Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
566 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 238 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C29 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C29 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C42×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 41. Q16×C29 | Q16 | direct (N × H) |
| 53. Q162 | C29 | direct (N × H) |
| 112. C29 | Q162 | direct (N × H) |
| 117. Q16 | Q16×C29 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×Q16×C29 | 2 |
| C29×#1713 | 2 |
| C29×#1676 | 2 |
| C29×#1633 | 2 |
| C8×Q16×C29 | 2 |
| Q162 | 29 |