Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
120 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8×C29 | D16 |
| commutator | C16 | C2×C8×C29 |
| Frattini, cosocle | C4×C16 | C22×C29 |
| socle | C2×C29 | #1331 |
| prime radical | C29×#508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C29 ◁ C2×C8×C29 ◁ C4×C8×C29 ◁ C4×C16×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C29×C32 | C2 | semidirect |
| 3. C29×#1373 | C2 | semidirect |
| 37. #7387 | C29 | direct (N × H) |
| 50. C4×C32 | C2×C29 | semidirect |
| 51. #1373 | C2×C29 | semidirect |
| 80. C29 | #7387 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C29×#2178 | 2 |
| C4×C29×C32 | 2 |
| C29×#1373 | 2 |
| #7387 | 29 |