Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
652 conjugacy classes of subgroups; only the 249 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 249 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×#124×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#124 | D53 |
| hypocenter | C53 | C4×#124 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×Dic53 | C2 | semidirect |
| 3. #16810530 | C2 | semidirect |
| 5. #16810516 | C2 | semidirect |
| 6. #16810489 | C2 | semidirect |
| 7. C22⋊C4×Dic53 | C2 | semidirect |
| 8. C42×Dic53 | C2 | semidirect |
| 27. #124×C53 | C4 | semidirect |
| 33. C2×Q8×C53 | C2×C4 | semidirect |
| 47. C22⋊C4×C53 | C2×C4 | semidirect |
| 48. C42×C53 | C2×C4 | semidirect |
| 105. Dic53 | #124 | direct (N × H) |
| 122. C53 | C4×#124 | semidirect |
| 126. #124 | Dic53 | direct (N × H) |
| 146. C2×Q8 | C2×Dic53 | semidirect |
| 165. C22⋊C4 | C2×Dic53 | semidirect |
| 171. C42 | C2×Dic53 | semidirect |