Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
900 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 900 conjugacy classes of subgroups — too many to draw. Only its 185 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, socle | C23×C53 | C24 |
| Fitting | C4×C22⋊C4×C53 | C2 |
| hypercenter | C4×C22⋊C4 | D53 |
| hypocenter | C53 | #2539 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133201 | C2 | semidirect |
| 2. C2×#13133191 | C2 | semidirect |
| 3. C2×#13133187 | C2 | semidirect |
| 4. C22×#11719921 | C2 | semidirect |
| 5. #16810560 | C2 | semidirect |
| 6. #16810533 | C2 | semidirect |
| 7. #16810525 | C2 | semidirect |
| 8. C4×C22⋊C4×C53 | C2 | semidirect |
| 11. C22×D212 | C22 | semidirect |
| 17. C2×C4⋊Dic53 | C22 | semidirect |
| 18. C2×C4⋊Dic53 | C22 | semidirect |
| 23. C2×#11719921 | C22 | semidirect |
| 24. C2×#11719921 | C22 | semidirect |
| 27. C2×C42×C53 | C22 | semidirect |
| 28. #95×C53 | C22 | semidirect |
| 37. C22×C4×C53 | D4 | semidirect |
| 53. C22⋊C4×C53 | D4 | semidirect |
| 137. C4×C22⋊C4 | D53 | semidirect |
| 138. C53 | #2539 | semidirect |
| 157. C2×C42 | C2×D53 | semidirect |
| 158. #95 | C2×D53 | semidirect |
| 179. C22×C4 | D212 | semidirect |
| 190. C22⋊C4 | D212 | semidirect |