Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
582 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 582 conjugacy classes of subgroups — too many to draw. Only its 149 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 | C2×#126×C53 | C2 |
| hypercenter | C2×#126 | D53 |
| hypocenter | C53 | #2852 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C42×D53 | C2 | semidirect |
| 2. #16810511 | C2 | semidirect |
| 3. #16810504 | C2 | semidirect |
| 5. #16810494 | C2 | semidirect |
| 6. #16810491 | C2 | semidirect |
| 7. C2×#126×C53 | C2 | semidirect |
| 36. C42×C53 | D4 | semidirect |
| 71. C2×D53 | #126 | semidirect |
| 83. C2×#126 | D53 | semidirect |
| 90. C53 | #2852 | semidirect |
| 123. C42 | C22⋊D53 | semidirect |