Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
704 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 216 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 | C22×D32 |
| commutator, cosocle | C2×C32 | C24×C5 |
| Frattini | C2 | C22×C5×D32 |
| Fitting | C32×C5×Q8 | C22 |
| socle | C2×C32×C5 | C24 |
| hypercenter | C5×Q8 | D32 |
| hypocenter | C32 | C5×H(5,𝔽2) |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C5 ◁ C5×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#3832 | C2 | semidirect |
| 2. C3×C5×#239 | C2 | semidirect |
| 3. C5×#3802 | C2 | semidirect |
| 4. C5×D3×D12 | C2 | semidirect |
| 5. C5×#3798 | C2 | semidirect |
| 7. C2×C5×D32 | C22 | semidirect |
| 8. C32×C5×Q8 | C22 | semidirect |
| 9. C5×C4⋊(C3⋊D3) | C22 | semidirect |
| 10. C4×C5×C3⋊D3 | C22 | semidirect |
| 11. C3×C5×D12 | C22 | semidirect |
| 12. C3×C4×C5×D3 | C22 | semidirect |
| 13. C5×#627 | C22 | semidirect |
| 14. C5×#626 | C22 | semidirect |
| 15. C5×#625 | C22 | semidirect |
| 16. #64320 | C5 | direct (N × H) |
| 20. C5×#239 | D3 | semidirect |
| 32. #3832 | C2×C5 | semidirect |
| 33. C3×#239 | C2×C5 | semidirect |
| 34. #3802 | C2×C5 | semidirect |
| 35. D3×D12 | C2×C5 | semidirect |
| 36. #3798 | C2×C5 | semidirect |
| 39. C3×C5×Q8 | C2×D3 | semidirect |
| 47. C5×D12 | C2×D3 | semidirect |
| 50. C4×C5×D3 | C2×D3 | semidirect |
| 61. C2×D32 | C22×C5 | semidirect |
| 62. C32×Q8 | C22×C5 | semidirect |
| 63. C4⋊(C3⋊D3) | C22×C5 | semidirect |
| 64. C4×C3⋊D3 | C22×C5 | semidirect |
| 65. C3×D12 | C22×C5 | semidirect |
| 66. C3×C4×D3 | C22×C5 | semidirect |
| 67. #627 | C22×C5 | semidirect |
| 68. #626 | C22×C5 | semidirect |
| 69. #625 | C22×C5 | semidirect |
| 84. #239 | C5×D3 | semidirect |
| 89. C32×C5 | H(5,𝔽2) | semidirect |
| 92. C5×Q8 | D32 | semidirect |
| 114. C3×Q8 | C2×C5×D3 | semidirect |
| 122. D12 | C2×C5×D3 | semidirect |
| 125. C4×D3 | C2×C5×D3 | semidirect |
| 142. C3×C5 | #1009 | semidirect |
| 157. C32 | C5×H(5,𝔽2) | semidirect |
| 160. Q8 | C5×D32 | semidirect |
| 173. C5 | #64320 | direct (N × H) |
| 181. C3 | C5×#1009 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#3832 | 2 |
| C3×C5×#239 | 2 |
| C5×#3802 | 2 |
| C5×D3×D12 | 2 |
| C5×#3798 | 2 |
| C5×#1009 | 3 |
| #64320 | 5 |