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 | C22×C72 | C22×C3 |
| hypercenter | C22 | D7×AGL(1,𝔽7) |
| hypocenter | C72 | C3×#41 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11924202 | C2 | semidirect |
| 2. C2×ΓL(1,𝔽8)×Dic7 | C2 | semidirect |
| 3. C7×(C22⋊D7)⋊C3 | C2 | semidirect |
| 4. #11924172 | C2 | semidirect |
| 5. #11924171 | C2 | semidirect |
| 6. (ΓL(1,𝔽8)⋊D7)⋊C4 | C2 | semidirect |
| 7. Dic7×AGL(1,𝔽7) | C2 | semidirect |
| 8. #11716108 | C3 | semidirect |
| 16. C22⋊(C7⋊D7) | C2×C3 | semidirect |
| 17. C7×C22⋊D7 | C2×C3 | semidirect |
| 18. C2×C7×Dic7 | C2×C3 | semidirect |
| 19. #87642 | C2×C3 | semidirect |
| 20. #87641 | C2×C3 | semidirect |
| 21. #87639 | C2×C3 | semidirect |
| 22. D7×Dic7 | C2×C3 | semidirect |
| 45. (C22⋊D7)⋊C3 | D7 | semidirect |
| 50. C7×ΓL(1,𝔽8) | #41 | semidirect |
| 77. C22⋊D7 | C3×D7 | semidirect |
| 79. C2×Dic7 | AGL(1,𝔽7) | semidirect |
| 86. C72 | C3×#41 | semidirect |
| 121. ΓL(1,𝔽8) | #1159 | semidirect |
| 149. C7 | #1157⋊C3 | semidirect |
| 150. C7 | C3×#1159 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11924202 | 2 |
| C2×ΓL(1,𝔽8)×Dic7 | 2 |
| C7×(C22⋊D7)⋊C3 | 2 |
| #11924172 | 2 |
| #11924171 | 2 |
| (ΓL(1,𝔽8)⋊D7)⋊C4 | 2 |
| Dic7×AGL(1,𝔽7) | 2 |
| #11716108 | 3 |
| C3×#1159 | 7 |
| #1157⋊C3 | 7 |