Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
292 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 292 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 | #547 |
| 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. #4084 | C2 | semidirect |
| 4. #4070 | C2 | semidirect |
| 5. D5×C4⋊C4 | C2 | semidirect |
| 6. #4029 | C2 | semidirect |
| 7. C4×D20 | C2 | semidirect |
| 8. #4015 | C2 | semidirect |
| 38. D20 | D4 | semidirect |
| 48. C4×D4 | D5 | semidirect |
| 60. C4×C5 | #41 | semidirect |
| 100. C5 | #547 | semidirect |
| 112. 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 |
| #4084 | 2 |
| #4070 | 2 |
| D5×C4⋊C4 | 2 |
| #4029 | 2 |
| C4×D20 | 2 |
| #4015 | 2 |
| #547 | 5 |