Finite Group DatabasebetaFinite Group DatabasebetaOrder 7056.
1160 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×#64424 |
| commutator, socle, cosocle | C2×C3×C72 | C23×C3 |
| Fitting | C3×C4×C72 | C22×C3 |
| hypercenter | C4 | D3×#64424 |
| hypocenter | C3×C72 | C2×C3×D4 |
| LCS | G ▷ C2×C3×C72 ▷ C3×C72 | |
| UCS | C1 ◁ C2 ◁ C4 | |
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 |
|---|---|
| #72245897 | 2 |
| #72245893 | 2 |
| D12×#3861 | 2 |
| C3×C4×#64424 | 2 |
| D12×C7⋊D7 | 3 |
| C2×D3×#64424 | 2 |
| D4×#64424 | 3 |
| D12×AGL(1,𝔽7) | 7 |