Subgroups — #65384

Order 320.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4147	C₂	1	1	yes	maximal, characteristic
2	C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#4088	C₂	1	2	yes	maximal, complemented
4	#3968	C₂	1	1	yes	maximal, characteristic, complemented
5	#3937	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
8	#713	—	2	2	no	—
9	Q₈×D₅	—	2	2	no	—
10	#709	—	2	2	no	—
11	#709	C₂²	1	2	yes	—
12	C₅×C₄⋊C₄	—	2	2	no	—
13	C₄²×C₅	C₂²	1	1	yes	characteristic
14	#689	—	2	2	no	—
15	#688	—	2	2	no	—
16	#681	C₂²	1	2	yes	—
17	#456	—	5	1	no	maximal, Sylow
18	C₅×Q₈	—	2	2	no	—
19	C₅×Q₈	—	2	2	no	—
20	C₂×C₄×C₅	—	2	1	no	—
21	C₂×C₄×C₅	—	2	1	no	—
22	C₂×C₄×C₅	D₄	1	2	yes	—
23	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²⋊D₅	—	2	2	no	—
25	D₂₀	—	2	2	no	—
26	D₂₀	D₄	1	2	yes	—
27	C₄×D₅	—	2	4	no	—
28	C₄×D₅	—	2	2	no	—
29	Dic₁₀	—	2	2	no	—
30	Dic₁₀	D₄	1	2	yes	—
31	#168	—	2	2	no	—
32	2₋¹⁺⁴	—	5	1	no	—
33	#137	—	5	2	no	—
34	C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
35	#104	—	5	2	no	—
36	#101	—	5	1	no	—
37	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂×D₅	—	2	2	no	—
39	C₄×C₅	—	2	2	no	—
40	C₄×C₅	—	2	2	no	—
41	C₄×C₅	—	4	1	no	—
42	C₄×C₅	C₂×D₄	1	2	yes	—
43	Dic₅	—	2	2	no	—
44	#41	—	5	2	no	—
45	#41	—	10	2	no	—
46	#41	—	10	2	no	—
47	C₂×Q₈	—	10	2	no	—
48	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
49	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
50	Q₁₆	—	10	2	no	—
51	SD₁₆	—	10	2	no	—
52	M₁₆	—	5	2	no	—
53	C₄⋊C₄	—	2	2	no	—
54	C₄²	C₂×D₅	1	1	yes	characteristic
55	C₂×C₅	—	2	1	no	—
56	C₂×C₅	#120	1	1	yes	characteristic
57	D₅	—	4	2	no	—
58	Q₈	—	2	2	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	5	2	no	—
61	Q₈	—	10	2	no	—
62	D₄	—	5	2	no	—
63	D₄	—	10	2	no	—
64	D₄	—	10	2	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	10	4	no	—
68	C₂×C₄	—	10	2	no	—
69	C₂×C₄	C₂²⋊D₅	1	2	yes	—
70	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
71	C₈	—	10	2	no	—
72	C₅	#456	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	10	2	no	—
74	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	10	2	no	—
79	C₄	D₄×D₅	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	20	2	no	—
82	C₂	#4083	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4083
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₄⋊Q₈	C₂
socle	C₂×C₅	#120
prime radical	#4147	C₂
hypercenter	C₄⋊Q₈	D₅
hypocenter	C₅	#456
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₅×C₄⋊Q₈	C₂	semidirect
3. #4088	C₂	semidirect
4. #3968	C₂	semidirect
5. #3937	C₂	semidirect
34. C₄⋊Q₈	D₅	semidirect
72. C₅	#456	semidirect

Maximal subgroups

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

subgroup	index
#4147	2
C₅×C₄⋊Q₈	2
#4088	2
#3968	2
#3937	2
#456	5