Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
498 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C23×D9 |
| commutator, cosocle | C2×C9 | C24×C5 |
| Frattini | C2×C3 | C23×C5×D3 |
| Fitting | C5×C9×#41 | C2 |
| socle | C2×C3×C5 | C23×D3 |
| hypercenter | C5×#41 | D9 |
| hypocenter | C9 | C5×H(5,𝔽2) |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2×C5 ◁ C5×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C9×#41 | C2 | semidirect |
| 2. C5×#3701 | C2 | semidirect |
| 3. C5×D4×D9 | C2 | semidirect |
| 4. C5×#3697 | C2 | semidirect |
| 5. C2×C5×D36 | C2 | semidirect |
| 7. C22×C5×D9 | C22 | semidirect |
| 8. C5×Q8×C9 | C22 | semidirect |
| 9. C5×D4×C9 | C22 | semidirect |
| 10. C2×C4×C5×C9 | C22 | semidirect |
| 11. C5×#611 | C22 | semidirect |
| 12. C5×D36 | C22 | semidirect |
| 13. C5×D36 | C22 | semidirect |
| 14. C4×C5×D9 | C22 | semidirect |
| 15. C5×Dic18 | C22 | semidirect |
| 16. #63715 | C5 | direct (N × H) |
| 29. C9×#41 | C2×C5 | semidirect |
| 30. #3701 | C2×C5 | semidirect |
| 31. D4×D9 | C2×C5 | semidirect |
| 32. #3697 | C2×C5 | semidirect |
| 33. C2×D36 | C2×C5 | semidirect |
| 49. C5×#41 | D9 | semidirect |
| 52. C22×D9 | C22×C5 | semidirect |
| 53. Q8×C9 | C22×C5 | semidirect |
| 54. D4×C9 | C22×C5 | semidirect |
| 55. C2×C4×C9 | C22×C5 | semidirect |
| 56. #611 | C22×C5 | semidirect |
| 57. D36 | C22×C5 | semidirect |
| 58. D36 | C22×C5 | semidirect |
| 59. C4×D9 | C22×C5 | semidirect |
| 60. Dic18 | C22×C5 | semidirect |
| 72. C5×C9 | H(5,𝔽2) | semidirect |
| 75. C5×Q8 | C2×D9 | semidirect |
| 79. C5×D4 | C2×D9 | semidirect |
| 81. C2×C4×C5 | C2×D9 | semidirect |
| 112. #41 | C5×D9 | semidirect |
| 125. C9 | C5×H(5,𝔽2) | semidirect |
| 128. Q8 | C2×C5×D9 | semidirect |
| 132. D4 | C2×C5×D9 | semidirect |
| 134. C2×C4 | C2×C5×D9 | semidirect |
| 138. C5 | #63715 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C9×#41 | 2 |
| C5×#3701 | 2 |
| C5×D4×D9 | 2 |
| C5×#3697 | 2 |
| C2×C5×D36 | 2 |
| C5×#1009 | 3 |
| #63715 | 5 |