Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#123 | C2 |
| prime radical | C2×D4×D5 | C2 |
| hypercenter | #123 | D5 |
| hypocenter | C5 | C2×#123 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C4×D5 | C2 | semidirect |
| 3. C5×#123 | C2 | semidirect |
| 4. #4081 | C2 | semidirect |
| 5. #4075 | C2 | semidirect |
| 6. #4040 | C2 | semidirect |
| 7. D5×C4⋊C4 | C2 | semidirect |
| 8. #4032 | C2 | semidirect |
| 9. #4029 | C2 | semidirect |
| 10. D5×C22⋊C4 | C2 | semidirect |
| 11. D5×C22⋊C4 | C2 | semidirect |
| 15. C22×C4×C5 | C22 | semidirect |
| 18. C22×Dic5 | C22 | semidirect |
| 25. C2×C4×D5 | C22 | semidirect |
| 26. C5×C4⋊C4 | C22 | semidirect |
| 27. C5×C22⋊C4 | C22 | semidirect |
| 28. C5×C22⋊C4 | C22 | semidirect |
| 29. C22⋊Dic5 | C22 | semidirect |
| 30. C22⋊Dic5 | C22 | semidirect |
| 31. #685 | C22 | semidirect |
| 32. #685 | C22 | semidirect |
| 33. #685 | C22 | semidirect |
| 34. C4⋊Dic5 | C22 | semidirect |
| 35. #683 | C22 | semidirect |
| 36. #683 | C22 | semidirect |
| 70. #123 | D5 | direct (N × H) |
| 106. C22×C4 | C2×D5 | semidirect |
| 110. C4⋊C4 | C2×D5 | semidirect |
| 116. C22⋊C4 | C2×D5 | semidirect |
| 117. C22⋊C4 | C2×D5 | semidirect |
| 124. D5 | #123 | direct (N × H) |
| 150. C5 | C2×#123 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×D5 | 2 |
| C22×C4×D5 | 2 |
| C5×#123 | 2 |
| #4081 | 2 |
| #4075 | 2 |
| #4040 | 2 |
| D5×C4⋊C4 | 2 |
| #4032 | 2 |
| #4029 | 2 |
| D5×C22⋊C4 | 2 |
| C2×#123 | 5 |