Subgroups — #30627028

Order 15625.

410 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×H(3,𝔽₅)⋊C₅	C₅	1	1	yes	maximal, characteristic
2	C₂₅×H(3,𝔽₅)	C₅	1	5	yes	maximal, complemented
3	C₅×C₅²⋊C₂₅	C₅	1	1	yes	maximal, characteristic, complemented
4	H(3,𝔽₅)⋊C₂₅	C₅	1	20	yes	maximal, complemented
5	#15691075	C₅	1	4	yes	maximal, complemented
6	C₅⁴	C₅²	1	1	yes	characteristic
7	C₅×H(3,𝔽₅)	C₅²	1	5	yes	—
8	C₅²×C₂₅	—	5	5	no	—
9	C₅²×C₂₅	C₅²	1	4	yes	—
10	C₅²×C₂₅	C₅²	1	1	yes	characteristic
11	H(3,𝔽₅)⋊C₅	C₂₅	1	25	yes	complemented
12	C₅²⋊C₂₅	—	5	5	no	—
13	C₅²⋊C₂₅	—	5	20	no	—
14	C₅²⋊C₂₅	C₅²	1	20	yes	—
15	C₅³	—	5	1	no	—
16	C₅³	—	5	5	no	—
17	C₅³	—	5	4	no	—
18	C₅³	—	25	1	no	—
19	C₅³	—	25	4	no	—
20	C₅³	C₅×C₂₅	1	5	yes	complemented
21	C₅³	C₅³	1	1	yes	characteristic
22	H(3,𝔽₅)	—	5	20	no	—
23	H(3,𝔽₅)	C₅×C₂₅	1	25	yes	complemented
24	C₅×C₂₅	—	5	4	no	—
25	C₅×C₂₅	—	5	20	no	—
26	C₅×C₂₅	—	25	1	no	—
27	C₅×C₂₅	—	25	25	no	—
28	C₅×C₂₅	—	25	4	no	—
29	C₅×C₂₅	H(3,𝔽₅)	1	5	yes	—
30	C₅²	—	5	1	no	—
31	C₅²	—	5	4	no	—
32	C₅²	—	5	25	no	—
33	C₅²	—	5	5	no	—
34	C₅²	—	25	5	no	—
35	C₅²	—	25	20	no	—
36	C₅²	—	25	5	no	—
37	C₅²	—	25	1	no	—
38	C₅²	—	25	20	no	—
39	C₅²	—	25	4	no	—
40	C₅²	C₅²×C₂₅	1	1	yes	characteristic
41	C₅²	C₅×H(3,𝔽₅)	1	1	yes	characteristic
42	C₅²	#10597837	1	4	yes	—
43	C₂₅	—	5	5	no	—
44	C₂₅	—	25	4	no	—
45	C₂₅	—	25	20	no	—
46	C₅	—	5	1	no	—
47	C₅	—	5	4	no	—
48	C₅	—	25	25	no	—
49	C₅	—	25	5	no	—
50	C₅	C₂₅×H(3,𝔽₅)	1	1	yes	characteristic
51	C₅	#15691123	1	1	yes	characteristic
52	C₅	#15691127	1	4	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)⋊C₅	C₅
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. C₂₅×H(3,𝔽₅)	C₅	semidirect
3. C₅×C₅²⋊C₂₅	C₅	semidirect
4. H(3,𝔽₅)⋊C₂₅	C₅	semidirect
5. #15691075	C₅	semidirect
11. H(3,𝔽₅)⋊C₅	C₂₅	semidirect
20. C₅³	C₅×C₂₅	semidirect
23. 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
C₅×H(3,𝔽₅)⋊C₅	5
C₂₅×H(3,𝔽₅)	5
C₅×C₅²⋊C₂₅	5
H(3,𝔽₅)⋊C₂₅	5
#15691075	5