Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
352 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 106 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 | #340 |
| commutator | C2×C8 | C42×C53 |
| Frattini, cosocle | C23×C8 | C22×C53 |
| prime radical | C24×C53 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ C23×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C2×#106×C53 | C4 | semidirect |
| 10. C53×C22⋊C16 | C4 | semidirect |
| 13. C53×#340 | C4 | semidirect |
| 91. #7262 | C53 | direct (N × H) |
| 122. C2×#106 | C4×C53 | semidirect |
| 125. C22⋊C16 | C4×C53 | semidirect |
| 128. #340 | C4×C53 | semidirect |
| 129. C53 | #7262 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2112 | 2 |
| C53×#1807 | 2 |
| C53×#1735 | 2 |
| #7262 | 53 |