Subgroups — C₂⁵×C₂₅×C₃₁ #72202098

Order 24800.

2244 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂₅×C₃₁	C₂	1	31	yes	maximal, direct product factor
2	C₂³×C₂₅×C₃₁	C₂²	1	155	yes	direct product factor
3	C₂⁵×C₅×C₃₁	C₅	1	1	yes	maximal, characteristic
4	C₂⁴×C₅×C₃₁	C₂×C₅	1	31	yes	—
5	C₂²×C₂₅×C₃₁	C₂³	1	155	yes	direct product factor
6	C₂×C₂₅×C₃₁	C₂⁴	1	31	yes	direct product factor
7	C₂³×C₅×C₃₁	C₂²×C₅	1	155	yes	—
8	C₂⁵×C₃₁	C₂₅	1	1	yes	characteristic, direct product factor, Hall
9	C₂⁵×C₂₅	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂₅×C₃₁	C₂⁵	1	1	yes	characteristic, direct product factor, Hall
11	C₂²×C₅×C₃₁	C₂³×C₅	1	155	yes	—
12	C₂⁴×C₃₁	C₂×C₂₅	1	31	yes	direct product factor
13	C₂⁴×C₂₅	C₂×C₃₁	1	31	yes	direct product factor
14	C₂×C₅×C₃₁	C₂⁴×C₅	1	31	yes	—
15	C₂³×C₃₁	C₂²×C₂₅	1	155	yes	direct product factor
16	C₂³×C₂₅	C₂²×C₃₁	1	155	yes	direct product factor
17	C₂⁵×C₅	C₅×C₃₁	1	1	yes	characteristic
18	C₅×C₃₁	C₂⁵×C₅	1	1	yes	characteristic
19	C₂²×C₃₁	C₂³×C₂₅	1	155	yes	direct product factor
20	C₂²×C₂₅	C₂³×C₃₁	1	155	yes	direct product factor
21	C₂⁴×C₅	C₂×C₅×C₃₁	1	31	yes	—
22	C₂×C₃₁	C₂⁴×C₂₅	1	31	yes	direct product factor
23	C₂×C₂₅	C₂⁴×C₃₁	1	31	yes	direct product factor
24	C₂³×C₅	C₂²×C₅×C₃₁	1	155	yes	—
25	C₂⁵	C₂₅×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
26	C₃₁	C₂⁵×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
27	C₂₅	C₂⁵×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
28	C₂²×C₅	C₂³×C₅×C₃₁	1	155	yes	—
29	C₂⁴	C₂×C₂₅×C₃₁	1	31	yes	direct product factor
30	C₂×C₅	C₂⁴×C₅×C₃₁	1	31	yes	—
31	C₂³	C₂²×C₂₅×C₃₁	1	155	yes	direct product factor
32	C₅	C₂⁵×C₅×C₃₁	1	1	yes	characteristic
33	C₂²	C₂³×C₂₅×C₃₁	1	155	yes	direct product factor
34	C₂	C₂⁴×C₂₅×C₃₁	1	31	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₅	C₂⁵×C₅×C₃₁
socle, prime radical	C₂⁵×C₅×C₃₁	C₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂⁴×C₂₅×C₃₁	C₂	direct (N × H)
2. C₂³×C₂₅×C₃₁	C₂²	direct (N × H)
5. C₂²×C₂₅×C₃₁	C₂³	direct (N × H)
6. C₂×C₂₅×C₃₁	C₂⁴	direct (N × H)
8. C₂⁵×C₃₁	C₂₅	direct (N × H)
9. C₂⁵×C₂₅	C₃₁	direct (N × H)
10. C₂₅×C₃₁	C₂⁵	direct (N × H)
12. C₂⁴×C₃₁	C₂×C₂₅	direct (N × H)
13. C₂⁴×C₂₅	C₂×C₃₁	direct (N × H)
15. C₂³×C₃₁	C₂²×C₂₅	direct (N × H)
16. C₂³×C₂₅	C₂²×C₃₁	direct (N × H)
19. C₂²×C₃₁	C₂³×C₂₅	direct (N × H)
20. C₂²×C₂₅	C₂³×C₃₁	direct (N × H)
22. C₂×C₃₁	C₂⁴×C₂₅	direct (N × H)
23. C₂×C₂₅	C₂⁴×C₃₁	direct (N × H)
25. C₂⁵	C₂₅×C₃₁	direct (N × H)
26. C₃₁	C₂⁵×C₂₅	direct (N × H)
27. C₂₅	C₂⁵×C₃₁	direct (N × H)
29. C₂⁴	C₂×C₂₅×C₃₁	direct (N × H)
31. C₂³	C₂²×C₂₅×C₃₁	direct (N × H)
33. C₂²	C₂³×C₂₅×C₃₁	direct (N × H)
34. C₂	C₂⁴×C₂₅×C₃₁	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₂₅×C₃₁	2
C₂⁵×C₅×C₃₁	5
C₂⁵×C₂₅	31

Subgroups — C25×C25×C31 #72202098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×C₂₅×C₃₁ #72202098