Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
326 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C22⋊C4 |
| commutator | C2×C4 | C4×C8×C53 |
| Frattini, cosocle | C23×C8 | C22×C53 |
| socle | C23×C53 | C22⋊C8 |
| prime radical | C24×C53 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ C23×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C53×C22⋊C16 | C4 | semidirect |
| 88. #7186 | C53 | direct (N × H) |
| 119. C22⋊C16 | C4×C53 | semidirect |
| 121. C53 | #7186 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×C22⋊C16 | 2 |
| C53×#1729 | 2 |
| #7186 | 53 |