Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
880 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 880 conjugacy classes of subgroups — too many to draw. Only its 229 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 | C53×#391 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #391 | D53 |
| hypocenter | C53 | #2311 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#13133174 | C2 | semidirect |
| 3. C2×C4⋊C4×D53 | C2 | semidirect |
| 4. #16810530 | C2 | semidirect |
| 5. #16810525 | C2 | semidirect |
| 6. #16810524 | C2 | semidirect |
| 7. #16810501 | C2 | semidirect |
| 8. #16810482 | C2 | semidirect |
| 9. C53×#391 | C2 | semidirect |
| 12. C2×#11719949 | C4 | semidirect |
| 36. C2×Q8×C53 | C2×C4 | semidirect |
| 50. C2×D212 | C2×C4 | semidirect |
| 56. C2×C4×D53 | C2×C4 | semidirect |
| 103. C4×D53 | C22⋊C4 | semidirect |
| 138. #391 | D53 | semidirect |
| 143. C53 | #2311 | semidirect |
| 171. C2×Q8 | C4×D53 | semidirect |