Finite Group DatabasebetaFinite Group DatabasebetaOrder 7424.
460 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C29 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C29 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C4×Q8×C29 ◁ C8×Q8×C29 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 23. C29×Q32 | Q8 | direct (N × H) |
| 46. Q8×Q32 | C29 | direct (N × H) |
| 50. Q8×C29 | Q32 | direct (N × H) |
| 91. Q32 | Q8×C29 | direct (N × H) |
| 99. C29 | Q8×Q32 | direct (N × H) |
| 118. Q8 | C29×Q32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×Q16×C29 | 2 |
| C29×#2252 | 2 |
| C29×#2227 | 2 |
| Q8×C16×C29 | 2 |
| C4×C29×Q32 | 2 |
| Q8×Q32 | 29 |