Finite Group DatabasebetaFinite Group Databasebeta← C52×C3⋊(ΓL(1,𝔽8)⋊D7) #36348023
Order 22050.
352 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C52 | C3⋊(ΓL(1,𝔽8)⋊D7) |
| commutator, cosocle, hypocenter | C3×C72 | C2×C3×C52 |
| Frattini | C1 | G |
| Fitting, socle | C3×C52×C72 | C2×C3 |
| LCS | G ▷ C3×C72 | |
| UCS | C1 ◁ C52 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C52×C7×ΓL(1,𝔽8) | 2 |
| C52×C3⋊(C7⋊D7) | 3 |
| C52×ΓL(1,𝔽8)⋊D7 | 3 |
| C5×C3⋊(ΓL(1,𝔽8)⋊D7) | 5 |
| C3×C52×D21 | 7 |
| C52×C3⋊AGL(1,𝔽7) | 7 |