Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C7 |
| prime radical | C2×C7×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ C7×#446 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#2209 | C2 | semidirect |
| 4. C7×#1690 | C2 | semidirect |
| 6. C7×#1338 | C2 | semidirect |
| 7. C7×#1334 | C2 | semidirect |
| 27. #12339 | C7 | direct (N × H) |
| 66. #2209 | C2×C7 | semidirect |
| 67. #1690 | C2×C7 | semidirect |
| 69. #1338 | C2×C7 | semidirect |
| 70. #1334 | C2×C7 | semidirect |
| 278. C7 | #12339 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3402 | 2 |
| C7×#2251 | 2 |
| C7×#2209 | 2 |
| C7×#1690 | 2 |
| C7×#1360 | 2 |
| C7×#1338 | 2 |
| C7×#1334 | 2 |
| #12339 | 7 |