Finite Group DatabasebetaFinite Group DatabasebetaOrder 2352.
168 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 55 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 | C2×D7×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C72 | C23×C3 |
| Fitting | C4×C72 | C22×C3 |
| hypercenter | C4 | D7×AGL(1,𝔽7) |
| hypocenter | C72 | C3×#41 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4⋊(ΓL(1,𝔽8)⋊D7) | C2 | semidirect |
| 2. ΓL(1,𝔽8)×Dic14 | C2 | semidirect |
| 3. C4×C7×AGL(1,𝔽7) | C2 | semidirect |
| 4. #11924171 | C2 | semidirect |
| 5. (ΓL(1,𝔽8)⋊D7)⋊C4 | C2 | semidirect |
| 6. #11716103 | C3 | semidirect |
| 12. C4⋊(C7⋊D7) | C2×C3 | semidirect |
| 13. C4×C7×D7 | C2×C3 | semidirect |
| 14. C7×Dic14 | C2×C3 | semidirect |
| 15. #87641 | C2×C3 | semidirect |
| 16. #87639 | C2×C3 | semidirect |
| 36. C4×AGL(1,𝔽7) | D7 | semidirect |
| 37. C7×ΓL(1,𝔽8) | #41 | semidirect |
| 62. C4×D7 | C3×D7 | semidirect |
| 63. Dic14 | AGL(1,𝔽7) | semidirect |
| 64. C72 | C3×#41 | semidirect |
| 90. ΓL(1,𝔽8) | #1157 | semidirect |
| 111. C7 | #66659 | semidirect |
| 112. C7 | C3×#1157 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4⋊(ΓL(1,𝔽8)⋊D7) | 2 |
| ΓL(1,𝔽8)×Dic14 | 2 |
| C4×C7×AGL(1,𝔽7) | 2 |
| #11924171 | 2 |
| (ΓL(1,𝔽8)⋊D7)⋊C4 | 2 |
| #11716103 | 3 |
| C3×#1157 | 7 |
| #66659 | 7 |