Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
126 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1404 |
| commutator | #125 | C2×C4×C53 |
| Frattini, cosocle | #491 | C22×C53 |
| prime radical | C22×C53 | AΣL(1,𝔽16) |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C53 ◁ C2×C4×C53 ◁ C42×C53 ◁ C53×#491 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 51. #7526 | C53 | direct (N × H) |
| 71. C53 | #7526 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#1663 | 2 |
| C53×#1413 | 2 |
| C53×#1356 | 2 |
| #7526 | 53 |