Group #10604026 — subgroups

Subgroups — #10604026

Order 640.

126 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65119	C₂	1	1	yes	maximal, characteristic
2	#64888	C₂	1	1	yes	maximal, characteristic
3	#64888	C₂	1	1	yes	maximal, characteristic
4	#64886	C₂	1	1	yes	maximal, characteristic
5	#64885	C₂	1	1	yes	maximal, characteristic
6	#64859	C₂	1	1	yes	maximal, characteristic
7	C₄⋊#674	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₁₀	C₄	1	1	yes	characteristic
9	C₂×#674	C₂²	1	1	yes	characteristic
10	C₂×#674	C₂²	1	1	yes	characteristic
11	C₂×#674	C₂²	1	1	yes	characteristic
12	C₂×#674	C₂²	1	1	yes	characteristic
13	C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
14	#3946	C₄	1	1	yes	characteristic
15	#3946	C₄	1	1	yes	characteristic
16	C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₈×Dic₅	C₂²	1	1	yes	characteristic
18	C₄×#168	C₂²	1	1	yes	characteristic
19	#2183	—	5	1	no	maximal, Sylow
20	C₂×Dic₁₀	C₈	1	1	yes	characteristic
21	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₄⋊Dic₅	C₈	1	1	yes	characteristic
25	#683	C₈	1	1	yes	characteristic
26	#683	C₈	1	1	yes	characteristic
27	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₂×#168	C₂×C₄	1	1	yes	characteristic
30	C₂×#168	C₂³	1	1	yes	characteristic
31	#674	—	2	1	no	—
32	#674	—	2	1	no	—
33	#674	—	2	1	no	—
34	#674	Q₈	1	2	yes	complemented
35	C₈×Q₈	—	5	1	no	—
36	C₄⋊C₁₆	—	5	1	no	—
37	C₄⋊C₁₆	—	5	1	no	—
38	C₄⋊C₁₆	—	5	1	no	—
39	#346	—	5	1	no	—
40	#346	—	5	1	no	—
41	C₄×C₁₆	—	5	1	no	—
42	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₅	C₂×C₈	1	1	yes	characteristic
46	C₂×Dic₅	C₂×C₈	1	1	yes	characteristic
47	C₂×Dic₅	C₂×C₈	1	1	yes	characteristic
48	C₂×Dic₅	C₂×C₈	1	1	yes	characteristic
49	Dic₁₀	—	2	2	no	—
50	C₅×C₈	—	2	1	no	—
51	C₅×C₈	—	2	1	no	—
52	#168	—	2	1	no	—
53	#168	C₂×Q₈	1	1	yes	characteristic
54	#168	#41	1	1	yes	characteristic
55	C₄×Q₈	—	5	1	no	—
56	C₂×C₁₆	—	5	1	no	—
57	C₂×C₁₆	—	5	1	no	—
58	C₂×C₁₆	—	5	1	no	—
59	C₂×C₁₆	—	5	1	no	—
60	C₄⋊C₈	—	5	1	no	—
61	C₄⋊C₈	—	5	1	no	—
62	C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₄×C₈	—	5	1	no	—
64	C₄×C₈	—	5	1	no	—
65	C₄×C₈	—	5	1	no	—
66	C₂²×C₅	C₂²×C₈	1	1	yes	characteristic
67	C₄×C₅	—	2	1	no	—
68	C₄×C₅	—	2	1	no	—
69	C₄×C₅	C₄×Q₈	1	1	yes	characteristic
70	C₄×C₅	#131	1	1	yes	characteristic
71	Dic₅	—	2	1	no	—
72	Dic₅	—	2	1	no	—
73	Dic₅	—	2	1	no	—
74	Dic₅	M₃₂	1	2	yes	—
75	C₂×Q₈	—	5	1	no	—
76	C₂×C₈	—	5	1	no	—
77	C₂×C₈	—	5	1	no	—
78	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
80	C₄⋊C₄	—	5	1	no	—
81	C₄⋊C₄	—	5	1	no	—
82	C₄⋊C₄	—	5	1	no	—
83	C₄²	—	5	1	no	—
84	C₄²	—	5	1	no	—
85	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
86	C₁₆	—	5	2	no	—
87	C₁₆	—	10	1	no	—
88	C₁₆	—	10	1	no	—
89	C₁₆	—	10	1	no	—
90	C₂×C₅	C₈×Q₈	1	1	yes	characteristic
91	C₂×C₅	C₂×M₃₂	1	1	yes	characteristic
92	C₂×C₅	#504	1	1	yes	characteristic
93	Q₈	—	10	2	no	—
94	C₂×C₄	—	5	1	no	—
95	C₂×C₄	—	5	1	no	—
96	C₂×C₄	—	5	1	no	—
97	C₂×C₄	—	5	1	no	—
98	C₂×C₄	D₅⋊C₈	1	1	yes	characteristic
99	C₂×C₄	D₅⋊C₈	1	1	yes	characteristic
100	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₈	—	5	1	no	—
104	C₈	—	5	1	no	—
105	C₈	—	10	1	no	—
106	C₅	#2183	1	1	yes	characteristic, complemented, Sylow
107	C₂²	C₂×D₅⋊C₈	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	5	2	no	—
111	C₄	—	10	1	no	—
112	C₄	—	10	1	no	—
113	C₄	—	10	1	no	—
114	C₄	#4131	1	1	yes	characteristic
115	C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
116	C₂	#65703	1	1	yes	characteristic
117	C₂	D₅⋊M₃₂	1	1	yes	characteristic
118	C₂	#65725	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#65119

C₂

yes

maximal, characteristic

#64888

C₂

yes

maximal, characteristic

#64888

C₂

yes

maximal, characteristic

#64886

C₂

yes

maximal, characteristic

#64885

C₂

yes

maximal, characteristic

#64859

C₂

yes

maximal, characteristic

C₄⋊#674

C₂

yes

maximal, characteristic

C₄×Dic₁₀

C₄

yes

characteristic

C₂×#674

C₂²

yes

characteristic

C₂×#674

C₂²

yes

characteristic

C₂×#674

C₂²

yes

characteristic

C₂×#674

C₂²

yes

characteristic

C₅×C₄⋊C₈

C₄

yes

characteristic

#3946

C₄

yes

characteristic

#3946

C₄

yes

characteristic

C₈×Dic₅

C₂²

yes

characteristic

C₈×Dic₅

C₂²

yes

characteristic

C₄×#168

C₂²

yes

characteristic

#2183

—

maximal, Sylow

C₂×Dic₁₀

C₈

yes

characteristic

C₂×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₅×C₈

C₂×C₄

yes

characteristic

C₄²×C₅

C₂×C₄

yes

characteristic

C₄⋊Dic₅

C₈

yes

characteristic

#683

C₈

yes

characteristic

#683

C₈

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₂×#168

C₂×C₄

yes

characteristic

C₂×#168

C₂³

yes

—

—

—

Q₈

yes

complemented

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#346

—

C₄×C₁₆

—

C₂×C₄×C₅

C₂×C₈

yes

characteristic

C₂×C₄×C₅

C₂×C₈

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₂×C₈

yes

characteristic

C₂×Dic₅

C₂×C₈

yes

characteristic

C₂×Dic₅

C₂×C₈

yes

characteristic

C₂×Dic₅

C₂×C₈

yes

characteristic

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

C₂×Q₈

yes

yes

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

AGL(1,𝔽₅)

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

C₂²×C₈

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄×Q₈

yes

characteristic

C₄×C₅

#131

yes

characteristic

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

M₃₂

yes

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂×C₅

C₈×Q₈

yes

characteristic

C₂×C₅

C₂×M₃₂

yes

characteristic

C₂×C₅

#504

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₅⋊C₈

yes

characteristic

C₂×C₄

D₅⋊C₈

yes

characteristic

100

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₅

#2183

yes

characteristic, complemented, Sylow

107

C₂²

C₂×D₅⋊C₈

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#4131

yes

characteristic

115

C₄

Q₈×AGL(1,𝔽₅)

yes

characteristic

116

C₂

#65703

yes

characteristic

117

C₂

D₅⋊M₃₂

yes

characteristic

118

C₂

#65725

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
commutator, socle, prime radical	C₂²×C₅	C₂²×C₈
Fitting	C₅×C₄⋊C₈	C₄
cosocle	C₂×#168	C₂³
hypercenter	C₄⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	#2183
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

commutator, socle, prime radical

C₂²×C₅

C₂²×C₈

Fitting

C₅×C₄⋊C₈

C₄

C₂³

C₄⋊C₈

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
34. #674	Q₈	semidirect
106. C₅	#2183	semidirect

H ≅ G/N

type

34. #674

Q₈

semidirect

106. C₅

#2183

semidirect

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

subgroup	index
#65119	2
#64888	2
#64886	2
#64885	2
#64859	2
C₄⋊#674	2
#2183	5

subgroup

index

Subgroups — #10604026

Conjugacy classes of subgroups

Complements

Maximal subgroups