Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
298 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 70 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 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C42×C53 |
| Frattini, cosocle | C2×#124 | C22×C53 |
| prime radical | C22×D4×C53 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C22×C4×C53 ◁ C2×#124×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. C53×#355 | C4 | semidirect |
| 14. C53×#329 | C4 | semidirect |
| 98. #7487 | C53 | direct (N × H) |
| 132. #355 | C4×C53 | semidirect |
| 135. #329 | C4×C53 | semidirect |
| 136. C53 | #7487 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#355 | 2 |
| C53×#1885 | 2 |
| C2×C53×#329 | 2 |
| #7487 | 53 |