Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
426 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C7⋊S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C23 |
| Frattini | C2 | C22×C7⋊S4 |
| Fitting | C22×C7×Q8 | D3 |
| socle | C23×C7 | S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×SL(2,𝔽3) | C2 | semidirect |
| 2. C2×C7⋊GL(2,𝔽3) | C2 | direct (N × H) |
| 5. C2×C7×SL(2,𝔽3) | C22 | semidirect |
| 6. C7⋊GL(2,𝔽3) | C22 | direct (N × H) |
| 7. C22×C7×Q8 | D3 | semidirect |
| 16. C7×SL(2,𝔽3) | C23 | semidirect |
| 19. C2×C7×Q8 | C2×D3 | semidirect |
| 26. C22×SL(2,𝔽3) | D7 | semidirect |
| 38. C7×Q8 | C22×D3 | semidirect |
| 45. C2×SL(2,𝔽3) | C2×D7 | semidirect |
| 50. C22×Q8 | D21 | semidirect |
| 55. C22×C7 | GL(2,𝔽3) | semidirect |
| 64. SL(2,𝔽3) | C22×D7 | semidirect |
| 68. C2×Q8 | C2×D21 | semidirect |
| 75. C2×C7 | C2×GL(2,𝔽3) | semidirect |
| 87. Q8 | C22×D21 | semidirect |
| 93. C7 | C22×GL(2,𝔽3) | semidirect |
| 99. C22 | C7⋊GL(2,𝔽3) | direct (N × H) |
| 105. C2 | C2×C7⋊GL(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 |
| C2×C7⋊GL(2,𝔽3) | 2 |
| C22×#1143 | 3 |
| C23×D21 | 4 |
| C22×GL(2,𝔽3) | 7 |