Subgroups — #66260

Order 320.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#713	C₂	1	1	yes	maximal, characteristic
2	#4133	C₂	1	8	yes	maximal, complemented
3	D₅⋊M₁₆	C₂	1	6	yes	maximal, complemented
4	C₂×C₅×Q₈	C₄	1	1	yes	characteristic
5	#713	C₄	1	4	yes	—
6	#713	C₂²	1	4	yes	—
7	C₂×D₂₀	C₄	1	3	yes	—
8	C₂×C₄×D₅	C₂²	1	3	yes	—
9	#704	C₂²	1	4	yes	—
10	#700	C₂²	1	12	yes	complemented
11	D₅⋊C₈	C₂²	1	12	yes	complemented
12	#568	—	5	1	no	maximal, Sylow
13	C₂²×D₅	C₂×C₄	1	3	yes	—
14	C₅×Q₈	C₂×C₄	1	4	yes	—
15	C₂×C₄×C₅	C₂×C₄	1	3	yes	—
16	C₂×Dic₅	C₂³	1	1	yes	characteristic
17	D₂₀	C₂×C₄	1	12	yes	—
18	C₄×D₅	C₂×C₄	1	6	yes	—
19	C₄×D₅	C₂³	1	6	yes	—
20	#170	C₂³	1	8	yes	—
21	C₂×#41	—	5	1	no	—
22	#131	—	5	8	no	—
23	C₂×M₁₆	—	5	6	no	—
24	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
25	C₂×D₅	—	4	3	no	—
26	C₂×D₅	C₂²×C₄	1	6	yes	—
27	C₄×C₅	C₂²×C₄	1	6	yes	—
28	Dic₅	C₂²×C₄	1	1	yes	characteristic
29	Dic₅	C₂⁴	1	1	yes	characteristic
30	#41	—	5	4	no	—
31	#41	—	5	4	no	—
32	C₂×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
33	C₂×D₄	—	5	3	no	—
34	C₂²×C₄	—	5	3	no	—
35	M₁₆	—	5	4	no	—
36	M₁₆	—	5	12	no	—
37	C₂×C₈	—	5	12	no	—
38	C₂×C₅	—	2	1	no	—
39	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
40	D₅	—	2	6	no	—
41	C₂³	—	5	3	no	—
42	Q₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
43	D₄	—	5	12	no	—
44	C₂×C₄	—	5	1	no	—
45	C₂×C₄	—	5	6	no	—
46	C₂×C₄	—	5	6	no	—
47	C₂×C₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
48	C₈	—	5	8	no	—
49	C₅	#568	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	5	6	no	—
51	C₂²	—	20	3	no	—
52	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₄	—	5	1	no	—
54	C₄	—	5	1	no	—
55	C₄	C₂²×AGL(1,𝔽₅)	1	6	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	10	6	no	—
58	C₂	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂³×C₄
Fitting	C₂×C₅×Q₈	C₄
prime radical	C₂×#713	C₂
cosocle	Dic₅	C₂⁴
hypercenter	C₂×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#568
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. #4133	C₂	semidirect
3. D₅⋊M₁₆	C₂	semidirect
10. #700	C₂²	semidirect
11. D₅⋊C₈	C₂²	semidirect
49. C₅	#568	semidirect

Maximal subgroups

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

subgroup	index
C₂×#713	2
#4133	2
D₅⋊M₁₆	2
#568	5