Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
1232 conjugacy classes of subgroups; only the 330 normal subgroups are shown.
This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 330 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | #1044 |
| commutator, hypocenter | C52 | C22×C42 |
| Frattini | C2 | C23×#1044 |
| Fitting | C22×C4×C52 | C4 |
| socle | C23×C52 | C2×C4 |
| prime radical | C23×C5⋊D5 | C22 |
| cosocle | C2×C5⋊D5 | C24 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C4×#1044 | C2 | direct (N × H) |
| 5. C22×#1044 | C4 | direct (N × H) |
| 7. C22×C4×C52 | C4 | semidirect |
| 8. C22×#1039 | C4 | semidirect |
| 9. C2×C4×C5⋊D5 | C4 | semidirect |
| 11. C4×#1044 | C22 | direct (N × H) |
| 17. C2×#1044 | C2×C4 | direct (N × H) |
| 19. C2×C4×C52 | C2×C4 | semidirect |
| 20. C2×#1039 | C2×C4 | semidirect |
| 21. C4×C5⋊D5 | C2×C4 | semidirect |
| 27. C22×C52 | C42 | semidirect |
| 29. C2×C5⋊D5 | C42 | semidirect |
| 33. #1044 | C22×C4 | direct (N × H) |
| 34. C4×C52 | C22×C4 | semidirect |
| 35. #1039 | C22×C4 | semidirect |
| 41. C22×C4×C5 | AGL(1,𝔽5) | semidirect |
| 50. C2×C52 | C2×C42 | semidirect |
| 52. C5⋊D5 | C2×C42 | semidirect |
| 65. C2×C4×C5 | C2×AGL(1,𝔽5) | semidirect |
| 75. C52 | C22×C42 | semidirect |
| 78. C22×C5 | C4×AGL(1,𝔽5) | semidirect |
| 90. C4×C5 | C22×AGL(1,𝔽5) | semidirect |
| 99. C22×C4 | #1044 | direct (N × H) |
| 103. C2×C5 | C2×C4×AGL(1,𝔽5) | semidirect |
| 118. C2×C4 | C2×#1044 | direct (N × H) |
| 120. C5 | C22×C4×AGL(1,𝔽5) | semidirect |
| 125. C22 | C4×#1044 | direct (N × H) |
| 129. C4 | C22×#1044 | direct (N × H) |
| 132. C2 | C2×C4×#1044 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#1044 | 2 |
| C22×C4×C5⋊D5 | 2 |
| C2×C4×#1044 | 2 |
| C22×C4×AGL(1,𝔽5) | 5 |