Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 258 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 | C2 | C22×C22⋊D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C5×H(5,𝔽2) | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | H(5,𝔽2) | D5 |
| hypocenter | C5 | #577 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ H(5,𝔽2) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×H(5,𝔽2) | C2 | semidirect |
| 2. #4150 | C2 | semidirect |
| 3. #4097 | C2 | semidirect |
| 4. #4096 | C2 | semidirect |
| 5. #4094 | C2 | semidirect |
| 6. C2×#687 | C2 | semidirect |
| 7. #4078 | C2 | semidirect |
| 15. #709 | C22 | semidirect |
| 16. C2×Dic10 | C22 | semidirect |
| 17. #689 | C22 | semidirect |
| 18. #688 | C22 | semidirect |
| 19. #687 | C22 | semidirect |
| 20. #687 | C22 | semidirect |
| 21. #686 | C22 | semidirect |
| 22. #681 | C22 | semidirect |
| 23. C2×#168 | C22 | semidirect |
| 43. H(5,𝔽2) | D5 | semidirect |
| 94. C5 | #577 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×H(5,𝔽2) | 2 |
| #4150 | 2 |
| #4097 | 2 |
| #4096 | 2 |
| #4094 | 2 |
| C2×#687 | 2 |
| #4078 | 2 |
| #577 | 5 |