Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
408 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 70 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 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23×C29 |
| prime radical | C29×#1598 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C23×C29 ◁ C2×C4⋊C4×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C29×#2785 | C2 | semidirect |
| 2. C29×#2040 | C2 | semidirect |
| 3. C29×#1604 | C2 | semidirect |
| 5. C29×#1354 | C2 | semidirect |
| 6. C29×#1345 | C2 | semidirect |
| 7. C29×#1324 | C2 | semidirect |
| 111. #12145 | C29 | direct (N × H) |
| 162. #2785 | C2×C29 | semidirect |
| 163. #2040 | C2×C29 | semidirect |
| 164. #1604 | C2×C29 | semidirect |
| 166. #1354 | C2×C29 | semidirect |
| 167. #1345 | C2×C29 | semidirect |
| 168. #1324 | C2×C29 | semidirect |
| 256. C29 | #12145 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#2785 | 2 |
| C29×#2040 | 2 |
| C29×#1604 | 2 |
| C29×#1598 | 2 |
| C29×#1354 | 2 |
| C29×#1345 | 2 |
| C29×#1324 | 2 |
| #12145 | 29 |