Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
648 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 90 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 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C23×C7 |
| prime radical | C2×C7×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C7×D4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#2804 | C2 | semidirect |
| 4. C7×#1393 | C2 | semidirect |
| 5. C7×#1390 | C2 | semidirect |
| 19. #13377 | C7 | direct (N × H) |
| 23. C7×H(5,𝔽2) | Q8 | semidirect |
| 53. #2804 | C2×C7 | semidirect |
| 56. #1393 | C2×C7 | semidirect |
| 57. #1390 | C2×C7 | semidirect |
| 175. H(5,𝔽2) | C7×Q8 | semidirect |
| 339. C7 | #13377 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2804 | 2 |
| C7×#1792 | 2 |
| C7×#1790 | 2 |
| C7×#1393 | 2 |
| C7×#1390 | 2 |
| #13377 | 7 |