Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
596 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C22×C7×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C7×#376 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#2645 | C2 | semidirect |
| 3. C7×#2088 | C2 | semidirect |
| 4. C7×#2066 | C2 | semidirect |
| 7. C7×#1843 | C2 | semidirect |
| 25. #32868 | C7 | direct (N × H) |
| 49. #2645 | C2×C7 | semidirect |
| 51. #2088 | C2×C7 | semidirect |
| 52. #2066 | C2×C7 | semidirect |
| 55. #1843 | C2×C7 | semidirect |
| 215. C7 | #32868 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2645 | 2 |
| C7×#2644 | 2 |
| C7×#2088 | 2 |
| C7×#2066 | 2 |
| C7×#2065 | 2 |
| C7×#2055 | 2 |
| C7×#1843 | 2 |
| #32868 | 7 |