Subgroups — #30627039

Order 15625.

174 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₂₅⋊C₅	C₅	1	4	yes	maximal
2	C₅×C₁₂₅⋊C₅	C₅	1	1	yes	maximal, characteristic
3	#15691099	C₅	1	1	yes	maximal, characteristic
4	C₅²×C₂₅	C₅²	1	1	yes	characteristic
5	C₁₂₅⋊C₅	—	5	4	no	—
6	C₁₂₅⋊C₅	—	5	1	no	—
7	C₁₂₅⋊C₅	—	5	4	no	—
8	C₁₂₅⋊C₅	—	5	8	no	—
9	C₁₂₅⋊C₅	—	5	4	no	—
10	C₁₂₅⋊C₅	C₂₅	1	20	yes	complemented
11	C₅×C₁₂₅	—	5	4	no	—
12	C₅×C₁₂₅	C₂₅	1	5	yes	complemented
13	C₂₅⋊C₂₅	—	5	1	no	—
14	C₂₅⋊C₂₅	—	5	2	no	—
15	C₂₅⋊C₂₅	C₂₅	1	5	yes	—
16	C₅²⋊C₂₅	—	5	1	no	—
17	C₂₅²	—	5	1	no	—
18	C₅³	5₋¹⁺²	1	1	yes	characteristic
19	C₅×C₂₅	—	5	5	no	—
20	C₅×C₂₅	C₅×C₂₅	1	4	yes	—
21	C₅×C₂₅	C₅×C₂₅	1	1	yes	characteristic
22	C₅×C₂₅	C₅×C₂₅	1	1	yes	characteristic
23	C₅×C₂₅	H(3,𝔽₅)	1	1	yes	characteristic
24	C₅×C₂₅	5₋¹⁺²	1	4	yes	—
25	C₅×C₂₅	5₋¹⁺²	1	8	yes	—
26	C₅×C₂₅	5₋¹⁺²	1	4	yes	—
27	C₅×C₂₅	5₋¹⁺²	1	4	yes	—
28	C₅×C₂₅	5₋¹⁺²	1	2	yes	—
29	C₅×C₂₅	5₋¹⁺²	1	1	yes	characteristic
30	C₁₂₅	—	5	20	no	—
31	C₁₂₅	—	5	5	no	—
32	C₅²	—	25	1	no	—
33	C₅²	C₅²⋊C₂₅	1	1	yes	characteristic
34	C₅²	C₂₅⋊C₂₅	1	4	yes	—
35	C₅²	C₂₅⋊C₂₅	1	1	yes	characteristic
36	C₂₅	—	25	5	no	—
37	C₂₅	C₂₅²	1	1	yes	characteristic
38	C₂₅	C₅²⋊C₂₅	1	4	yes	—
39	C₂₅	C₅²⋊C₂₅	1	1	yes	characteristic
40	C₂₅	C₂₅⋊C₂₅	1	4	yes	—
41	C₂₅	C₂₅⋊C₂₅	1	8	yes	—
42	C₂₅	C₂₅⋊C₂₅	1	2	yes	—
43	C₂₅	C₂₅⋊C₂₅	1	4	yes	—
44	C₂₅	C₂₅⋊C₂₅	1	1	yes	characteristic
45	C₅	—	5	1	no	—
46	C₅	—	5	4	no	—
47	C₅	—	5	1	no	—
48	C₅	#15691059	1	1	yes	characteristic

	subgroup	G/H
center	C₂₅	C₂₅⋊C₂₅
commutator	C₂₅	C₂₅²
Frattini, cosocle	C₅²×C₂₅	C₅²
socle	C₅	#15691059
prime radical	C₅³	5₋¹⁺²
LCS	G ▷ C₂₅ ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₂₅ ◁ C₅²×C₂₅ ◁ G
LECS	G ▷ C₅²×C₂₅ ▷ C₅ ▷ C₁

Complements

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

N	H ≅ G/N	type
10. C₁₂₅⋊C₅	C₂₅	semidirect
12. C₅×C₁₂₅	C₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₁₂₅⋊C₅	5
#15691099	5