Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
236 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 236 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 | #4086 | C2 |
| hypercenter | #123 | D5 |
| hypocenter | C5 | #559 |
| 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. #4081 | C2 | semidirect |
| 4. C4×C22⋊D5 | C2 | semidirect |
| 5. #4040 | C2 | semidirect |
| 6. #4033 | C2 | semidirect |
| 7. #4031 | C2 | semidirect |
| 8. #4031 | C2 | semidirect |
| 9. #4027 | C2 | semidirect |
| 54. #123 | D5 | semidirect |
| 115. C5 | #559 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#123 | 2 |
| #4086 | 2 |
| #4081 | 2 |
| C4×C22⋊D5 | 2 |
| #4040 | 2 |
| #4033 | 2 |
| #4031 | 2 |
| #4027 | 2 |
| #559 | 5 |