Subgroups — #65532

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	#4081	C₂	1	3	yes	maximal, complemented
3	#3966	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×Dic₅	C₂²	1	1	yes	characteristic, complemented
8	C₂²⋊Dic₅	—	2	3	no	—
9	C₂²⋊Dic₅	C₂²	1	3	yes	complemented
10	#683	—	2	3	no	—
11	#458	—	5	1	no	maximal, 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	D₄	1	3	yes	—
20	C₂×Dic₅	—	2	3	no	—
21	C₂×Dic₅	—	2	3	no	—
22	H(5,𝔽₂)	D₅	1	1	yes	characteristic
23	#123	—	5	3	no	—
24	#99	—	5	3	no	—
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	Dic₅	—	4	3	no	—
32	Dic₅	—	4	1	no	—
33	#41	—	2	3	no	—
34	C₂×D₄	—	2	3	no	—
35	C₂×D₄	C₂×D₅	1	3	yes	—
36	C₂²×C₄	—	5	1	no	—
37	C₄⋊C₄	—	10	3	no	—
38	C₂²⋊C₄	—	5	3	no	—
39	C₂²⋊C₄	—	10	3	no	—
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	C₂²⋊D₅	1	3	yes	—
45	C₂³	C₂²×D₅	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	—	10	3	no	—
51	C₂×C₄	—	10	3	no	—
52	C₂×C₄	C₂²⋊D₅	1	3	yes	—
53	C₅	#458	1	1	yes	characteristic, complemented, Sylow
54	C₂²	—	2	3	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	4	3	no	—
57	C₂²	—	8	1	no	—
58	C₂²	C₂×C₂²⋊D₅	1	3	yes	—
59	C₄	—	2	3	no	—
60	C₄	—	20	3	no	—
61	C₄	—	20	1	no	—
62	C₂	—	2	3	no	—
63	C₂	—	4	3	no	—
64	C₂	#4099	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4099
commutator, cosocle	C₂³×C₅	C₂³
Frattini	C₂³	C₂²×D₅
Fitting, prime radical	C₅×H(5,𝔽₂)	C₂
socle	C₂×C₅	#120
hypercenter	H(5,𝔽₂)	D₅
hypocenter	C₅	#458
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
2. #4081	C₂	semidirect
3. #3966	C₂	semidirect
7. C₂²×Dic₅	C₂²	semidirect
9. C₂²⋊Dic₅	C₂²	semidirect
53. C₅	#458	semidirect

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
#4081	2
#3966	2
#458	5