Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
350 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C7 | C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C2×C3×C7 |
| Frattini | C2 | C2×C7×C22⋊A4 |
| Fitting | C22×C7×#41 | C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C4×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C22⋊SL(2,𝔽3) | C2 | semidirect |
| 2. C22×C7×#41 | C3 | semidirect |
| 10. #5948 | C7 | direct (N × H) |
| 17. C7×#41 | A4 | semidirect |
| 24. C22⋊SL(2,𝔽3) | C2×C7 | semidirect |
| 27. C22×#41 | C3×C7 | semidirect |
| 54. C22×C7 | #41⋊C3 | semidirect |
| 65. #41 | C7×A4 | semidirect |
| 91. C7 | #5948 | direct (N × H) |
| 98. C22 | C7×#41⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C22⋊SL(2,𝔽3) | 2 |
| C22×C7×#41 | 3 |
| C7×#41⋊C3 | 4 |
| C4×C7×A4 | 4 |
| #5948 | 7 |