Subgroups — C₅×#15691126 #30627025

Order 15625.

2672 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×5₋¹⁺²	C₅	1	4	yes	maximal, complemented
2	C₅²×5₋¹⁺²	C₅	1	1	yes	maximal, characteristic, complemented
3	C₅²×H(3,𝔽₅)	C₅	1	1	yes	maximal, characteristic
4	#15691126	C₅	1	125	yes	maximal, direct product factor
5	C₅×#10597833	C₅	1	25	yes	maximal, complemented
6	C₅⁴	—	5	1	no	—
7	C₅⁴	C₅²	1	1	yes	characteristic
8	C₅×5₋¹⁺²	—	5	80	no	—
9	C₅×5₋¹⁺²	—	5	20	no	—
10	C₅×5₋¹⁺²	—	5	12	no	—
11	C₅×5₋¹⁺²	—	5	4	no	—
12	C₅×5₋¹⁺²	—	5	4	no	—
13	C₅×5₋¹⁺²	—	5	1	no	—
14	C₅×5₋¹⁺²	C₅²	1	20	yes	complemented
15	C₅×5₋¹⁺²	C₅²	1	100	yes	complemented
16	C₅×5₋¹⁺²	C₅²	1	25	yes	complemented
17	C₅×H(3,𝔽₅)	—	5	20	no	—
18	C₅×H(3,𝔽₅)	—	5	4	no	—
19	C₅×H(3,𝔽₅)	C₅²	1	5	yes	—
20	C₅×H(3,𝔽₅)	C₅²	1	25	yes	—
21	C₅²×C₂₅	—	5	4	no	—
22	C₅²×C₂₅	C₅²	1	5	yes	complemented
23	#10597833	C₅²	1	625	yes	complemented
24	C₅³	—	5	25	no	—
25	C₅³	—	5	5	no	—
26	C₅³	—	25	5	no	—
27	C₅³	—	25	25	no	—
28	C₅³	H(3,𝔽₅)	1	4	yes	—
29	C₅³	H(3,𝔽₅)	1	1	yes	characteristic
30	C₅³	H(3,𝔽₅)	1	20	yes	—
31	C₅³	C₅³	1	1	yes	characteristic
32	C₅³	C₅³	1	5	yes	—
33	5₋¹⁺²	—	5	80	no	—
34	5₋¹⁺²	—	5	300	no	—
35	5₋¹⁺²	—	5	100	no	—
36	5₋¹⁺²	—	5	100	no	—
37	5₋¹⁺²	—	5	25	no	—
38	5₋¹⁺²	C₅³	1	100	yes	complemented
39	H(3,𝔽₅)	—	5	20	no	—
40	H(3,𝔽₅)	—	5	100	no	—
41	H(3,𝔽₅)	C₅³	1	25	yes	—
42	C₅×C₂₅	—	5	100	no	—
43	C₅×C₂₅	—	5	20	no	—
44	C₅×C₂₅	—	5	20	no	—
45	C₅×C₂₅	—	5	5	no	—
46	C₅×C₂₅	C₅³	1	25	yes	complemented
47	C₅²	—	5	5	no	—
48	C₅²	—	5	20	no	—
49	C₅²	—	5	5	no	—
50	C₅²	—	5	25	no	—
51	C₅²	—	25	20	no	—
52	C₅²	—	25	5	no	—
53	C₅²	—	25	125	no	—
54	C₅²	—	25	25	no	—
55	C₅²	C₅×H(3,𝔽₅)	1	20	yes	—
56	C₅²	C₅×H(3,𝔽₅)	1	4	yes	—
57	C₅²	C₅×H(3,𝔽₅)	1	5	yes	—
58	C₅²	C₅×H(3,𝔽₅)	1	1	yes	characteristic
59	C₅²	C₅⁴	1	1	yes	characteristic
60	C₂₅	—	5	100	no	—
61	C₂₅	—	5	25	no	—
62	C₅	—	5	5	no	—
63	C₅	—	5	20	no	—
64	C₅	—	5	4	no	—
65	C₅	—	5	1	no	—
66	C₅	—	25	25	no	—
67	C₅	#15691126	1	5	yes	direct product factor
68	C₅	C₅²×H(3,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₅²	C₅×H(3,𝔽₅)
commutator, Frattini, cosocle	C₅²	C₅⁴
prime radical	C₅²×H(3,𝔽₅)	C₅
LCS	G ▷ C₅² ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₅² ◁ C₅⁴ ◁ G
LECS	G ▷ C₅² ▷ C₅ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₅²×5₋¹⁺²	C₅	semidirect
2. C₅²×5₋¹⁺²	C₅	semidirect
4. #15691126	C₅	direct (N × H)
5. C₅×#10597833	C₅	semidirect
14. C₅×5₋¹⁺²	C₅²	semidirect
15. C₅×5₋¹⁺²	C₅²	semidirect
16. C₅×5₋¹⁺²	C₅²	semidirect
22. C₅²×C₂₅	C₅²	semidirect
23. #10597833	C₅²	semidirect
38. 5₋¹⁺²	C₅³	semidirect
46. C₅×C₂₅	C₅³	semidirect
67. C₅	#15691126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×5₋¹⁺²	5
C₅²×H(3,𝔽₅)	5
#15691126	5
C₅×#10597833	5

Subgroups — C5×#15691126 #30627025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#15691126 #30627025