Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
416 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D7⋊A4 |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Fitting | C7×H(5,𝔽2) | C2×C3 |
| socle | C2×C7 | C23×A4 |
| hypercenter | Q8 | D7⋊A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10621615 | C2 | semidirect |
| 2. D7⋊(#41⋊C3) | C2 | semidirect |
| 3. #10621539 | C2 | semidirect |
| 4. D7×H(5,𝔽2) | C3 | semidirect |
| 5. #41⋊ΓL(1,𝔽8) | C22 | semidirect |
| 6. D7⋊SL(2,𝔽3) | C22 | semidirect |
| 7. #66665 | C22 | semidirect |
| 9. C7×H(5,𝔽2) | C2×C3 | semidirect |
| 11. #6553 | C2×C3 | semidirect |
| 17. C7⋊SL(2,𝔽3) | C23 | semidirect |
| 76. H(5,𝔽2) | AGL(1,𝔽7) | semidirect |
| 117. D7 | #994 | semidirect |
| 141. C7 | C2×#994 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10621615 | 2 |
| D7⋊(#41⋊C3) | 2 |
| #10621539 | 2 |
| D7×H(5,𝔽2) | 3 |
| Q8×AGL(1,𝔽7) | 4 |
| C2×#994 | 7 |