Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
740 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 740 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C17 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C17 |
| prime radical | C24×C17 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C17 ◁ C23×C4×C17 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×Q16×C17 | C2 | semidirect |
| 4. C17×#451 | C2 | direct (N × H) |
| 5. C2×C17×#103 | C2 | semidirect |
| 16. C2×Q16×C17 | C22 | semidirect |
| 23. C17×#103 | C22 | semidirect |
| 56. C23×C17 | Q16 | semidirect |
| 80. C2×#451 | C17 | direct (N × H) |
| 86. C22×C17 | C2×Q16 | semidirect |
| 96. C22×Q16 | C2×C17 | semidirect |
| 98. #451 | C2×C17 | direct (N × H) |
| 99. C2×#103 | C2×C17 | semidirect |
| 102. C2×C17 | #451 | direct (N × H) |
| 115. C2×Q16 | C22×C17 | semidirect |
| 122. #103 | C22×C17 | semidirect |
| 124. C17 | C2×#451 | direct (N × H) |
| 156. C23 | Q16×C17 | semidirect |
| 185. C22 | C2×Q16×C17 | semidirect |
| 195. C2 | C17×#451 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8×C17 | 2 |
| C22×Q16×C17 | 2 |
| C2×C17×C22⋊Q8 | 2 |
| C17×#451 | 2 |
| C2×C17×#103 | 2 |
| C2×C17×C22⋊C8 | 2 |
| C2×#451 | 17 |