Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1488 conjugacy classes of subgroups; only the 762 normal subgroups are shown.
This group has 1488 conjugacy classes of subgroups — too many to draw. Only its 762 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×#6157 |
| commutator | C2×C53 | C24×C4 |
| Frattini | C2 | C24×#6157 |
| Fitting | C22×Q8×C53 | C4 |
| socle | C22×C53 | C23×C4 |
| hypercenter | C22×Q8 | #6157 |
| hypocenter | C53 | C2×#519 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811691 | C2 | direct (N × H) |
| 2. C2×Q8×#6157 | C2 | semidirect |
| 5. Q8×#6157 | C22 | semidirect |
| 10. C22×Dic106 | C4 | semidirect |
| 11. C2×Q8×D53 | C4 | semidirect |
| 12. C2×Q8×D53 | C4 | semidirect |
| 16. C22×Q8×C53 | C4 | semidirect |
| 20. C2×Q8×C53 | C2×C4 | semidirect |
| 24. Q8×D53 | C2×C4 | semidirect |
| 31. C2×Dic106 | C2×C4 | semidirect |
| 67. C2×C53 | #519 | semidirect |
| 70. D53 | #519 | semidirect |
| 76. C53 | C2×#519 | semidirect |
| 82. C22×Q8 | #6157 | semidirect |
| 96. C2×Q8 | C2×#6157 | semidirect |
| 141. C2 | #16811691 | direct (N × H) |