Finite Group DatabasebetaFinite Group DatabasebetaOrder 480.
132 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 30 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 | A4×AGL(1,𝔽5) |
| commutator, hypocenter | C22×C5 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C5 | C3×C4 |
| prime radical | C2×D5×A4 | C2 |
| cosocle | C22×D5 | C22×C3 |
| LCS | G ▷ C22×C5 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A4×AGL(1,𝔽5) | C2 | direct (N × H) |
| 3. C23×AGL(1,𝔽5) | C3 | semidirect |
| 4. C2×C5×A4 | C4 | semidirect |
| 6. D5×A4 | C4 | semidirect |
| 13. C22×AGL(1,𝔽5) | C2×C3 | semidirect |
| 15. C5×A4 | C2×C4 | semidirect |
| 19. C23×C5 | C3×C4 | semidirect |
| 24. C22×D5 | C3×C4 | semidirect |
| 31. C2×AGL(1,𝔽5) | A4 | direct (N × H) |
| 38. C2×A4 | AGL(1,𝔽5) | direct (N × H) |
| 42. C22×C5 | C2×C3×C4 | semidirect |
| 53. AGL(1,𝔽5) | C2×A4 | direct (N × H) |
| 61. A4 | C2×AGL(1,𝔽5) | direct (N × H) |
| 65. C2×C5 | C4×A4 | semidirect |
| 68. D5 | C4×A4 | semidirect |
| 76. C23 | C3×AGL(1,𝔽5) | semidirect |
| 86. C5 | C2×C4×A4 | semidirect |
| 98. C22 | C2×C3×AGL(1,𝔽5) | semidirect |
| 108. C2 | A4×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×A4 | 2 |
| A4×AGL(1,𝔽5) | 2 |
| C23×AGL(1,𝔽5) | 3 |
| C2×C3×AGL(1,𝔽5) | 4 |
| C2×C4×A4 | 5 |