Subgroups — #65203

Order 320.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4065	C₂	1	1	yes	maximal, characteristic, complemented
2	D₅×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#3960	C₂	1	1	yes	maximal, characteristic, complemented
5	#3958	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈₀	C₂	1	1	yes	maximal, characteristic, complemented
7	#3930	C₂	1	1	yes	maximal, characteristic, complemented
8	#713	—	2	1	no	—
9	D₄×D₅	—	2	1	no	—
10	C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
11	C₅×D₈	C₂²	1	1	yes	characteristic
12	#688	—	2	1	no	—
13	#686	—	2	1	no	—
14	D₄₀	C₂²	1	1	yes	characteristic
15	D₄₀	C₂²	1	1	yes	characteristic, complemented
16	C₈×D₅	C₂²	1	1	yes	characteristic
17	C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
18	#672	C₂²	1	1	yes	characteristic, complemented
19	#509	—	5	1	no	maximal, Sylow
20	C₂²×D₅	—	4	1	no	—
21	C₅×Q₈	—	2	1	no	—
22	C₅×D₄	—	2	1	no	—
23	C₂²⋊D₅	—	4	1	no	—
24	D₂₀	—	2	1	no	—
25	D₂₀	—	2	1	no	—
26	D₂₀	—	4	1	no	—
27	C₄×D₅	—	4	1	no	—
28	C₄×D₅	D₄	1	1	yes	characteristic
29	C₅×C₈	C₂³	1	1	yes	characteristic
30	#168	D₄	1	1	yes	characteristic
31	#135	—	5	1	no	—
32	C₂×D₈	—	5	1	no	—
33	SD₃₂	—	5	1	no	—
34	SD₃₂	D₅	1	1	yes	characteristic, complemented
35	D₁₆	—	5	1	no	—
36	D₁₆	—	5	1	no	—
37	M₃₂	—	5	1	no	—
38	C₂²×C₅	—	4	1	no	—
39	C₂×D₅	—	4	1	no	—
40	C₂×D₅	—	4	1	no	—
41	C₂×D₅	—	8	2	no	—
42	C₂×D₅	D₈	1	1	yes	characteristic
43	C₄×C₅	—	4	1	no	—
44	C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	Dic₅	D₈	1	1	yes	characteristic
46	#41	—	10	1	no	—
47	C₂×D₄	—	10	1	no	—
48	Q₁₆	C₂×D₅	1	1	yes	characteristic, complemented
49	SD₁₆	—	10	1	no	—
50	D₈	—	5	1	no	—
51	D₈	—	5	1	no	—
52	D₈	—	10	1	no	—
53	D₈	C₂×D₅	1	1	yes	characteristic
54	C₂×C₈	—	5	1	no	—
55	C₁₆	—	5	1	no	—
56	C₁₆	C₂×D₅	1	1	yes	characteristic, complemented
57	C₂×C₅	—	8	1	no	—
58	C₂×C₅	C₂×D₈	1	1	yes	characteristic
59	D₅	—	2	1	no	—
60	D₅	—	8	1	no	—
61	D₅	—	8	1	no	—
62	C₂³	—	20	1	no	—
63	Q₈	—	2	1	no	—
64	D₄	—	2	1	no	—
65	D₄	—	10	1	no	—
66	D₄	—	10	1	no	—
67	D₄	—	20	1	no	—
68	D₄	—	20	1	no	—
69	C₂×C₄	—	5	1	no	—
70	C₂×C₄	—	20	1	no	—
71	C₈	—	5	1	no	—
72	C₈	C₂²×D₅	1	1	yes	characteristic
73	C₅	#509	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	4	1	no	—
75	C₂²	—	5	1	no	—
76	C₂²	—	20	1	no	—
77	C₂²	—	20	1	no	—
78	C₂²	—	40	2	no	—
79	C₄	—	4	1	no	—
80	C₄	—	5	1	no	—
81	C₄	D₄×D₅	1	1	yes	characteristic
82	C₂	—	8	1	no	—
83	C₂	—	10	1	no	—
84	C₂	—	40	1	no	—
85	C₂	—	40	1	no	—
86	C₂	D₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₅×D₈
commutator, cosocle	C₅×C₈	C₂³
Frattini	C₈	C₂²×D₅
Fitting	C₅×SD₃₂	C₂
socle	C₂×C₅	C₂×D₈
hypercenter	SD₃₂	D₅
hypocenter	C₅	#509
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #4065	C₂	semidirect
2. D₅×D₈	C₂	semidirect
3. C₅×SD₃₂	C₂	semidirect
4. #3960	C₂	semidirect
5. #3958	C₂	semidirect
6. D₈₀	C₂	semidirect
7. #3930	C₂	semidirect
10. C₅×Q₁₆	C₂²	semidirect
15. D₄₀	C₂²	semidirect
17. C₅×C₁₆	C₂²	semidirect
18. #672	C₂²	semidirect
34. SD₃₂	D₅	semidirect
48. Q₁₆	C₂×D₅	semidirect
56. C₁₆	C₂×D₅	semidirect
73. C₅	#509	semidirect

Maximal subgroups

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

subgroup	index
#4065	2
D₅×D₈	2
C₅×SD₃₂	2
#3960	2
#3958	2
D₈₀	2
#3930	2
#509	5