Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
174 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×A4×D7 |
| commutator, cosocle | C2×C7×Q8 | C22×C3 |
| Fitting | C2×C7×#41 | C2×C3 |
| socle | C22×C7 | C22×A4 |
| hypercenter | C2×C4 | A4×D7 |
| hypocenter | C7×Q8 | C3×D4 |
| LCS | G ▷ C2×C7×Q8 ▷ C7×Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#41⋊C3 | C2 | semidirect |
| 2. C2×#66662 | C2 | semidirect |
| 3. SL(2,𝔽3)×Dic7 | C2 | semidirect |
| 4. #90525 | C3 | semidirect |
| 20. C7×SL(2,𝔽3) | D4 | semidirect |
| 38. C2×#41⋊C3 | D7 | semidirect |
| 90. Dic7 | #41⋊C3 | semidirect |
| 94. SL(2,𝔽3) | C22⋊D7 | semidirect |
| 140. C7 | C4⋊(#41⋊C3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#41⋊C3 | 2 |
| C2×#66662 | 2 |
| SL(2,𝔽3)×Dic7 | 2 |
| #90525 | 3 |
| C3×Dic7⋊C4 | 4 |
| C4⋊(#41⋊C3) | 7 |