Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
924 conjugacy classes of subgroups; only the 201 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 201 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×#121×C53 | C2 |
| hypercenter | C2×#121 | D53 |
| hypocenter | C53 | #2856 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133213 | C2 | semidirect |
| 2. C2×#13133191 | C2 | semidirect |
| 3. C2×#13133189 | C2 | semidirect |
| 4. C22×C4×Dic53 | C2 | semidirect |
| 5. #16810526 | C2 | semidirect |
| 6. #16810524 | C2 | semidirect |
| 7. #16810523 | C2 | semidirect |
| 8. #16810492 | C2 | semidirect |
| 9. C2×#121×C53 | C2 | semidirect |
| 42. C22×C4×C53 | D4 | semidirect |
| 53. C22×Dic53 | D4 | semidirect |
| 69. C4×Dic53 | D4 | semidirect |
| 118. C22×C53 | #127 | semidirect |
| 131. C4×C53 | #121 | semidirect |
| 134. Dic53 | #121 | semidirect |
| 147. C2×#121 | D53 | semidirect |
| 153. C53 | #2856 | semidirect |
| 200. C22×C4 | C22⋊D53 | semidirect |
| 267. C22 | #13133213 | semidirect |
| 278. C4 | #13133189 | semidirect |