Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C5×#135 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | #135 | D5 |
| hypocenter | C5 | #578 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #4150 | C2 | semidirect |
| 2. C5×#135 | C2 | semidirect |
| 3. #4097 | C2 | semidirect |
| 4. D5×Q16 | C2 | semidirect |
| 5. #4061 | C2 | semidirect |
| 6. #4058 | C2 | semidirect |
| 7. C2×Dic20 | C2 | semidirect |
| 8. #4047 | C2 | semidirect |
| 52. #135 | D5 | semidirect |
| 106. C5 | #578 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #4150 | 2 |
| C5×#135 | 2 |
| #4097 | 2 |
| D5×Q16 | 2 |
| #4061 | 2 |
| #4058 | 2 |
| C2×Dic20 | 2 |
| #4047 | 2 |
| #578 | 5 |