Finite Group DatabasebetaFinite Group DatabasebetaOrder 30492.
616 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 616 conjugacy classes of subgroups — too many to draw. Only its 238 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C112 | D3×AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C3×C7 | C22×C3×C112 |
| Frattini | C1 | G |
| Fitting, socle | C3×C7×C112 | C22×C3 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C112 | |
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 |
|---|---|
| D3×C112×ΓL(1,𝔽8) | 2 |
| C112×C3⋊AGL(1,𝔽7) | 2 |
| C3×C112×AGL(1,𝔽7) | 2 |
| D3×C112×D7 | 3 |
| C2×C112×AGL(1,𝔽7) | 3 |
| C2×C3×D3×C112 | 7 |
| D3×C11×AGL(1,𝔽7) | 11 |