 | no | — |
| 141 | C2×SD16 | — | 14 | 2 | no | — |
| 142 | C2×SD16 | — | 14 | 2 | no | — |
| 143 | C2×SD16 | — | 14 | 2 | no | — |
| 144 | C2×SD16 | — | 28 | 2 | no | — |
| 145 | C2×SD16 | — | 28 | 2 | no | — |
| 146 | C2×SD16 | — | 28 | 2 | no | — |
| 147 | C2×SD16 | — | 28 | 2 | no | — |
| 148 | C22×C8 | — | 7 | 2 | no | — |
| 149 | C4⋊Q8 | — | 7 | 1 | no | — |
| 150 | C4⋊Q8 | — | 14 | 1 | no | — |
| 151 | C4⋊Q8 | — | 28 | 1 | no | — |
| 152 | C4⋊Q8 | C4×D7 | 1 | 1 | yes | characteristic |
| 153 | #127 | — | 7 | 1 | no | — |
| 154 | #127 | — | 14 | 1 | no | — |
| 155 | #127 | — | 28 | 1 | no | — |
| 156 | #127 | C4×D7 | 1 | 1 | yes | characteristic |
| 157 | C2×C4⋊C4 | — | 28 | 1 | no | — |
| 158 | C2×C42 | — | 7 | 1 | no | — |
| 159 | M32 | — | 2 | 2 | no | — |
| 160 | M32 | — | 14 | 2 | no | — |
| 161 | M32 | — | 28 | 2 | no | — |
| 162 | C2×C16 | — | 28 | 2 | no | — |
| 163 | C4×C8 | — | 7 | 2 | no | — |
| 164 | C4×C8 | — | 7 | 1 | no | — |
| 165 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 166 | C22×C7 | — | 8 | 2 | no | — |
| 167 | C22×C7 | — | 8 | 1 | no | — |
| 168 | C22×C7 | C2×#100 | 1 | 1 | yes | characteristic |
| 169 | C2×D7 | — | 2 | 2 | no | — |
| 170 | C2×D7 | — | 8 | 2 | no | — |
| 171 | C2×D7 | — | 8 | 1 | no | — |
| 172 | C2×D7 | — | 16 | 2 | no | — |
| 173 | C2×D7 | — | 16 | 2 | no | — |
| 174 | C2×D7 | #331 | 1 | 1 | yes | characteristic |
| 175 | C2×D7 | #331 | 1 | 1 | yes | characteristic |
| 176 | C4×C7 | — | 2 | 2 | no | — |
| 177 | C4×C7 | — | 8 | 1 | no | — |
| 178 | C4×C7 | C2×#102 | 1 | 2 | yes | — |
| 179 | Dic7 | — | 2 | 2 | no | — |
| 180 | Dic7 | — | 8 | 1 | no | — |
| 181 | Dic7 | #331 | 1 | 2 | yes | — |
| 182 | C24 | — | 28 | 1 | no | — |
| 183 | C2×Q8 | — | 2 | 1 | no | — |
| 184 | C2×Q8 | — | 14 | 1 | no | — |
| 185 | C2×Q8 | — | 28 | 1 | no | — |
| 186 | C2×Q8 | — | 28 | 2 | no | — |
| 187 | C2×Q8 | — | 28 | 2 | no | — |
| 188 | C2×D4 | — | 2 | 1 | no | — |
| 189 | C2×D4 | — | 4 | 1 | no | — |
| 190 | C2×D4 | — | 14 | 1 | no | — |
| 191 | C2×D4 | — | 28 | 1 | no | — |
| 192 | C2×D4 | — | 28 | 2 | no | — |
| 193 | C2×D4 | — | 28 | 2 | no | — |
| 194 | C2×D4 | — | 28 | 1 | no | — |
| 195 | C2×D4 | — | 28 | 2 | no | — |
| 196 | C2×D4 | — | 56 | 4 | no | — |
| 197 | C22×C4 | — | 7 | 1 | no | — |
| 198 | C22×C4 | — | 14 | 1 | no | — |
| 199 | C22×C4 | — | 28 | 1 | no | — |
| 200 | SD16 | — | 4 | 2 | no | — |
| 201 | SD16 | — | 4 | 2 | no | — |
| 202 | SD16 | — | 28 | 2 | no | — |
| 203 | SD16 | — | 28 | 2 | no | — |
| 204 | SD16 | — | 28 | 2 | no | — |
| 205 | SD16 | — | 28 | 2 | no | — |
| 206 | SD16 | — | 28 | 2 | no | — |
| 207 | SD16 | — | 28 | 2 | no | — |
| 208 | C2×C8 | — | 7 | 2 | no | — |
| 209 | C2×C8 | — | 7 | 2 | no | — |
| 210 | C2×C8 | — | 7 | 2 | no | — |
| 211 | C2×C8 | — | 7 | 2 | no | — |
| 212 | C2×C8 | — | 7 | 2 | no | — |
| 213 | C2×C8 | D4×D7 | 1 | 2 | yes | — |
| 214 | C4⋊C4 | — | 4 | 1 | no | — |
| 215 | C4⋊C4 | — | 28 | 1 | no | — |
| 216 | C4⋊C4 | — | 28 | 2 | no | — |
| 217 | C42 | — | 7 | 1 | no | — |
| 218 | C42 | — | 14 | 1 | no | — |
| 219 | C42 | C2×C4×D7 | 1 | 1 | yes | characteristic |
| 220 | C16 | — | 4 | 2 | no | — |
| 221 | C16 | — | 28 | 2 | no | — |
| 222 | C2×C7 | — | 2 | 1 | no | — |
| 223 | C2×C7 | — | 16 | 1 | no | — |
| 224 | C2×C7 | C2×#331 | 1 | 1 | yes | characteristic |
| 225 | D7 | — | 2 | 1 | no | — |
| 226 | D7 | — | 16 | 1 | no | — |
| 227 | D7 | #1359 | 1 | 2 | yes | direct product factor |
| 228 | C23 | — | 4 | 1 | no | — |
| 229 | C23 | — | 7 | 1 | no | — |
| 230 | C23 | — | 28 | 1 | no | — |
| 231 | C23 | — | 56 | 1 | no | — |
| 232 | C23 | — | 56 | 1 | no | — |
| 233 | C23 | — | 56 | 4 | no | — |
| 234 | Q8 | — | 4 | 2 | no | — |
| 235 | Q8 | — | 28 | 2 | no | — |
| 236 | Q8 | — | 56 | 2 | no | — |
| 237 | D4 | — | 4 | 2 | no | — |
| 238 | D4 | — | 8 | 2 | no | — |
| 239 | D4 | — | 28 | 2 | no | — |
| 240 | D4 | — | 56 | 2 | no | — |
| 241 | D4 | — | 56 | 2 | no | — |
| 242 | D4 | — | 56 | 4 | no | — |
| 243 | C2×C4 | — | 2 | 1 | no | — |
| 244 | C2×C4 | — | 4 | 1 | no | — |
| 245 | C2×C4 | — | 7 | 1 | no | — |
| 246 | C2×C4 | — | 7 | 2 | no | — |
| 247 | C2×C4 | — | 7 | 2 | no | — |
| 248 | C2×C4 | — | 14 | 4 | no | — |
| 249 | C2×C4 | — | 14 | 1 | no | — |
| 250 | C2×C4 | — | 28 | 1 | no | — |
| 251 | C2×C4 | — | 56 | 1 | no | — |
| 252 | C2×C4 | — | 56 | 1 | no | — |
| 253 | C2×C4 | D7×C22⋊C4 | 1 | 1 | yes | characteristic |
| 254 | C8 | — | 7 | 2 | no | — |
| 255 | C8 | — | 7 | 2 | no | — |
| 256 | C8 | D7×D8 | 1 | 2 | yes | — |
| 257 | C8 | D7×SD16 | 1 | 2 | yes | — |
| 258 | C7 | C2×#1359 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 259 | C22 | — | 7 | 1 | no | — |
| 260 | C22 | — | 7 | 1 | no | — |
| 261 | C22 | — | 8 | 2 | no | — |
| 262 | C22 | — | 8 | 1 | no | — |
| 263 | C22 | — | 14 | 2 | no | — |
| 264 | C22 | — | 56 | 2 | no | — |
| 265 | C22 | — | 56 | 1 | no | — |
| 266 | C22 | — | 112 | 2 | no | — |
| 267 | C22 | — | 112 | 2 | no | — |
| 268 | C22 | D7×#100 | 1 | 1 | yes | characteristic |
| 269 | C4 | — | 2 | 2 | no | — |
| 270 | C4 | — | 7 | 2 | no | — |
| 271 | C4 | — | 8 | 1 | no | — |
| 272 | C4 | — | 14 | 2 | no | — |
| 273 | C4 | — | 56 | 1 | no | — |
| 274 | C4 | D7×#102 | 1 | 2 | yes | — |
| 275 | C2 | — | 2 | 1 | no | — |
| 276 | C2 | — | 7 | 2 | no | — |
| 277 | C2 | — | 14 | 1 | no | — |
| 278 | C2 | — | 16 | 1 | no | — |
| 279 | C2 | — | 112 | 1 | no | — |
| 280 | C2 | D7×#331 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. D7×#364 | C2 | semidirect |
| 3. C7×#1359 | C2 | semidirect |
| 4. #11725593 | C2 | semidirect |
| 5. #11725376 | C2 | semidirect |
| 16. C7×#364 | C22 | semidirect |
| 18. #89258 | C22 | semidirect |
| 53. #1359 | D7 | direct (N × H) |
| 106. #364 | C2×D7 | semidirect |
| 227. D7 | #1359 | direct (N × H) |
| 258. C7 | C2×#1359 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D7×#492 | 2 |
| D7×#364 | 2 |
| C7×#1359 | 2 |
| #11725593 | 2 |
| #11725376 | 2 |
| C2×#1359 | 7 |