Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
780 conjugacy classes of subgroups; only the 325 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 325 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 | C2×C53 | C22×C42 |
| Fitting | C4×C4⋊C4×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C4×C4⋊C4 | D53 |
| hypocenter | C53 | #2275 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4⋊C4×Dic53 | C2 | semidirect |
| 4. C4×#11719921 | C2 | semidirect |
| 5. #16810470 | C2 | semidirect |
| 6. #16810469 | C2 | semidirect |
| 7. #95×D53 | C2 | semidirect |
| 8. C4×C4⋊C4×C53 | C2 | semidirect |
| 12. C4×D212 | C4 | semidirect |
| 13. #13133179 | C4 | semidirect |
| 40. C2×C4×D53 | C2×C4 | semidirect |
| 41. C2×C4×D53 | C2×C4 | semidirect |
| 42. C4⋊C4×C53 | C2×C4 | semidirect |
| 44. C42×C53 | C2×C4 | semidirect |
| 45. #11719921 | C2×C4 | semidirect |
| 46. #11719921 | C2×C4 | semidirect |
| 47. C4⋊Dic53 | C2×C4 | semidirect |
| 49. C4×Dic53 | C2×C4 | semidirect |
| 74. D212 | C42 | semidirect |
| 104. C4×C4⋊C4 | D53 | semidirect |
| 109. C53 | #2275 | semidirect |
| 144. C4⋊C4 | C4×D53 | semidirect |
| 150. C42 | C4×D53 | semidirect |