Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
686 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 686 conjugacy classes of subgroups — too many to draw. Only its 254 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #127 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C22×C53 | #392 |
| prime radical | C24×C53 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. C53×#338 | C4 | semidirect |
| 64. #11337 | C53 | direct (N × H) |
| 88. #338 | C4×C53 | semidirect |
| 90. C53 | #11337 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2911 | 2 |
| C2×C53×#338 | 2 |
| C53×#1729 | 2 |
| #11337 | 53 |