Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
3464 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 3464 conjugacy classes of subgroups — too many to draw. Only its 354 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×#6157 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | C25×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C25 | #6157 |
| hypocenter | C53 | C2×#379 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C25 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C22⋊#6157 | C2 | semidirect |
| 2. #16811564 | C2 | direct (N × H) |
| 4. C23×#6157 | C22 | semidirect |
| 6. C2×C22⋊#6157 | C22 | semidirect |
| 8. C24×D53 | C4 | semidirect |
| 9. C24×D53 | C4 | semidirect |
| 12. C25×C53 | C4 | semidirect |
| 15. C24×C53 | C2×C4 | semidirect |
| 24. C23×D53 | C2×C4 | semidirect |
| 29. C22×#6157 | C23 | semidirect |
| 34. C23×C53 | C22⋊C4 | semidirect |
| 46. C22×D53 | C22⋊C4 | semidirect |
| 47. C22×D53 | C22⋊C4 | semidirect |
| 59. C22×C53 | C2×C22⋊C4 | semidirect |
| 67. C2×D53 | C2×C22⋊C4 | semidirect |
| 73. C2×C53 | #379 | semidirect |
| 76. D53 | #379 | semidirect |
| 81. C53 | C2×#379 | semidirect |
| 87. C25 | #6157 | semidirect |
| 105. C24 | C2×#6157 | semidirect |
| 128. C23 | C22⋊#6157 | semidirect |
| 146. C22 | C2×C22⋊#6157 | semidirect |
| 153. C2 | #16811564 | direct (N × H) |