Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1072 conjugacy classes of subgroups; only the 469 normal subgroups are shown.
This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 469 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D53 |
| commutator | C2×C53 | C23×C8 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C24×C4×C53 | C2 |
| socle | C24×C53 | C2×C4 |
| hypercenter | C24×C4 | D53 |
| hypocenter | C53 | C22×C22⋊C8 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#88322 | C2 | semidirect |
| 2. C2×C22⋊#88322 | C2 | direct (N × H) |
| 5. C22×#88322 | C22 | semidirect |
| 6. C22⋊#88322 | C22 | direct (N × H) |
| 15. C24×C53 | C8 | semidirect |
| 28. C2×#88322 | C23 | semidirect |
| 34. C23×C53 | C2×C8 | semidirect |
| 50. C22×C53 | C22⋊C8 | semidirect |
| 53. C22×C53 | C22×C8 | semidirect |
| 61. C2×C53 | C2×C22⋊C8 | semidirect |
| 68. C53 | C22×C22⋊C8 | semidirect |
| 80. C24 | #88322 | semidirect |
| 98. C23 | C2×#88322 | semidirect |
| 115. C22 | C22⋊#88322 | direct (N × H) |
| 118. C22 | C22×#88322 | semidirect |
| 125. C2 | C2×C22⋊#88322 | direct (N × H) |