Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
560 conjugacy classes of subgroups; only the 233 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 233 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 | C22×C53 | C23×C4 |
| Fitting | C2×C4×D4×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×C4×D4 | D53 |
| hypocenter | C53 | #2357 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16810516 | C2 | semidirect |
| 3. #16810489 | C2 | semidirect |
| 4. #16810486 | C2 | semidirect |
| 5. C4⋊C4×Dic53 | C2 | semidirect |
| 6. C4×C4⋊Dic53 | C2 | semidirect |
| 7. C4×C22⋊Dic53 | C2 | semidirect |
| 22. C4×D4×C53 | C4 | semidirect |
| 106. C53 | #2357 | semidirect |
| 109. C4×D4 | Dic53 | semidirect |