Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1240 conjugacy classes of subgroups; only the 501 normal subgroups are shown.
This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 501 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C2×C4⋊Q8×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×C4⋊Q8 | D53 |
| hypocenter | C53 | C2×#555 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811339 | C2 | direct (N × H) |
| 2. C2×#13133206 | C2 | semidirect |
| 3. C2×#13133192 | C2 | semidirect |
| 4. C2×C4×D212 | C2 | semidirect |
| 5. C2×#13133175 | C2 | semidirect |
| 6. C2×C4⋊Q8×C53 | C2 | semidirect |
| 10. #13133206 | C22 | semidirect |
| 11. #13133192 | C22 | semidirect |
| 12. C4×D212 | C22 | semidirect |
| 13. #13133175 | C22 | semidirect |
| 19. C4⋊Q8×C53 | C22 | semidirect |
| 70. C2×C53 | #555 | semidirect |
| 75. C2×C4⋊Q8 | D53 | semidirect |
| 80. C53 | C2×#555 | semidirect |
| 85. C4⋊Q8 | C2×D53 | semidirect |
| 144. C2 | #16811339 | direct (N × H) |