Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
680 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C33×C37 | H(3,𝔽3) |
| commutator | C32 | C32×C9×C37 |
| Frattini, cosocle | C33 | C33×C37 |
| prime radical | C32×H(3,𝔽3)×C37 | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33×C37 ◁ C34×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C33×C9×C37 | C3 | semidirect |
| 3. C3×C37×C32⋊C9 | C3 | semidirect |
| 4. C37×H(3,𝔽3)⋊C9 | C3 | direct (N × H) |
| 9. C3×H(3,𝔽3)×C37 | C9 | semidirect |
| 15. C32×C9×C37 | C32 | semidirect |
| 17. C37×C32⋊C9 | C32 | semidirect |
| 33. H(3,𝔽3)×C37 | C3×C9 | semidirect |
| 40. C3×H(3,𝔽3)⋊C9 | C37 | direct (N × H) |
| 65. C33×C9 | C3×C37 | semidirect |
| 66. C3×C32⋊C9 | C3×C37 | semidirect |
| 67. H(3,𝔽3)⋊C9 | C3×C37 | direct (N × H) |
| 72. C3×C37 | H(3,𝔽3)⋊C9 | direct (N × H) |
| 81. C3×H(3,𝔽3) | C9×C37 | semidirect |
| 87. C32×C9 | C32×C37 | semidirect |
| 89. C32⋊C9 | C32×C37 | semidirect |
| 90. C37 | C3×H(3,𝔽3)⋊C9 | direct (N × H) |
| 106. H(3,𝔽3) | C3×C9×C37 | semidirect |
| 140. C3 | C37×H(3,𝔽3)⋊C9 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×H(3,𝔽3)×C37 | 3 |
| C33×C9×C37 | 3 |
| C3×C37×C32⋊C9 | 3 |
| C37×H(3,𝔽3)⋊C9 | 3 |
| C3×H(3,𝔽3)⋊C9 | 37 |