Subgroups — #65405

Order 320.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	#4070	C₂	1	2	yes	maximal
3	C₂×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#3950	C₂	1	1	yes	maximal, characteristic, complemented
5	#3948	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₀	C₂²	1	2	yes	—
8	C₂×C₅×C₈	—	2	2	no	—
9	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₅	—	2	2	no	—
12	C₄⋊Dic₅	C₂²	1	2	yes	—
13	#683	—	2	2	no	—
14	Dic₂₀	—	2	2	no	—
15	#467	—	5	1	no	maximal, Sylow
16	C₂³×C₅	D₄	1	1	yes	characteristic
17	C₂×C₄×C₅	—	2	1	no	—
18	C₂×C₄×C₅	—	2	1	no	—
19	C₂×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂×Dic₅	—	2	2	no	—
22	C₂×Dic₅	—	2	2	no	—
23	Dic₁₀	—	2	2	no	—
24	Dic₁₀	—	2	2	no	—
25	C₅×C₈	—	2	1	no	—
26	C₅×C₈	D₄	1	2	yes	—
27	C₂×Q₁₆	—	5	1	no	—
28	C₂²×C₈	D₅	1	1	yes	characteristic
29	C₂²⋊Q₈	—	5	2	no	—
30	#107	—	5	1	no	—
31	#103	—	5	2	no	—
32	C₂²×C₅	—	2	1	no	—
33	C₂²×C₅	—	2	1	no	—
34	C₂²×C₅	Q₁₆	1	2	yes	complemented
35	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
36	C₄×C₅	—	2	1	no	—
37	C₄×C₅	C₂×D₄	1	1	yes	characteristic
38	C₄×C₅	#41	1	1	yes	characteristic
39	Dic₅	—	4	2	no	—
40	Dic₅	—	4	2	no	—
41	C₂×Q₈	—	5	2	no	—
42	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
43	Q₁₆	—	10	2	no	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	C₂×D₅	1	1	yes	characteristic
46	C₂×C₈	C₂×D₅	1	1	yes	characteristic
47	C₄⋊C₄	—	5	2	no	—
48	C₄⋊C₄	—	10	2	no	—
49	C₂²⋊C₄	—	10	2	no	—
50	C₂×C₅	—	2	2	no	—
51	C₂×C₅	#121	1	1	yes	characteristic
52	C₂×C₅	C₂×Q₁₆	1	1	yes	characteristic
53	C₂×C₅	#135	1	1	yes	characteristic
54	C₂³	D₂₀	1	1	yes	characteristic
55	Q₈	—	10	2	no	—
56	Q₈	—	10	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	10	2	no	—
60	C₂×C₄	—	10	2	no	—
61	C₂×C₄	D₂₀	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	C₂²⋊D₅	1	2	yes	—
65	C₅	#467	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	Dic₂₀	1	2	yes	complemented
69	C₂²	C₂×D₂₀	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	20	2	no	—
72	C₄	—	20	2	no	—
73	C₄	#709	1	1	yes	characteristic
74	C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
75	C₂	—	2	2	no	—
76	C₂	#4050	1	1	yes	characteristic
77	C₂	C₂×Dic₂₀	1	1	yes	characteristic
78	C₂	#4076	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₂²×C₅×C₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂³×C₅	D₄
hypercenter	C₂²×C₈	D₅
hypocenter	C₅	#467
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×Dic₂₀	C₂	semidirect
4. #3950	C₂	semidirect
5. #3948	C₂	semidirect
34. C₂²×C₅	Q₁₆	semidirect
65. C₅	#467	semidirect
68. C₂²	Dic₂₀	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₈	2
#4070	2
C₂×Dic₂₀	2
#3950	2
#3948	2
#467	5