Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
4638 conjugacy classes of subgroups; only the 3000 normal subgroups are shown.
This group has 4638 conjugacy classes of subgroups — too many to draw. Only its 3000 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C4 | #6157 |
| commutator, hypocenter | C53 | C23×C4×C8 |
| Frattini | C22 | C24×#6157 |
| Fitting | C24×C4×C53 | C4 |
| socle, prime radical | C25×C53 | C2×C4 |
| cosocle | C2×Dic53 | C25 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C4×#88324 | C2 | direct (N × H) |
| 4. C23×#88324 | C4 | direct (N × H) |
| 6. C2×C4×#88324 | C22 | direct (N × H) |
| 12. C22×#88324 | C2×C4 | direct (N × H) |
| 14. C4×#88324 | C23 | direct (N × H) |
| 15. C23×Dic53 | C8 | semidirect |
| 23. C23×C4×C53 | C8 | semidirect |
| 29. C22×C4×C53 | C2×C8 | semidirect |
| 32. C22×Dic53 | C2×C8 | semidirect |
| 37. C2×#88324 | C22×C4 | direct (N × H) |
| 41. C23×C53 | C4×C8 | semidirect |
| 46. C2×C4×C53 | C22×C8 | semidirect |
| 49. C2×Dic53 | C22×C8 | semidirect |
| 54. #88324 | C23×C4 | direct (N × H) |
| 56. C22×C53 | C2×C4×C8 | semidirect |
| 61. C4×C53 | C23×C8 | semidirect |
| 63. Dic53 | C23×C8 | semidirect |
| 68. C2×C53 | C22×C4×C8 | semidirect |
| 80. C53 | C23×C4×C8 | semidirect |
| 87. C23×C4 | #88324 | direct (N × H) |
| 104. C22×C4 | C2×#88324 | direct (N × H) |
| 110. C23 | C4×#88324 | direct (N × H) |
| 121. C2×C4 | C22×#88324 | direct (N × H) |
| 124. C22 | C2×C4×#88324 | direct (N × H) |
| 132. C4 | C23×#88324 | direct (N × H) |
| 133. C2 | C22×C4×#88324 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×#88324 | 2 |
| C22×C4×#88324 | 2 |
| C23×C4×Dic53 | 2 |
| C23×C4×C8 | 53 |