Finite Group DatabasebetaFinite Group DatabasebetaOrder 2160.
100 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 100 conjugacy classes of subgroups — too many to draw. Only its 16 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, cosocle, hypocenter | C5×PSU(3,𝔽2) | C2×C3 |
| Fitting, socle | C32×C5 | C2×SL(2,𝔽3) |
| LCS | G ▷ C5×PSU(3,𝔽2) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×PSU(3,𝔽2)⋊C3 | C2 | semidirect |
| 2. D5×PSU(3,𝔽2) | C3 | semidirect |
| 6. C5×PSU(3,𝔽2) | C2×C3 | semidirect |
| 12. PSU(3,𝔽2)⋊C3 | D5 | direct (N × H) |
| 21. C3⋊D15 | SL(2,𝔽3) | semidirect |
| 27. C32×D5 | SL(2,𝔽3) | semidirect |
| 31. PSU(3,𝔽2) | C3×D5 | semidirect |
| 40. C32×C5 | C2×SL(2,𝔽3) | semidirect |
| 75. D5 | PSU(3,𝔽2)⋊C3 | direct (N × H) |
| 78. C32 | D5×SL(2,𝔽3) | semidirect |
| 89. C5 | C2×PSU(3,𝔽2)⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×PSU(3,𝔽2)⋊C3 | 2 |
| D5×PSU(3,𝔽2) | 3 |
| D5×(C3⋊D3)⋊C3 | 4 |
| C2×PSU(3,𝔽2)⋊C3 | 5 |
| D5×SL(2,𝔽3) | 9 |