Group #10601756 — subgroups

Subgroups — #10601756

Order 640.

146 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₅×#446	C₂	1	1	yes	maximal, characteristic
2	#65320	C₂	1	1	yes	maximal, characteristic
3	#65310	C₂	1	1	yes	maximal, characteristic
4	#64969	C₂	1	1	yes	maximal, characteristic
5	#64706	C₂	1	1	yes	maximal, characteristic
6	#64701	C₂	1	1	yes	maximal, characteristic
7	#64678	C₂	1	1	yes	maximal, characteristic
8	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂×#689	—	2	1	no	—
10	#4015	C₂²	1	1	yes	characteristic
11	#4015	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊C₈	—	2	1	no	—
13	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₅×#97	—	2	1	no	—
15	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#3967	—	2	1	no	—
17	#3949	—	2	1	no	—
18	#3949	—	2	1	no	—
19	#3942	—	2	1	no	—
20	#3939	—	2	1	no	—
21	C₄⋊#168	C₂²	1	1	yes	characteristic
22	C₄⋊#168	C₂²	1	1	yes	characteristic
23	#1692	—	5	1	no	maximal, Sylow
24	C₂×C₅×Q₈	D₄	1	1	yes	characteristic
25	C₂×Dic₁₀	—	2	1	no	—
26	C₂×Dic₁₀	—	2	1	no	—
27	C₂×C₅×C₈	—	2	1	no	—
28	C₂×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂×C₅×C₈	D₄	1	1	yes	characteristic
30	C₅×C₄⋊C₄	—	2	1	no	—
31	C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
32	C₄²×C₅	—	2	1	no	—
33	C₄²×C₅	C₂³	1	1	yes	characteristic
34	#689	—	4	2	no	—
35	C₄⋊Dic₅	—	2	1	no	—
36	C₄⋊Dic₅	—	2	1	no	—
37	C₄⋊Dic₅	—	2	1	no	—
38	C₄⋊Dic₅	—	2	1	no	—
39	C₂×#168	—	2	1	no	—
40	C₂×#168	—	2	1	no	—
41	#498	—	5	1	no	—
42	#477	—	5	1	no	—
43	#462	—	5	1	no	—
44	#446	D₅	1	1	yes	characteristic
45	#335	—	5	1	no	—
46	#333	—	5	1	no	—
47	#332	—	5	1	no	—
48	C₅×Q₈	—	2	1	no	—
49	C₅×Q₈	—	2	1	no	—
50	C₂×C₄×C₅	—	2	1	no	—
51	C₂×C₄×C₅	—	2	1	no	—
52	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅	#41	1	1	yes	characteristic
55	C₂×Dic₅	—	4	1	no	—
56	C₂×Dic₅	—	4	1	no	—
57	Dic₁₀	—	4	1	no	—
58	Dic₁₀	—	4	1	no	—
59	Dic₁₀	—	8	1	no	—
60	C₅×C₈	—	2	1	no	—
61	C₅×C₈	—	4	1	no	—
62	C₅×C₈	Q₁₆	1	2	yes	complemented
63	#168	—	4	1	no	—
64	#168	—	4	1	no	—
65	C₂×Q₁₆	—	10	1	no	—
66	C₄⋊Q₈	—	5	1	no	—
67	C₄⋊Q₈	—	5	1	no	—
68	C₄×Q₈	C₂×D₅	1	1	yes	characteristic
69	#107	—	10	1	no	—
70	#106	—	10	1	no	—
71	#106	—	10	1	no	—
72	C₄⋊C₈	—	2	1	no	—
73	C₄⋊C₈	—	5	1	no	—
74	C₄⋊C₈	—	5	1	no	—
75	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
76	#103	—	10	1	no	—
77	#103	—	10	1	no	—
78	#97	—	2	1	no	—
79	C₄×C₈	C₂×D₅	1	1	yes	characteristic
80	C₂²×C₅	#121	1	1	yes	characteristic
81	C₄×C₅	—	2	1	no	—
82	C₄×C₅	—	4	1	no	—
83	C₄×C₅	—	4	1	no	—
84	C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
85	C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₄×C₅	#137	1	1	yes	characteristic
87	C₄×C₅	#137	1	1	yes	characteristic
88	Dic₅	—	8	1	no	—
89	Dic₅	—	8	1	no	—
90	C₂×Q₈	—	10	1	no	—
91	C₂×Q₈	—	10	1	no	—
92	C₂×Q₈	D₂₀	1	1	yes	characteristic
93	Q₁₆	—	20	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	10	1	no	—
96	C₂×C₈	—	10	1	no	—
97	C₂×C₈	C₂²⋊D₅	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₅	1	1	yes	characteristic
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	10	1	no	—
101	C₄⋊C₄	—	10	1	no	—
102	C₄⋊C₄	—	10	1	no	—
103	C₄⋊C₄	—	10	1	no	—
104	C₄⋊C₄	D₂₀	1	1	yes	characteristic
105	C₄²	—	2	1	no	—
106	C₄²	C₂²×D₅	1	1	yes	characteristic
107	C₂×C₅	#462	1	1	yes	characteristic
108	C₂×C₅	#468	1	1	yes	characteristic
109	C₂×C₅	#471	1	1	yes	characteristic
110	Q₈	—	2	1	no	—
111	Q₈	—	2	1	no	—
112	Q₈	—	20	1	no	—
113	Q₈	—	20	1	no	—
114	Q₈	—	40	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	20	1	no	—
118	C₂×C₄	—	20	1	no	—
119	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
120	C₂×C₄	#709	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	20	1	no	—
125	C₈	—	20	1	no	—
126	C₈	#689	1	2	yes	complemented
127	C₅	#1692	1	1	yes	characteristic, complemented, Sylow
128	C₂²	#4076	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	4	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	40	1	no	—
133	C₄	—	40	1	no	—
134	C₄	#4054	1	1	yes	characteristic
135	C₄	#4055	1	1	yes	characteristic
136	C₄	C₂×#689	1	1	yes	characteristic
137	C₄	#4097	1	1	yes	characteristic
138	C₂	#65320	1	1	yes	characteristic
139	C₂	#65423	1	1	yes	characteristic
140	C₂	#65430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#446

C₂

yes

maximal, characteristic

#65320

C₂

yes

maximal, characteristic

#65310

C₂

yes

maximal, characteristic

#64969

C₂

yes

maximal, characteristic

#64706

C₂

yes

maximal, characteristic

#64701

C₂

yes

maximal, characteristic

#64678

C₂

yes

maximal, characteristic

C₄×C₅×Q₈

C₂²

yes

characteristic

C₂×#689

—

#4015

C₂²

yes

characteristic

#4015

C₂²

yes

characteristic

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

C₂²

yes

characteristic

C₅×#97

—

C₄×C₅×C₈

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄⋊#168

C₂²

yes

characteristic

#1692

—

maximal, Sylow

C₂×C₅×Q₈

D₄

yes

characteristic

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

D₄

yes

characteristic

C₂×C₅×C₈

D₄

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

D₄

yes

characteristic

C₄²×C₅

—

C₄²×C₅

C₂³

yes

characteristic

#689

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

—

D₅

yes

—

—

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

#41

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

Q₁₆

yes

complemented

#168

—

#168

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₅

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅

yes

—

—

—

C₄×C₈

C₂×D₅

yes

characteristic

C₂²×C₅

#121

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×Q₁₆

yes

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₂₀

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅

yes

characteristic

C₂×C₈

C₂²⋊D₅

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

D₂₀

yes

characteristic

105

C₄²

—

106

C₄²

C₂²×D₅

yes

characteristic

107

C₂×C₅

#462

yes

characteristic

108

C₂×C₅

#468

yes

characteristic

109

C₂×C₅

#471

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×D₂₀

yes

characteristic

120

C₂×C₄

#709

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

#689

yes

complemented

127

C₅

#1692

yes

characteristic, complemented, Sylow

128

C₂²

#4076

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#4054

yes

characteristic

135

C₄

#4055

yes

characteristic

136

C₄

C₂×#689

yes

characteristic

137

C₄

#4097

yes

characteristic

138

C₂

#65320

yes

characteristic

139

C₂

#65423

yes

characteristic

140

C₂

#65430

yes

characteristic

	subgroup	G/H
center	C₂²	#4076
commutator, cosocle	C₄²×C₅	C₂³
Frattini	C₄²	C₂²×D₅
Fitting	C₅×#446	C₂
socle, prime radical	C₂²×C₅	#121
hypercenter	#446	D₅
hypocenter	C₅	#1692
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #446

subgroup

G/H

center

C₂²

#4076

commutator, cosocle

C₄²×C₅

C₂³

Frattini

C₄²

C₂²×D₅

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #446

N	H ≅ G/N	type
62. C₅×C₈	Q₁₆	semidirect
126. C₈	#689	semidirect
127. C₅	#1692	semidirect

H ≅ G/N

type

62. C₅×C₈

Q₁₆

semidirect

126. C₈

#689

semidirect

127. C₅

#1692

semidirect

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

subgroup	index
C₅×#446	2
#65320	2
#65310	2
#64969	2
#64706	2
#64701	2
#64678	2
#1692	5

subgroup

index

Subgroups — #10601756

Conjugacy classes of subgroups

Complements

Maximal subgroups