Finite Group DatabasebetaFinite Group Databasebeta← C2×AGL(1,𝔽7)×C3⋊S4 #72244888
Order 6048.
1502 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | AGL(1,𝔽7)×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×C7×A4 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C7 | C2×C3×D3 |
| LCS | G ▷ C3×C7×A4 | |
| UCS | C1 ◁ C2 | |
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 |
|---|---|
| C2×D7×C3⋊S4 | 3 |
| C2×C3⋊(A4⋊AGL(1,𝔽7)) | 2 |
| C2×ΓL(1,𝔽8)×C3⋊S4 | 2 |
| C2×C3×A4×AGL(1,𝔽7) | 2 |
| C2×#67×AGL(1,𝔽7) | 3 |
| AGL(1,𝔽7)×C3⋊S4 | 2 |
| C2×S4×AGL(1,𝔽7) | 3 |
| C2×C3⋊D3×AGL(1,𝔽7) | 4 |
| C22×C3×C3⋊S4 | 7 |