Finite Group DatabasebetaFinite Group DatabasebetaOrder 480.
112 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C5×A4 | C2×C4 |
| Fitting, socle | C22×C5 | C4×D3 |
| prime radical | D5×S4 | C2 |
| cosocle | D5×A4 | C22 |
| LCS | G ▷ C5×A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A4×AGL(1,𝔽5) | C2 | semidirect |
| 3. A4⋊AGL(1,𝔽5) | C2 | semidirect |
| 6. C5⋊S4 | C4 | semidirect |
| 7. C5×S4 | C4 | semidirect |
| 11. C22×AGL(1,𝔽5) | D3 | semidirect |
| 17. C5×A4 | C2×C4 | semidirect |
| 43. S4 | AGL(1,𝔽5) | direct (N × H) |
| 46. C22×C5 | C4×D3 | semidirect |
| 55. AGL(1,𝔽5) | S4 | direct (N × H) |
| 67. A4 | C2×AGL(1,𝔽5) | semidirect |
| 91. C5 | C4×S4 | semidirect |
| 98. C22 | D3×AGL(1,𝔽5) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×S4 | 2 |
| A4×AGL(1,𝔽5) | 2 |
| A4⋊AGL(1,𝔽5) | 2 |
| D4×AGL(1,𝔽5) | 3 |
| D3×AGL(1,𝔽5) | 4 |
| C4×S4 | 5 |