Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
476 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 200 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C4⋊#6157 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C4 | C23×#6157 |
| Fitting | #131×C53 | C4 |
| socle | C2×C53 | C22×C4⋊C4 |
| hypercenter | #131 | #6157 |
| hypocenter | C53 | #2913 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16811634 | C2 | semidirect |
| 3. #16811633 | C2 | semidirect |
| 4. C2×#13133286 | C2 | semidirect |
| 15. #13133225 | C4 | semidirect |
| 16. #13133224 | C4 | semidirect |
| 17. #13133223 | C4 | semidirect |
| 20. #131×C53 | C4 | semidirect |
| 34. M16×C53 | C2×C4 | semidirect |
| 35. C2×C8×C53 | C2×C4 | semidirect |
| 36. #11719917 | C2×C4 | semidirect |
| 37. C2×#88322 | C2×C4 | semidirect |
| 38. #11719912 | C2×C4 | semidirect |
| 39. C8×D53 | C2×C4 | semidirect |
| 40. C8×D53 | C2×C4 | semidirect |
| 84. C53 | #2913 | semidirect |
| 89. #131 | #6157 | semidirect |
| 110. M16 | C2×#6157 | semidirect |
| 116. C2×C8 | C2×#6157 | semidirect |