Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
712 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2842 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C23×C7×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#2091 | C2 | semidirect |
| 2. C7×#2069 | C2 | semidirect |
| 5. C7×#430 | C22 | semidirect |
| 8. C7×#338 | C22 | semidirect |
| 9. #32875 | C7 | direct (N × H) |
| 19. #2091 | C2×C7 | semidirect |
| 20. #2069 | C2×C7 | semidirect |
| 35. #430 | C22×C7 | semidirect |
| 38. #338 | C22×C7 | semidirect |
| 102. C7 | #32875 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2091 | 2 |
| C7×#2069 | 2 |
| C7×#1783 | 2 |
| #32875 | 7 |