Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
482 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4×C29 |
| Frattini, cosocle | C4⋊Q8 | C23×C29 |
| socle | C2×C29 | C2×AΣL(1,𝔽16) |
| prime radical | C29×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C2×C4×C29 ◁ C22×C4×C29 ◁ C29×#532 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C29×#2132 | C2 | semidirect |
| 3. C29×#1543 | C2 | semidirect |
| 4. C29×#1405 | C2 | semidirect |
| 16. C29×#356 | C22 | semidirect |
| 18. C29×#333 | C22 | semidirect |
| 106. #13572 | C29 | direct (N × H) |
| 150. #2132 | C2×C29 | semidirect |
| 151. #1543 | C2×C29 | semidirect |
| 152. #1405 | C2×C29 | semidirect |
| 182. #356 | C22×C29 | semidirect |
| 184. #333 | C22×C29 | semidirect |
| 225. C29 | #13572 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#3018 | 2 |
| C29×#2132 | 2 |
| C29×#1543 | 2 |
| C29×#1405 | 2 |
| #13572 | 29 |