Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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 | C4×C5×D4 | C2 |
| hypercenter | C4×D4 | D5 |
| hypocenter | C5 | #550 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#709 | C2 | semidirect |
| 2. C4×C5×D4 | C2 | semidirect |
| 3. #4086 | C2 | semidirect |
| 4. #4076 | C2 | semidirect |
| 5. #4039 | C2 | semidirect |
| 6. #4030 | C2 | semidirect |
| 7. #4020 | C2 | semidirect |
| 8. C4×Dic10 | C2 | semidirect |
| 43. Dic10 | D4 | semidirect |
| 51. C4×D4 | D5 | semidirect |
| 62. C4×C5 | #41 | semidirect |
| 104. C5 | #550 | semidirect |
| 116. C4 | #709 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#709 | 2 |
| C4×C5×D4 | 2 |
| #4086 | 2 |
| #4076 | 2 |
| #4039 | 2 |
| #4030 | 2 |
| #4020 | 2 |
| C4×Dic10 | 2 |
| #550 | 5 |