Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
704 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 237 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C4×C4⋊C4×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C4×C4⋊C4 | D53 |
| hypocenter | C53 | #2349 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16810521 | C2 | semidirect |
| 4. #16810494 | C2 | semidirect |
| 5. #16810489 | C2 | semidirect |
| 6. #16810482 | C2 | semidirect |
| 7. #95×D53 | C2 | semidirect |
| 8. C4×C4⋊C4×C53 | C2 | semidirect |
| 13. C4×D212 | C4 | semidirect |
| 42. C2×C4×D53 | C2×C4 | semidirect |
| 45. C42×C53 | C2×C4 | semidirect |
| 47. #11719921 | C2×C4 | semidirect |
| 49. C4⋊Dic53 | C2×C4 | semidirect |
| 75. D212 | C4⋊C4 | semidirect |
| 108. C4×C4⋊C4 | D53 | semidirect |
| 110. C53 | #2349 | semidirect |
| 152. C42 | C4×D53 | semidirect |