Subgroups — #30627409

Order 15625.

2424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	5₋¹⁺⁴	C₅	1	5	yes	maximal, complemented
2	C₅×#10597837	C₅	1	1	yes	maximal, characteristic, complemented
3	#15691127	C₅	1	100	yes	maximal, complemented
4	#15691126	C₅	1	10	yes	maximal, complemented
5	#15691125	C₅	1	5	yes	maximal, complemented
6	#15691124	C₅	1	4	yes	maximal, complemented
7	#15691123	C₅	1	1	yes	maximal, characteristic
8	#15691122	C₅	1	25	yes	maximal, complemented
9	C₅×#10597832	C₅	1	5	yes	maximal, complemented
10	#10597837	—	5	25	no	—
11	#10597837	—	5	100	no	—
12	#10597837	C₅²	1	125	yes	complemented
13	C₅×5₋¹⁺²	—	5	25	no	—
14	C₅×5₋¹⁺²	C₅²	1	20	yes	—
15	C₅×5₋¹⁺²	C₅²	1	1	yes	characteristic
16	C₅×5₋¹⁺²	C₅²	1	2	yes	—
17	C₅×5₋¹⁺²	C₅²	1	1	yes	characteristic
18	C₅×5₋¹⁺²	C₅²	1	20	yes	complemented
19	C₅×H(3,𝔽₅)	C₅²	1	1	yes	characteristic
20	C₅×H(3,𝔽₅)	C₅²	1	5	yes	—
21	C₅²×C₂₅	C₅²	1	1	yes	characteristic
22	C₅²×C₂₅	C₅²	1	5	yes	complemented
23	#10597833	C₅²	1	250	yes	complemented
24	#10597832	C₅²	1	25	yes	complemented
25	#10597832	C₅²	1	100	yes	complemented
26	#10597832	C₅²	1	125	yes	complemented
27	#10597831	C₅²	1	100	yes	complemented
28	H(3,𝔽₅)⋊C₅	C₅²	1	25	yes	—
29	C₅³	—	5	5	no	—
30	C₅³	C₅³	1	5	yes	—
31	C₅³	C₅³	1	1	yes	characteristic
32	5₋¹⁺²	—	5	20	no	—
33	5₋¹⁺²	—	5	60	no	—
34	5₋¹⁺²	—	5	5	no	—
35	5₋¹⁺²	—	5	10	no	—
36	5₋¹⁺²	—	5	500	no	—
37	5₋¹⁺²	—	5	100	no	—
38	5₋¹⁺²	H(3,𝔽₅)	1	25	yes	—
39	5₋¹⁺²	C₅³	1	100	yes	—
40	H(3,𝔽₅)	—	5	5	no	—
41	H(3,𝔽₅)	—	5	20	no	—
42	H(3,𝔽₅)	C₅³	1	25	yes	—
43	C₅×C₂₅	—	5	5	no	—
44	C₅×C₂₅	—	5	20	no	—
45	C₅×C₂₅	—	5	25	no	—
46	C₅×C₂₅	—	5	125	no	—
47	C₅×C₂₅	C₅³	1	1	yes	characteristic
48	C₅×C₂₅	C₅³	1	4	yes	—
49	C₅×C₂₅	C₅³	1	5	yes	—
50	C₅×C₂₅	C₅³	1	10	yes	—
51	C₅×C₂₅	C₅³	1	5	yes	—
52	C₅²	—	5	5	no	—
53	C₅²	—	5	25	no	—
54	C₅²	—	25	1	no	—
55	C₅²	—	25	5	no	—
56	C₅²	—	125	5	no	—
57	C₅²	C₅×H(3,𝔽₅)	1	5	yes	—
58	C₅²	C₅⁴	1	1	yes	characteristic
59	C₂₅	—	5	1	no	—
60	C₂₅	—	5	100	no	—
61	C₂₅	—	5	4	no	—
62	C₂₅	—	5	5	no	—
63	C₂₅	—	5	10	no	—
64	C₂₅	C₅×H(3,𝔽₅)	1	25	yes	—
65	C₅	—	5	1	no	—
66	C₅	—	5	5	no	—
67	C₅	—	25	5	no	—
68	C₅	—	25	25	no	—
69	C₅	C₅²×H(3,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₅	C₅²×H(3,𝔽₅)
commutator, Frattini, cosocle	C₅²	C₅⁴
prime radical	#15691123	C₅
LCS	G ▷ C₅² ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₅ ◁ C₅×5₋¹⁺² ◁ G
LECS	G ▷ C₅² ▷ C₅ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. 5₋¹⁺⁴	C₅	semidirect
2. C₅×#10597837	C₅	semidirect
3. #15691127	C₅	semidirect
4. #15691126	C₅	semidirect
5. #15691125	C₅	semidirect
6. #15691124	C₅	semidirect
8. #15691122	C₅	semidirect
9. C₅×#10597832	C₅	semidirect
12. #10597837	C₅²	semidirect
18. C₅×5₋¹⁺²	C₅²	semidirect
22. C₅²×C₂₅	C₅²	semidirect
23. #10597833	C₅²	semidirect
24. #10597832	C₅²	semidirect
25. #10597832	C₅²	semidirect
26. #10597832	C₅²	semidirect
27. #10597831	C₅²	semidirect

Maximal subgroups

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

subgroup	index
5₋¹⁺⁴	5
C₅×#10597837	5
#15691127	5
#15691126	5
#15691125	5
#15691124	5
#15691123	5
#15691122	5
C₅×#10597832	5