Finite Group DatabasebetaFinite Group DatabasebetaOrder 2000.
172 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D5×AGL(1,𝔽5) |
| commutator, hypocenter | C52 | C42×C5 |
| Frattini | C2 | C5×D5×AGL(1,𝔽5) |
| Fitting, socle | C2×C53 | C2×C4 |
| prime radical | C2×C52×D5 | C22 |
| cosocle | C2×C5×D5 | C22×C5 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C2×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C5×#1042 | C4 | semidirect |
| 6. C52×AGL(1,𝔽5) | C4 | semidirect |
| 7. C5×#1039 | C4 | semidirect |
| 8. C52×Dic5 | C4 | semidirect |
| 10. Dic5×AGL(1,𝔽5) | C5 | direct (N × H) |
| 25. C53 | C42 | semidirect |
| 30. #1042 | C4×C5 | semidirect |
| 33. C5×AGL(1,𝔽5) | Dic5 | direct (N × H) |
| 34. C5×AGL(1,𝔽5) | C4×C5 | semidirect |
| 37. #1039 | C4×C5 | semidirect |
| 43. C5×Dic5 | C4×C5 | semidirect |
| 44. C5×Dic5 | AGL(1,𝔽5) | direct (N × H) |
| 77. C52 | C4×Dic5 | semidirect |
| 78. C52 | C42×C5 | semidirect |
| 79. C52 | C4×AGL(1,𝔽5) | semidirect |
| 85. AGL(1,𝔽5) | C5×Dic5 | direct (N × H) |
| 98. Dic5 | C5×AGL(1,𝔽5) | direct (N × H) |
| 122. C5 | C4×C5×Dic5 | semidirect |
| 123. C5 | Dic5×AGL(1,𝔽5) | direct (N × H) |
| 124. C5 | C4×C5×AGL(1,𝔽5) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#1042 | 2 |
| C2×C52×AGL(1,𝔽5) | 2 |
| C5×D5×Dic5 | 2 |
| C4×C5×AGL(1,𝔽5) | 5 |
| Dic5×AGL(1,𝔽5) | 5 |
| C4×C5×Dic5 | 5 |