Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
492 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 166 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 | #127×C53 | C4 |
| hypercenter | #127 | #6157 |
| hypocenter | C53 | #2935 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133304 | C2 | semidirect |
| 2. #16811658 | C2 | semidirect |
| 3. #16811654 | C2 | semidirect |
| 4. D4×#88324 | C2 | semidirect |
| 5. D53⋊(C4⋊C8) | C2 | semidirect |
| 6. #16811597 | C2 | semidirect |
| 7. C4×#11719936 | C2 | semidirect |
| 49. #11719936 | D4 | semidirect |
| 92. C4×C53 | #131 | semidirect |
| 112. C53 | #2935 | semidirect |
| 201. C4 | #13133304 | semidirect |