Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
178 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×C5⋊S4 |
| commutator, cosocle, hypocenter | C5×SL(2,𝔽3) | C22×C3 |
| Frattini | C2 | C2×C3×C5⋊S4 |
| Fitting | C3×C5×#41 | D3 |
| socle | C2×C3×C5 | C2×S4 |
| hypercenter | C3×C4 | C5⋊S4 |
| LCS | G ▷ C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C5×#41⋊C3 | C2 | semidirect |
| 2. C3×C5⋊GL(2,𝔽3) | C2 | semidirect |
| 3. #91956 | C3 | direct (N × H) |
| 6. C3×C5×SL(2,𝔽3) | C22 | semidirect |
| 8. C3×C5×#41 | D3 | semidirect |
| 11. C5×#41⋊C3 | C2×C3 | semidirect |
| 12. C5⋊GL(2,𝔽3) | C2×C3 | semidirect |
| 19. C3×#41⋊C3 | D5 | semidirect |
| 29. C5×SL(2,𝔽3) | C22×C3 | semidirect |
| 35. C5×#41 | C3×D3 | semidirect |
| 41. C3×SL(2,𝔽3) | C2×D5 | semidirect |
| 50. C3×#41 | D15 | semidirect |
| 53. #41⋊C3 | C3×D5 | semidirect |
| 83. SL(2,𝔽3) | C2×C3×D5 | semidirect |
| 92. #41 | C3×D15 | semidirect |
| 99. C3×C5 | #986 | semidirect |
| 125. C5 | C3×#986 | semidirect |
| 133. C3 | #91956 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C5×#41⋊C3 | 2 |
| C3×C5⋊GL(2,𝔽3) | 2 |
| #91956 | 3 |
| C3×#4095 | 3 |
| C3×D60 | 4 |
| C3×#986 | 5 |