Subgroups — #65351

Order 320.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4147	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	#4078	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×D₄	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×C₂²⋊C₄	—	2	2	no	—
13	C₄²×C₅	C₂²	1	1	yes	characteristic
14	#687	—	2	2	no	—
15	#686	—	2	2	no	—
16	#681	C₂²	1	2	yes	—
17	#454	—	5	1	no	maximal, Sylow
18	C₂³×C₅	—	2	1	no	—
19	C₅×Q₈	—	2	2	no	—
20	C₅×D₄	—	2	2	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	AGL(1,ℤ/8)	—	5	2	no	—
34	#124	D₅	1	1	yes	characteristic, complemented
35	#104	—	5	2	no	—
36	#101	—	5	1	no	—
37	C₂²×C₅	—	4	2	no	—
38	C₂²×C₅	—	4	1	no	—
39	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂×D₅	—	2	2	no	—
41	C₄×C₅	—	2	2	no	—
42	C₄×C₅	—	4	1	no	—
43	C₄×C₅	C₂×D₄	1	2	yes	—
44	Dic₅	—	2	2	no	—
45	#41	—	5	2	no	—
46	#41	—	10	2	no	—
47	#41	—	10	2	no	—
48	C₂×Q₈	—	10	2	no	—
49	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
50	C₂×D₄	C₂×D₅	1	1	yes	characteristic
51	SD₁₆	—	10	2	no	—
52	D₈	—	10	2	no	—
53	M₁₆	—	5	2	no	—
54	C₂²⋊C₄	—	2	2	no	—
55	C₄²	C₂×D₅	1	1	yes	characteristic
56	C₂×C₅	—	2	1	no	—
57	C₂×C₅	—	8	1	no	—
58	C₂×C₅	#120	1	1	yes	characteristic
59	D₅	—	4	2	no	—
60	C₂³	—	2	1	no	—
61	Q₈	—	2	2	no	—
62	Q₈	—	5	2	no	—
63	Q₈	—	10	2	no	—
64	D₄	—	2	2	no	—
65	D₄	—	5	2	no	—
66	D₄	—	10	2	no	—
67	D₄	—	10	2	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	10	4	no	—
70	C₂×C₄	—	10	2	no	—
71	C₂×C₄	C₂²⋊D₅	1	2	yes	—
72	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
73	C₈	—	10	2	no	—
74	C₅	#454	1	1	yes	characteristic, complemented, Sylow
75	C₂²	—	4	2	no	—
76	C₂²	—	4	1	no	—
77	C₂²	—	10	2	no	—
78	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	4	1	no	—
81	C₄	—	10	2	no	—
82	C₄	D₄×D₅	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	8	1	no	—
85	C₂	—	20	2	no	—
86	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×#124	C₂
socle	C₂×C₅	#120
hypercenter	#124	D₅
hypocenter	C₅	#454
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #124

Complements

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

N	H ≅ G/N	type
1. #4147	C₂	semidirect
2. C₅×#124	C₂	semidirect
3. #4078	C₂	semidirect
4. #3968	C₂	semidirect
5. #3937	C₂	semidirect
34. #124	D₅	semidirect
74. C₅	#454	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×#124	2
#4078	2
#3968	2
#3937	2
#454	5