Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
534 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 534 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C4×C4⋊C4×C53 | C2 |
| hypercenter | C4×C4⋊C4 | D53 |
| hypocenter | C53 | #2662 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133211 | C2 | semidirect |
| 2. C2×#13133205 | C2 | semidirect |
| 3. #16810545 | C2 | semidirect |
| 4. #16810544 | C2 | semidirect |
| 5. #16810541 | C2 | semidirect |
| 6. #16810540 | C2 | semidirect |
| 7. #16810536 | C2 | semidirect |
| 9. #16810525 | C2 | semidirect |
| 10. #16810514 | C2 | semidirect |
| 11. C4×C4⋊C4×C53 | C2 | semidirect |
| 57. C42×C53 | D4 | semidirect |
| 154. C4×C4⋊C4 | D53 | semidirect |
| 155. C53 | #2662 | semidirect |
| 214. C42 | D212 | semidirect |