Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1022 conjugacy classes of subgroups; only the 680 normal subgroups are shown.
This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 680 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C8 | #6157 |
| commutator, hypocenter | C53 | C22×C4×C16 |
| Frattini | C2×C4 | C23×#6157 |
| Fitting | C23×C8×C53 | C4 |
| socle, prime radical | C24×C53 | C2×C8 |
| cosocle | C2×#88322 | C24 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C23×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#16811509 | C2 | direct (N × H) |
| 4. C22×#11719910 | C4 | semidirect |
| 6. #16811509 | C22 | direct (N × H) |
| 12. C2×#11719910 | C2×C4 | semidirect |
| 30. C22×Dic53 | C16 | semidirect |
| 40. #11719910 | C22×C4 | semidirect |
| 49. C2×Dic53 | C2×C16 | semidirect |
| 55. C22×C53 | C4×C16 | semidirect |
| 62. Dic53 | C22×C16 | semidirect |
| 66. C2×C53 | C2×C4×C16 | semidirect |
| 77. C53 | C22×C4×C16 | semidirect |
| 120. C22 | #16811509 | direct (N × H) |
| 128. C2 | C2×#16811509 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#11719910 | 2 |
| C2×#16811509 | 2 |
| C22×C8×Dic53 | 2 |
| C22×C4×C16 | 53 |