Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
248 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 86 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 | H(3,ℤ/4) |
| commutator | C2×C8 | C42×C53 |
| Frattini, cosocle | C2×#131 | C22×C53 |
| socle | C2×C53 | #1289 |
| prime radical | C2×#41×C53 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C2×M32×C53 | C4 | semidirect |
| 6. C22×C16×C53 | C4 | semidirect |
| 13. C53×#338 | C4 | semidirect |
| 92. #7227 | C53 | direct (N × H) |
| 115. C2×M32 | C4×C53 | semidirect |
| 116. C22×C16 | C4×C53 | semidirect |
| 123. #338 | C4×C53 | semidirect |
| 125. C53 | #7227 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2113 | 2 |
| C53×#1813 | 2 |
| C53×#1812 | 2 |
| #7227 | 53 |