Finite Group DatabasebetaFinite Group DatabasebetaOrder 960.
304 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23×C5 |
| Frattini | C2 | C22×C5×S4 |
| Fitting | C5×2−1+4 | D3 |
| hypercenter | C5×D4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C5 ◁ C5×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×2−1+4⋊C3 | C2 | semidirect |
| 2. C5×#986 | C2 | semidirect |
| 3. C5×#985 | C2 | semidirect |
| 4. C5×#983 | C2 | semidirect |
| 5. C2×C5×GL(2,𝔽3) | C2 | semidirect |
| 8. C5×#41⋊C3 | C22 | semidirect |
| 9. C2×C5×SL(2,𝔽3) | C22 | semidirect |
| 10. C5×GL(2,𝔽3) | C22 | semidirect |
| 11. C5×GL(2,𝔽3) | C22 | semidirect |
| 12. C5×#226 | C22 | semidirect |
| 13. #5926 | C5 | direct (N × H) |
| 14. C5×2−1+4 | D3 | semidirect |
| 31. 2−1+4⋊C3 | C2×C5 | semidirect |
| 32. #986 | C2×C5 | semidirect |
| 33. #985 | C2×C5 | semidirect |
| 34. #983 | C2×C5 | semidirect |
| 35. C2×GL(2,𝔽3) | C2×C5 | semidirect |
| 41. C5×#41 | C2×D3 | semidirect |
| 43. C2×C5×Q8 | C2×D3 | semidirect |
| 70. #41⋊C3 | C22×C5 | semidirect |
| 71. C2×SL(2,𝔽3) | C22×C5 | semidirect |
| 72. GL(2,𝔽3) | C22×C5 | semidirect |
| 73. GL(2,𝔽3) | C22×C5 | semidirect |
| 74. #226 | C22×C5 | semidirect |
| 98. 2−1+4 | C5×D3 | semidirect |
| 134. #41 | C2×C5×D3 | semidirect |
| 136. C2×Q8 | C2×C5×D3 | semidirect |
| 194. C5 | #5926 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×2−1+4⋊C3 | 2 |
| C5×#986 | 2 |
| C5×#985 | 2 |
| C5×#983 | 2 |
| C2×C5×GL(2,𝔽3) | 2 |
| C5×#577 | 3 |
| C5×D3×D4 | 4 |
| #5926 | 5 |