Finite Group DatabasebetaFinite Group DatabasebetaOrder 5504.
670 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 136 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×C43 | #120 |
| commutator | C23 | C22×C4×C43 |
| Frattini, cosocle | C24 | C23×C43 |
| prime radical | C2×H(5,𝔽2)×C43 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C43 ◁ C24×C43 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#99×C43 | C2 | semidirect |
| 3. C43×#386 | C2 | semidirect |
| 5. H(5,𝔽2)×C43 | C4 | semidirect |
| 9. C23×C4×C43 | C22 | semidirect |
| 11. C2×C22⋊C4×C43 | C22 | semidirect |
| 43. C23×C43 | C22⋊C4 | semidirect |
| 60. C2×C4×C43 | C22⋊C4 | semidirect |
| 77. #1790 | C43 | direct (N × H) |
| 85. C2×#99 | C2×C43 | semidirect |
| 86. #386 | C2×C43 | semidirect |
| 87. C43 | #1790 | direct (N × H) |
| 89. H(5,𝔽2) | C4×C43 | semidirect |
| 93. C23×C4 | C22×C43 | semidirect |
| 95. C2×C22⋊C4 | C22×C43 | semidirect |
| 127. C23 | C22⋊C4×C43 | semidirect |
| 144. C2×C4 | C22⋊C4×C43 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽2)×C43 | 2 |
| C2×#99×C43 | 2 |
| C43×#386 | 2 |
| #1790 | 43 |