Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
360 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | D212 |
| commutator | C4×C53 | C2×C42 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C43×C53 | C2 |
| socle | C22×C53 | C2×C22⋊C4 |
| hypercenter | C43 | D53 |
| hypocenter | C53 | C4×#104 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C43 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×#11719917 | C2 | semidirect |
| 4. #16810383 | C2 | semidirect |
| 5. C43×C53 | C2 | semidirect |
| 8. C4×Dic106 | C4 | semidirect |
| 9. C4×D212 | C4 | semidirect |
| 14. #13133145 | C4 | direct (N × H) |
| 33. C42×C53 | C2×C4 | semidirect |
| 43. #11719917 | C2×C4 | semidirect |
| 60. D212 | C42 | semidirect |
| 64. Dic106 | C42 | semidirect |
| 73. C4×C53 | #104 | semidirect |
| 87. C43 | D53 | semidirect |
| 89. C53 | C4×#104 | semidirect |
| 126. C42 | C4×D53 | semidirect |
| 159. C4 | #13133145 | direct (N × H) |