Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
196 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×#6157 |
| commutator, socle | C2×C53 | C2×C4×C8 |
| Fitting | M32×C53 | C4 |
| hypercenter | M32 | #6157 |
| hypocenter | C53 | C4×M32 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C8 ◁ M32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #16811510 | C2 | semidirect |
| 4. C16×#6157 | C2 | semidirect |
| 8. #13133292 | C4 | semidirect |
| 9. #13133291 | C4 | semidirect |
| 16. #13133157 | C4 | semidirect |
| 17. #13133156 | C4 | semidirect |
| 19. M32×C53 | C4 | semidirect |
| 35. #11719910 | C2×C4 | semidirect |
| 36. C16×C53 | C2×C4 | semidirect |
| 37. #11719908 | C2×C4 | semidirect |
| 59. #6157 | M32 | direct (N × H) |
| 74. C53 | C4×M32 | semidirect |
| 82. M32 | #6157 | direct (N × H) |
| 106. C16 | C2×#6157 | semidirect |