Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
680 conjugacy classes of subgroups; only the 469 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 469 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D53 |
| commutator | C2×C53 | C22×C42 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C23×C8×C53 | C2 |
| socle | C24×C53 | C2×C4 |
| hypercenter | C23×C8 | D53 |
| hypocenter | C53 | C22×#97 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#13133127 | C2 | direct (N × H) |
| 6. C22×#88322 | C4 | semidirect |
| 10. #13133127 | C22 | direct (N × H) |
| 12. C22×C8×C53 | C4 | semidirect |
| 21. C2×C8×C53 | C2×C4 | semidirect |
| 25. C2×#88322 | C2×C4 | semidirect |
| 37. C8×C53 | C22×C4 | semidirect |
| 38. #88322 | C22×C4 | semidirect |
| 39. C22×C53 | #97 | semidirect |
| 48. C2×C53 | C2×#97 | semidirect |
| 55. C53 | C22×#97 | semidirect |
| 60. C22×C8 | Dic53 | semidirect |
| 74. C2×C8 | C2×Dic53 | semidirect |
| 89. C8 | C22×Dic53 | semidirect |
| 90. C22 | #13133127 | direct (N × H) |
| 98. C2 | C2×#13133127 | direct (N × H) |