Group #10713349 — subgroups

Subgroups — #10713349

Order 768.

450 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 143 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	#78458	C₂	1	1	yes	maximal, characteristic
2	C₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#4515	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#13191	—	3	1	no	maximal, Sylow
6	C₂×#901	C₂²	1	2	yes	—
7	#5477	C₄	1	1	yes	characteristic
8	C₂×C₃×M₃₂	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	#5110	C₄	1	2	yes	—
11	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
12	#4924	C₄	1	1	yes	characteristic
13	#4688	—	2	2	no	—
14	C₈×D₁₂	—	2	2	no	—
15	C₃×C₄⋊C₁₆	—	2	2	no	—
16	#4515	—	2	8	no	—
17	C₄×C₃⋊C₁₆	—	2	2	no	—
18	#2931	—	3	1	no	—
19	#2150	D₃	1	1	yes	characteristic, complemented
20	C₂×#350	—	3	4	no	—
21	C₂×C₄×C₁₆	—	3	1	no	—
22	C₂×#235	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
26	#930	—	2	2	no	—
27	C₂×#208	C₂×C₄	1	2	yes	—
28	#901	—	2	4	no	—
29	#901	—	2	4	no	—
30	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
31	C₂×D₃×C₈	—	2	2	no	—
32	#875	C₈	1	2	yes	—
33	#874	C₈	1	1	yes	characteristic
34	C₄×D₁₂	—	2	2	no	—
35	#869	C₈	1	1	yes	characteristic
36	C₃×M₃₂	—	2	4	no	—
37	C₂×C₃×C₁₆	—	2	2	no	—
38	C₃×C₄×C₈	D₄	1	2	yes	—
39	C₃×C₄×C₈	D₄	1	2	yes	—
40	(C₂×D₃)⋊C₈	—	2	4	no	—
41	C₂×C₃⋊C₁₆	—	2	2	no	—
42	C₂×C₃⋊C₁₆	—	2	2	no	—
43	C₂×C₃⋊C₁₆	D₄	1	4	yes	complemented
44	C₄⋊(C₃⋊C₈)	—	2	2	no	—
45	C₂×#131	—	3	2	no	—
46	#532	—	3	1	no	—
47	C₂×M₃₂	C₂×D₃	1	2	yes	—
48	C₂²×C₁₆	—	3	2	no	—
49	#436	—	6	2	no	—
50	C₈×D₄	—	6	2	no	—
51	#423	—	3	1	no	—
52	#408	—	3	2	no	—
53	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	C₄⋊C₁₆	—	2	2	no	—
55	#350	—	6	8	no	—
56	C₄×C₁₆	—	6	2	no	—
57	C₂²×C₃×C₄	M₁₆	1	2	yes	—
58	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂×#67	—	2	2	no	—
60	#235	—	2	4	no	—
61	#235	—	2	4	no	—
62	C₂×D₁₂	—	2	2	no	—
63	C₂×D₁₂	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₃	—	2	2	no	—
65	C₂×Dic₆	C₂×C₈	1	2	yes	—
66	C₂×C₃×C₈	—	2	2	no	—
67	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
70	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₈	#41	1	2	yes	—
72	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
73	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
74	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
75	(C₂×D₃)⋊C₄	—	2	4	no	—
76	C₄⋊Dic₃	—	2	2	no	—
77	#208	—	2	4	no	—
78	C₂×C₃⋊C₈	—	2	2	no	—
79	ΓL(1,𝔽₂₅)	—	4	4	no	—
80	D₃×C₈	—	4	2	no	—
81	D₃×C₈	—	4	2	no	—
82	C₃×C₁₆	—	4	2	no	—
83	C₃⋊C₁₆	—	2	4	no	—
84	C₂×#41	—	3	2	no	—
85	#131	—	6	4	no	—
86	#131	—	6	4	no	—
87	C₂×M₁₆	—	3	2	no	—
88	C₂×M₁₆	—	6	2	no	—
89	C₂²×C₈	—	6	2	no	—
90	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
92	C₄⋊Q₈	—	3	1	no	—
93	#127	—	3	1	no	—
94	#124	—	3	2	no	—
95	#121	—	6	2	no	—
96	C₄×D₄	—	6	2	no	—
97	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
98	M₃₂	—	2	4	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	3	4	no	—
101	C₂×C₁₆	—	6	2	no	—
102	C₂×C₁₆	—	6	2	no	—
103	C₄⋊C₈	—	6	2	no	—
104	C₂²⋊C₈	—	6	4	no	—
105	C₄×C₈	D₁₂	1	2	yes	—
106	C₄×C₈	#67	1	2	yes	—
107	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂²×D₃	—	2	2	no	—
109	C₂×C₃×C₄	—	2	4	no	—
110	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
111	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
112	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
114	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₃×C₄	#131	1	2	yes	—
116	#67	—	4	4	no	—
117	C₂×Dic₃	—	2	2	no	—
118	D₁₂	—	2	4	no	—
119	D₁₂	—	4	4	no	—
120	C₄×D₃	—	4	2	no	—
121	C₄×D₃	—	4	2	no	—
122	Dic₆	—	2	4	no	—
123	C₃×C₈	—	2	2	no	—
124	C₃×C₈	#121	1	2	yes	—
125	C₃×C₈	#124	1	1	yes	characteristic
126	C₃×C₈	#127	1	1	yes	characteristic
127	C₃⋊C₈	—	4	2	no	—
128	#41	—	6	4	no	—
129	#41	—	6	4	no	—
130	C₂×Q₈	—	3	2	no	—
131	C₂×D₄	—	3	2	no	—
132	C₂×D₄	—	6	2	no	—
133	C₂×D₄	—	6	2	no	—
134	C₂²×C₄	—	6	2	no	—
135	C₂²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
136	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
137	M₁₆	—	6	4	no	—
138	M₁₆	—	12	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	6	2	no	—
141	C₂×C₈	—	12	2	no	—
142	C₂×C₈	—	12	2	no	—
143	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
144	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
145	C₂×C₈	D₃×D₄	1	2	yes	—
146	C₂×C₈	#239	1	2	yes	—
147	C₂×C₈	C₂×#67	1	1	yes	characteristic
148	C₄⋊C₄	—	6	2	no	—
149	C₂²⋊C₄	—	6	4	no	—
150	C₄²	D₃×C₈	1	1	yes	characteristic
151	C₄²	D₃×C₈	1	1	yes	characteristic
152	C₄²	(C₂×D₃)⋊C₄	1	2	yes	—
153	C₁₆	—	4	2	no	—
154	C₁₆	—	6	4	no	—
155	C₂²×C₃	—	2	2	no	—
156	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₂²×C₃	#436	1	2	yes	—
158	C₂×D₃	—	4	4	no	—
159	C₂×D₃	—	4	2	no	—
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	#350	1	4	yes	complemented
162	C₃×C₄	#390	1	1	yes	characteristic
163	C₃×C₄	#408	1	1	yes	characteristic
164	C₃×C₄	C₈×D₄	1	2	yes	—
165	Dic₃	—	4	2	no	—
166	C₂³	—	6	2	no	—
167	C₂³	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
168	Q₈	—	6	4	no	—
169	D₄	—	6	4	no	—
170	D₄	—	12	4	no	—
171	D₄	—	12	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	6	2	no	—
174	C₂×C₄	—	12	2	no	—
175	C₂×C₄	—	12	2	no	—
176	C₂×C₄	(C₂×D₃)⋊C₈	1	2	yes	—
177	C₂×C₄	(C₂×D₃)⋊C₈	1	2	yes	—
178	C₂×C₄	#893	1	2	yes	—
179	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
180	C₂×C₄	#907	1	2	yes	—
181	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	12	2	no	—
184	C₈	#895	1	2	yes	—
185	C₈	#940	1	1	yes	characteristic
186	C₈	#947	1	1	yes	characteristic
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	#1954	1	1	yes	characteristic
189	C₂×C₃	C₂×#350	1	2	yes	—
190	D₃	—	8	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	12	4	no	—
193	C₂²	—	12	2	no	—
194	C₂²	#4838	1	2	yes	—
195	C₂²	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	12	2	no	—
198	C₄	#4515	1	4	yes	complemented
199	C₄	#4834	1	2	yes	—
200	C₄	#4988	1	1	yes	characteristic
201	C₄	#5129	1	1	yes	characteristic
202	C₃	#13191	1	1	yes	characteristic, complemented, Sylow
203	C₂	—	2	2	no	—
204	C₂	—	24	2	no	—
205	C₂	#70470	1	1	yes	characteristic
206	C₂	C₂×#4515	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#78458

C₂

yes

maximal, characteristic

C₃×#2150

C₂

yes

maximal, characteristic, complemented

C₂×#4515

C₂

yes

maximal, complemented

C₂×C₄×C₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#13191

—

maximal, Sylow

C₂×#901

C₂²

yes

—

#5477

C₄

yes

characteristic

C₂×C₃×M₃₂

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

#5110

C₄

yes

—

C₂²×C₃⋊C₁₆

C₂²

yes

—

#4924

C₄

yes

characteristic

#4688

—

C₈×D₁₂

—

C₃×C₄⋊C₁₆

—

#4515

—

C₄×C₃⋊C₁₆

—

#2931

—

#2150

D₃

yes

characteristic, complemented

C₂×#350

—

C₂×C₄×C₁₆

—

C₂×#235

C₂×C₄

yes

—

C₂²×C₃×C₈

C₂×C₄

yes

characteristic

C₂²×C₃×C₈

C₂³

yes

characteristic

C₂×C₃×C₄²

C₂×C₄

yes

characteristic

#930

—

C₂×#208

C₂×C₄

yes

—

#901

—

#901

—

C₂×ΓL(1,𝔽₂₅)

—

C₂×D₃×C₈

—

#875

C₈

yes

—

#874

C₈

yes

characteristic

C₄×D₁₂

—

#869

C₈

yes

characteristic

C₃×M₃₂

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

D₄

yes

—

(C₂×D₃)⋊C₈

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

D₄

yes

complemented

C₄⋊(C₃⋊C₈)

—

C₂×#131

—

#532

—

C₂×M₃₂

C₂×D₃

yes

—

C₂²×C₁₆

—

#436

—

C₈×D₄

—

#423

—

#408

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

C₄⋊C₁₆

—

#350

—

C₄×C₁₆

—

C₂²×C₃×C₄

M₁₆

yes

—

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×#67

—

#235

—

#235

—

C₂×D₁₂

—

C₂×D₁₂

C₂×C₈

yes

—

C₂×C₄×D₃

—

C₂×Dic₆

C₂×C₈

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

C₂×D₄

yes

—

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

#41

yes

—

C₃×C₄²

C₂²⋊C₄

yes

—

C₃×C₄²

C₂×C₈

yes

characteristic

C₃×C₄²

C₂×C₈

yes

characteristic

(C₂×D₃)⋊C₄

—

C₄⋊Dic₃

—

#208

—

C₂×C₃⋊C₈

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

D₃×C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₄×D₃

yes

characteristic

C₂²×C₈

C₂²×D₃

yes

characteristic

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₄×D₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₄⋊C₈

—

104

C₂²⋊C₈

—

105

C₄×C₈

D₁₂

yes

—

106

C₄×C₈

#67

yes

—

107

C₂³×C₃

C₂×M₁₆

yes

characteristic

108

C₂²×D₃

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

C₂²⋊C₈

yes

—

111

C₂×C₃×C₄

C₂²⋊C₈

yes

—

112

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

113

C₂×C₃×C₄

C₄×D₄

yes

—

114

C₂×C₃×C₄

C₂²×C₈

yes

characteristic

115

C₂×C₃×C₄

#131

yes

—

116

#67

—

117

C₂×Dic₃

—

118

D₁₂

—

119

D₁₂

—

120

C₄×D₃

—

121

C₄×D₃

—

122

Dic₆

—

123

C₃×C₈

—

124

C₃×C₈

#121

yes

—

125

C₃×C₈

#124

yes

characteristic

126

C₃×C₈

#127

yes

characteristic

127

C₃⋊C₈

—

128

#41

—

129

#41

—

130

C₂×Q₈

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂²×C₄

—

135

C₂²×C₄

ΓL(1,𝔽₂₅)

yes

—

136

C₂²×C₄

C₂×C₄×D₃

yes

characteristic

137

M₁₆

—

138

M₁₆

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

144

C₂×C₈

C₂×D₁₂

yes

characteristic

145

C₂×C₈

D₃×D₄

yes

—

146

C₂×C₈

#239

yes

—

147

C₂×C₈

C₂×#67

yes

characteristic

148

C₄⋊C₄

—

149

C₂²⋊C₄

—

150

C₄²

D₃×C₈

yes

characteristic

151

C₄²

D₃×C₈

yes

characteristic

152

C₄²

(C₂×D₃)⋊C₄

yes

—

153

C₁₆

—

154

C₁₆

—

155

C₂²×C₃

—

156

C₂²×C₃

C₂×C₂²⋊C₈

yes

characteristic

157

C₂²×C₃

#436

yes

—

158

C₂×D₃

—

159

C₂×D₃

—

160

C₃×C₄

—

161

C₃×C₄

#350

yes

complemented

162

C₃×C₄

#390

yes

characteristic

163

C₃×C₄

#408

yes

characteristic

164

C₃×C₄

C₈×D₄

yes

—

165

Dic₃

—

166

C₂³

—

167

C₂³

C₂×ΓL(1,𝔽₂₅)

yes

characteristic

168

Q₈

—

169

D₄

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

(C₂×D₃)⋊C₈

yes

—

177

C₂×C₄

(C₂×D₃)⋊C₈

yes

—

178

C₂×C₄

#893

yes

—

179

C₂×C₄

C₂×D₃×C₈

yes

characteristic

180

C₂×C₄

#907

yes

—

181

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

182

C₈

—

183

C₈

—

184

C₈

#895

yes

—

185

C₈

#940

yes

characteristic

186

C₈

#947

yes

characteristic

187

C₂×C₃

—

188

C₂×C₃

#1954

yes

characteristic

189

C₂×C₃

C₂×#350

yes

—

190

D₃

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

#4838

yes

—

195

C₂²

C₂×(C₂×D₃)⋊C₈

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

#4515

yes

complemented

199

C₄

#4834

yes

—

200

C₄

#4988

yes

characteristic

201

C₄

#5129

yes

characteristic

202

C₃

#13191

yes

characteristic, complemented, Sylow

203

C₂

—

204

C₂

—

205

C₂

#70470

yes

characteristic

206

C₂

C₂×#4515

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#2150	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₈
prime radical	#5477	C₄
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#2150	D₃
hypocenter	C₃	#13191
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×#67

commutator

C₂×C₃×C₄

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₃

Fitting

C₃×#2150

C₂

socle

C₂²×C₃

C₂×C₂²⋊C₈

prime radical

#5477

C₄

cosocle

C₂²×C₃×C₈

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
2. C₃×#2150	C₂	semidirect
3. C₂×#4515	C₂	semidirect
4. C₂×C₄×C₃⋊C₁₆	C₂	semidirect
19. #2150	D₃	semidirect
43. C₂×C₃⋊C₁₆	D₄	semidirect
161. C₃×C₄	#350	semidirect
198. C₄	#4515	semidirect
202. C₃	#13191	semidirect

H ≅ G/N

type

2. C₃×#2150

C₂

semidirect

3. C₂×#4515

C₂

semidirect

4. C₂×C₄×C₃⋊C₁₆

C₂

semidirect

19. #2150

D₃

semidirect

43. C₂×C₃⋊C₁₆

D₄

semidirect

161. C₃×C₄

#350

semidirect

198. C₄

#4515

semidirect

202. C₃

#13191

semidirect

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

subgroup	index
#78458	2
C₃×#2150	2
C₂×#4515	2
C₂×C₄×C₃⋊C₁₆	2
#13191	3

subgroup

index

C₂×C₄×C₃⋊C₁₆

Subgroups — #10713349

Conjugacy classes of subgroups

Complements

Maximal subgroups