Finite Group DatabasebetaFinite Group DatabasebetaOrder 16380.
240 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | AGL(1,𝔽7)×D65 |
| commutator, cosocle, hypocenter | C5×C7×C13 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C7×C13 | C22×C3 |
| LCS | G ▷ C5×C7×C13 | |
| UCS | C1 ◁ C3 | |
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×ΓL(1,𝔽8)×D65 | 2 |
| C3×C5⋊(ΓL(1,𝔽8)⋊D13) | 2 |
| C3×C5×C13×AGL(1,𝔽7) | 2 |
| C3×D7×D65 | 3 |
| AGL(1,𝔽7)×D65 | 3 |
| C3×D13×AGL(1,𝔽7) | 5 |
| C2×C32×D65 | 7 |
| C3×D5×AGL(1,𝔽7) | 13 |