Finite Group DatabasebetaFinite Group DatabasebetaOrder 2016.
554 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 554 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C3×C7 | C23×C3 |
| Fitting | C23×C3×C7 | C22×C3 |
| hypercenter | C23 | D3×AGL(1,𝔽7) |
| hypocenter | C3×C7 | C3×#120 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
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 |
|---|---|
| C23×D3×ΓL(1,𝔽8) | 2 |
| C2×#4268⋊C3 | 2 |
| C2×#4247⋊C3 | 2 |
| C3×(C22⋊Dic7)⋊C3 | 2 |
| (C2×D3)⋊(ΓL(1,𝔽8)⋊C4) | 2 |
| #10621132 | 3 |
| #10620789 | 3 |
| C3×#882 | 7 |