Finite Group DatabasebetaFinite Group DatabasebetaOrder 3200.
12000 conjugacy classes of subgroups; only the 2326 normal subgroups are shown.
This group has 12000 conjugacy classes of subgroups — too many to draw. Only its 2326 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | #1043 |
| commutator, hypocenter | C52 | C23×C42 |
| Frattini | C2 | C24×#1043 |
| Fitting | C23×C4×C52 | C4 |
| socle | C24×C52 | C2×C4 |
| prime radical | C24×C5⋊D5 | C22 |
| cosocle | C2×C5⋊D5 | C25 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C22×C4×#1043 | C2 | direct (N × H) |
| 5. C23×#1043 | C4 | direct (N × H) |
| 7. C23×C4×C52 | C4 | semidirect |
| 8. C23×#1039 | C4 | semidirect |
| 9. C22×C4×C5⋊D5 | C4 | semidirect |
| 11. C2×C4×#1043 | C22 | direct (N × H) |
| 17. C22×#1043 | C2×C4 | direct (N × H) |
| 19. C22×C4×C52 | C2×C4 | semidirect |
| 20. C22×#1039 | C2×C4 | semidirect |
| 21. C2×C4×C5⋊D5 | C2×C4 | semidirect |
| 23. C4×#1043 | C23 | direct (N × H) |
| 27. C23×C52 | C42 | semidirect |
| 29. C22×C5⋊D5 | C42 | semidirect |
| 33. C2×#1043 | C22×C4 | direct (N × H) |
| 35. C2×C4×C52 | C22×C4 | semidirect |
| 36. C2×#1039 | C22×C4 | semidirect |
| 37. C4×C5⋊D5 | C22×C4 | semidirect |
| 42. C23×C4×C5 | AGL(1,𝔽5) | semidirect |
| 48. C22×C52 | C2×C42 | semidirect |
| 50. C2×C5⋊D5 | C2×C42 | semidirect |
| 54. #1043 | C23×C4 | direct (N × H) |
| 55. C4×C52 | C23×C4 | semidirect |
| 56. #1039 | C23×C4 | semidirect |
| 63. C22×C4×C5 | C2×AGL(1,𝔽5) | semidirect |
| 71. C2×C52 | C22×C42 | semidirect |
| 73. C5⋊D5 | C22×C42 | semidirect |
| 75. C23×C5 | C4×AGL(1,𝔽5) | semidirect |
| 83. C2×C4×C5 | C22×AGL(1,𝔽5) | semidirect |
| 93. C23×C4 | #1043 | direct (N × H) |
| 95. C52 | C23×C42 | semidirect |
| 96. C22×C5 | C2×C4×AGL(1,𝔽5) | semidirect |
| 103. C4×C5 | C23×AGL(1,𝔽5) | semidirect |
| 114. C22×C4 | C2×#1043 | direct (N × H) |
| 116. C2×C5 | C22×C4×AGL(1,𝔽5) | semidirect |
| 124. C23 | C4×#1043 | direct (N × H) |
| 131. C2×C4 | C22×#1043 | direct (N × H) |
| 132. C5 | C23×C4×AGL(1,𝔽5) | semidirect |
| 137. C22 | C2×C4×#1043 | direct (N × H) |
| 141. C4 | C23×#1043 | direct (N × H) |
| 144. C2 | C22×C4×#1043 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×#1043 | 2 |
| C23×C4×C5⋊D5 | 2 |
| C22×C4×#1043 | 2 |
| C23×C4×AGL(1,𝔽5) | 5 |