Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
456 conjugacy classes of subgroups; only the 305 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 305 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D53 |
| commutator | C2×C53 | C23×C8 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C4×#41×C53 | C2 |
| socle | C22×C53 | C23×C4 |
| hypercenter | C4×#41 | D53 |
| hypocenter | C53 | C8×#41 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C42 ◁ C4×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×#88322 | C2 | semidirect |
| 2. D4×#88322 | C2 | semidirect |
| 3. #16810676 | C2 | semidirect |
| 4. C2×C4×#88322 | C2 | semidirect |
| 19. #41×C53 | C8 | semidirect |
| 36. Q8×C53 | C2×C8 | semidirect |
| 37. D4×C53 | C2×C8 | semidirect |
| 40. C2×C4×C53 | C2×C8 | semidirect |
| 48. #88322 | #41 | direct (N × H) |
| 69. C53 | C8×#41 | semidirect |
| 83. #41 | #88322 | direct (N × H) |
| 100. Q8 | C2×#88322 | semidirect |
| 101. D4 | C2×#88322 | semidirect |
| 104. C2×C4 | C2×#88322 | semidirect |