Subgroups — #65732

Order 320.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4131	C₂	1	2	yes	maximal, complemented
2	#4058	C₂	1	1	yes	maximal, characteristic
3	#4008	C₂	1	2	yes	maximal, complemented
4	#3996	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
6	#711	C₂²	1	2	yes	—
7	#704	—	2	2	no	—
8	C₂×#170	—	2	2	no	—
9	C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
10	#700	C₂²	1	2	yes	—
11	D₅⋊C₈	C₂²	1	1	yes	characteristic
12	C₅×D₈	C₄	1	1	yes	characteristic, complemented
13	#687	C₄	1	2	yes	complemented
14	Dic₂₀	C₄	1	1	yes	characteristic, complemented
15	C₈×D₅	C₂²	1	1	yes	characteristic
16	#443	—	5	1	no	maximal, Sylow
17	C₂×AGL(1,𝔽₅)	—	2	1	no	—
18	C₅×D₄	C₂×C₄	1	2	yes	—
19	C₂²⋊D₅	—	2	2	no	—
20	C₂×Dic₅	—	2	2	no	—
21	C₄×D₅	C₂³	1	1	yes	characteristic
22	Dic₁₀	C₂×C₄	1	2	yes	—
23	#170	—	2	2	no	—
24	#170	D₄	1	1	yes	characteristic
25	#170	D₄	1	1	yes	characteristic
26	C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	#168	C₂×C₄	1	1	yes	characteristic
28	#135	—	5	1	no	—
29	#131	—	5	2	no	—
30	#108	—	5	1	no	—
31	#104	—	5	2	no	—
32	C₄×C₈	—	5	1	no	—
33	C₂²×C₅	—	2	2	no	—
34	C₂×D₅	#41	1	1	yes	characteristic
35	AGL(1,𝔽₅)	—	2	2	no	—
36	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
37	Dic₅	—	2	2	no	—
38	Dic₅	C₂×D₄	1	1	yes	characteristic
39	#41	—	5	2	no	—
40	Q₁₆	—	5	1	no	—
41	SD₁₆	—	5	2	no	—
42	D₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
43	M₁₆	—	5	2	no	—
44	M₁₆	—	10	2	no	—
45	C₂×C₈	—	5	1	no	—
46	C₂×C₈	—	5	1	no	—
47	C₂×C₈	—	10	2	no	—
48	C₄²	—	5	1	no	—
49	C₂×C₅	—	4	2	no	—
50	C₂×C₅	C₄×D₄	1	1	yes	characteristic
51	D₅	—	2	1	no	—
52	Q₈	—	5	2	no	—
53	D₄	—	10	2	no	—
54	D₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
55	C₂×C₄	—	5	1	no	—
56	C₂×C₄	—	10	2	no	—
57	C₂×C₄	—	10	1	no	—
58	C₈	—	5	1	no	—
59	C₈	—	5	1	no	—
60	C₈	—	5	1	no	—
61	C₈	—	10	2	no	—
62	C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₅	#443	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	2	2	no	—
65	C₂²	—	5	1	no	—
66	C₄	—	5	1	no	—
67	C₄	—	10	2	no	—
68	C₄	—	10	2	no	—
69	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₂	—	4	2	no	—
71	C₂	—	10	1	no	—
72	C₂	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₄
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×D₈	C₄
socle	C₂×C₅	C₄×D₄
prime radical	#4058	C₂
cosocle	C₄×D₅	C₂³
hypercenter	D₈	AGL(1,𝔽₅)
hypocenter	C₅	#443
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #4131	C₂	semidirect
3. #4008	C₂	semidirect
4. #3996	C₂	semidirect
5. C₈×AGL(1,𝔽₅)	C₂	semidirect
12. C₅×D₈	C₄	semidirect
13. #687	C₄	semidirect
14. Dic₂₀	C₄	semidirect
42. D₈	AGL(1,𝔽₅)	semidirect
63. C₅	#443	semidirect

Maximal subgroups

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

subgroup	index
#4131	2
#4058	2
#4008	2
#3996	2
C₈×AGL(1,𝔽₅)	2
#443	5