Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
270 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 184 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 | C2×C4×C32 |
| Frattini | C2×C8 | C22×#6157 |
| Fitting | C22×C16×C53 | C4 |
| socle, prime radical | C23×C53 | C2×C16 |
| cosocle | C2×#11719908 | C23 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C22×C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20289820 | C2 | direct (N × H) |
| 4. C2×#13133267 | C4 | semidirect |
| 11. #13133267 | C2×C4 | semidirect |
| 40. C2×Dic53 | C32 | semidirect |
| 53. Dic53 | C2×C32 | semidirect |
| 57. C2×C53 | C4×C32 | semidirect |
| 67. C53 | C2×C4×C32 | semidirect |
| 110. C2 | #20289820 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#13133267 | 2 |
| #20289820 | 2 |
| C2×C16×Dic53 | 2 |
| C2×C4×C32 | 53 |