Finite Group DatabasebetaFinite Group DatabasebetaOrder 3360.
204 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D7×S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C22×C5 |
| Frattini | C2 | C5×D7×S4 |
| Fitting | C5×C7×Q8 | C2×D3 |
| socle | C2×C5×C7 | C2×S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×D7×SL(2,𝔽3) | C2 | semidirect |
| 2. C5×C7⋊GL(2,𝔽3) | C2 | semidirect |
| 3. C5×C7×GL(2,𝔽3) | C2 | semidirect |
| 6. C5×C7×SL(2,𝔽3) | C22 | semidirect |
| 7. D7×GL(2,𝔽3) | C5 | direct (N × H) |
| 8. C5×Q8×D7 | D3 | semidirect |
| 20. D7×SL(2,𝔽3) | C2×C5 | semidirect |
| 21. C7⋊GL(2,𝔽3) | C2×C5 | semidirect |
| 22. C7×GL(2,𝔽3) | C2×C5 | semidirect |
| 24. C5×C7×Q8 | C2×D3 | semidirect |
| 34. C5×GL(2,𝔽3) | D7 | direct (N × H) |
| 41. C7×SL(2,𝔽3) | C22×C5 | semidirect |
| 50. C5×SL(2,𝔽3) | C2×D7 | semidirect |
| 51. Q8×D7 | C5×D3 | semidirect |
| 74. C5×D7 | GL(2,𝔽3) | direct (N × H) |
| 80. C7×Q8 | C2×C5×D3 | semidirect |
| 90. GL(2,𝔽3) | C5×D7 | direct (N × H) |
| 97. C5×Q8 | D3×D7 | semidirect |
| 104. C5×C7 | C2×GL(2,𝔽3) | semidirect |
| 117. SL(2,𝔽3) | C2×C5×D7 | semidirect |
| 136. D7 | C5×GL(2,𝔽3) | direct (N × H) |
| 147. Q8 | C5×D3×D7 | semidirect |
| 154. C7 | C2×C5×GL(2,𝔽3) | semidirect |
| 159. C5 | D7×GL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×D7×SL(2,𝔽3) | 2 |
| C5×C7⋊GL(2,𝔽3) | 2 |
| C5×C7×GL(2,𝔽3) | 2 |
| C5×D7×SD16 | 3 |
| C2×C5×D3×D7 | 4 |
| D7×GL(2,𝔽3) | 5 |
| C2×C5×GL(2,𝔽3) | 7 |