Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
328 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #342 |
| commutator | C24 | C42×C53 |
| Frattini, cosocle | C2×C4×D4 | C22×C53 |
| prime radical | C22×D4×C53 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4×D4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 112. #7321 | C53 | direct (N × H) |
| 157. C53 | #7321 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#1870 | 2 |
| C53×#1800 | 2 |
| C53×#1465 | 2 |
| #7321 | 53 |