Subgroups — #10597833⋊C₂₅ #30627036

Order 15625.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#10597833	C₅	1	1	yes	maximal, characteristic
2	C₂₅×5₋¹⁺²	C₅	1	4	yes	maximal, complemented
3	C₂₅×H(3,𝔽₅)	C₅	1	1	yes	maximal, characteristic
4	C₅×C₂₅⋊C₂₅	C₅	1	1	yes	maximal, characteristic, complemented
5	5₋¹⁺²⋊C₂₅	C₅	1	16	yes	maximal, complemented
6	H(3,𝔽₅)⋊C₂₅	C₅	1	4	yes	maximal
7	#15691077	C₅	1	4	yes	maximal, complemented
8	C₅×5₋¹⁺²	C₅²	1	4	yes	—
9	C₅×H(3,𝔽₅)	C₅²	1	1	yes	characteristic
10	C₅²×C₂₅	—	5	1	no	—
11	C₅²×C₂₅	C₅²	1	1	yes	characteristic
12	C₅²×C₂₅	C₅²	1	4	yes	—
13	C₅²×C₂₅	C₅²	1	1	yes	characteristic
14	#10597833	C₂₅	1	25	yes	complemented
15	C₂₅⋊C₂₅	—	5	4	no	—
16	C₂₅⋊C₂₅	—	5	1	no	—
17	C₂₅⋊C₂₅	—	5	12	no	—
18	C₂₅⋊C₂₅	—	5	4	no	—
19	C₅²⋊C₂₅	—	5	4	no	—
20	C₅²⋊C₂₅	C₅²	1	16	yes	—
21	C₅²⋊C₂₅	C₅²	1	4	yes	—
22	C₂₅²	—	5	4	no	—
23	C₅³	—	5	1	no	—
24	C₅³	C₅³	1	1	yes	characteristic
25	5₋¹⁺²	—	5	16	no	—
26	5₋¹⁺²	C₅×C₂₅	1	20	yes	complemented
27	H(3,𝔽₅)	—	5	4	no	—
28	H(3,𝔽₅)	C₅×C₂₅	1	5	yes	—
29	C₅×C₂₅	—	5	4	no	—
30	C₅×C₂₅	—	5	1	no	—
31	C₅×C₂₅	—	5	4	no	—
32	C₅×C₂₅	—	5	16	no	—
33	C₅×C₂₅	—	5	4	no	—
34	C₅×C₂₅	—	5	4	no	—
35	C₅×C₂₅	—	25	1	no	—
36	C₅×C₂₅	—	25	5	no	—
37	C₅×C₂₅	—	25	4	no	—
38	C₅×C₂₅	C₅×C₂₅	1	5	yes	complemented
39	C₅×C₂₅	H(3,𝔽₅)	1	4	yes	—
40	C₅×C₂₅	H(3,𝔽₅)	1	1	yes	characteristic
41	C₅²	—	5	1	no	—
42	C₅²	—	5	4	no	—
43	C₅²	—	5	5	no	—
44	C₅²	—	25	1	no	—
45	C₅²	—	25	4	no	—
46	C₅²	C₅²×C₂₅	1	1	yes	characteristic
47	C₅²	C₅×H(3,𝔽₅)	1	1	yes	characteristic
48	C₅²	#10597837	1	4	yes	—
49	C₂₅	—	5	5	no	—
50	C₂₅	—	5	20	no	—
51	C₂₅	—	5	1	no	—
52	C₂₅	—	5	4	no	—
53	C₂₅	—	25	4	no	—
54	C₂₅	—	25	16	no	—
55	C₂₅	—	25	4	no	—
56	C₅	—	5	1	no	—
57	C₅	—	5	4	no	—
58	C₅	—	25	5	no	—
59	C₅	C₂₅×H(3,𝔽₅)	1	1	yes	characteristic
60	C₅	#15691126	1	1	yes	characteristic
61	C₅	#15691127	1	4	yes	—

	subgroup	G/H
center, socle	C₅²	C₅×H(3,𝔽₅)
commutator	C₅²	C₅²×C₂₅
Frattini, cosocle	C₅³	C₅³
prime radical	C₅×H(3,𝔽₅)	C₅²
LCS	G ▷ C₅² ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₅² ◁ C₅²×C₂₅ ◁ G
LECS	G ▷ C₅³ ▷ C₅ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂₅×5₋¹⁺²	C₅	semidirect
4. C₅×C₂₅⋊C₂₅	C₅	semidirect
5. 5₋¹⁺²⋊C₂₅	C₅	semidirect
7. #15691077	C₅	semidirect
14. #10597833	C₂₅	semidirect
26. 5₋¹⁺²	C₅×C₂₅	semidirect
38. C₅×C₂₅	C₅×C₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₅×#10597833	5
C₂₅×5₋¹⁺²	5
C₂₅×H(3,𝔽₅)	5
C₅×C₂₅⋊C₂₅	5
5₋¹⁺²⋊C₂₅	5
H(3,𝔽₅)⋊C₂₅	5
#15691077	5

Subgroups — #10597833⋊C25 #30627036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10597833⋊C₂₅ #30627036