Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1880 conjugacy classes of subgroups; only the 645 normal subgroups are shown.
This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 645 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C22×C42×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C22×C42 | D53 |
| hypocenter | C53 | C22×#124 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Dic106 | C2 | semidirect |
| 3. C2×#13133205 | C2 | direct (N × H) |
| 4. C22×#11719921 | C2 | semidirect |
| 5. C22×C42×C53 | C2 | semidirect |
| 8. C22×Dic106 | C22 | semidirect |
| 12. #13133205 | C22 | direct (N × H) |
| 13. C2×#11719921 | C22 | semidirect |
| 16. C2×C42×C53 | C22 | semidirect |
| 30. C2×Dic106 | C23 | semidirect |
| 31. C42×C53 | C23 | semidirect |
| 32. #11719921 | C23 | semidirect |
| 48. C22×C53 | #124 | semidirect |
| 59. C2×C53 | C2×#124 | semidirect |
| 67. C22×C42 | D53 | semidirect |
| 68. C53 | C22×#124 | semidirect |
| 79. C2×C42 | C2×D53 | semidirect |
| 95. C42 | C22×D53 | semidirect |
| 114. C22 | #13133205 | direct (N × H) |
| 122. C2 | C2×#13133205 | direct (N × H) |