Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
318 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 46 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×D4 | C2×C8×C29 |
| Frattini, cosocle | #408 | C22×C29 |
| socle | C2×C29 | #1316 |
| prime radical | C29×#585 | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C2×C4×C29 ◁ C22×C4×C29 ◁ C29×#408 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C29×#1321 | C2 | semidirect |
| 3. C29×#1315 | C2 | semidirect |
| 67. #7346 | C29 | direct (N × H) |
| 100. #1321 | C2×C29 | semidirect |
| 101. #1315 | C2×C29 | semidirect |
| 147. C29 | #7346 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#1794 | 2 |
| C29×#1321 | 2 |
| C29×#1315 | 2 |
| #7346 | 29 |