Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
384 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C17 | C22 |
| commutator | C2 | C2×C4×C8×C17 |
| Frattini, cosocle | C22×C4 | C23×C17 |
| socle | C23×C17 | C22×C4 |
| prime radical | C24×C17 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C17 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C4×C8×C17 | C2 | semidirect |
| 5. C17×#336 | C2 | semidirect |
| 8. C22×C8×C17 | C4 | semidirect |
| 14. C17×C22⋊C8 | C4 | semidirect |
| 33. C22⋊C4×C17 | C8 | direct (N × H) |
| 48. C2×C4×C17 | C2×C8 | semidirect |
| 55. C8×C17 | C22⋊C4 | direct (N × H) |
| 56. C8×C22⋊C4 | C17 | direct (N × H) |
| 60. C22×C17 | C4×C8 | semidirect |
| 72. C2×C4×C8 | C2×C17 | semidirect |
| 74. #336 | C2×C17 | semidirect |
| 82. C22×C8 | C4×C17 | semidirect |
| 88. C22⋊C8 | C4×C17 | semidirect |
| 91. C17 | C8×C22⋊C4 | direct (N × H) |
| 108. C22⋊C4 | C8×C17 | direct (N × H) |
| 123. C2×C4 | C2×C8×C17 | semidirect |
| 130. C8 | C22⋊C4×C17 | direct (N × H) |
| 134. C22 | C4×C8×C17 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C8×C17 | 2 |
| C2×C17×C22⋊C8 | 2 |
| C2×C4×C8×C17 | 2 |
| C4×C22⋊C4×C17 | 2 |
| C17×#336 | 2 |
| C8×C22⋊C4 | 17 |