Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1342 conjugacy classes of subgroups; only the 811 normal subgroups are shown.
This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 811 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C82 | D53 |
| commutator, hypocenter | C53 | C22×C82 |
| Frattini | C42 | C23×D53 |
| Fitting | C2×C82×C53 | C2 |
| socle | C23×C53 | C2×C42 |
| prime radical | C23×D53 | C42 |
| cosocle | C42×C53 | C24 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C2×C82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C8×#88322 | C2 | semidirect |
| 3. C82×D53 | C2 | direct (N × H) |
| 4. C2×C82×C53 | C2 | semidirect |
| 11. C8×#88322 | C22 | semidirect |
| 13. C82×C53 | C22 | semidirect |
| 16. C2×C8×D53 | C8 | direct (N × H) |
| 34. C2×C8×C53 | C2×C8 | semidirect |
| 40. C2×#88322 | C2×C8 | semidirect |
| 42. C8×D53 | C2×C8 | direct (N × H) |
| 53. C8×C53 | C22×C8 | semidirect |
| 54. #88322 | C22×C8 | semidirect |
| 58. C2×D53 | C82 | direct (N × H) |
| 66. C2×C82 | D53 | direct (N × H) |
| 67. C2×C53 | C2×C82 | semidirect |
| 69. D53 | C2×C82 | direct (N × H) |
| 78. C82 | C2×D53 | direct (N × H) |
| 79. C53 | C22×C82 | semidirect |
| 103. C2×C8 | C8×D53 | direct (N × H) |
| 120. C8 | C2×C8×D53 | direct (N × H) |
| 129. C2 | C82×D53 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C8×D53 | 2 |
| C2×C8×#88322 | 2 |
| C82×D53 | 2 |
| C2×C82×C53 | 2 |
| C22×C82 | 53 |