Subgroups — #30626939

Order 15625.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₂₅⋊C₅	C₅	1	1	yes	maximal, characteristic
2	C₂₅×5₋¹⁺²	C₅	1	1	yes	maximal, characteristic
3	C₂₅⋊C₁₂₅	C₅	1	4	yes	maximal, complemented
4	C₂₅⋊C₁₂₅	C₅	1	16	yes	maximal, complemented
5	C₅²⋊C₁₂₅	C₅	1	4	yes	maximal
6	#15691073	C₅	1	4	yes	maximal, complemented
7	C₂₅×C₁₂₅	C₅	1	1	yes	maximal, characteristic, complemented
8	C₅×5₋¹⁺²	C₂₅	1	1	yes	characteristic
9	C₅²×C₂₅	C₅²	1	1	yes	characteristic
10	C₁₂₅⋊C₅	—	5	5	no	—
11	C₅×C₁₂₅	C₅²	1	4	yes	—
12	C₅×C₁₂₅	C₅²	1	16	yes	—
13	C₅×C₁₂₅	C₅²	1	4	yes	—
14	C₅×C₁₂₅	C₅²	1	1	yes	characteristic
15	C₂₅⋊C₂₅	C₂₅	1	16	yes	—
16	C₂₅⋊C₂₅	C₂₅	1	4	yes	—
17	C₅²⋊C₂₅	C₂₅	1	4	yes	—
18	C₂₅²	C₅²	1	1	yes	characteristic
19	C₂₅²	C₅²	1	4	yes	—
20	C₅³	C₅×C₂₅	1	1	yes	characteristic
21	5₋¹⁺²	—	5	5	no	—
22	C₅×C₂₅	—	5	1	no	—
23	C₅×C₂₅	—	5	4	no	—
24	C₅×C₂₅	C₅×C₂₅	1	4	yes	—
25	C₅×C₂₅	C₅×C₂₅	1	16	yes	—
26	C₅×C₂₅	C₅×C₂₅	1	4	yes	—
27	C₅×C₂₅	C₅×C₂₅	1	4	yes	—
28	C₅×C₂₅	C₅×C₂₅	1	1	yes	characteristic
29	C₅×C₂₅	C₅³	1	1	yes	characteristic
30	C₁₂₅	—	5	4	no	—
31	C₁₂₅	—	5	4	no	—
32	C₁₂₅	—	5	16	no	—
33	C₁₂₅	5₋¹⁺²	1	5	yes	complemented
34	C₅²	—	5	1	no	—
35	C₅²	—	5	1	no	—
36	C₅²	—	5	4	no	—
37	C₅²	C₅²×C₂₅	1	1	yes	characteristic
38	C₂₅	—	5	4	no	—
39	C₂₅	—	5	4	no	—
40	C₂₅	—	5	4	no	—
41	C₂₅	—	5	16	no	—
42	C₂₅	C₁₂₅⋊C₅	1	5	yes	complemented
43	C₂₅	C₅×5₋¹⁺²	1	1	yes	characteristic
44	C₂₅	#10597837	1	4	yes	—
45	C₅	—	25	1	no	—
46	C₅	C₂₅×5₋¹⁺²	1	1	yes	characteristic
47	C₅	C₅×C₁₂₅⋊C₅	1	1	yes	characteristic
48	C₅	#15691117	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₅⋊C₁₂₅	C₅	semidirect
4. C₂₅⋊C₁₂₅	C₅	semidirect
6. #15691073	C₅	semidirect
7. C₂₅×C₁₂₅	C₅	semidirect
33. C₁₂₅	5₋¹⁺²	semidirect
42. C₂₅	C₁₂₅⋊C₅	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
C₂₅×5₋¹⁺²	5
C₂₅⋊C₁₂₅	5
C₅²⋊C₁₂₅	5
#15691073	5
C₂₅×C₁₂₅	5