Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
224 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133213 |
| commutator | C42×C53 | C23 |
| Frattini | C42 | C22×D53 |
| Fitting | C53×#500 | C2 |
| socle | C22×C53 | #127 |
| hypercenter | #500 | D53 |
| hypocenter | C53 | #1706 |
| LCS | G ▷ C42×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #500 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16810997 | C2 | semidirect |
| 3. #16810971 | C2 | semidirect |
| 4. C8×#88322 | C2 | semidirect |
| 5. C53×#500 | C2 | semidirect |
| 36. C8×C53 | SD16 | semidirect |
| 38. #88322 | D8 | semidirect |
| 52. #500 | D53 | semidirect |
| 57. C53 | #1706 | semidirect |
| 90. C8 | #11719924 | semidirect |