Finite Group DatabasebetaFinite Group DatabasebetaOrder 480.
332 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C23×D5 |
| commutator, cosocle | C2×C5 | C24×C3 |
| Frattini | C2 | C23×C3×D5 |
| Fitting | C3×C5×#41 | C2 |
| socle | C2×C3×C5 | C24 |
| hypercenter | C3×#41 | D5 |
| hypocenter | C5 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C3 ◁ C3×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C5×#41 | C2 | semidirect |
| 2. C3×#713 | C2 | semidirect |
| 3. C3×D4×D5 | C2 | semidirect |
| 4. C3×#709 | C2 | semidirect |
| 5. C2×C3×D20 | C2 | semidirect |
| 6. #4149 | C3 | direct (N × H) |
| 7. C22×C3×D5 | C22 | semidirect |
| 8. C3×C5×Q8 | C22 | semidirect |
| 9. C3×C5×D4 | C22 | semidirect |
| 10. C2×C3×C4×C5 | C22 | semidirect |
| 11. C3×C22⋊D5 | C22 | semidirect |
| 12. C3×D20 | C22 | semidirect |
| 13. C3×D20 | C22 | semidirect |
| 14. C3×C4×D5 | C22 | semidirect |
| 15. C3×Dic10 | C22 | semidirect |
| 17. C5×#41 | C2×C3 | semidirect |
| 18. #713 | C2×C3 | semidirect |
| 19. D4×D5 | C2×C3 | semidirect |
| 20. #709 | C2×C3 | semidirect |
| 21. C2×D20 | C2×C3 | semidirect |
| 30. C3×#41 | D5 | semidirect |
| 33. C22×D5 | C22×C3 | semidirect |
| 34. C5×Q8 | C22×C3 | semidirect |
| 35. C5×D4 | C22×C3 | semidirect |
| 36. C2×C4×C5 | C22×C3 | semidirect |
| 37. C22⋊D5 | C22×C3 | semidirect |
| 38. D20 | C22×C3 | semidirect |
| 39. D20 | C22×C3 | semidirect |
| 40. C4×D5 | C22×C3 | semidirect |
| 41. Dic10 | C22×C3 | semidirect |
| 48. C3×Q8 | C2×D5 | semidirect |
| 52. C3×D4 | C2×D5 | semidirect |
| 54. C2×C3×C4 | C2×D5 | semidirect |
| 63. #41 | C3×D5 | semidirect |
| 66. C3×C5 | H(5,𝔽2) | semidirect |
| 78. Q8 | C2×C3×D5 | semidirect |
| 82. D4 | C2×C3×D5 | semidirect |
| 84. C2×C4 | C2×C3×D5 | semidirect |
| 88. C5 | C3×H(5,𝔽2) | semidirect |
| 95. C3 | #4149 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C5×#41 | 2 |
| C3×#713 | 2 |
| C3×D4×D5 | 2 |
| C3×#709 | 2 |
| C2×C3×D20 | 2 |
| #4149 | 3 |
| C3×H(5,𝔽2) | 5 |