Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1090 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 1090 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, socle | C22×C53 | #393 |
| commutator | C24 | C22×C4×C53 |
| Frattini, cosocle | C25 | C23×C53 |
| prime radical | C25×C53 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C25×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C53×#543 | C4 | semidirect |
| 16. C23×C4×C53 | Q8 | semidirect |
| 33. C2×C22⋊C4×C53 | Q8 | semidirect |
| 96. #11394 | C53 | direct (N × H) |
| 132. #543 | C4×C53 | semidirect |
| 139. C53 | #11394 | direct (N × H) |
| 144. C23×C4 | Q8×C53 | semidirect |
| 161. 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 |
| #11394 | 53 |