Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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×#126 | C2 |
| prime radical | C2×C22⋊D5 | C22 |
| hypercenter | #126 | D5 |
| hypocenter | C5 | #563 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#126 | C2 | semidirect |
| 2. #4044 | C2 | semidirect |
| 3. #4043 | C2 | semidirect |
| 4. #4042 | C2 | semidirect |
| 5. #4036 | C2 | semidirect |
| 6. #4035 | C2 | semidirect |
| 7. #4034 | C2 | semidirect |
| 11. #4025 | C2 | semidirect |
| 12. #4024 | C2 | semidirect |
| 13. #4024 | C2 | semidirect |
| 14. #4022 | C2 | semidirect |
| 15. #4015 | C2 | semidirect |
| 78. #126 | D5 | semidirect |
| 169. C5 | #563 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#126 | 2 |
| #4044 | 2 |
| #4043 | 2 |
| #4042 | 2 |
| #4036 | 2 |
| #4035 | 2 |
| #4034 | 2 |
| #4032 | 2 |
| #4031 | 2 |
| #4029 | 2 |
| #4025 | 2 |
| #4024 | 2 |
| #4022 | 2 |
| #4015 | 2 |
| #563 | 5 |