Subgroups — C₂²×C₅²×C₁₂₅ #72190849

Order 12500.

880 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×C₂₅	C₅	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₁₂₅	C₅	1	30	yes	maximal, direct product factor
3	C₂×C₅²×C₁₂₅	C₂	1	3	yes	maximal, direct product factor
4	C₅²×C₁₂₅	C₂²	1	1	yes	characteristic, direct product factor, Sylow
5	C₂×C₅²×C₂₅	C₂×C₅	1	3	yes	—
6	C₂×C₅×C₁₂₅	C₂×C₅	1	90	yes	direct product factor
7	C₅²×C₂₅	C₂²×C₅	1	1	yes	characteristic
8	C₅×C₁₂₅	C₂²×C₅	1	30	yes	direct product factor
9	C₂²×C₅³	C₂₅	1	1	yes	characteristic
10	C₂²×C₅×C₂₅	C₂₅	1	24	yes	—
11	C₂²×C₅×C₂₅	C₅²	1	6	yes	—
12	C₂²×C₁₂₅	C₅²	1	25	yes	direct product factor
13	C₂×C₅³	C₂×C₂₅	1	3	yes	—
14	C₂×C₅×C₂₅	C₂×C₂₅	1	72	yes	—
15	C₂×C₅×C₂₅	C₂×C₅²	1	18	yes	—
16	C₂×C₁₂₅	C₂×C₅²	1	75	yes	direct product factor
17	C₅³	C₂²×C₂₅	1	1	yes	characteristic
18	C₅×C₂₅	C₂²×C₂₅	1	24	yes	—
19	C₅×C₂₅	C₂²×C₅²	1	6	yes	—
20	C₁₂₅	C₂²×C₅²	1	25	yes	direct product factor
21	C₂²×C₅²	C₁₂₅	1	25	yes	direct product factor
22	C₂²×C₅²	C₅×C₂₅	1	6	yes	—
23	C₂²×C₂₅	C₅×C₂₅	1	24	yes	—
24	C₂²×C₂₅	C₅³	1	1	yes	characteristic
25	C₂×C₅²	C₂×C₁₂₅	1	75	yes	direct product factor
26	C₂×C₅²	C₂×C₅×C₂₅	1	18	yes	—
27	C₂×C₂₅	C₂×C₅×C₂₅	1	72	yes	—
28	C₂×C₂₅	C₂×C₅³	1	3	yes	—
29	C₅²	C₂²×C₁₂₅	1	25	yes	direct product factor
30	C₅²	C₂²×C₅×C₂₅	1	6	yes	—
31	C₂₅	C₂²×C₅×C₂₅	1	24	yes	—
32	C₂₅	C₂²×C₅³	1	1	yes	characteristic
33	C₂²×C₅	C₅×C₁₂₅	1	30	yes	direct product factor
34	C₂²×C₅	C₅²×C₂₅	1	1	yes	characteristic
35	C₂×C₅	C₂×C₅×C₁₂₅	1	90	yes	direct product factor
36	C₂×C₅	C₂×C₅²×C₂₅	1	3	yes	—
37	C₅	C₂²×C₅×C₁₂₅	1	30	yes	direct product factor
38	C₅	C₂²×C₅²×C₂₅	1	1	yes	characteristic
39	C₂²	C₅²×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
40	C₂	C₂×C₅²×C₁₂₅	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₅	C₂²×C₅³
socle, prime radical	C₂²×C₅³	C₂₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×C₁₂₅	C₅	direct (N × H)
3. C₂×C₅²×C₁₂₅	C₂	direct (N × H)
4. C₅²×C₁₂₅	C₂²	direct (N × H)
6. C₂×C₅×C₁₂₅	C₂×C₅	direct (N × H)
8. C₅×C₁₂₅	C₂²×C₅	direct (N × H)
12. C₂²×C₁₂₅	C₅²	direct (N × H)
16. C₂×C₁₂₅	C₂×C₅²	direct (N × H)
20. C₁₂₅	C₂²×C₅²	direct (N × H)
21. C₂²×C₅²	C₁₂₅	direct (N × H)
25. C₂×C₅²	C₂×C₁₂₅	direct (N × H)
29. C₅²	C₂²×C₁₂₅	direct (N × H)
33. C₂²×C₅	C₅×C₁₂₅	direct (N × H)
35. C₂×C₅	C₂×C₅×C₁₂₅	direct (N × H)
37. C₅	C₂²×C₅×C₁₂₅	direct (N × H)
39. C₂²	C₅²×C₁₂₅	direct (N × H)
40. C₂	C₂×C₅²×C₁₂₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅²×C₂₅	5
C₂²×C₅×C₁₂₅	5
C₂×C₅²×C₁₂₅	2

Subgroups — C22×C52×C125 #72190849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅²×C₁₂₅ #72190849