Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
1606 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×C5⋊S4 |
| commutator, cosocle, hypocenter | C5×SL(2,𝔽3) | C24 |
| Frattini | C2 | C23×C5⋊S4 |
| Fitting | C23×C5×Q8 | D3 |
| socle | C23×C5 | C2×S4 |
| hypercenter | C24 | C5⋊S4 |
| LCS | G ▷ C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C5×SL(2,𝔽3) | C2 | semidirect |
| 2. C2×#91953 | C2 | direct (N × H) |
| 3. C22×C5⋊GL(2,𝔽3) | C2 | semidirect |
| 4. C22×C5⋊#226 | C2 | semidirect |
| 8. C22×C5×SL(2,𝔽3) | C22 | semidirect |
| 9. #91953 | C22 | direct (N × H) |
| 10. C2×C5⋊GL(2,𝔽3) | C22 | semidirect |
| 11. C2×C5⋊#226 | C22 | semidirect |
| 13. C23×C5×Q8 | D3 | semidirect |
| 27. C2×C5×SL(2,𝔽3) | C23 | semidirect |
| 28. C5⋊GL(2,𝔽3) | C23 | semidirect |
| 29. C5⋊#226 | C23 | semidirect |
| 30. C23×SL(2,𝔽3) | D5 | semidirect |
| 37. C22×C5×Q8 | C2×D3 | semidirect |
| 67. C22×SL(2,𝔽3) | C2×D5 | semidirect |
| 79. C2×C5×Q8 | C22×D3 | semidirect |
| 103. C23×Q8 | D15 | semidirect |
| 123. C2×SL(2,𝔽3) | C22×D5 | semidirect |
| 156. C22×Q8 | C2×D15 | semidirect |
| 185. C22×C5 | #983 | semidirect |
| 206. C2×Q8 | C22×D15 | semidirect |
| 234. C2×C5 | C2×#983 | semidirect |
| 265. C5 | C22×#983 | semidirect |
| 269. C22 | #91953 | direct (N × H) |
| 279. C2 | C2×#91953 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C5×SL(2,𝔽3) | 2 |
| C2×#91953 | 2 |
| C22×C5⋊GL(2,𝔽3) | 2 |
| C22×C5⋊#226 | 2 |
| C22×#4088 | 3 |
| C22×#1220 | 4 |
| C22×#983 | 5 |