Finite Group DatabasebetaFinite Group DatabasebetaOrder 320.
280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 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 | C22×C5×Q8 | C2 |
| prime radical | C2×C22⋊D5 | C22 |
| hypercenter | C22×Q8 | D5 |
| hypocenter | C5 | D4×Q8 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C5×Q8 | C2 | semidirect |
| 2. C2×Q8×D5 | C2 | semidirect |
| 3. #4092 | C2 | semidirect |
| 4. Q8×Dic5 | C2 | semidirect |
| 5. #4090 | C2 | semidirect |
| 7. #4070 | C2 | semidirect |
| 26. C5×Q8 | D4 | semidirect |
| 30. C22⋊D5 | Q8 | direct (N × H) |
| 37. C22×Q8 | D5 | semidirect |
| 45. C22×C5 | C2×Q8 | semidirect |
| 49. C2×D5 | C2×Q8 | semidirect |
| 54. Dic5 | C2×Q8 | semidirect |
| 78. Q8 | C22⋊D5 | direct (N × H) |
| 87. C5 | D4×Q8 | semidirect |
| 93. C22 | Q8×D5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C5×Q8 | 2 |
| C2×Q8×D5 | 2 |
| #4092 | 2 |
| Q8×Dic5 | 2 |
| #4090 | 2 |
| C4×C22⋊D5 | 2 |
| #4070 | 2 |
| D4×Q8 | 5 |