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, hypercenter | C4 | D7×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C72 | C23×C3 |
| Frattini | C2 | C2×D7×AGL(1,𝔽7) |
| Fitting | C4×C72 | C22×C3 |
| hypocenter | C72 | C3×#41 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ 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. C4×D7×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C4×C7×AGL(1,𝔽7) | C2 | semidirect |
| 4. #11924172 | C2 | semidirect |
| 5. #11924171 | C2 | semidirect |
| 6. #11924170 | C2 | semidirect |
| 7. #11924169 | C2 | semidirect |
| 8. #11716102 | C3 | semidirect |
| 16. C4×C7⋊D7 | C2×C3 | semidirect |
| 17. C4×C7×D7 | C2×C3 | semidirect |
| 18. C4×C7×D7 | C2×C3 | semidirect |
| 19. #87642 | C2×C3 | semidirect |
| 20. #87641 | C2×C3 | semidirect |
| 21. #87641 | C2×C3 | semidirect |
| 22. #87640 | C2×C3 | semidirect |
| 48. C4×AGL(1,𝔽7) | D7 | semidirect |
| 50. C7×ΓL(1,𝔽8) | #41 | semidirect |
| 82. C4×D7 | AGL(1,𝔽7) | semidirect |
| 83. C4×D7 | C3×D7 | semidirect |
| 86. C72 | C3×#41 | semidirect |
| 121. ΓL(1,𝔽8) | #1157 | semidirect |
| 149. C7 | #1157⋊C3 | semidirect |
| 150. 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 |
| C4×D7×ΓL(1,𝔽8) | 2 |
| C4×C7×AGL(1,𝔽7) | 2 |
| #11924172 | 2 |
| #11924171 | 2 |
| #11924170 | 2 |
| #11924169 | 2 |
| #11716102 | 3 |
| C3×#1157 | 7 |
| #1157⋊C3 | 7 |