Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
348 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 348 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×#127 | C22×C53 |
| prime radical | C2×#127×C53 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C22×C4×C53 ◁ C2×#127×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C53×#518 | C4 | semidirect |
| 8. C2×#100×C53 | C4 | semidirect |
| 11. C53×#353 | C4 | semidirect |
| 14. C53×#331 | C4 | semidirect |
| 109. #7485 | C53 | direct (N × H) |
| 145. #518 | C4×C53 | semidirect |
| 148. C2×#100 | C4×C53 | semidirect |
| 151. #353 | C4×C53 | semidirect |
| 154. #331 | C4×C53 | semidirect |
| 155. C53 | #7485 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#353 | 2 |
| C53×#1887 | 2 |
| C2×C53×#331 | 2 |
| #7485 | 53 |