Subgroups — #64989

Order 320.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4050	C₂	1	1	yes	maximal, characteristic
2	#4047	C₂	1	2	yes	maximal, complemented
3	C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#3951	C₂	1	1	yes	maximal, characteristic, complemented
5	#3937	C₂	1	2	yes	maximal, complemented
6	#709	C₂²	1	2	yes	—
7	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₄²×C₅	C₂²	1	1	yes	characteristic
10	#681	C₂²	1	2	yes	—
11	Dic₂₀	C₄	1	1	yes	characteristic, complemented
12	D₄₀	C₄	1	1	yes	characteristic, complemented
13	#677	C₄	1	2	yes	complemented
14	#676	—	2	2	no	—
15	C₈×D₅	—	2	2	no	—
16	#443	—	5	1	no	maximal, Sylow
17	C₂×C₄×C₅	—	2	1	no	—
18	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂²⋊D₅	—	2	2	no	—
20	D₂₀	C₂×C₄	1	2	yes	—
21	C₄×D₅	—	2	2	no	—
22	Dic₁₀	C₂×C₄	1	2	yes	—
23	C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₅×C₈	D₄	1	1	yes	characteristic
26	C₅×C₈	D₄	1	1	yes	characteristic
27	#168	—	2	2	no	—
28	#135	—	5	1	no	—
29	#131	—	5	2	no	—
30	#108	—	5	1	no	—
31	#104	—	5	2	no	—
32	C₄×C₈	D₅	1	1	yes	characteristic, complemented
33	C₂²×C₅	#41	1	1	yes	characteristic
34	C₂×D₅	—	2	2	no	—
35	C₄×C₅	—	2	2	no	—
36	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₅	C₂×D₄	1	1	yes	characteristic
38	Dic₅	—	2	2	no	—
39	#41	—	5	2	no	—
40	Q₁₆	—	5	1	no	—
41	SD₁₆	—	5	2	no	—
42	D₈	—	5	1	no	—
43	M₁₆	—	5	2	no	—
44	M₁₆	—	10	2	no	—
45	C₂×C₈	—	10	2	no	—
46	C₂×C₈	C₂×D₅	1	1	yes	characteristic
47	C₂×C₈	C₂×D₅	1	1	yes	characteristic
48	C₄²	C₂×D₅	1	1	yes	characteristic
49	C₂×C₅	—	2	1	no	—
50	C₂×C₅	C₄×D₄	1	1	yes	characteristic
51	D₅	—	4	2	no	—
52	Q₈	—	5	2	no	—
53	D₄	—	5	2	no	—
54	D₄	—	10	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	10	2	no	—
57	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
58	C₈	—	10	2	no	—
59	C₈	C₄×D₅	1	1	yes	characteristic
60	C₈	C₄×D₅	1	1	yes	characteristic
61	C₈	D₂₀	1	1	yes	characteristic
62	C₈	D₂₀	1	1	yes	characteristic
63	C₅	#443	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	10	2	no	—
65	C₂²	#709	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	10	2	no	—
68	C₄	C₂×C₄×D₅	1	1	yes	characteristic
69	C₄	C₂×D₂₀	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	20	2	no	—
72	C₂	C₄×D₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₂₀
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×C₈	C₂
socle	C₂×C₅	C₄×D₄
prime radical	#4050	C₂
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₄×C₈	D₅
hypocenter	C₅	#443
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #4047	C₂	semidirect
3. C₄×C₅×C₈	C₂	semidirect
4. #3951	C₂	semidirect
5. #3937	C₂	semidirect
11. Dic₂₀	C₄	semidirect
12. D₄₀	C₄	semidirect
13. #677	C₄	semidirect
32. C₄×C₈	D₅	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
#4050	2
#4047	2
C₄×C₅×C₈	2
#3951	2
#3937	2
#443	5