Finite Group DatabasebetaFinite Group DatabasebetaOrder 1680.
160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 160 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×D5×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C5×C7 | C23×C3 |
| Fitting | C22×C5×C7 | C22×C3 |
| hypercenter | C22 | D5×AGL(1,𝔽7) |
| hypocenter | C5×C7 | C3×#41 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. (C22⋊D35)⋊C3 | C2 | semidirect |
| 2. ΓL(1,𝔽8)×C22⋊D5 | C2 | semidirect |
| 3. C2×C5×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 4. #63308⋊C3 | C2 | semidirect |
| 5. #63307⋊C3 | C2 | semidirect |
| 6. #63304⋊C3 | C2 | semidirect |
| 7. D5×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 8. #10587955 | C3 | semidirect |
| 17. C22⋊D35 | C2×C3 | semidirect |
| 18. C7×C22⋊D5 | C2×C3 | semidirect |
| 19. C2×C5×Dic7 | C2×C3 | semidirect |
| 20. #63308 | C2×C3 | semidirect |
| 21. #63307 | C2×C3 | semidirect |
| 22. #63304 | C2×C3 | semidirect |
| 23. D5×Dic7 | C2×C3 | semidirect |
| 33. C2×ΓL(1,𝔽8)⋊C4 | D5 | semidirect |
| 51. C5×ΓL(1,𝔽8) | #41 | semidirect |
| 75. C2×Dic7 | C3×D5 | semidirect |
| 85. C22⋊D5 | AGL(1,𝔽7) | semidirect |
| 90. C5×C7 | C3×#41 | semidirect |
| 109. ΓL(1,𝔽8) | #709 | semidirect |
| 141. C7 | C3×#709 | semidirect |
| 146. C5 | #1159⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| (C22⋊D35)⋊C3 | 2 |
| ΓL(1,𝔽8)×C22⋊D5 | 2 |
| C2×C5×ΓL(1,𝔽8)⋊C4 | 2 |
| #63308⋊C3 | 2 |
| #63307⋊C3 | 2 |
| #63304⋊C3 | 2 |
| D5×ΓL(1,𝔽8)⋊C4 | 2 |
| #10587955 | 3 |
| #1159⋊C3 | 5 |
| C3×#709 | 7 |