Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
612 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 205 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 | C2×C4×Q8×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×C4×Q8 | D53 |
| hypocenter | C53 | #2326 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16810530 | C2 | semidirect |
| 5. #16810514 | C2 | semidirect |
| 6. #16810494 | C2 | semidirect |
| 7. #16810492 | C2 | semidirect |
| 8. #16810469 | C2 | semidirect |
| 9. C2×C4×Q8×C53 | C2 | semidirect |
| 12. #13133206 | C4 | semidirect |
| 33. C2×Q8×C53 | C2×C4 | semidirect |
| 53. #11719921 | C2×C4 | semidirect |
| 57. C4×Dic53 | C2×C4 | semidirect |
| 113. C2×C4×Q8 | D53 | semidirect |
| 121. C53 | #2326 | semidirect |
| 145. C2×Q8 | C4×D53 | semidirect |