Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1504 conjugacy classes of subgroups; only the 770 normal subgroups are shown.
This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 770 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×#6157 |
| commutator | C2×C53 | C24×C4 |
| Frattini | C2 | C24×#6157 |
| Fitting | C22×Q8×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C22×Q8 | #6157 |
| hypocenter | C53 | C22×#131 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133305 | C2 | direct (N × H) |
| 2. C22×#11719936 | C2 | semidirect |
| 3. C22×D53⋊C8 | C2 | semidirect |
| 5. #13133305 | C22 | direct (N × H) |
| 6. C2×#11719936 | C22 | semidirect |
| 7. C2×D53⋊C8 | C22 | semidirect |
| 26. #11719936 | C23 | semidirect |
| 27. D53⋊C8 | C23 | semidirect |
| 40. C22×C53 | #131 | semidirect |
| 48. C2×C53 | C2×#131 | semidirect |
| 55. C53 | C22×#131 | semidirect |
| 93. C22 | #13133305 | direct (N × H) |
| 99. C2 | C2×#13133305 | direct (N × H) |