Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
146 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 18 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×C7 | #1020 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C2×C7 |
| Frattini | C2 | C7×#1020 |
| Fitting | C22×C7×Q8 | D3 |
| socle | C23×C7 | S4 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C22⋊SL(2,𝔽3) | C2 | semidirect |
| 5. C22×C7×Q8 | D3 | semidirect |
| 10. #5931 | C7 | direct (N × H) |
| 25. C22⋊SL(2,𝔽3) | C2×C7 | semidirect |
| 34. C7×Q8 | S4 | semidirect |
| 47. C22×Q8 | D3×C7 | semidirect |
| 57. C22×C7 | GL(2,𝔽3) | semidirect |
| 87. Q8 | C7×S4 | semidirect |
| 96. C7 | #5931 | direct (N × H) |
| 104. C22 | C7×GL(2,𝔽3) | 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 |
| C7×#448 | 3 |
| C2×C7×S4 | 4 |
| C7×GL(2,𝔽3) | 4 |
| #5931 | 7 |