Group #10604797 — subgroups

Subgroups — #10604797

Order 640.

300 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 70 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×#4058	C₂	1	1	yes	maximal, characteristic
2	#65776	C₂	1	2	yes	maximal, complemented
3	C₂×#4008	C₂	1	2	yes	maximal, complemented
4	C₂×#3996	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#711	C₂²	1	2	yes	—
7	C₂×C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
8	C₂×#700	C₂²	1	2	yes	—
9	C₂×D₅⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×D₈	C₄	1	1	yes	characteristic, complemented
11	C₂×#687	C₄	1	2	yes	complemented
12	#4058	—	2	2	no	—
13	#4058	—	2	2	no	—
14	C₂×Dic₂₀	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
16	C₂²⋊#170	—	2	4	no	—
17	#4008	—	2	4	no	—
18	#3996	—	2	2	no	—
19	C₈×AGL(1,𝔽₅)	—	2	2	no	—
20	#1974	—	5	1	no	maximal, Sylow
21	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
22	C₂×C₅×D₄	C₂×C₄	1	2	yes	—
23	C₂×C₂²⋊D₅	—	2	2	no	—
24	C₂²×Dic₅	—	2	2	no	—
25	#711	—	2	2	no	—
26	#711	—	2	2	no	—
27	#711	—	2	2	no	—
28	#711	—	2	2	no	—
29	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
31	C₂×#170	—	2	2	no	—
32	C₂×#170	D₄	1	2	yes	—
33	C₄×AGL(1,𝔽₅)	—	2	1	no	—
34	C₄×AGL(1,𝔽₅)	D₄	1	2	yes	—
35	#700	—	2	2	no	—
36	#700	—	2	2	no	—
37	D₅⋊C₈	—	2	1	no	—
38	D₅⋊C₈	—	2	1	no	—
39	C₅×D₈	—	2	2	no	—
40	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	#687	—	2	4	no	—
42	C₂×#168	C₂×C₄	1	1	yes	characteristic
43	Dic₂₀	—	2	2	no	—
44	C₈×D₅	D₄	1	2	yes	—
45	C₈×D₅	D₄	1	2	yes	—
46	C₂×#135	—	5	1	no	—
47	C₂×#108	—	5	1	no	—
48	C₂×#104	—	5	2	no	—
49	#408	—	5	2	no	—
50	C₂×C₄×C₈	—	5	1	no	—
51	C₂³×C₅	—	2	2	no	—
52	C₂²×D₅	#41	1	1	yes	characteristic
53	C₂×AGL(1,𝔽₅)	—	2	4	no	—
54	C₂×AGL(1,𝔽₅)	—	2	2	no	—
55	C₅×D₄	—	2	2	no	—
56	C₅×D₄	—	2	2	no	—
57	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²⋊D₅	—	4	2	no	—
59	C₂²⋊D₅	—	4	2	no	—
60	C₂×Dic₅	—	2	2	no	—
61	C₂×Dic₅	—	4	2	no	—
62	C₂×Dic₅	—	4	2	no	—
63	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
64	C₄×D₅	C₂×D₄	1	1	yes	characteristic
65	C₄×D₅	C₂×D₄	1	1	yes	characteristic
66	C₄×D₅	C₂×D₄	1	1	yes	characteristic
67	C₄×D₅	#41	1	1	yes	characteristic
68	Dic₁₀	—	2	2	no	—
69	Dic₁₀	—	2	2	no	—
70	#170	—	2	2	no	—
71	#170	—	4	2	no	—
72	C₅×C₈	C₂²⋊C₄	1	2	yes	—
73	#168	C₂²⋊C₄	1	2	yes	—
74	C₂×#41	—	5	2	no	—
75	#135	—	10	2	no	—
76	#135	—	10	2	no	—
77	C₂×Q₁₆	—	5	1	no	—
78	C₂×SD₁₆	—	5	2	no	—
79	C₂×D₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
80	C₂×M₁₆	—	5	2	no	—
81	C₂²×C₈	—	5	1	no	—
82	C₂²×C₈	—	5	1	no	—
83	C₂×C₄²	—	5	1	no	—
84	#108	—	10	2	no	—
85	#104	—	10	4	no	—
86	C₂²⋊C₈	—	10	4	no	—
87	C₄×C₈	—	10	2	no	—
88	C₂²×C₅	—	4	2	no	—
89	C₂²×C₅	—	4	4	no	—
90	C₂²×C₅	C₄×D₄	1	1	yes	characteristic
91	C₂×D₅	—	2	2	no	—
92	C₂×D₅	#121	1	1	yes	characteristic
93	C₂×D₅	#124	1	1	yes	characteristic
94	AGL(1,𝔽₅)	—	2	4	no	—
95	C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₅	C₄×D₄	1	1	yes	characteristic
97	Dic₅	—	4	2	no	—
98	Dic₅	#121	1	1	yes	characteristic
99	Dic₅	#127	1	1	yes	characteristic
100	#41	—	10	2	no	—
101	#41	—	10	2	no	—
102	#41	—	10	2	no	—
103	#41	—	10	2	no	—
104	C₂×Q₈	—	5	2	no	—
105	C₂×D₄	—	10	2	no	—
106	C₂×D₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
107	C₂²×C₄	—	5	1	no	—
108	C₂²×C₄	—	10	2	no	—
109	C₂²×C₄	—	10	1	no	—
110	Q₁₆	—	10	2	no	—
111	SD₁₆	—	10	4	no	—
112	D₈	—	2	2	no	—
113	M₁₆	—	10	2	no	—
114	M₁₆	—	10	2	no	—
115	C₂×C₈	—	5	1	no	—
116	C₂×C₈	—	5	2	no	—
117	C₂×C₈	—	5	2	no	—
118	C₂×C₈	—	5	2	no	—
119	C₂×C₈	—	10	1	no	—
120	C₂×C₈	—	10	1	no	—
121	C₂×C₈	—	10	2	no	—
122	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
123	C₄²	—	5	2	no	—
124	C₄²	—	10	1	no	—
125	C₂×C₅	—	8	2	no	—
126	C₂×C₅	#390	1	1	yes	characteristic
127	C₂×C₅	#443	1	2	yes	—
128	D₅	—	2	1	no	—
129	D₅	—	2	1	no	—
130	C₂³	—	2	2	no	—
131	C₂³	—	5	1	no	—
132	Q₈	—	10	2	no	—
133	Q₈	—	10	2	no	—
134	D₄	—	2	2	no	—
135	D₄	—	2	2	no	—
136	D₄	—	20	2	no	—
137	D₄	—	20	2	no	—
138	C₂×C₄	—	5	1	no	—
139	C₂×C₄	—	5	1	no	—
140	C₂×C₄	—	5	1	no	—
141	C₂×C₄	—	5	1	no	—
142	C₂×C₄	—	5	1	no	—
143	C₂×C₄	—	10	2	no	—
144	C₂×C₄	—	10	4	no	—
145	C₂×C₄	—	10	2	no	—
146	C₂×C₄	—	20	2	no	—
147	C₂×C₄	—	20	2	no	—
148	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
149	C₈	—	5	2	no	—
150	C₈	—	10	2	no	—
151	C₈	—	20	2	no	—
152	C₈	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
153	C₅	#1974	1	1	yes	characteristic, complemented, Sylow
154	C₂²	—	4	4	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	5	1	no	—
157	C₂²	—	5	1	no	—
158	C₂²	—	10	2	no	—
159	C₂²	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
160	C₄	—	5	1	no	—
161	C₄	—	5	1	no	—
162	C₄	—	10	4	no	—
163	C₄	—	20	2	no	—
164	C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
165	C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₂	—	8	2	no	—
167	C₂	—	10	1	no	—
168	C₂	—	10	1	no	—
169	C₂	#65732	1	2	yes	—
170	C₂	#65780	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4058

C₂

yes

maximal, characteristic

#65776

C₂

yes

maximal, complemented

C₂×#4008

C₂

yes

maximal, complemented

C₂×#3996

C₂

yes

maximal, characteristic, complemented

C₂×C₈×AGL(1,𝔽₅)

C₂

yes

maximal, characteristic, complemented

C₂×#711

C₂²

yes

—

C₂×C₄×AGL(1,𝔽₅)

C₂²

yes

characteristic

C₂×#700

C₂²

yes

—

C₂×D₅⋊C₈

C₂²

yes

characteristic

C₂×C₅×D₈

C₄

yes

characteristic, complemented

C₂×#687

C₄

yes

complemented

#4058

—

#4058

—

C₂×Dic₂₀

C₄

yes

characteristic, complemented

C₂×C₈×D₅

C₂²

yes

characteristic

C₂²⋊#170

—

—

—

—

—

C₂²×AGL(1,𝔽₅)

—

C₂×C₅×D₄

C₂×C₄

yes

—

C₂×C₂²⋊D₅

—

C₂²×Dic₅

—

—

—

—

—

C₂×C₄×D₅

C₂³

yes

characteristic

C₂×Dic₁₀

C₂×C₄

yes

—

C₂×#170

—

C₂×#170

D₄

yes

—

C₄×AGL(1,𝔽₅)

—

C₄×AGL(1,𝔽₅)

D₄

yes

—

#700

—

#700

—

D₅⋊C₈

—

D₅⋊C₈

—

C₅×D₈

—

C₂×C₅×C₈

C₂×C₄

yes

characteristic

#687

—

C₂×#168

C₂×C₄

yes

characteristic

Dic₂₀

—

C₈×D₅

D₄

yes

—

C₈×D₅

D₄

yes

—

—

—

—

—

C₂×C₄×C₈

—

C₂³×C₅

—

C₂²×D₅

yes

—

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×D₄

yes

characteristic

C₄×D₅

C₂×D₄

yes

characteristic

C₄×D₅

C₂×D₄

yes

characteristic

C₄×D₅

C₂×D₄

yes

characteristic

C₄×D₅

#41

yes

characteristic

Dic₁₀

—

Dic₁₀

—

#170

—

#170

—

C₅×C₈

C₂²⋊C₄

yes

—

#168

C₂²⋊C₄

yes

—

C₂×#41

—

#135

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

AGL(1,𝔽₅)

yes

characteristic, complemented

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

#104

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₄×D₄

yes

characteristic

C₂×D₅

—

C₂×D₅

yes

C₂×D₅

yes

—

C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅

C₄×D₄

yes

characteristic

Dic₅

—

Dic₅

yes

Dic₅

yes

100

—

101

#41

—

102

#41

—

103

#41

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

C₂×AGL(1,𝔽₅)

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

Q₁₆

—

111

SD₁₆

—

112

D₈

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

123

C₄²

—

124

C₄²

—

125

C₂×C₅

—

126

C₂×C₅

#390

yes

characteristic

127

C₂×C₅

#443

yes

—

128

D₅

—

129

D₅

—

130

C₂³

—

131

C₂³

—

132

Q₈

—

133

Q₈

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

C₂²⋊AGL(1,𝔽₅)

yes

—

153

C₅

#1974

yes

characteristic, complemented, Sylow

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

D₄×AGL(1,𝔽₅)

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

165

C₄

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

#65732

yes

—

170

C₂

#65780

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×D₈	C₄
socle	C₂²×C₅	C₄×D₄
prime radical	C₂×#4058	C₂
cosocle	C₂×C₄×D₅	C₂³
hypercenter	C₂×D₈	AGL(1,𝔽₅)
hypocenter	C₅	#1974
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₂×C₅×D₈

C₄

socle

C₂²×C₅

C₄×D₄

prime radical

C₂×#4058

C₂

cosocle

C₂×C₄×D₅

C₂³

C₂×D₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
2. #65776	C₂	semidirect
3. C₂×#4008	C₂	semidirect
4. C₂×#3996	C₂	semidirect
5. C₂×C₈×AGL(1,𝔽₅)	C₂	semidirect
10. C₂×C₅×D₈	C₄	semidirect
11. C₂×#687	C₄	semidirect
14. C₂×Dic₂₀	C₄	semidirect
79. C₂×D₈	AGL(1,𝔽₅)	semidirect
153. C₅	#1974	semidirect

H ≅ G/N

type

2. #65776

C₂

semidirect

3. C₂×#4008

C₂

semidirect

4. C₂×#3996

C₂

semidirect

5. C₂×C₈×AGL(1,𝔽₅)

C₂

semidirect

10. C₂×C₅×D₈

C₄

semidirect

11. C₂×#687

C₄

semidirect

14. C₂×Dic₂₀

C₄

semidirect

79. C₂×D₈

AGL(1,𝔽₅)

semidirect

153. C₅

#1974

semidirect

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

subgroup	index
C₂×#4058	2
#65776	2
C₂×#4008	2
C₂×#3996	2
C₂×C₈×AGL(1,𝔽₅)	2
#1974	5

subgroup

index

C₂×C₈×AGL(1,𝔽₅)

Subgroups — #10604797

Conjugacy classes of subgroups

Complements

Maximal subgroups