Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
730 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 142 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×C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C23 |
| Frattini, cosocle | C2×C42 | C23×C23 |
| prime radical | C22×#41×C23 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C23 ◁ C22×C42×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×#1728 | C2 | semidirect |
| 3. C23×#1272 | C2 | semidirect |
| 4. C23×#1270 | C2 | semidirect |
| 51. C2×Q8×C23 | M16 | semidirect |
| 58. C2×D4×C23 | M16 | semidirect |
| 91. #7631 | C23 | direct (N × H) |
| 136. #1728 | C2×C23 | semidirect |
| 137. #1272 | C2×C23 | semidirect |
| 138. #1270 | C2×C23 | semidirect |
| 209. C23 | #7631 | direct (N × H) |
| 217. C2×Q8 | M16×C23 | semidirect |
| 224. C2×D4 | M16×C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#2297 | 2 |
| C23×#1728 | 2 |
| C23×#1272 | 2 |
| C23×#1270 | 2 |
| #7631 | 23 |