Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
372 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #99 |
| commutator | C2×C4 | C4×C8×C53 |
| Frattini, cosocle | C2×C4×D4 | C22×C53 |
| socle | C22×C53 | #342 |
| prime radical | C22×D4×C53 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C2×C4×D4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C2×C4⋊C8×C53 | C4 | semidirect |
| 10. C2×C4×C8×C53 | C4 | semidirect |
| 13. C53×#324 | C4 | semidirect |
| 31. #99×C53 | C8 | semidirect |
| 115. #7205 | C53 | direct (N × H) |
| 151. C2×C4⋊C8 | C4×C53 | semidirect |
| 156. C2×C4×C8 | C4×C53 | semidirect |
| 159. #324 | C4×C53 | semidirect |
| 161. C53 | #7205 | direct (N × H) |
| 178. #99 | C8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#1954 | 2 |
| C4×#99×C53 | 2 |
| C53×#1459 | 2 |
| #7205 | 53 |