Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1102 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 1102 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, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C24×C4×C53 | C2 |
| hypercenter | C24×C4 | D53 |
| hypocenter | C53 | #2849 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133200 | C2 | semidirect |
| 3. #16810502 | C2 | semidirect |
| 4. #16810496 | C2 | semidirect |
| 5. #16810486 | C2 | semidirect |
| 6. #16810479 | C2 | semidirect |
| 7. C24×C4×C53 | C2 | semidirect |
| 12. C2×#11719919 | C22 | semidirect |
| 17. C2×#11719921 | C22 | semidirect |
| 18. #13133124 | C22 | semidirect |
| 35. C22×C4×C53 | D4 | semidirect |
| 84. C22×C53 | #123 | semidirect |
| 102. C24×C4 | D53 | semidirect |
| 109. C53 | #2849 | semidirect |
| 144. C22×C4 | C22⋊D53 | semidirect |
| 189. C22 | #13133200 | semidirect |