Group #10713709 — subgroups

Subgroups — #10713709

Order 768.

386 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 155 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×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#4461	C₂	1	1	yes	maximal, characteristic
4	#67804	C₂	1	4	yes	maximal
5	#13264	—	3	1	no	maximal, Sylow
6	C₂×C₄×Dic₆	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×Dic₃⋊C₈	C₄	1	2	yes	—
10	C₂×C₈×Dic₃	C₂²	1	2	yes	—
11	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
13	C₈×Dic₆	—	2	2	no	—
14	C₈×Dic₆	—	2	2	no	—
15	C₃×C₄⋊C₁₆	—	2	2	no	—
16	#4461	—	2	2	no	—
17	C₂×C₈×Q₈	—	3	1	no	—
18	C₂×C₄⋊C₁₆	D₃	1	1	yes	characteristic
19	C₂×#346	—	3	1	no	—
20	#1374	—	3	4	no	—
21	C₂²×Dic₆	C₈	1	1	yes	characteristic
22	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄⋊Dic₃	C₈	1	1	yes	characteristic
26	C₂×Dic₃⋊C₄	C₈	1	2	yes	—
27	C₂×C₄×Dic₃	C₂×C₄	1	2	yes	—
28	C₂²×C₃⋊C₈	C₂×C₄	1	2	yes	—
29	C₄×Dic₆	—	2	2	no	—
30	C₄×Dic₆	—	2	2	no	—
31	C₂×C₃×C₁₆	—	2	2	no	—
32	C₂×C₃×C₁₆	—	2	4	no	—
33	C₃×C₄×C₈	D₄	1	2	yes	—
34	C₃×C₄×C₈	D₄	1	2	yes	—
35	Dic₃⋊C₈	—	2	4	no	—
36	C₈×Dic₃	—	2	2	no	—
37	C₈×Dic₃	—	2	2	no	—
38	C₂×C₃⋊C₁₆	—	2	2	no	—
39	C₂×C₃⋊C₁₆	—	2	2	no	—
40	C₂×C₃⋊C₁₆	Q₈	1	4	yes	complemented
41	C₄⋊(C₃⋊C₈)	—	2	2	no	—
42	C₂×C₄×Q₈	—	3	1	no	—
43	C₂²×C₁₆	—	3	2	no	—
44	C₂²×C₁₆	C₂×D₃	1	2	yes	—
45	C₈×Q₈	—	6	2	no	—
46	C₈×Q₈	—	6	2	no	—
47	C₂×C₄⋊C₈	—	3	2	no	—
48	C₂×C₄⋊C₈	—	3	1	no	—
49	C₂×C₄×C₈	—	3	2	no	—
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	C₄⋊C₁₆	—	2	2	no	—
52	#346	—	6	2	no	—
53	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₃	C₂×C₈	1	2	yes	—
57	C₂²×Dic₃	C₂×C₈	1	2	yes	—
58	C₂×Dic₆	—	2	2	no	—
59	C₂×Dic₆	—	2	4	no	—
60	C₂×C₃×C₈	—	2	2	no	—
61	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
62	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
65	C₂×C₃×C₈	#41	1	2	yes	—
66	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
67	C₃×C₄²	M₁₆	1	2	yes	—
68	C₄⋊Dic₃	—	2	2	no	—
69	Dic₃⋊C₄	—	2	4	no	—
70	C₄×Dic₃	—	2	2	no	—
71	C₄×Dic₃	—	2	2	no	—
72	C₂×C₃⋊C₈	—	2	2	no	—
73	C₂×C₃⋊C₈	—	2	4	no	—
74	C₃×C₁₆	—	4	2	no	—
75	C₃⋊C₁₆	—	2	4	no	—
76	C₂²×Q₈	—	3	1	no	—
77	C₂²×C₈	—	3	2	no	—
78	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
79	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
80	C₄×Q₈	—	6	2	no	—
81	C₄×Q₈	—	6	2	no	—
82	C₂×C₄⋊C₄	—	3	2	no	—
83	C₂×C₄⋊C₄	—	3	1	no	—
84	C₂×C₄²	—	3	2	no	—
85	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
86	C₂×C₁₆	—	2	4	no	—
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	3	4	no	—
89	C₂×C₁₆	—	6	2	no	—
90	C₂×C₁₆	—	6	2	no	—
91	C₄⋊C₈	—	6	4	no	—
92	C₄⋊C₈	—	6	2	no	—
93	C₄×C₈	—	6	2	no	—
94	C₄×C₈	—	6	2	no	—
95	C₄×C₈	D₁₂	1	2	yes	—
96	C₄×C₈	#67	1	2	yes	—
97	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
101	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
102	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
104	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₃×C₄	#131	1	2	yes	—
106	C₂×Dic₃	—	2	2	no	—
107	C₂×Dic₃	—	2	2	no	—
108	C₂×Dic₃	—	2	2	no	—
109	C₂×Dic₃	—	2	4	no	—
110	C₂×Dic₃	M₃₂	1	4	yes	—
111	Dic₆	—	2	8	no	—
112	C₃×C₈	—	2	2	no	—
113	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
114	C₃×C₈	#124	1	1	yes	characteristic
115	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
116	C₃⋊C₈	—	4	2	no	—
117	C₂×Q₈	—	6	4	no	—
118	C₂×Q₈	—	6	2	no	—
119	C₂²×C₄	—	3	2	no	—
120	C₂²×C₄	—	3	2	no	—
121	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
122	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	6	4	no	—
126	C₂×C₈	—	6	2	no	—
127	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
129	C₂×C₈	#237	1	2	yes	—
130	C₂×C₈	D₃×Q₈	1	2	yes	—
131	C₂×C₈	C₂×#67	1	1	yes	characteristic
132	C₄⋊C₄	—	6	4	no	—
133	C₄⋊C₄	—	6	2	no	—
134	C₄²	—	6	2	no	—
135	C₄²	—	6	2	no	—
136	C₄²	ΓL(1,𝔽₂₅)	1	2	yes	—
137	C₄²	(C₂×D₃)⋊C₄	1	2	yes	—
138	C₁₆	—	4	2	no	—
139	C₁₆	—	6	4	no	—
140	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₃	C₈×Q₈	1	2	yes	—
142	C₂²×C₃	C₂×M₃₂	1	2	yes	—
143	C₂²×C₃	#504	1	2	yes	—
144	C₃×C₄	—	2	2	no	—
145	C₃×C₄	—	2	2	no	—
146	C₃×C₄	#391	1	1	yes	characteristic
147	C₃×C₄	#408	1	1	yes	characteristic
148	C₃×C₄	#446	1	2	yes	—
149	Dic₃	—	2	4	no	—
150	Dic₃	—	4	2	no	—
151	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
152	Q₈	—	6	8	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	3	4	no	—
156	C₂×C₄	—	6	2	no	—
157	C₂×C₄	—	6	2	no	—
158	C₂×C₄	—	6	4	no	—
159	C₂×C₄	—	6	2	no	—
160	C₂×C₄	(C₂×D₃)⋊C₈	1	2	yes	—
161	C₂×C₄	(C₂×D₃)⋊C₈	1	2	yes	—
162	C₂×C₄	#887	1	2	yes	—
163	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
164	C₂×C₄	#907	1	2	yes	—
165	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	12	2	no	—
168	C₈	#897	1	2	yes	—
169	C₈	#936	1	1	yes	characteristic
170	C₈	#944	1	1	yes	characteristic
171	C₂×C₃	#1984	1	1	yes	characteristic
172	C₂×C₃	#2112	1	1	yes	characteristic
173	C₂×C₃	#2113	1	1	yes	characteristic
174	C₂×C₃	#2183	1	4	yes	—
175	C₂²	#4830	1	2	yes	—
176	C₂²	#4901	1	2	yes	—
177	C₂²	D₃×M₃₂	1	2	yes	—
178	C₂²	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	6	4	no	—
182	C₄	—	12	2	no	—
183	C₄	#4831	1	2	yes	—
184	C₄	#4973	1	1	yes	characteristic
185	C₄	#5129	1	1	yes	characteristic
186	C₃	#13264	1	1	yes	characteristic, complemented, Sylow
187	C₂	#69282	1	4	yes	—
188	C₂	#70472	1	1	yes	characteristic
189	C₂	#71409	1	1	yes	characteristic
190	C₂	#71431	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₆

C₂

yes

maximal, characteristic

C₂×C₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#4461

C₂

yes

maximal, characteristic

#67804

C₂

yes

maximal

#13264

—

maximal, Sylow

C₂×C₄×Dic₆

C₄

yes

characteristic

C₂²×C₃×C₁₆

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×Dic₃⋊C₈

C₄

yes

—

C₂×C₈×Dic₃

C₂²

yes

—

C₂²×C₃⋊C₁₆

C₂²

yes

—

C₂×C₄⋊(C₃⋊C₈)

C₄

yes

characteristic

C₈×Dic₆

—

C₈×Dic₆

—

C₃×C₄⋊C₁₆

—

#4461

—

C₂×C₈×Q₈

—

C₂×C₄⋊C₁₆

D₃

yes

characteristic

C₂×#346

—

#1374

—

C₂²×Dic₆

C₈

yes

characteristic

C₂²×C₃×C₈

C₂×C₄

yes

characteristic

C₂²×C₃×C₈

C₂³

yes

characteristic

C₂×C₃×C₄²

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₃

C₈

yes

characteristic

C₂×Dic₃⋊C₄

C₈

yes

—

C₂×C₄×Dic₃

C₂×C₄

yes

—

C₂²×C₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₆

—

C₄×Dic₆

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

D₄

yes

—

Dic₃⋊C₈

—

C₈×Dic₃

—

C₈×Dic₃

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

Q₈

yes

complemented

C₄⋊(C₃⋊C₈)

—

C₂×C₄×Q₈

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₃

yes

—

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

C₄⋊C₁₆

—

#346

—

C₂²×C₃×C₄

C₂×C₈

yes

characteristic

C₂²×C₃×C₄

C₂×C₈

yes

characteristic

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂²×Dic₃

C₂×C₈

yes

—

C₂²×Dic₃

C₂×C₈

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

C₂×Q₈

yes

—

C₂×C₃×C₈

#41

yes

—

C₃×C₄²

C₂²⋊C₄

yes

—

C₃×C₄²

M₁₆

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₃

yes

characteristic

C₂²×C₈

C₂²×D₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₄×D₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₁₂

yes

—

C₄×C₈

#67

yes

—

C₂³×C₃

C₂²×C₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

C₂²⋊C₈

yes

—

101

C₂×C₃×C₄

C₂²⋊C₈

yes

—

102

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

103

C₂×C₃×C₄

C₄×Q₈

yes

—

104

C₂×C₃×C₄

C₂×M₁₆

yes

characteristic

105

C₂×C₃×C₄

#131

yes

—

106

C₂×Dic₃

—

107

C₂×Dic₃

—

108

C₂×Dic₃

—

109

C₂×Dic₃

—

110

C₂×Dic₃

M₃₂

yes

—

111

Dic₆

—

112

C₃×C₈

—

113

C₃×C₈

C₂²⋊Q₈

yes

—

114

C₃×C₈

#124

yes

characteristic

115

C₃×C₈

C₄⋊Q₈

yes

characteristic

116

C₃⋊C₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

D₃×C₈

yes

characteristic

122

C₂²×C₄

D₃×C₈

yes

characteristic

123

C₂²×C₄

C₂×C₄×D₃

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

128

C₂×C₈

C₂×D₁₂

yes

characteristic

129

C₂×C₈

#237

yes

—

130

C₂×C₈

D₃×Q₈

yes

—

131

C₂×C₈

C₂×#67

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

ΓL(1,𝔽₂₅)

yes

—

137

C₄²

(C₂×D₃)⋊C₄

yes

—

138

C₁₆

—

139

C₁₆

—

140

C₂²×C₃

C₂×C₂²⋊C₈

yes

characteristic

141

C₂²×C₃

C₈×Q₈

yes

—

142

C₂²×C₃

C₂×M₃₂

yes

—

143

C₂²×C₃

#504

yes

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

#391

yes

characteristic

147

C₃×C₄

#408

yes

characteristic

148

C₃×C₄

#446

yes

—

149

Dic₃

—

150

Dic₃

—

151

C₂³

C₂×D₃×C₈

yes

characteristic

152

Q₈

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

(C₂×D₃)⋊C₈

yes

—

161

C₂×C₄

(C₂×D₃)⋊C₈

yes

—

162

C₂×C₄

#887

yes

—

163

C₂×C₄

C₂×ΓL(1,𝔽₂₅)

yes

characteristic

164

C₂×C₄

#907

yes

—

165

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

166

C₈

—

167

C₈

—

168

C₈

#897

yes

—

169

C₈

#936

yes

characteristic

170

C₈

#944

yes

characteristic

171

C₂×C₃

#1984

yes

characteristic

172

C₂×C₃

#2112

yes

characteristic

173

C₂×C₃

#2113

yes

characteristic

174

C₂×C₃

#2183

yes

—

175

C₂²

#4830

yes

—

176

C₂²

#4901

yes

—

177

C₂²

D₃×M₃₂

yes

—

178

C₂²

C₂×(C₂×D₃)⋊C₈

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

#4831

yes

—

184

C₄

#4973

yes

characteristic

185

C₄

#5129

yes

characteristic

186

C₃

#13264

yes

characteristic, complemented, Sylow

187

C₂

#69282

yes

—

188

C₂

#70472

yes

characteristic

189

C₂

#71409

yes

characteristic

190

C₂

#71431

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₃
commutator, socle, prime radical	C₂³×C₃	C₂²×C₈
Fitting	C₂×C₃×C₄⋊C₁₆	C₂
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₃
hypocenter	C₃	#13264
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₃

commutator, socle, prime radical

C₂³×C₃

C₂²×C₈

Fitting

C₂×C₃×C₄⋊C₁₆

C₂

cosocle

C₂²×C₃×C₈

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
40. C₂×C₃⋊C₁₆	Q₈	semidirect
186. C₃	#13264	semidirect

H ≅ G/N

type

40. C₂×C₃⋊C₁₆

Q₈

semidirect

186. C₃

#13264

semidirect

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

subgroup	index
C₂×C₈×Dic₆	2
C₂×C₃×C₄⋊C₁₆	2
C₂×#4461	2
#67804	2
#13264	3

subgroup

index

C₂×C₈×Dic₆

C₂×C₃×C₄⋊C₁₆

C₂×#4461

#67804

#13264

Subgroups — #10713709

Conjugacy classes of subgroups

Complements

Maximal subgroups