Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
820 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 250 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #393 |
| commutator | C24 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C2×C53 | #1446 |
| prime radical | C24×C53 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 20. C2×M16×C53 | Q8 | semidirect |
| 22. C22×C8×C53 | Q8 | semidirect |
| 79. #11395 | C53 | direct (N × H) |
| 106. C53 | #11395 | direct (N × H) |
| 115. C2×M16 | Q8×C53 | semidirect |
| 117. C22×C8 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3458 | 2 |
| C2×C53×#344 | 2 |
| #11395 | 53 |