Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
634 conjugacy classes of subgroups; only the 286 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 286 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C29 |
| prime radical | C23×C29 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C22×C4×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 28. C29×#108 | Q8 | semidirect |
| 37. C4×C8×C29 | Q8 | semidirect |
| 58. #28824 | C29 | direct (N × H) |
| 113. #108 | Q8×C29 | semidirect |
| 122. C4×C8 | Q8×C29 | semidirect |
| 123. C29 | #28824 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#3163 | 2 |
| C29×#2060 | 2 |
| C29×#1951 | 2 |
| C29×#1844 | 2 |
| C29×#1772 | 2 |
| #28824 | 29 |