Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
542 conjugacy classes of subgroups; only the 331 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 331 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×C32 | D53 |
| commutator, hypocenter | C53 | C23×C32 |
| Frattini | C16 | C23×D53 |
| Fitting | C22×C32×C53 | C2 |
| socle | C23×C53 | C2×C16 |
| prime radical | C23×D53 | C16 |
| cosocle | C16×C53 | C24 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C22×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#13133123 | C2 | semidirect |
| 3. C2×C32×D53 | C2 | direct (N × H) |
| 4. C22×C32×C53 | C2 | semidirect |
| 9. C2×#13133123 | C22 | semidirect |
| 10. C32×D53 | C22 | direct (N × H) |
| 12. C2×C32×C53 | C22 | semidirect |
| 20. #13133123 | C23 | semidirect |
| 24. C32×C53 | C23 | semidirect |
| 39. C22×D53 | C32 | direct (N × H) |
| 50. C22×C53 | C2×C32 | semidirect |
| 52. C2×D53 | C2×C32 | direct (N × H) |
| 60. C22×C32 | D53 | direct (N × H) |
| 61. C2×C53 | C22×C32 | semidirect |
| 63. D53 | C22×C32 | direct (N × H) |
| 71. C2×C32 | C2×D53 | direct (N × H) |
| 72. C53 | C23×C32 | semidirect |
| 84. C32 | C22×D53 | direct (N × H) |
| 111. C22 | C32×D53 | direct (N × H) |
| 117. C2 | C2×C32×D53 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C16×D53 | 2 |
| C22×#13133123 | 2 |
| C2×C32×D53 | 2 |
| C22×C32×C53 | 2 |
| C23×C32 | 53 |