Subgroups — #30627034

Order 15625.

482 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#10597837	C₅	1	1	yes	maximal, characteristic
2	#15691099	C₅	1	4	yes	maximal, complemented
3	C₅×C₂₅⋊C₂₅	C₅	1	1	yes	maximal, characteristic, complemented
4	5₋¹⁺²⋊C₂₅	C₅	1	20	yes	maximal, complemented
5	H(3,𝔽₅)⋊C₂₅	C₅	1	5	yes	maximal
6	#10597837	—	5	20	no	—
7	#10597837	C₂₅	1	25	yes	complemented
8	C₅×5₋¹⁺²	—	5	4	no	—
9	C₅×5₋¹⁺²	C₅²	1	4	yes	—
10	C₅×H(3,𝔽₅)	C₅²	1	1	yes	characteristic
11	C₅²×C₂₅	—	5	1	no	—
12	C₅²×C₂₅	C₅²	1	5	yes	—
13	C₅²×C₂₅	C₅²	1	1	yes	characteristic
14	C₂₅⋊C₂₅	—	5	4	no	—
15	C₂₅⋊C₂₅	—	5	1	no	—
16	C₂₅⋊C₂₅	—	5	12	no	—
17	C₂₅⋊C₂₅	—	5	4	no	—
18	C₅²⋊C₂₅	C₅²	1	20	yes	—
19	C₂₅²	—	5	4	no	—
20	C₅³	—	5	1	no	—
21	C₅³	C₅³	1	1	yes	characteristic
22	5₋¹⁺²	—	5	16	no	—
23	5₋¹⁺²	—	5	100	no	—
24	5₋¹⁺²	C₅×C₂₅	1	20	yes	—
25	H(3,𝔽₅)	—	5	4	no	—
26	H(3,𝔽₅)	C₅×C₂₅	1	5	yes	—
27	C₅×C₂₅	—	5	4	no	—
28	C₅×C₂₅	—	5	25	no	—
29	C₅×C₂₅	—	5	20	no	—
30	C₅×C₂₅	—	5	5	no	—
31	C₅×C₂₅	—	25	5	no	—
32	C₅×C₂₅	C₅×C₂₅	1	5	yes	—
33	C₅×C₂₅	H(3,𝔽₅)	1	4	yes	—
34	C₅×C₂₅	H(3,𝔽₅)	1	1	yes	characteristic
35	C₅×C₂₅	5₋¹⁺²	1	20	yes	—
36	C₅²	—	5	5	no	—
37	C₅²	—	5	5	no	—
38	C₅²	—	25	5	no	—
39	C₅²	C₅²×C₂₅	1	1	yes	characteristic
40	C₅²	C₅×H(3,𝔽₅)	1	1	yes	characteristic
41	C₅²	C₅×5₋¹⁺²	1	4	yes	—
42	C₂₅	—	5	20	no	—
43	C₂₅	—	25	20	no	—
44	C₂₅	—	25	5	no	—
45	C₂₅	C₅²⋊C₂₅	1	5	yes	—
46	C₂₅	C₅²⋊C₂₅	1	20	yes	—
47	C₅	—	5	1	no	—
48	C₅	—	5	4	no	—
49	C₅	—	25	5	no	—
50	C₅	C₅×C₅²⋊C₂₅	1	1	yes	characteristic
51	C₅	#15691127	1	5	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. #15691099	C₅	semidirect
3. C₅×C₂₅⋊C₂₅	C₅	semidirect
4. 5₋¹⁺²⋊C₂₅	C₅	semidirect
7. #10597837	C₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₅×#10597837	5
#15691099	5
C₅×C₂₅⋊C₂₅	5
5₋¹⁺²⋊C₂₅	5
H(3,𝔽₅)⋊C₂₅	5