Subgroups — #30627026

Order 15625.

772 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×H(3,𝔽₅)	C₅	1	1	yes	maximal, characteristic
2	C₅×C₅²⋊C₂₅	C₅	1	1	yes	maximal, characteristic, complemented
3	C₅×C₅²⋊C₂₅	C₅	1	4	yes	maximal, complemented
4	H(3,𝔽₅)⋊C₂₅	C₅	1	25	yes	maximal, complemented
5	C₅⁴	—	5	1	no	—
6	C₅⁴	C₅²	1	1	yes	characteristic
7	C₅×H(3,𝔽₅)	—	5	4	no	—
8	C₅×H(3,𝔽₅)	—	5	20	no	—
9	C₅×H(3,𝔽₅)	C₂₅	1	25	yes	complemented
10	C₅×H(3,𝔽₅)	C₅²	1	5	yes	—
11	C₅²×C₂₅	—	5	4	no	—
12	C₅²×C₂₅	C₅²	1	5	yes	complemented
13	C₅²⋊C₂₅	—	5	1	no	—
14	C₅²⋊C₂₅	—	5	4	no	—
15	C₅²⋊C₂₅	—	5	4	no	—
16	C₅²⋊C₂₅	—	5	12	no	—
17	C₅²⋊C₂₅	C₅²	1	20	yes	complemented
18	C₅³	—	5	25	no	—
19	C₅³	—	5	5	no	—
20	C₅³	—	25	1	no	—
21	C₅³	—	25	2	no	—
22	C₅³	—	25	2	no	—
23	C₅³	—	25	5	no	—
24	C₅³	—	25	10	no	—
25	C₅³	—	25	10	no	—
26	C₅³	C₅×C₂₅	1	5	yes	—
27	C₅³	H(3,𝔽₅)	1	1	yes	characteristic
28	C₅³	H(3,𝔽₅)	1	4	yes	—
29	C₅³	5₋¹⁺²	1	20	yes	—
30	C₅³	C₅³	1	1	yes	characteristic
31	H(3,𝔽₅)	—	5	100	no	—
32	H(3,𝔽₅)	—	5	20	no	—
33	H(3,𝔽₅)	C₅×C₂₅	1	25	yes	complemented
34	C₅×C₂₅	—	5	5	no	—
35	C₅×C₂₅	—	5	20	no	—
36	C₅×C₂₅	—	25	20	no	—
37	C₅×C₂₅	—	25	5	no	—
38	C₅²	—	5	4	no	—
39	C₅²	—	5	1	no	—
40	C₅²	—	5	1	no	—
41	C₅²	—	5	2	no	—
42	C₅²	—	5	8	no	—
43	C₅²	—	5	2	no	—
44	C₅²	—	5	2	no	—
45	C₅²	—	5	8	no	—
46	C₅²	—	5	2	no	—
47	C₅²	—	5	25	no	—
48	C₅²	—	25	5	no	—
49	C₅²	—	25	20	no	—
50	C₅²	—	25	125	no	—
51	C₅²	—	25	5	no	—
52	C₅²	—	25	10	no	—
53	C₅²	—	25	10	no	—
54	C₅²	C₅²⋊C₂₅	1	20	yes	—
55	C₅²	C₅²⋊C₂₅	1	5	yes	—
56	C₅²	C₅²×C₂₅	1	1	yes	characteristic
57	C₅²	C₅×H(3,𝔽₅)	1	1	yes	characteristic
58	C₅²	C₅×5₋¹⁺²	1	4	yes	—
59	C₂₅	—	25	5	no	—
60	C₂₅	—	25	20	no	—
61	C₅	—	5	5	no	—
62	C₅	—	5	20	no	—
63	C₅	—	5	1	no	—
64	C₅	—	5	4	no	—
65	C₅	—	25	25	no	—
66	C₅	C₅×C₅²⋊C₂₅	1	1	yes	characteristic
67	C₅	#15691123	1	1	yes	characteristic
68	C₅	#15691125	1	2	yes	—
69	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	C₅²×H(3,𝔽₅)	C₅
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
2. C₅×C₅²⋊C₂₅	C₅	semidirect
3. C₅×C₅²⋊C₂₅	C₅	semidirect
4. H(3,𝔽₅)⋊C₂₅	C₅	semidirect
9. C₅×H(3,𝔽₅)	C₂₅	semidirect
12. C₅²×C₂₅	C₅²	semidirect
17. C₅²⋊C₂₅	C₅²	semidirect
33. H(3,𝔽₅)	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×H(3,𝔽₅)	5
C₅×C₅²⋊C₂₅	5
H(3,𝔽₅)⋊C₂₅	5