Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
376 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 50 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 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C8×C29 |
| Frattini, cosocle | #390 | C22×C29 |
| prime radical | C2×C29×H(5,𝔽2) | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C23×C29 ◁ C22×C4×C29 ◁ C29×#390 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C29×#1318 | C2 | semidirect |
| 3. C29×#1270 | C2 | semidirect |
| 14. C29×H(5,𝔽2) | C8 | semidirect |
| 71. #7344 | C29 | direct (N × H) |
| 114. #1318 | C2×C29 | semidirect |
| 115. #1270 | C2×C29 | semidirect |
| 157. H(5,𝔽2) | C8×C29 | semidirect |
| 173. C29 | #7344 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#1792 | 2 |
| C29×#1318 | 2 |
| C29×#1270 | 2 |
| #7344 | 29 |