Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C25 | #120 |
| commutator | C2×C4 | C23×C25 |
| Frattini, cosocle | C2×C4×C5 | C23×C5 |
| socle | C2×C5 | C5×#120 |
| prime radical | C5×#453 | C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C25 ◁ C2×C4×C25 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×H(5,𝔽2) | C2 | semidirect |
| 2. C25×AGL(1,ℤ/8) | C2 | semidirect |
| 3. C25×#127 | C2 | semidirect |
| 4. C25×#104 | C2 | semidirect |
| 5. C25×#100 | C2 | semidirect |
| 12. C2×D4×C25 | C22 | semidirect |
| 15. M16×C25 | C22 | semidirect |
| 16. C42×C25 | C22 | semidirect |
| 21. Q8×C25 | D4 | semidirect |
| 27. D4×C25 | D4 | semidirect |
| 59. #453 | C25 | direct (N × H) |
| 80. H(5,𝔽2) | C2×C25 | semidirect |
| 81. AGL(1,ℤ/8) | C2×C25 | semidirect |
| 82. #127 | C2×C25 | semidirect |
| 83. #104 | C2×C25 | semidirect |
| 84. #100 | C2×C25 | semidirect |
| 85. C25 | #453 | direct (N × H) |
| 102. C2×D4 | C22×C25 | semidirect |
| 105. M16 | C22×C25 | semidirect |
| 106. C42 | C22×C25 | semidirect |
| 115. Q8 | D4×C25 | semidirect |
| 121. D4 | D4×C25 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×H(5,𝔽2) | 2 |
| C25×AGL(1,ℤ/8) | 2 |
| C25×#127 | 2 |
| C25×#104 | 2 |
| C25×#100 | 2 |
| C5×#453 | 5 |