Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
712 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 258 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×#6157 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | C2×C42×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C42 | #6157 |
| hypocenter | C53 | C2×#382 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811567 | C2 | direct (N × H) |
| 11. C2×C4×Dic53 | C4 | semidirect |
| 12. C2×C4×Dic53 | C4 | semidirect |
| 13. C42×D53 | C4 | semidirect |
| 15. C2×C42×C53 | C4 | semidirect |
| 32. C42×C53 | C2×C4 | semidirect |
| 33. C4×Dic53 | C2×C4 | semidirect |
| 34. C4×Dic53 | C2×C4 | semidirect |
| 73. C2×C53 | #382 | semidirect |
| 76. D53 | #382 | semidirect |
| 83. C53 | C2×#382 | semidirect |
| 93. C2×C42 | #6157 | semidirect |
| 113. C42 | C2×#6157 | semidirect |
| 154. C2 | #16811567 | direct (N × H) |