Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
858 conjugacy classes of subgroups; only the 454 normal subgroups are shown.
This group has 858 conjugacy classes of subgroups — too many to draw. Only its 454 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42×C53 | C42 |
| commutator | C4 | C43×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C22×C53 | C2×#95 |
| prime radical | C24×C53 | C2×Q8 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C53×#338 | C4 | direct (N × H) |
| 49. C4×#338 | C53 | direct (N × H) |
| 55. C4×C53 | #338 | direct (N × H) |
| 69. #338 | C4×C53 | direct (N × H) |
| 70. C53 | C4×#338 | direct (N × H) |
| 116. C4 | C53×#338 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×M16×C53 | 2 |
| C2×C53×#338 | 2 |
| C4×#338 | 53 |