Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
816 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C2×C22⋊C4×C53 | C4 |
| hypercenter | C2×C22⋊C4 | #6157 |
| hypocenter | C53 | #2315 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811570 | C2 | semidirect |
| 3. #16811565 | C2 | semidirect |
| 6. C2×C4⋊#6157 | C22 | semidirect |
| 10. (C2×#6157)⋊C4 | C22 | semidirect |
| 13. C2×C22⋊Dic53 | C4 | semidirect |
| 14. C2×#11719921 | C4 | semidirect |
| 15. C22⋊C4×D53 | C4 | semidirect |
| 17. C2×C22⋊C4×C53 | C4 | semidirect |
| 29. C22×C4×C53 | C2×C4 | semidirect |
| 30. C22×Dic53 | C2×C4 | semidirect |
| 32. C2×C4×D53 | C2×C4 | semidirect |
| 37. C22⋊C4×C53 | C2×C4 | semidirect |
| 38. C22⋊Dic53 | C2×C4 | semidirect |
| 39. #11719921 | C2×C4 | semidirect |
| 41. C23×C53 | C4⋊C4 | semidirect |
| 50. C22×D53 | C4⋊C4 | semidirect |
| 51. C22×D53 | C4⋊C4 | semidirect |
| 103. C53 | #2315 | semidirect |
| 115. C2×C22⋊C4 | #6157 | semidirect |
| 132. C22×C4 | C2×#6157 | semidirect |
| 138. C22⋊C4 | C2×#6157 | semidirect |
| 155. C23 | C4⋊#6157 | semidirect |