Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1358 conjugacy classes of subgroups; only the 270 normal subgroups are shown.
This group has 1358 conjugacy classes of subgroups — too many to draw. Only its 270 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #120 |
| commutator | C24 | C22×C4×C53 |
| Frattini, cosocle | C25 | C23×C53 |
| prime radical | C25×C53 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C25×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C53×#543 | C4 | semidirect |
| 19. C23×C4×C53 | Q8 | semidirect |
| 30. C2×C22⋊C4×C53 | Q8 | semidirect |
| 104. #11390 | C53 | direct (N × H) |
| 144. #543 | C4×C53 | semidirect |
| 150. C53 | #11390 | direct (N × H) |
| 158. C23×C4 | Q8×C53 | semidirect |
| 169. C2×C22⋊C4 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#543 | 2 |
| C2×C53×#342 | 2 |
| C53×C22⋊#95 | 2 |
| #11390 | 53 |