Finite Group DatabasebetaFinite Group DatabasebetaOrder 4704.
540 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | #64425 |
| commutator, hypocenter | C72 | C2×C3×C42 |
| Frattini | C22 | C22×#64425 |
| Fitting | C42×C72 | C2×C3 |
| socle, cosocle | C22×C72 | C23×C3 |
| prime radical | C22×#64425 | C22 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4×#10596875 | C2 | semidirect |
| 2. C42×#3862 | C2 | semidirect |
| 4. C42×C7⋊D7 | C3 | semidirect |
| 8. C4×#64425 | C4 | direct (N × H) |
| 10. C42×C72 | C2×C3 | semidirect |
| 11. C4×C7⋊Dic7 | C2×C3 | semidirect |
| 15. C4×#3862 | C2×C4 | semidirect |
| 16. #10596875 | C2×C4 | semidirect |
| 20. C4×C7⋊D7 | C3×C4 | semidirect |
| 25. #64425 | C42 | direct (N × H) |
| 30. C4×C72 | C2×C3×C4 | semidirect |
| 31. C7⋊Dic7 | C2×C3×C4 | semidirect |
| 36. #3862 | C2×C42 | semidirect |
| 40. C42×C7 | AGL(1,𝔽7) | semidirect |
| 44. C7⋊D7 | C3×C42 | semidirect |
| 58. C72 | C2×C3×C42 | semidirect |
| 70. C4×C7 | C4×AGL(1,𝔽7) | semidirect |
| 80. C42 | #64425 | direct (N × H) |
| 94. C7 | C42×AGL(1,𝔽7) | semidirect |
| 100. C4 | C4×#64425 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#10596875 | 2 |
| C42×#3862 | 2 |
| C2×C4×#64425 | 2 |
| C42×C7⋊D7 | 3 |
| C42×AGL(1,𝔽7) | 7 |