Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1968 conjugacy classes of subgroups; only the 645 normal subgroups are shown.
This group has 1968 conjugacy classes of subgroups — too many to draw. Only its 645 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C23×D4×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C23×D4 | D53 |
| hypocenter | C53 | C22×#123 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C23×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Dic53 | C2 | semidirect |
| 2. C2×#13133198 | C2 | direct (N × H) |
| 3. C22×#11719919 | C2 | semidirect |
| 4. C22×C22⋊Dic53 | C2 | semidirect |
| 5. C22×C22⋊Dic53 | C2 | semidirect |
| 10. C23×Dic53 | C22 | semidirect |
| 11. #13133198 | C22 | direct (N × H) |
| 12. C2×#11719919 | C22 | semidirect |
| 13. C2×C22⋊Dic53 | C22 | semidirect |
| 14. C2×C22⋊Dic53 | C22 | semidirect |
| 36. C22×Dic53 | C23 | semidirect |
| 37. C22⋊Dic53 | C23 | semidirect |
| 38. C22⋊Dic53 | C23 | semidirect |
| 39. #11719919 | C23 | semidirect |
| 64. C22×C53 | #123 | semidirect |
| 75. C2×C53 | C2×#123 | semidirect |
| 84. C53 | C22×#123 | semidirect |
| 142. C22 | #13133198 | direct (N × H) |
| 152. C2 | C2×#13133198 | direct (N × H) |