Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1358 conjugacy classes of subgroups; only the 270 normal subgroups are shown.
This group has 1358 conjugacy classes of subgroups — too many to draw. Only its 270 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 | #120 |
| commutator | C24 | C22×C4×C7 |
| Frattini, cosocle | C25 | C23×C7 |
| prime radical | C25×C7 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C25×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C7×#543 | C4 | semidirect |
| 12. #11390 | C7 | direct (N × H) |
| 20. C23×C4×C7 | Q8 | semidirect |
| 31. C2×C7×C22⋊C4 | Q8 | semidirect |
| 72. #543 | C4×C7 | semidirect |
| 123. C23×C4 | C7×Q8 | semidirect |
| 134. C2×C22⋊C4 | C7×Q8 | semidirect |
| 242. C7 | #11390 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#543 | 2 |
| C2×C7×#342 | 2 |
| C7×C22⋊#95 | 2 |
| #11390 | 7 |