Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
238 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 238 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 |
| hypercenter | #123 | D5 |
| hypocenter | C5 | #553 |
| 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 |
| 2. #4085 | C2 | semidirect |
| 3. #4084 | C2 | semidirect |
| 4. C4×C22⋊D5 | C2 | semidirect |
| 5. C4×C22⋊D5 | C2 | semidirect |
| 6. #4044 | C2 | semidirect |
| 7. #4039 | C2 | semidirect |
| 8. D5×C4⋊C4 | C2 | semidirect |
| 9. #4035 | C2 | semidirect |
| 10. #4030 | C2 | semidirect |
| 11. #4030 | C2 | semidirect |
| 12. #4029 | C2 | semidirect |
| 13. #4027 | C2 | semidirect |
| 14. #4025 | C2 | semidirect |
| 15. #4023 | C2 | semidirect |
| 83. #123 | D5 | semidirect |
| 187. C5 | #553 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#123 | 2 |
| #4085 | 2 |
| #4084 | 2 |
| C4×C22⋊D5 | 2 |
| #4044 | 2 |
| #4039 | 2 |
| D5×C4⋊C4 | 2 |
| #4035 | 2 |
| #4030 | 2 |
| #4029 | 2 |
| #4027 | 2 |
| #4025 | 2 |
| #4023 | 2 |
| #553 | 5 |