Finite Group DatabasebetaFinite Group DatabasebetaOrder 1080.
272 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 72 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×C5 | D3×C3⋊D3 |
| commutator, hypocenter | C33 | C2×C4×C5 |
| Frattini | C2 | C5×D3×C3⋊D3 |
| Fitting, socle | C2×C33×C5 | C22 |
| prime radical | C2×C5×C3⋊(C3⋊D3) | C2 |
| cosocle | C2×C33 | C22×C5 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C5×C3⋊Dic3 | C2 | semidirect |
| 3. C32×C5×Dic3 | C2 | semidirect |
| 7. C5×C3⋊(C3⋊D3) | C4 | semidirect |
| 8. #6289 | C5 | direct (N × H) |
| 13. C5×C3⋊Dic3 | D3 | semidirect |
| 15. C3×C5×Dic3 | D3 | semidirect |
| 16. C33×C5 | C2×C4 | semidirect |
| 20. C3×C3⋊Dic3 | C2×C5 | semidirect |
| 21. C32×Dic3 | C2×C5 | semidirect |
| 36. C5×Dic3 | C3⋊D3 | semidirect |
| 38. C3⋊(C3⋊D3) | C4×C5 | semidirect |
| 40. C32×C5 | C4×D3 | semidirect |
| 41. C32×C5 | C4×D3 | semidirect |
| 47. C3⋊Dic3 | C5×D3 | semidirect |
| 49. C3×Dic3 | C5×D3 | semidirect |
| 56. C33 | C2×C4×C5 | semidirect |
| 69. C3×C5 | #624 | semidirect |
| 70. C3×C5 | C4×C3⋊D3 | semidirect |
| 77. Dic3 | C5×C3⋊D3 | semidirect |
| 81. C32 | C4×C5×D3 | semidirect |
| 82. C32 | C4×C5×D3 | semidirect |
| 90. C5 | #6289 | direct (N × H) |
| 95. C3 | C5×#624 | semidirect |
| 96. C3 | C4×C5×C3⋊D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×C3⋊(C3⋊D3) | 2 |
| C3×C5×C3⋊Dic3 | 2 |
| C32×C5×Dic3 | 2 |
| C4×C5×C3⋊D3 | 3 |
| C5×#624 | 3 |
| #6289 | 5 |