Subgroups — C₄²×C₅×C₁₂₅ #72188673

Order 10000.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₂₅	C₂	1	3	yes	maximal
2	C₂²×C₅×C₁₂₅	C₂²	1	1	yes	characteristic
3	C₄×C₅×C₁₂₅	C₄	1	6	yes	direct product factor
4	C₄²×C₅×C₂₅	C₅	1	1	yes	maximal, characteristic
5	C₄²×C₁₂₅	C₅	1	5	yes	maximal, direct product factor
6	C₂×C₅×C₁₂₅	C₂×C₄	1	3	yes	—
7	C₂×C₄×C₅×C₂₅	C₂×C₅	1	3	yes	—
8	C₂×C₄×C₁₂₅	C₂×C₅	1	15	yes	—
9	C₅×C₁₂₅	C₄²	1	1	yes	characteristic, direct product factor, Sylow
10	C₂²×C₅×C₂₅	C₂²×C₅	1	1	yes	characteristic
11	C₄×C₅×C₂₅	C₄×C₅	1	6	yes	—
12	C₂²×C₁₂₅	C₂²×C₅	1	5	yes	—
13	C₄×C₁₂₅	C₄×C₅	1	30	yes	direct product factor
14	C₄²×C₅²	C₂₅	1	1	yes	characteristic
15	C₄²×C₂₅	C₂₅	1	4	yes	—
16	C₄²×C₂₅	C₅²	1	1	yes	characteristic
17	C₂×C₅×C₂₅	C₂×C₄×C₅	1	3	yes	—
18	C₂×C₁₂₅	C₂×C₄×C₅	1	15	yes	—
19	C₂×C₄×C₅²	C₂×C₂₅	1	3	yes	—
20	C₂×C₄×C₂₅	C₂×C₂₅	1	12	yes	—
21	C₂×C₄×C₂₅	C₂×C₅²	1	3	yes	—
22	C₅×C₂₅	C₄²×C₅	1	1	yes	characteristic
23	C₁₂₅	C₄²×C₅	1	5	yes	direct product factor
24	C₂²×C₅²	C₂²×C₂₅	1	1	yes	characteristic
25	C₄×C₅²	C₄×C₂₅	1	6	yes	—
26	C₂²×C₂₅	C₂²×C₂₅	1	4	yes	—
27	C₂²×C₂₅	C₂²×C₅²	1	1	yes	characteristic
28	C₄×C₂₅	C₄×C₂₅	1	24	yes	—
29	C₄×C₂₅	C₄×C₅²	1	6	yes	—
30	C₄²×C₅	C₁₂₅	1	5	yes	direct product factor
31	C₄²×C₅	C₅×C₂₅	1	1	yes	characteristic
32	C₂×C₅²	C₂×C₄×C₂₅	1	3	yes	—
33	C₂×C₂₅	C₂×C₄×C₂₅	1	12	yes	—
34	C₂×C₂₅	C₂×C₄×C₅²	1	3	yes	—
35	C₂×C₄×C₅	C₂×C₁₂₅	1	15	yes	—
36	C₂×C₄×C₅	C₂×C₅×C₂₅	1	3	yes	—
37	C₅²	C₄²×C₂₅	1	1	yes	characteristic
38	C₂₅	C₄²×C₂₅	1	4	yes	—
39	C₂₅	C₄²×C₅²	1	1	yes	characteristic
40	C₂²×C₅	C₂²×C₁₂₅	1	5	yes	—
41	C₂²×C₅	C₂²×C₅×C₂₅	1	1	yes	characteristic
42	C₄×C₅	C₄×C₁₂₅	1	30	yes	direct product factor
43	C₄×C₅	C₄×C₅×C₂₅	1	6	yes	—
44	C₄²	C₅×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₅	C₂×C₄×C₁₂₅	1	15	yes	—
46	C₂×C₅	C₂×C₄×C₅×C₂₅	1	3	yes	—
47	C₂×C₄	C₂×C₅×C₁₂₅	1	3	yes	—
48	C₅	C₄²×C₁₂₅	1	5	yes	direct product factor
49	C₅	C₄²×C₅×C₂₅	1	1	yes	characteristic
50	C₂²	C₂²×C₅×C₁₂₅	1	1	yes	characteristic
51	C₄	C₄×C₅×C₁₂₅	1	6	yes	direct product factor
52	C₂	C₂×C₄×C₅×C₁₂₅	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₂₅	C₂²×C₅²
socle, prime radical	C₂²×C₅²	C₂²×C₂₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×C₁₂₅	C₄	direct (N × H)
5. C₄²×C₁₂₅	C₅	direct (N × H)
9. C₅×C₁₂₅	C₄²	direct (N × H)
13. C₄×C₁₂₅	C₄×C₅	direct (N × H)
23. C₁₂₅	C₄²×C₅	direct (N × H)
30. C₄²×C₅	C₁₂₅	direct (N × H)
42. C₄×C₅	C₄×C₁₂₅	direct (N × H)
44. C₄²	C₅×C₁₂₅	direct (N × H)
48. C₅	C₄²×C₁₂₅	direct (N × H)
51. C₄	C₄×C₅×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
C₂×C₄×C₅×C₁₂₅	2
C₄²×C₅×C₂₅	5
C₄²×C₁₂₅	5

Subgroups — C42×C5×C125 #72188673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₅×C₁₂₅ #72188673