Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1024 conjugacy classes of subgroups; only the 469 normal subgroups are shown.
This group has 1024 conjugacy classes of subgroups — too many to draw. Only its 469 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C2×#124×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#124 | D53 |
| hypocenter | C53 | C2×#563 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811373 | C2 | direct (N × H) |
| 2. C2×#13133215 | C2 | semidirect |
| 3. C2×#13133210 | C2 | semidirect |
| 4. C2×#13133209 | C2 | semidirect |
| 6. C2×#13133194 | C2 | semidirect |
| 10. #13133215 | C22 | semidirect |
| 11. #13133210 | C22 | semidirect |
| 12. #13133209 | C22 | semidirect |
| 14. #13133194 | C22 | semidirect |
| 83. C2×C53 | #563 | semidirect |
| 93. C53 | C2×#563 | semidirect |
| 168. C2 | #16811373 | direct (N × H) |