Subgroups — C₅×#15691123 #30627021

Order 15625.

3672 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×H(3,𝔽₅)	C₅	1	1	yes	maximal, characteristic, complemented
2	C₅²×H(3,𝔽₅)	C₅	1	5	yes	maximal, complemented
3	#15691123	C₅	1	125	yes	maximal, direct product factor
4	C₅×H(3,𝔽₅)⋊C₅	C₅	1	25	yes	maximal, complemented
5	C₅⁴	—	5	5	no	—
6	C₅⁴	C₅²	1	1	yes	characteristic
7	C₅⁴	C₅²	1	5	yes	complemented
8	C₅×H(3,𝔽₅)	—	5	20	no	—
9	C₅×H(3,𝔽₅)	—	5	100	no	—
10	C₅×H(3,𝔽₅)	—	5	5	no	—
11	C₅×H(3,𝔽₅)	—	5	20	no	—
12	C₅×H(3,𝔽₅)	C₅²	1	25	yes	complemented
13	C₅×H(3,𝔽₅)	C₅²	1	25	yes	complemented
14	C₅×H(3,𝔽₅)	C₅²	1	125	yes	complemented
15	H(3,𝔽₅)⋊C₅	C₅²	1	625	yes	complemented
16	C₅³	—	5	20	no	—
17	C₅³	—	5	125	no	—
18	C₅³	—	5	5	no	—
19	C₅³	—	5	25	no	—
20	C₅³	—	25	5	no	—
21	C₅³	—	25	25	no	—
22	C₅³	—	25	125	no	—
23	C₅³	H(3,𝔽₅)	1	5	yes	—
24	C₅³	H(3,𝔽₅)	1	20	yes	—
25	C₅³	C₅³	1	1	yes	characteristic
26	C₅³	C₅³	1	5	yes	—
27	C₅³	C₅³	1	25	yes	complemented
28	H(3,𝔽₅)	—	5	125	no	—
29	H(3,𝔽₅)	—	5	100	no	—
30	H(3,𝔽₅)	—	5	500	no	—
31	H(3,𝔽₅)	C₅³	1	125	yes	complemented
32	C₅²	—	5	25	no	—
33	C₅²	—	5	5	no	—
34	C₅²	—	5	25	no	—
35	C₅²	—	5	125	no	—
36	C₅²	—	25	20	no	—
37	C₅²	—	25	5	no	—
38	C₅²	—	25	25	no	—
39	C₅²	—	25	100	no	—
40	C₅²	—	25	625	no	—
41	C₅²	—	25	25	no	—
42	C₅²	—	25	125	no	—
43	C₅²	C₅×H(3,𝔽₅)	1	4	yes	—
44	C₅²	C₅×H(3,𝔽₅)	1	25	yes	—
45	C₅²	C₅×H(3,𝔽₅)	1	1	yes	characteristic
46	C₅²	C₅⁴	1	1	yes	characteristic
47	C₅	—	5	25	no	—
48	C₅	—	5	4	no	—
49	C₅	—	5	1	no	—
50	C₅	—	25	25	no	—
51	C₅	—	25	125	no	—
52	C₅	#15691123	1	5	yes	direct product factor
53	C₅	C₅²×H(3,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₅²	C₅×H(3,𝔽₅)
commutator, Frattini, cosocle	C₅²	C₅⁴
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×H(3,𝔽₅)	C₅	semidirect
2. C₅²×H(3,𝔽₅)	C₅	semidirect
3. #15691123	C₅	direct (N × H)
4. C₅×H(3,𝔽₅)⋊C₅	C₅	semidirect
7. C₅⁴	C₅²	semidirect
12. C₅×H(3,𝔽₅)	C₅²	semidirect
13. C₅×H(3,𝔽₅)	C₅²	semidirect
14. C₅×H(3,𝔽₅)	C₅²	semidirect
15. H(3,𝔽₅)⋊C₅	C₅²	semidirect
27. C₅³	C₅³	semidirect
31. H(3,𝔽₅)	C₅³	semidirect
52. C₅	#15691123	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×H(3,𝔽₅)	5
#15691123	5
C₅×H(3,𝔽₅)⋊C₅	5

Subgroups — C5×#15691123 #30627021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#15691123 #30627021