Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
888 conjugacy classes of subgroups; only the 429 normal subgroups are shown.
This group has 888 conjugacy classes of subgroups — too many to draw. Only its 429 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 | C23×C4 |
| Frattini | C23 | C23×D53 |
| Fitting | C22×C4⋊C4×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C22×C4⋊C4 | D53 |
| hypocenter | C53 | C2×#384 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C22×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #16810492 | C2 | direct (N × H) |
| 12. C2×C4×Dic53 | C4 | semidirect |
| 36. C4×Dic53 | C2×C4 | semidirect |
| 46. C2×C4×C53 | C4⋊C4 | semidirect |
| 53. C2×Dic53 | C4⋊C4 | semidirect |
| 68. C4×C53 | C2×C4⋊C4 | semidirect |
| 70. Dic53 | C2×C4⋊C4 | semidirect |
| 72. C2×C53 | #384 | semidirect |
| 83. C53 | C2×#384 | semidirect |
| 131. C2×C4 | #11719919 | semidirect |
| 148. C4 | C2×#11719919 | semidirect |
| 149. C2 | #16810492 | direct (N × H) |