Group #10600432 — subgroups

Subgroups — #10600432

Order 640.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂×#4016	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#390	C₂	1	1	yes	maximal, characteristic
3	C₂×#3964	C₂	1	2	yes	maximal
4	#64749	C₂	1	2	yes	maximal, complemented
5	#64744	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
9	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₂×#683	—	2	2	no	—
12	C₂²×#168	C₂²	1	2	yes	—
13	#4016	—	2	4	no	—
14	#3964	—	2	4	no	—
15	#2025	—	5	1	no	maximal, Sylow
16	C₂⁴×C₅	—	2	1	no	—
17	C₂×C₅×D₄	—	2	2	no	—
18	C₂×C₅×D₄	—	2	2	no	—
19	C₂×C₅×D₄	—	2	1	no	—
20	C₂×C₅×D₄	Q₈	1	2	yes	—
21	C₂²×C₄×C₅	—	2	1	no	—
22	C₂²×C₄×C₅	D₄	1	2	yes	—
23	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅	—	2	2	no	—
25	C₅×C₄⋊C₄	—	2	2	no	—
26	C₅×C₂²⋊C₄	—	4	4	no	—
27	C₄²×C₅	—	2	2	no	—
28	C₄⋊Dic₅	—	2	2	no	—
29	C₄⋊Dic₅	D₄	1	4	yes	—
30	#683	—	2	8	no	—
31	C₂×#168	—	2	2	no	—
32	C₂×#168	—	2	2	no	—
33	C₂×#168	—	2	2	no	—
34	C₂×#125	—	5	1	no	—
35	C₂×#102	—	5	2	no	—
36	#390	D₅	1	1	yes	characteristic
37	#340	—	5	2	no	—
38	#336	—	5	1	no	—
39	C₂³×C₅	—	2	2	no	—
40	C₂³×C₅	—	4	1	no	—
41	C₂³×C₅	—	4	1	no	—
42	C₂³×C₅	—	4	2	no	—
43	C₂³×C₅	—	4	2	no	—
44	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
45	C₅×D₄	—	2	4	no	—
46	C₅×D₄	—	4	2	no	—
47	C₂×C₄×C₅	—	2	2	no	—
48	C₂×C₄×C₅	—	2	2	no	—
49	C₂×C₄×C₅	—	2	2	no	—
50	C₂×C₄×C₅	—	4	1	no	—
51	C₂×C₄×C₅	—	4	2	no	—
52	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅	#41	1	1	yes	characteristic
55	C₂×C₄×C₅	#41	1	2	yes	—
56	C₂×Dic₅	—	2	4	no	—
57	C₂×Dic₅	—	4	2	no	—
58	C₂×Dic₅	—	4	2	no	—
59	#168	—	4	2	no	—
60	C₂²×D₄	C₂×D₅	1	1	yes	characteristic
61	C₂²×C₈	—	5	2	no	—
62	#125	—	10	4	no	—
63	C₂×C₄⋊C₄	—	5	2	no	—
64	C₂×C₄⋊C₄	—	10	2	no	—
65	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
66	C₂×C₂²⋊C₄	—	2	2	no	—
67	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
68	#102	—	10	4	no	—
69	C₂²×C₅	—	4	2	no	—
70	C₂²×C₅	—	4	4	no	—
71	C₂²×C₅	—	8	2	no	—
72	C₂²×C₅	—	8	1	no	—
73	C₂²×C₅	—	8	1	no	—
74	C₂²×C₅	#120	1	1	yes	characteristic
75	C₂²×C₅	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₅	#124	1	1	yes	characteristic
77	C₂²×C₅	#135	1	2	yes	—
78	C₂²×C₅	AGL(1,ℤ/8)	1	2	yes	—
79	C₄×C₅	—	4	2	no	—
80	C₄×C₅	—	8	1	no	—
81	C₄×C₅	C₂²⋊Q₈	1	2	yes	—
82	C₄×C₅	#124	1	2	yes	—
83	Dic₅	—	4	4	no	—
84	C₂⁴	—	2	1	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	Dic₁₀	1	2	yes	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	10	2	no	—
91	C₂²×C₄	C₂²⋊D₅	1	2	yes	—
92	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
93	C₂×C₈	—	10	2	no	—
94	C₂×C₈	—	10	2	no	—
95	C₂×C₈	—	10	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	5	4	no	—
98	C₄⋊C₄	—	10	8	no	—
99	C₄⋊C₄	—	10	2	no	—
100	C₂²⋊C₄	—	4	4	no	—
101	C₄²	—	2	2	no	—
102	C₂×C₅	—	8	2	no	—
103	C₂×C₅	#393	1	1	yes	characteristic
104	C₂×C₅	#449	1	2	yes	—
105	C₂×C₅	#478	1	2	yes	—
106	C₂×C₅	#488	1	1	yes	characteristic
107	C₂×C₅	#490	1	1	yes	characteristic
108	C₂³	—	2	2	no	—
109	C₂³	—	4	2	no	—
110	C₂³	—	4	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	1	no	—
113	C₂³	C₂×C₂²⋊D₅	1	1	yes	characteristic
114	D₄	—	2	4	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	10	4	no	—
122	C₂×C₄	—	20	2	no	—
123	C₂×C₄	—	20	2	no	—
124	C₂×C₄	C₂×Dic₁₀	1	1	yes	characteristic
125	C₂×C₄	#709	1	1	yes	characteristic
126	C₂×C₄	D₄×D₅	1	2	yes	—
127	C₂×C₄	#711	1	2	yes	—
128	C₈	—	20	2	no	—
129	C₅	#2025	1	1	yes	characteristic, complemented, Sylow
130	C₂²	—	4	4	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	8	2	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	#4070	1	1	yes	characteristic
136	C₂²	#4078	1	2	yes	—
137	C₂²	#4082	1	1	yes	characteristic
138	C₂²	#4083	1	1	yes	characteristic
139	C₂²	#4096	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	8	1	no	—
142	C₄	—	20	4	no	—
143	C₄	#4024	1	2	yes	—
144	C₄	#4031	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	#65236	1	1	yes	characteristic
147	C₂	#65298	1	2	yes	—
148	C₂	#65326	1	2	yes	—
149	C₂	#65345	1	1	yes	characteristic
150	C₂	#65360	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4016

C₂

yes

maximal, characteristic, complemented

C₅×#390

C₂

yes

maximal, characteristic

C₂×#3964

C₂

yes

maximal

#64749

C₂

yes

maximal, complemented

#64744

C₂

yes

maximal, characteristic, complemented

C₂²×C₅×D₄

C₂²

yes

characteristic

C₂×C₅×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₅×C₂²⋊C₄

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×#683

—

C₂²×#168

C₂²

yes

—

—

—

—

C₂⁴×C₅

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

Q₈

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

D₄

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

D₄

yes

—

—

—

—

—

—

—

D₅

yes

characteristic

#340

—

#336

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂×D₄

yes

characteristic

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

#168

—

C₂²×D₄

C₂×D₅

yes

characteristic

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅

yes

characteristic

#102

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#120

yes

characteristic

C₂²×C₅

C₂²⋊Q₈

yes

C₂²×C₅

yes

C₂²×C₅

yes

—

C₂²×C₅

AGL(1,ℤ/8)

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂²⋊Q₈

yes

—

C₄×C₅

#124

yes

—

Dic₅

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₁₀

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₂×C₅

—

103

C₂×C₅

#393

yes

characteristic

104

C₂×C₅

#449

yes

—

105

C₂×C₅

#478

yes

—

106

C₂×C₅

#488

yes

characteristic

107

C₂×C₅

#490

yes

characteristic

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×C₂²⋊D₅

yes

characteristic

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂×Dic₁₀

yes

characteristic

125

C₂×C₄

#709

yes

characteristic

126

C₂×C₄

D₄×D₅

yes

—

127

C₂×C₄

#711

yes

—

128

C₈

—

129

C₅

#2025

yes

characteristic, complemented, Sylow

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#4070

yes

characteristic

136

C₂²

#4078

yes

—

137

C₂²

#4082

yes

characteristic

138

C₂²

#4083

yes

characteristic

139

C₂²

#4096

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#4024

yes

—

144

C₄

#4031

yes

—

145

C₂

—

146

C₂

#65236

yes

characteristic

147

C₂

#65298

yes

—

148

C₂

#65326

yes

—

149

C₂

#65345

yes

characteristic

150

C₂

#65360

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅
commutator, cosocle	C₂²×C₄×C₅	C₂³
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₅×#390	C₂
socle	C₂³×C₅	C₂×D₄
prime radical	C₂²×C₅×D₄	C₂²
hypercenter	#390	D₅
hypocenter	C₅	#2025
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅

commutator, cosocle

C₂²×C₄×C₅

C₂³

Frattini

C₂²×C₄

C₂²×D₅

Fitting

C₅×#390

C₂

socle

C₂³×C₅

C₂×D₄

prime radical

C₂²×C₅×D₄

C₂²

D₅

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#4016	C₂	semidirect
4. #64749	C₂	semidirect
5. #64744	C₂	semidirect
129. C₅	#2025	semidirect

H ≅ G/N

type

1. C₂×#4016

C₂

semidirect

4. #64749

C₂

semidirect

5. #64744

C₂

semidirect

129. C₅

#2025

semidirect

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

subgroup	index
C₂×#4016	2
C₅×#390	2
C₂×#3964	2
#64749	2
#64744	2
#2025	5

subgroup

index

Subgroups — #10600432

Conjugacy classes of subgroups

Complements

Maximal subgroups