Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
124 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 48 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 | C22×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting, prime radical | C2×C5×D4 | C4 |
| cosocle | C2×Dic5 | C23 |
| hypercenter | C2×D4 | AGL(1,𝔽5) |
| hypocenter | C5 | #435 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#704 | C2 | semidirect |
| 2. C22×#170 | C2 | semidirect |
| 4. C22⋊#170 | C2 | semidirect |
| 5. C22⋊#170 | C2 | semidirect |
| 6. #4004 | C2 | semidirect |
| 7. #4002 | C2 | semidirect |
| 37. #170 | D4 | semidirect |
| 50. C22×C5 | M16 | semidirect |
| 92. C5 | #435 | semidirect |
| 98. C22 | #704 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#704 | 2 |
| C22×#170 | 2 |
| D4×Dic5 | 2 |
| C22⋊#170 | 2 |
| #4004 | 2 |
| #4002 | 2 |
| #435 | 5 |