Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
566 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 138 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 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C24×C7 | D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C23×C4×C7 ◁ C23×C7×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×Q32 | C2 | semidirect |
| 3. C2×C7×#358 | C2 | semidirect |
| 5. C7×#1380 | C2 | semidirect |
| 19. #13306 | C7 | direct (N × H) |
| 47. C22×Q32 | C2×C7 | semidirect |
| 49. C2×#358 | C2×C7 | semidirect |
| 51. #1380 | C2×C7 | semidirect |
| 103. C23×C7 | Q32 | semidirect |
| 215. C23 | C7×Q32 | semidirect |
| 240. C7 | #13306 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×Q32 | 2 |
| C2×C7×#467 | 2 |
| C2×C7×#358 | 2 |
| C2×C7×C22⋊C16 | 2 |
| C7×#1380 | 2 |
| #13306 | 7 |