Finite Group DatabasebetaFinite Group Databasebeta← C2×C7×C22⋊SL(2,𝔽3) #13062319
Order 1344.
350 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C7 | C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C2×C3×C7 |
| Frattini | C2 | C2×C7×C22⋊A4 |
| Fitting | C23×C7×Q8 | C3 |
| socle | C24×C7 | A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C22×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C22⋊SL(2,𝔽3) | C2 | direct (N × H) |
| 2. C23×C7×Q8 | C3 | semidirect |
| 8. C22×C7×Q8 | C2×C3 | semidirect |
| 10. C2×C22⋊SL(2,𝔽3) | C7 | direct (N × H) |
| 20. C2×C7×Q8 | A4 | semidirect |
| 25. C22⋊SL(2,𝔽3) | C2×C7 | direct (N × H) |
| 28. C23×Q8 | C3×C7 | semidirect |
| 32. C23×C7 | SL(2,𝔽3) | semidirect |
| 37. C7×Q8 | C2×A4 | semidirect |
| 49. C22×Q8 | C2×C3×C7 | semidirect |
| 58. C22×C7 | C2×SL(2,𝔽3) | semidirect |
| 72. C2×Q8 | C7×A4 | semidirect |
| 80. C2×C7 | C22⋊SL(2,𝔽3) | direct (N × H) |
| 87. C23 | C7×SL(2,𝔽3) | semidirect |
| 92. Q8 | C2×C7×A4 | semidirect |
| 97. C7 | C2×C22⋊SL(2,𝔽3) | direct (N × H) |
| 107. C22 | C2×C7×SL(2,𝔽3) | semidirect |
| 115. C2 | C7×C22⋊SL(2,𝔽3) | direct (N × H) |
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 |
| C23×C7×Q8 | 3 |
| C22×C7×A4 | 4 |
| C2×C7×SL(2,𝔽3) | 4 |
| C2×C22⋊SL(2,𝔽3) | 7 |