Finite Group DatabasebetaFinite Group Databasebeta← C2×C22⋊(C7⋊SL(2,𝔽3)) #13062324
Order 1344.
350 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 42 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 | C22⋊(C22⋊ΓL(1,𝔽8)) |
| commutator, cosocle, hypocenter | C22×C7×Q8 | C2×C3 |
| Frattini | C2 | C2×C22⋊(C22⋊ΓL(1,𝔽8)) |
| Fitting | C23×C7×Q8 | C3 |
| socle | C24×C7 | A4 |
| LCS | G ▷ C22×C7×Q8 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22⋊(C7⋊SL(2,𝔽3)) | C2 | direct (N × H) |
| 2. C23×C7×Q8 | C3 | semidirect |
| 8. C22×C7×Q8 | C2×C3 | semidirect |
| 20. C2×C7×Q8 | A4 | semidirect |
| 28. C23×Q8 | ΓL(1,𝔽8) | semidirect |
| 32. C23×C7 | SL(2,𝔽3) | semidirect |
| 37. C7×Q8 | C2×A4 | semidirect |
| 49. C22×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 58. C22×C7 | C2×SL(2,𝔽3) | semidirect |
| 72. C2×Q8 | C22⋊ΓL(1,𝔽8) | semidirect |
| 80. C2×C7 | C22⋊SL(2,𝔽3) | semidirect |
| 87. C23 | C7⋊SL(2,𝔽3) | semidirect |
| 92. Q8 | C2×C22⋊ΓL(1,𝔽8) | semidirect |
| 97. C7 | C2×C22⋊SL(2,𝔽3) | semidirect |
| 107. C22 | C2×C7⋊SL(2,𝔽3) | semidirect |
| 115. C2 | C22⋊(C7⋊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 |
|---|---|
| C22⋊(C7⋊SL(2,𝔽3)) | 2 |
| C23×C7×Q8 | 3 |
| C22×C22⋊ΓL(1,𝔽8) | 4 |
| C2×C7⋊SL(2,𝔽3) | 4 |
| C2×C22⋊SL(2,𝔽3) | 7 |