Subgroups — #30627031

Order 15625.

572 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₅²	C₅×H(3,𝔽₅)
commutator	C₅²	C₅²×C₂₅
Frattini, cosocle	C₅³	C₅³
prime radical	C₅⁴	C₅²
LCS	G ▷ C₅² ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₅² ◁ C₅⁴ ◁ G
LECS	G ▷ C₅³ ▷ C₅ ▷ C₁

Complements

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

N	H ≅ G/N	type
5. 5₋¹⁺²⋊C₂₅	C₅	semidirect
10. C₅×5₋¹⁺²	C₂₅	semidirect
39. 5₋¹⁺²	C₅×C₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₅²×5₋¹⁺²	5
C₅×C₅²⋊C₂₅	5
5₋¹⁺²⋊C₂₅	5