Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
168 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C23 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C23 |
| prime radical | C23×C23 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C22×C8×C23 ◁ C22×C16×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C23×#1424 | C2 | semidirect |
| 4. C23×#1423 | C2 | semidirect |
| 5. C23×#1416 | C2 | semidirect |
| 6. C23×#1416 | C2 | semidirect |
| 54. #13716 | C23 | direct (N × H) |
| 71. #1424 | C2×C23 | semidirect |
| 72. #1423 | C2×C23 | semidirect |
| 73. #1416 | C2×C23 | semidirect |
| 74. #1416 | C2×C23 | semidirect |
| 119. C23 | #13716 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#2216 | 2 |
| C23×#2158 | 2 |
| C23×#1424 | 2 |
| C23×#1423 | 2 |
| C23×#1416 | 2 |
| C23×C22⋊C32 | 2 |
| #13716 | 23 |