Finite Group DatabasebetaFinite Group DatabasebetaOrder 8400.
304 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #72218916 |
| commutator, hypocenter | C52×C7 | C3×C42 |
| Fitting, socle | C2×C52×C7 | C2×C3×C4 |
| prime radical | C2×ΓL(1,𝔽8)×C5⋊D5 | C22 |
| cosocle | C2×C7×C5⋊D5 | C22×C3 |
| LCS | G ▷ C52×C7 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #72243058 | C3 | semidirect |
| 6. ΓL(1,𝔽8)×#1044 | C4 | semidirect |
| 7. ΓL(1,𝔽8)⋊#1043 | C4 | semidirect |
| 8. C5×ΓL(1,𝔽8)⋊Dic5 | C4 | semidirect |
| 22. C7×#1044 | C3×C4 | semidirect |
| 23. C7⋊#1043 | C3×C4 | semidirect |
| 24. C5×Dic35 | C3×C4 | semidirect |
| 29. C52×ΓL(1,𝔽8) | C42 | semidirect |
| 35. ΓL(1,𝔽8)⋊Dic5 | AGL(1,𝔽5) | semidirect |
| 60. C52×C7 | C3×C42 | semidirect |
| 72. Dic35 | C3×AGL(1,𝔽5) | semidirect |
| 80. C5×ΓL(1,𝔽8) | C4×AGL(1,𝔽5) | semidirect |
| 82. #1044 | ΓL(1,𝔽8)⋊C4 | semidirect |
| 119. C5×C7 | C3×C4×AGL(1,𝔽5) | semidirect |
| 130. C52 | C4×ΓL(1,𝔽8)⋊C4 | semidirect |
| 134. ΓL(1,𝔽8) | #87830 | semidirect |
| 161. C7 | C3×#87830 | semidirect |
| 166. C5 | AGL(1,𝔽5)×ΓL(1,𝔽8)⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72243122 | 2 |
| C2×ΓL(1,𝔽8)×#1044 | 2 |
| #72243058 | 3 |
| C2×ΓL(1,𝔽8)⋊#1043 | 2 |
| AGL(1,𝔽5)×ΓL(1,𝔽8)⋊C4 | 5 |
| C3×#87830 | 7 |