Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
162 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C9⋊AGL(1,𝔽5) | C2 | semidirect |
| 3. C2×C9×AGL(1,𝔽5) | C2 | semidirect |
| 4. D9×AGL(1,𝔽5) | C2 | direct (N × H) |
| 6. C2×D45 | C4 | semidirect |
| 7. C2×C5×D9 | C4 | semidirect |
| 9. D5×D9 | C4 | semidirect |
| 11. C9⋊AGL(1,𝔽5) | C22 | semidirect |
| 12. C9×AGL(1,𝔽5) | C22 | semidirect |
| 18. C2×C5×C9 | C2×C4 | semidirect |
| 19. D45 | C2×C4 | semidirect |
| 20. C9×D5 | C2×C4 | semidirect |
| 22. C5×D9 | C2×C4 | semidirect |
| 36. C5×C9 | C22×C4 | semidirect |
| 40. C2×AGL(1,𝔽5) | D9 | direct (N × H) |
| 45. C2×D9 | AGL(1,𝔽5) | direct (N × H) |
| 63. AGL(1,𝔽5) | C2×D9 | direct (N × H) |
| 66. C2×C9 | C2×AGL(1,𝔽5) | semidirect |
| 68. D9 | C2×AGL(1,𝔽5) | direct (N × H) |
| 79. C2×C5 | C4×D9 | semidirect |
| 81. D5 | C4×D9 | semidirect |
| 83. C9 | C22×AGL(1,𝔽5) | semidirect |
| 93. C5 | C2×C4×D9 | semidirect |
| 106. C2 | D9×AGL(1,𝔽5) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D5×D9 | 2 |
| C2×C9⋊AGL(1,𝔽5) | 2 |
| C2×C9×AGL(1,𝔽5) | 2 |
| D9×AGL(1,𝔽5) | 2 |
| C2×D3×AGL(1,𝔽5) | 3 |
| C2×C4×D9 | 5 |