Subgroups — C₅×#458 #65621

Order 320.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₅×#123	C₂	1	3	yes	maximal, complemented
3	C₅×#99	C₂	1	3	yes	maximal, complemented
4	C₅×#41	—	2	3	no	—
5	C₂×C₅×D₄	—	2	3	no	—
6	C₂×C₅×D₄	C₂²	1	3	yes	—
7	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
8	C₅×C₄⋊C₄	—	2	3	no	—
9	C₅×C₂²⋊C₄	—	2	3	no	—
10	C₅×C₂²⋊C₄	C₂²	1	3	yes	complemented
11	#458	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂³×C₅	—	2	1	no	—
13	C₂³×C₅	D₄	1	3	yes	—
14	C₂³×C₅	C₂³	1	1	yes	characteristic
15	C₅×Q₈	—	2	1	no	—
16	C₅×D₄	—	2	3	no	—
17	C₅×D₄	—	2	6	no	—
18	C₂×C₄×C₅	—	2	3	no	—
19	C₂×C₄×C₅	—	2	3	no	—
20	C₂×C₄×C₅	—	2	3	no	—
21	C₂×C₄×C₅	D₄	1	3	yes	—
22	H(5,𝔽₂)	C₂×C₅	1	1	yes	characteristic
23	#123	C₂×C₅	1	3	yes	complemented
24	#99	C₂×C₅	1	3	yes	complemented
25	C₂²×C₅	—	2	3	no	—
26	C₂²×C₅	—	4	1	no	—
27	C₂²×C₅	—	4	3	no	—
28	C₂²×C₅	—	8	1	no	—
29	C₂²×C₅	C₂×D₄	1	3	yes	—
30	C₄×C₅	—	2	3	no	—
31	C₄×C₅	—	4	3	no	—
32	C₄×C₅	—	4	1	no	—
33	#41	—	2	3	no	—
34	C₂×D₄	—	2	3	no	—
35	C₂×D₄	C₂²×C₅	1	3	yes	—
36	C₂²×C₄	C₂²×C₅	1	1	yes	characteristic, complemented
37	C₄⋊C₄	—	2	3	no	—
38	C₂²⋊C₄	—	2	3	no	—
39	C₂²⋊C₄	C₂²×C₅	1	3	yes	complemented
40	C₂×C₅	—	2	3	no	—
41	C₂×C₅	—	4	3	no	—
42	C₂×C₅	#120	1	1	yes	characteristic
43	C₂³	—	2	1	no	—
44	C₂³	C₅×D₄	1	3	yes	—
45	C₂³	C₂³×C₅	1	1	yes	characteristic
46	Q₈	—	2	1	no	—
47	D₄	—	2	3	no	—
48	D₄	—	2	6	no	—
49	C₂×C₄	—	2	3	no	—
50	C₂×C₄	—	2	3	no	—
51	C₂×C₄	—	2	3	no	—
52	C₂×C₄	C₅×D₄	1	3	yes	—
53	C₅	#458	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²	—	2	3	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	4	3	no	—
57	C₂²	—	8	1	no	—
58	C₂²	C₂×C₅×D₄	1	3	yes	—
59	C₄	—	2	3	no	—
60	C₄	—	4	3	no	—
61	C₄	—	4	1	no	—
62	C₂	—	2	3	no	—
63	C₂	—	4	3	no	—
64	C₂	C₅×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	#120
commutator, Frattini, cosocle	C₂³	C₂³×C₅
prime radical	C₅×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#123	C₂	semidirect
3. C₅×#99	C₂	semidirect
7. C₂²×C₄×C₅	C₂²	semidirect
10. C₅×C₂²⋊C₄	C₂²	semidirect
11. #458	C₅	direct (N × H)
23. #123	C₂×C₅	semidirect
24. #99	C₂×C₅	semidirect
36. C₂²×C₄	C₂²×C₅	semidirect
39. C₂²⋊C₄	C₂²×C₅	semidirect
53. C₅	#458	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×H(5,𝔽₂)	2
C₅×#123	2
C₅×#99	2
#458	5

Subgroups — C5×#458 #65621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#458 #65621