Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
368 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 368 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 | C22×D4×C29 | D4 |
| 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 |
|---|---|---|
| 2. C29×#2060 | C2 | semidirect |
| 3. C29×#1617 | C2 | semidirect |
| 5. C29×#1353 | C2 | semidirect |
| 7. C29×#1325 | C2 | semidirect |
| 105. #12238 | C29 | direct (N × H) |
| 147. #2060 | C2×C29 | semidirect |
| 148. #1617 | C2×C29 | semidirect |
| 150. #1353 | C2×C29 | semidirect |
| 152. #1325 | C2×C29 | semidirect |
| 236. C29 | #12238 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#2795 | 2 |
| C29×#2060 | 2 |
| C29×#1617 | 2 |
| C29×#1599 | 2 |
| C29×#1353 | 2 |
| C29×#1345 | 2 |
| C29×#1325 | 2 |
| #12238 | 29 |