Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
542 conjugacy classes of subgroups; only the 320 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 320 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C16 | #6157 |
| commutator, hypocenter | C53 | C23×C32 |
| Frattini | C8 | C23×#6157 |
| Fitting | C22×C16×C53 | C4 |
| socle | C23×C53 | C2×C16 |
| prime radical | C23×D53 | C16 |
| cosocle | #11719908 | C24 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C22×C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D53⋊C32 | C2 | direct (N × H) |
| 2. C22×#13133267 | C2 | semidirect |
| 4. D53⋊C32 | C22 | direct (N × H) |
| 5. C2×#13133267 | C22 | semidirect |
| 11. #13133267 | C23 | semidirect |
| 36. C22×D53 | C32 | semidirect |
| 47. C22×C53 | C2×C32 | semidirect |
| 49. C2×D53 | C2×C32 | semidirect |
| 57. C2×C53 | C22×C32 | semidirect |
| 59. D53 | C22×C32 | semidirect |
| 67. C53 | C23×C32 | semidirect |
| 105. C22 | D53⋊C32 | direct (N × H) |
| 111. C2 | C2×D53⋊C32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D53⋊C32 | 2 |
| C22×#13133267 | 2 |
| C22×C16×D53 | 2 |
| C23×C32 | 53 |