Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
992 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22⋊#6157 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | C25×C53 | C4 |
| socle | C23×C53 | C22⋊C4 |
| hypercenter | C25 | #6157 |
| hypocenter | C53 | C22×#100 |
| LCS | G ▷ C22×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C25 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#11719940 | C2 | semidirect |
| 2. C2×#13133277 | C2 | direct (N × H) |
| 5. C2×#11719940 | C22 | semidirect |
| 7. #13133277 | C22 | direct (N × H) |
| 35. #11719940 | C23 | semidirect |
| 68. C22×C53 | #100 | semidirect |
| 82. C2×C53 | C2×#100 | semidirect |
| 88. C53 | C22×#100 | semidirect |
| 157. C22 | #13133277 | direct (N × H) |
| 167. C2 | C2×#13133277 | direct (N × H) |