Finite Group DatabasebetaFinite Group Databasebeta← C2×C3×D3×AGL(1,𝔽5) #10624333
Order 720.
232 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 84 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×C3 | D3×AGL(1,𝔽5) |
| commutator, hypocenter | C3×C5 | C22×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×C3×D3×D5 | C2 |
| cosocle | C3×D5 | C23×C3 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×ΓL(1,𝔽16) | C2 | semidirect |
| 3. C2×C32×AGL(1,𝔽5) | C2 | semidirect |
| 4. C3×D3×AGL(1,𝔽5) | C2 | direct (N × H) |
| 6. C2×D3×AGL(1,𝔽5) | C3 | direct (N × H) |
| 7. C2×C3×D15 | C4 | semidirect |
| 8. C2×C3×C5×D3 | C4 | semidirect |
| 10. C3×D3×D5 | C4 | semidirect |
| 12. C3×ΓL(1,𝔽16) | C22 | semidirect |
| 13. C32×AGL(1,𝔽5) | C22 | semidirect |
| 17. C2×ΓL(1,𝔽16) | C2×C3 | semidirect |
| 21. C2×C3×AGL(1,𝔽5) | D3 | direct (N × H) |
| 22. C2×C3×AGL(1,𝔽5) | C2×C3 | semidirect |
| 23. D3×AGL(1,𝔽5) | C2×C3 | direct (N × H) |
| 24. C2×C32×C5 | C2×C4 | semidirect |
| 25. C3×D15 | C2×C4 | semidirect |
| 26. C3×C5×D3 | C2×C4 | semidirect |
| 27. C32×D5 | C2×C4 | semidirect |
| 35. C2×D15 | C3×C4 | semidirect |
| 36. C2×C5×D3 | C3×C4 | semidirect |
| 43. D3×D5 | C3×C4 | semidirect |
| 45. ΓL(1,𝔽16) | C22×C3 | semidirect |
| 48. C3×AGL(1,𝔽5) | C2×D3 | direct (N × H) |
| 49. C3×AGL(1,𝔽5) | C22×C3 | semidirect |
| 52. C32×C5 | C22×C4 | semidirect |
| 56. C2×AGL(1,𝔽5) | C3×D3 | direct (N × H) |
| 61. C2×C3×D3 | AGL(1,𝔽5) | direct (N × H) |
| 66. C2×C3×C5 | C4×D3 | semidirect |
| 67. C2×C3×C5 | C2×C3×C4 | semidirect |
| 68. D15 | C2×C3×C4 | semidirect |
| 72. C3×D5 | C4×D3 | semidirect |
| 73. C3×D5 | C2×C3×C4 | semidirect |
| 76. C5×D3 | C2×C3×C4 | semidirect |
| 92. AGL(1,𝔽5) | C2×C3×D3 | direct (N × H) |
| 95. C2×C32 | C2×AGL(1,𝔽5) | semidirect |
| 97. C3×D3 | C2×AGL(1,𝔽5) | direct (N × H) |
| 100. C3×C5 | C2×C4×D3 | semidirect |
| 101. C3×C5 | C22×C3×C4 | semidirect |
| 112. C2×D3 | C3×AGL(1,𝔽5) | direct (N × H) |
| 119. C2×C5 | C3×C4×D3 | semidirect |
| 121. D5 | C3×C4×D3 | semidirect |
| 123. C32 | C22×AGL(1,𝔽5) | semidirect |
| 137. C2×C3 | D3×AGL(1,𝔽5) | direct (N × H) |
| 138. C2×C3 | C2×C3×AGL(1,𝔽5) | semidirect |
| 140. D3 | C2×C3×AGL(1,𝔽5) | direct (N × H) |
| 141. C5 | C2×C3×C4×D3 | semidirect |
| 150. C3 | C2×D3×AGL(1,𝔽5) | direct (N × H) |
| 151. C3 | C22×C3×AGL(1,𝔽5) | semidirect |
| 156. C2 | C3×D3×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×C3×D3×D5 | 2 |
| C2×C3×ΓL(1,𝔽16) | 2 |
| C2×C32×AGL(1,𝔽5) | 2 |
| C3×D3×AGL(1,𝔽5) | 2 |
| C22×C3×AGL(1,𝔽5) | 3 |
| C2×D3×AGL(1,𝔽5) | 3 |
| C2×C3×C4×D3 | 5 |