Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
730 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 120 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 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3×C5 |
| Frattini | C22 | C23×C5×A4 |
| Fitting | C5×#3500 | C3 |
| hypercenter | C5×#124 | A4 |
| hypocenter | Q8 | C2×C3×C5×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×2−1+4⋊C3 | C2 | semidirect |
| 2. C2×C5×#994 | C2 | semidirect |
| 3. C5×#548⋊C3 | C2 | semidirect |
| 4. C4×C5×#41⋊C3 | C2 | semidirect |
| 5. C5×#3500 | C3 | semidirect |
| 10. C22×C5×SL(2,𝔽3) | C22 | semidirect |
| 12. C4×C5×SL(2,𝔽3) | C22 | semidirect |
| 13. #3500⋊C3 | C5 | direct (N × H) |
| 15. C2×C5×H(5,𝔽2) | C2×C3 | semidirect |
| 16. C5×#548 | C2×C3 | semidirect |
| 22. C4×C5×#41 | C2×C3 | semidirect |
| 26. C5×#41⋊C3 | D4 | semidirect |
| 31. C2×2−1+4⋊C3 | C2×C5 | semidirect |
| 32. C2×#994 | C2×C5 | semidirect |
| 33. #548⋊C3 | C2×C5 | semidirect |
| 34. C4×#41⋊C3 | C2×C5 | semidirect |
| 65. #3500 | C3×C5 | semidirect |
| 77. C22×SL(2,𝔽3) | C22×C5 | semidirect |
| 79. C4×SL(2,𝔽3) | C22×C5 | semidirect |
| 125. C2×H(5,𝔽2) | C2×C3×C5 | semidirect |
| 126. #548 | C2×C3×C5 | semidirect |
| 132. C4×#41 | C2×C3×C5 | semidirect |
| 142. #41⋊C3 | C5×D4 | semidirect |
| 329. C5 | #3500⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×2−1+4⋊C3 | 2 |
| C2×C5×#994 | 2 |
| C5×#548⋊C3 | 2 |
| C4×C5×#41⋊C3 | 2 |
| C5×#3500 | 3 |
| C3×C5×#124 | 4 |
| #3500⋊C3 | 5 |