Subgroups — C₄×C₅²×C₁₂₅ #72190841

Order 12500.

528 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅²×C₂₅	C₅	1	1	yes	maximal, characteristic
2	C₄×C₅×C₁₂₅	C₅	1	30	yes	maximal, direct product factor
3	C₂×C₅²×C₁₂₅	C₂	1	1	yes	maximal, characteristic
4	C₅²×C₁₂₅	C₄	1	1	yes	characteristic, direct product factor, Sylow
5	C₂×C₅²×C₂₅	C₂×C₅	1	1	yes	characteristic
6	C₂×C₅×C₁₂₅	C₂×C₅	1	30	yes	—
7	C₅²×C₂₅	C₄×C₅	1	1	yes	characteristic
8	C₅×C₁₂₅	C₄×C₅	1	30	yes	direct product factor
9	C₄×C₅³	C₂₅	1	1	yes	characteristic
10	C₄×C₅×C₂₅	C₂₅	1	24	yes	—
11	C₄×C₅×C₂₅	C₅²	1	6	yes	—
12	C₄×C₁₂₅	C₅²	1	25	yes	direct product factor
13	C₂×C₅³	C₂×C₂₅	1	1	yes	characteristic
14	C₂×C₅×C₂₅	C₂×C₂₅	1	24	yes	—
15	C₂×C₅×C₂₅	C₂×C₅²	1	6	yes	—
16	C₂×C₁₂₅	C₂×C₅²	1	25	yes	—
17	C₅³	C₄×C₂₅	1	1	yes	characteristic
18	C₅×C₂₅	C₄×C₂₅	1	24	yes	—
19	C₅×C₂₅	C₄×C₅²	1	6	yes	—
20	C₁₂₅	C₄×C₅²	1	25	yes	direct product factor
21	C₄×C₅²	C₁₂₅	1	25	yes	direct product factor
22	C₄×C₅²	C₅×C₂₅	1	6	yes	—
23	C₄×C₂₅	C₅×C₂₅	1	24	yes	—
24	C₄×C₂₅	C₅³	1	1	yes	characteristic
25	C₂×C₅²	C₂×C₁₂₅	1	25	yes	—
26	C₂×C₅²	C₂×C₅×C₂₅	1	6	yes	—
27	C₂×C₂₅	C₂×C₅×C₂₅	1	24	yes	—
28	C₂×C₂₅	C₂×C₅³	1	1	yes	characteristic
29	C₅²	C₄×C₁₂₅	1	25	yes	direct product factor
30	C₅²	C₄×C₅×C₂₅	1	6	yes	—
31	C₂₅	C₄×C₅×C₂₅	1	24	yes	—
32	C₂₅	C₄×C₅³	1	1	yes	characteristic
33	C₄×C₅	C₅×C₁₂₅	1	30	yes	direct product factor
34	C₄×C₅	C₅²×C₂₅	1	1	yes	characteristic
35	C₂×C₅	C₂×C₅×C₁₂₅	1	30	yes	—
36	C₂×C₅	C₂×C₅²×C₂₅	1	1	yes	characteristic
37	C₅	C₄×C₅×C₁₂₅	1	30	yes	direct product factor
38	C₅	C₄×C₅²×C₂₅	1	1	yes	characteristic
39	C₄	C₅²×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
40	C₂	C₂×C₅²×C₁₂₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₂₅	C₂×C₅³
socle, prime radical	C₂×C₅³	C₂×C₂₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×C₁₂₅	C₅	direct (N × H)
4. C₅²×C₁₂₅	C₄	direct (N × H)
8. C₅×C₁₂₅	C₄×C₅	direct (N × H)
12. C₄×C₁₂₅	C₅²	direct (N × H)
20. C₁₂₅	C₄×C₅²	direct (N × H)
21. C₄×C₅²	C₁₂₅	direct (N × H)
29. C₅²	C₄×C₁₂₅	direct (N × H)
33. C₄×C₅	C₅×C₁₂₅	direct (N × H)
37. C₅	C₄×C₅×C₁₂₅	direct (N × H)
39. C₄	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
C₄×C₅²×C₂₅	5
C₄×C₅×C₁₂₅	5
C₂×C₅²×C₁₂₅	2

Subgroups — C4×C52×C125 #72190841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅²×C₁₂₅ #72190841