Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
476 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#427 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #427 | D53 |
| hypocenter | C53 | #2913 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #427 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16810666 | C2 | semidirect |
| 4. #16810662 | C2 | semidirect |
| 5. #16810660 | C2 | semidirect |
| 6. C2×#13133151 | C2 | semidirect |
| 7. #107×D53 | C2 | semidirect |
| 8. C53×#427 | C2 | semidirect |
| 10. #13133223 | C4 | semidirect |
| 38. C2×C8×C53 | C2×C4 | semidirect |
| 50. #11719917 | C2×C4 | semidirect |
| 52. #11719912 | C2×C4 | semidirect |
| 53. C8×D53 | C2×C4 | semidirect |
| 99. #427 | D53 | semidirect |
| 102. C53 | #2913 | semidirect |
| 136. C2×C8 | C4×D53 | semidirect |