Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
948 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 948 conjugacy classes of subgroups — too many to draw. Only its 185 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×#127×C53 | C2 |
| hypercenter | C2×#127 | D53 |
| hypocenter | C53 | #2541 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133213 | C2 | semidirect |
| 2. C2×#13133213 | C2 | semidirect |
| 3. C2×#13133189 | C2 | semidirect |
| 4. C2×D4×Dic53 | C2 | semidirect |
| 5. #16810531 | C2 | semidirect |
| 6. #16810526 | C2 | semidirect |
| 7. #16810525 | C2 | semidirect |
| 8. C4×#11719919 | C2 | semidirect |
| 9. C2×#127×C53 | C2 | semidirect |
| 43. C2×D4×C53 | D4 | semidirect |
| 67. #11719919 | D4 | semidirect |
| 69. C4×Dic53 | D4 | semidirect |
| 130. C4×C53 | #121 | semidirect |
| 134. Dic53 | #127 | semidirect |
| 146. C2×#127 | D53 | semidirect |
| 153. C53 | #2541 | semidirect |
| 197. C2×D4 | C22⋊D53 | semidirect |
| 278. C4 | #13133189 | semidirect |