Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
760 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C23 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4×C23 |
| Frattini, cosocle | C2×#41 | C23×C23 |
| socle | C2×C23 | #1790 |
| prime radical | C23×#3529 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C23 ◁ C2×C4×C23 ◁ C2×#41×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×#2126 | C2 | semidirect |
| 3. C23×#1884 | C2 | semidirect |
| 4. C23×#568 | C22 | semidirect |
| 5. C23×#563 | C4 | semidirect |
| 7. C23×H(3,𝔽4) | C4 | semidirect |
| 13. C23×#410 | C22 | semidirect |
| 84. #12772 | C23 | direct (N × H) |
| 111. #2126 | C2×C23 | semidirect |
| 112. #1884 | C2×C23 | semidirect |
| 126. #568 | C22×C23 | semidirect |
| 127. #563 | C4×C23 | semidirect |
| 129. H(3,𝔽4) | C4×C23 | semidirect |
| 135. #410 | C22×C23 | semidirect |
| 173. C23 | #12772 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#3529 | 2 |
| C23×#2126 | 2 |
| C23×#1884 | 2 |
| #12772 | 23 |