Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
1474 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1474 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24×C5 |
| Frattini | C2 | C23×C5×S4 |
| Fitting | C2×C5×H(5,𝔽2) | D3 |
| hypercenter | C2×C5×Q8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×#994 | C2 | semidirect |
| 2. C5×#5924 | C2 | direct (N × H) |
| 3. C2×C5×#985 | C2 | semidirect |
| 4. C2×C5×#984 | C2 | semidirect |
| 7. C5×#994 | C22 | semidirect |
| 9. C5×#985 | C22 | semidirect |
| 10. C5×#984 | C22 | semidirect |
| 11. C2×C5×GL(2,𝔽3) | C22 | semidirect |
| 12. C2×C5×#226 | C22 | semidirect |
| 13. C2×#5924 | C5 | direct (N × H) |
| 15. C2×C5×H(5,𝔽2) | D3 | semidirect |
| 29. C5×GL(2,𝔽3) | C23 | semidirect |
| 30. C5×#226 | C23 | semidirect |
| 31. C2×#994 | C2×C5 | semidirect |
| 32. #5924 | C2×C5 | direct (N × H) |
| 33. C2×#985 | C2×C5 | semidirect |
| 34. C2×#984 | C2×C5 | semidirect |
| 38. C5×H(5,𝔽2) | C2×D3 | semidirect |
| 69. #994 | C22×C5 | semidirect |
| 71. #985 | C22×C5 | semidirect |
| 72. #984 | C22×C5 | semidirect |
| 73. C2×GL(2,𝔽3) | C22×C5 | semidirect |
| 74. C2×#226 | C22×C5 | semidirect |
| 111. C2×H(5,𝔽2) | C5×D3 | semidirect |
| 130. GL(2,𝔽3) | C23×C5 | semidirect |
| 131. #226 | C23×C5 | semidirect |
| 160. H(5,𝔽2) | C2×C5×D3 | semidirect |
| 245. C2×C5 | #5924 | direct (N × H) |
| 275. C5 | C2×#5924 | direct (N × H) |
| 289. C2 | C5×#5924 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#994 | 2 |
| C5×#5924 | 2 |
| C2×C5×#985 | 2 |
| C2×C5×#984 | 2 |
| C2×C5×#577 | 3 |
| C2×C5×D3×Q8 | 4 |
| C2×#5924 | 5 |