Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
578 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 578 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133187 |
| commutator | C23×C53 | C22×C4 |
| Frattini | C24 | C22×D53 |
| Fitting | C2×#99×C53 | C2 |
| socle | C22×C53 | #120 |
| hypercenter | C2×#99 | D53 |
| hypocenter | C53 | #1889 |
| LCS | G ▷ C23×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#99 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811098 | C2 | semidirect |
| 3. #16810478 | C2 | semidirect |
| 4. #16810389 | C2 | semidirect |
| 5. C2×#99×C53 | C2 | semidirect |
| 9. #13133185 | C4 | semidirect |
| 27. C23×D53 | C2×C4 | semidirect |
| 38. C22×Dic53 | D4 | semidirect |
| 45. C22⋊Dic53 | C2×C4 | semidirect |
| 54. C23×C53 | C22⋊C4 | semidirect |
| 78. C2×C4×C53 | C22⋊C4 | semidirect |
| 122. C2×#99 | D53 | semidirect |
| 125. C53 | #1889 | semidirect |
| 186. C23 | #11719921 | semidirect |
| 204. C2×C4 | #11719921 | semidirect |