Finite Group DatabasebetaFinite Group DatabasebetaOrder 2000.
140 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C53 | C42 |
| prime radical, cosocle | C5×D52 | C22 |
| LCS | G ▷ C53 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #92561 | C4 | semidirect |
| 6. #92560 | C4 | semidirect |
| 7. #92559 | C4 | semidirect |
| 8. C5×#1042 | C4 | semidirect |
| 9. C5×#1042 | C4 | semidirect |
| 10. C52×AGL(1,𝔽5) | C4 | semidirect |
| 26. C53 | C42 | semidirect |
| 40. #1042 | Dic5 | semidirect |
| 41. #1042 | AGL(1,𝔽5) | direct (N × H) |
| 45. C5×AGL(1,𝔽5) | Dic5 | semidirect |
| 46. C5×AGL(1,𝔽5) | AGL(1,𝔽5) | semidirect |
| 79. C52 | C4×Dic5 | semidirect |
| 80. C52 | C4×AGL(1,𝔽5) | semidirect |
| 81. C52 | C4×AGL(1,𝔽5) | semidirect |
| 93. AGL(1,𝔽5) | #1042 | direct (N × H) |
| 126. C5 | Dic5×AGL(1,𝔽5) | semidirect |
| 127. C5 | C4×#1042 | semidirect |
| 128. C5 | AGL(1,𝔽5)2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×#1042 | 2 |
| D5⋊#1042 | 2 |
| C5×D5×AGL(1,𝔽5) | 2 |
| AGL(1,𝔽5)2 | 5 |
| C4×#1042 | 5 |
| Dic5×AGL(1,𝔽5) | 5 |