Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
248 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 95 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 | #4083 | C2 |
| hypercenter | #123 | D5 |
| hypocenter | C5 | #551 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#123 | C2 | semidirect |
| 3. #4082 | C2 | semidirect |
| 4. C4×C22⋊D5 | C2 | semidirect |
| 5. #4042 | C2 | semidirect |
| 6. #4038 | C2 | semidirect |
| 7. #4029 | C2 | semidirect |
| 8. D5×C22⋊C4 | C2 | semidirect |
| 9. #4025 | C2 | semidirect |
| 12. C22×C4×C5 | C22 | semidirect |
| 16. C2×Dic10 | C22 | semidirect |
| 17. C5×C4⋊C4 | C22 | semidirect |
| 19. C5×C22⋊C4 | C22 | semidirect |
| 22. C22⋊Dic5 | C22 | semidirect |
| 26. C4⋊Dic5 | C22 | semidirect |
| 27. #683 | C22 | semidirect |
| 28. #683 | C22 | semidirect |
| 29. C4×Dic5 | C22 | semidirect |
| 30. C4×Dic5 | C22 | semidirect |
| 52. #123 | D5 | semidirect |
| 79. C22×C4 | C2×D5 | semidirect |
| 83. C4⋊C4 | C2×D5 | semidirect |
| 91. C22⋊C4 | C2×D5 | semidirect |
| 116. C5 | #551 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#123 | 2 |
| #4083 | 2 |
| #4082 | 2 |
| C4×C22⋊D5 | 2 |
| #4042 | 2 |
| #4038 | 2 |
| #4029 | 2 |
| D5×C22⋊C4 | 2 |
| #4025 | 2 |
| #551 | 5 |