Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
386 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C7 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C2×C3×C7 |
| Frattini | C2 | C2×C7×C22⋊A4 |
| Fitting | C2×C7×H(5,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C22×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#997 | C2 | direct (N × H) |
| 2. C2×C7×H(5,𝔽2) | C3 | semidirect |
| 7. C7×H(5,𝔽2) | C2×C3 | semidirect |
| 10. C2×#997 | C7 | direct (N × H) |
| 15. C24×C7 | A4 | semidirect |
| 20. C2×C7×Q8 | A4 | semidirect |
| 28. #997 | C2×C7 | direct (N × H) |
| 31. C2×H(5,𝔽2) | C3×C7 | semidirect |
| 38. C23×C7 | C2×A4 | semidirect |
| 40. C7×Q8 | C2×A4 | semidirect |
| 54. H(5,𝔽2) | C2×C3×C7 | semidirect |
| 72. C24 | C7×A4 | semidirect |
| 77. C2×Q8 | C7×A4 | semidirect |
| 87. C2×C7 | #997 | direct (N × H) |
| 97. C23 | C2×C7×A4 | semidirect |
| 99. Q8 | C2×C7×A4 | semidirect |
| 107. C7 | C2×#997 | direct (N × H) |
| 123. C2 | C7×#997 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#997 | 2 |
| C2×C7×H(5,𝔽2) | 3 |
| C22×C7×A4 | 4 |
| C2×C7×SL(2,𝔽3) | 4 |
| C2×#997 | 7 |