Subgroups — #65731

Order 320.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
2	D₅×D₈	C₂	1	1	yes	maximal, characteristic
3	#4007	C₂	1	2	yes	maximal, complemented
4	#3993	C₂	1	1	yes	maximal, characteristic, complemented
5	#3992	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
7	D₄×D₅	C₂²	1	2	yes	—
8	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
9	C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
10	C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
11	D₅⋊C₈	C₂²	1	1	yes	characteristic
12	C₅×D₈	C₄	1	1	yes	characteristic, complemented
13	#686	C₄	1	2	yes	complemented
14	D₄₀	C₄	1	1	yes	characteristic, complemented
15	C₈×D₅	C₂²	1	1	yes	characteristic
16	#442	—	5	1	no	maximal, Sylow
17	C₂²×D₅	—	2	2	no	—
18	C₂×AGL(1,𝔽₅)	—	2	2	no	—
19	C₂×AGL(1,𝔽₅)	—	4	4	no	—
20	C₂×AGL(1,𝔽₅)	D₄	1	2	yes	—
21	C₅×D₄	C₂×C₄	1	2	yes	complemented
22	C₂²⋊D₅	—	2	2	no	—
23	D₂₀	C₂×C₄	1	2	yes	complemented
24	C₄×D₅	C₂³	1	1	yes	characteristic
25	#170	—	2	1	no	—
26	C₅×C₈	C₂×C₄	1	1	yes	characteristic, complemented
27	#168	C₂×C₄	1	1	yes	characteristic, complemented
28	C₂×D₈	—	5	1	no	—
29	C₄×D₄	—	5	2	no	—
30	#106	—	5	1	no	—
31	#102	—	5	2	no	—
32	#97	—	5	1	no	—
33	C₂²×C₅	—	2	2	no	—
34	C₂×D₅	—	2	2	no	—
35	C₂×D₅	—	4	2	no	—
36	C₂×D₅	—	4	2	no	—
37	C₂×D₅	C₂×D₄	1	1	yes	characteristic
38	AGL(1,𝔽₅)	—	2	2	no	—
39	AGL(1,𝔽₅)	—	4	2	no	—
40	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	Dic₅	#41	1	1	yes	characteristic
42	C₂×D₄	—	5	2	no	—
43	C₂²×C₄	—	10	2	no	—
44	D₈	—	5	2	no	—
45	D₈	—	5	1	no	—
46	D₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
47	C₂×C₈	—	5	1	no	—
48	C₂×C₈	—	5	1	no	—
49	C₄⋊C₄	—	5	2	no	—
50	C₂²⋊C₄	—	10	2	no	—
51	C₄²	—	5	1	no	—
52	C₂×C₅	—	4	2	no	—
53	C₂×C₅	C₄×D₄	1	1	yes	characteristic
54	D₅	—	4	2	no	—
55	D₅	AGL(1,ℤ/8)	1	2	yes	—
56	C₂³	—	10	2	no	—
57	D₄	—	5	2	no	—
58	D₄	—	10	2	no	—
59	D₄	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
60	C₂×C₄	—	5	1	no	—
61	C₂×C₄	—	5	2	no	—
62	C₂×C₄	—	10	2	no	—
63	C₂×C₄	—	20	4	no	—
64	C₈	—	5	1	no	—
65	C₈	—	10	1	no	—
66	C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
67	C₅	#442	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	2	2	no	—
69	C₂²	—	5	1	no	—
70	C₂²	—	10	2	no	—
71	C₂²	—	20	2	no	—
72	C₂²	—	20	2	no	—
73	C₄	—	5	1	no	—
74	C₄	—	10	2	no	—
75	C₄	—	20	2	no	—
76	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	—	5	2	no	—
79	C₂	—	20	2	no	—
80	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	D₅×D₈	C₂
cosocle	C₄×D₅	C₂³
hypercenter	D₈	AGL(1,𝔽₅)
hypocenter	C₅	#442
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. D₄×AGL(1,𝔽₅)	C₂	semidirect
3. #4007	C₂	semidirect
4. #3993	C₂	semidirect
5. #3992	C₂	semidirect
12. C₅×D₈	C₄	semidirect
13. #686	C₄	semidirect
14. D₄₀	C₄	semidirect
21. C₅×D₄	C₂×C₄	semidirect
23. D₂₀	C₂×C₄	semidirect
26. C₅×C₈	C₂×C₄	semidirect
27. #168	C₂×C₄	semidirect
46. D₈	AGL(1,𝔽₅)	semidirect
59. D₄	C₂×AGL(1,𝔽₅)	semidirect
66. C₈	C₂×AGL(1,𝔽₅)	semidirect
67. C₅	#442	semidirect

Maximal subgroups

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

subgroup	index
D₄×AGL(1,𝔽₅)	2
D₅×D₈	2
#4007	2
#3993	2
#3992	2
#442	5