Group #13246752 — subgroups

Subgroups — #13246752

Order 1792.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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×Q₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#1467	C₂	1	1	yes	maximal, characteristic
3	#11729379	C₂	1	1	yes	maximal, characteristic
4	#11728548	C₂	1	2	yes	maximal
5	#11725561	C₂	1	2	yes	maximal
6	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₇×#432	C₂²	1	2	yes	—
8	Q₈×C₇⋊C₈	—	2	2	no	—
9	Q₈×C₇⋊C₈	—	2	2	no	—
10	#89698	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₇⋊C₈)	C₄	1	2	yes	—
12	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇⋊C₈	C₄	1	2	yes	—
14	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₇×#324	—	2	2	no	—
16	#89323	—	2	2	no	—
17	#89275	—	2	2	no	—
18	#89274	—	2	2	no	—
19	#8623	—	7	1	no	maximal, Sylow
20	C₂²×C₇×Q₈	Q₈	1	1	yes	characteristic
21	C₄×C₇×Q₈	—	2	2	no	—
22	C₄×C₇×Q₈	—	2	2	no	—
23	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
24	C₂×C₇×C₄⋊C₄	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
27	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₇×C₄⋊C₈	—	2	2	no	—
31	C₇×C₂²⋊C₈	—	2	2	no	—
32	C₇×#97	—	2	2	no	—
33	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₇⋊C₈)	—	2	4	no	—
36	C₄⋊(C₇⋊C₈)	—	2	2	no	—
37	#6377	—	2	2	no	—
38	C₄×C₇⋊C₈	—	2	2	no	—
39	C₄×C₇⋊C₈	—	2	2	no	—
40	C₄×C₇⋊C₈	D₄	1	2	yes	—
41	C₄×C₇⋊C₈	D₄	1	2	yes	—
42	C₂×C₈×Q₈	—	7	1	no	—
43	#1568	—	7	2	no	—
44	#1467	—	7	1	no	—
45	#1467	D₇	1	1	yes	characteristic
46	#1290	—	7	2	no	—
47	C₂×C₇×Q₈	—	2	4	no	—
48	C₂×C₇×Q₈	—	2	2	no	—
49	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₇×C₈	—	4	2	no	—
55	C₇×C₄⋊C₄	—	2	4	no	—
56	C₇×C₄⋊C₄	—	2	2	no	—
57	C₄²×C₇	—	2	2	no	—
58	C₄²×C₇	—	2	2	no	—
59	C₄²×C₇	C₂×D₄	1	2	yes	—
60	C₄²×C₇	#41	1	2	yes	—
61	C₂×C₇⋊C₈	—	2	2	no	—
62	C₂×C₇⋊C₈	—	2	2	no	—
63	C₂×C₇⋊C₈	—	4	1	no	—
64	C₂×C₇⋊C₈	—	4	2	no	—
65	C₂×C₇⋊C₈	—	4	2	no	—
66	C₂×C₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₇⋊C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
70	C₈×Q₈	—	14	2	no	—
71	C₈×Q₈	—	14	2	no	—
72	#432	—	7	2	no	—
73	#432	C₂×D₇	1	2	yes	—
74	C₂×C₄⋊C₈	—	7	2	no	—
75	C₂×C₄⋊C₈	—	7	1	no	—
76	C₂×C₄×C₈	—	7	1	no	—
77	C₂×C₄×C₈	—	7	2	no	—
78	#335	—	14	2	no	—
79	#334	—	14	2	no	—
80	#324	—	2	2	no	—
81	#324	—	14	2	no	—
82	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₇×Q₈	—	2	8	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	4	1	no	—
91	C₂×C₄×C₇	—	4	2	no	—
92	C₂×C₄×C₇	#108	1	4	yes	—
93	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₇	#117	1	1	yes	characteristic
95	C₂×C₄×C₇	#121	1	2	yes	—
96	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
97	C₇×C₈	—	8	2	no	—
98	C₇⋊C₈	—	2	1	no	—
99	C₇⋊C₈	—	2	2	no	—
100	C₇⋊C₈	—	4	2	no	—
101	C₇⋊C₈	—	8	2	no	—
102	C₇⋊C₈	#101	1	2	yes	—
103	C₂²×Q₈	Dic₁₄	1	1	yes	characteristic
104	C₂²×C₈	—	7	1	no	—
105	C₂²×C₈	—	7	1	no	—
106	C₂²×C₈	—	7	2	no	—
107	C₄×Q₈	—	2	2	no	—
108	C₄×Q₈	—	2	2	no	—
109	C₂×C₄⋊C₄	Dic₁₄	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂²⋊D₇	1	2	yes	—
111	C₂×C₄²	C₄×D₇	1	2	yes	—
112	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	14	4	no	—
115	C₄⋊C₈	—	14	2	no	—
116	C₄⋊C₈	—	14	2	no	—
117	C₂²⋊C₈	—	2	2	no	—
118	C₂²⋊C₈	—	14	2	no	—
119	#97	—	2	2	no	—
120	#97	—	14	2	no	—
121	C₄×C₈	—	7	2	no	—
122	C₄×C₈	—	7	2	no	—
123	C₄×C₈	—	14	2	no	—
124	C₄×C₈	—	14	2	no	—
125	C₂²×C₇	—	2	1	no	—
126	C₂²×C₇	—	2	1	no	—
127	C₂²×C₇	#380	1	1	yes	characteristic
128	C₂²×C₇	C₂×#108	1	2	yes	—
129	C₄×C₇	—	2	1	no	—
130	C₄×C₇	—	2	4	no	—
131	C₄×C₇	—	2	2	no	—
132	C₄×C₇	—	2	2	no	—
133	C₄×C₇	—	4	2	no	—
134	C₄×C₇	C₂×#101	1	1	yes	characteristic
135	C₄×C₇	#413	1	1	yes	characteristic
136	C₂×Q₈	—	2	2	no	—
137	C₂×Q₈	—	2	4	no	—
138	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
139	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
140	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	7	1	no	—
145	C₂×C₈	—	7	1	no	—
146	C₂×C₈	—	7	2	no	—
147	C₂×C₈	—	14	2	no	—
148	C₂×C₈	—	14	2	no	—
149	C₂×C₈	—	28	1	no	—
150	C₂×C₈	—	28	2	no	—
151	C₂×C₈	—	28	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	D₄×D₇	1	2	yes	—
157	C₄²	#1159	1	2	yes	—
158	C₂×C₇	—	2	2	no	—
159	C₂×C₇	#1801	1	1	yes	characteristic
160	C₂×C₇	#1810	1	1	yes	characteristic
161	C₂×C₇	#1812	1	1	yes	characteristic
162	C₂³	C₂×Dic₇⋊C₄	1	1	yes	characteristic
163	Q₈	—	2	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	#6396	1	4	yes	—
173	C₂×C₄	#6440	1	1	yes	characteristic
174	C₂×C₄	#6441	1	2	yes	—
175	C₂×C₄	D₇×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₄	#6502	1	2	yes	—
177	C₈	—	7	2	no	—
178	C₈	—	8	2	no	—
179	C₈	—	14	1	no	—
180	C₈	—	14	2	no	—
181	C₈	—	28	2	no	—
182	C₈	—	56	2	no	—
183	C₇	#8623	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	#89714	1	1	yes	characteristic
187	C₂²	C₂×#6396	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	4	2	no	—
193	C₄	D₇×#101	1	1	yes	characteristic
194	C₄	#89523	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	#11727736	1	1	yes	characteristic
197	C₂	#11727795	1	1	yes	characteristic
198	C₂	#11727851	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₇⋊C₈

C₂

yes

maximal, characteristic

C₇×#1467

C₂

yes

maximal, characteristic

#11729379

C₂

yes

maximal, characteristic

#11728548

C₂

yes

maximal

#11725561

C₂

yes

maximal

C₂×C₄×C₇×Q₈

C₂²

yes

characteristic

C₇×#432

C₂²

yes

—

Q₈×C₇⋊C₈

—

Q₈×C₇⋊C₈

—

#89698

C₂²

yes

—

C₂×C₄⋊(C₇⋊C₈)

C₄

yes

—

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄×C₇⋊C₈

C₄

yes

—

C₂×C₄×C₇⋊C₈

C₂²

yes

—

—

—

—

—

C₂²×C₇×Q₈

Q₈

yes

characteristic

C₄×C₇×Q₈

—

C₄×C₇×Q₈

—

C₂×C₇×C₄⋊C₄

D₄

yes

—

C₂×C₇×C₄⋊C₄

Q₈

yes

characteristic

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇⋊C₈

C₂×C₄

yes

—

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₇×#97

—

C₂²⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

#6377

—

C₄×C₇⋊C₈

—

C₄×C₇⋊C₈

—

C₄×C₇⋊C₈

D₄

yes

—

C₄×C₇⋊C₈

D₄

yes

—

C₂×C₈×Q₈

—

#1568

—

#1467

—

#1467

D₇

yes

characteristic

#1290

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₄⋊C₄

yes

—

C₂²×C₄×C₇

C₄⋊C₄

yes

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

C₂×Q₈

yes

characteristic

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

#41

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₄×Q₈

C₂×D₇

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

#432

—

#432

C₂×D₇

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

—

—

—

—

C₂³×C₇

C₂×C₄⋊C₄

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#108

yes

—

C₂×C₄×C₇

C₂×C₂²⋊C₄

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

100

C₇⋊C₈

—

101

C₇⋊C₈

—

102

C₇⋊C₈

#101

yes

—

103

C₂²×Q₈

Dic₁₄

yes

characteristic

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₄×Q₈

—

108

C₄×Q₈

—

109

C₂×C₄⋊C₄

Dic₁₄

yes

characteristic

110

C₂×C₄⋊C₄

C₂²⋊D₇

yes

—

111

C₂×C₄²

C₄×D₇

yes

—

112

C₂×C₄²

C₂²×D₇

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₂²⋊C₈

—

118

C₂²⋊C₈

—

119

#97

—

120

#97

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₂²×C₇

—

126

C₂²×C₇

—

127

C₂²×C₇

#380

yes

characteristic

128

C₂²×C₇

C₂×#108

yes

—

129

C₄×C₇

—

130

C₄×C₇

—

131

C₄×C₇

—

132

C₄×C₇

—

133

C₄×C₇

—

134

C₄×C₇

C₂×#101

yes

characteristic

135

C₄×C₇

#413

yes

characteristic

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂²×C₄

Dic₇⋊C₄

yes

—

139

C₂²×C₄

Dic₇⋊C₄

yes

—

140

C₂²×C₄

C₂×Dic₁₄

yes

characteristic

141

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

142

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

—

155

C₄²

—

156

C₄²

D₄×D₇

yes

—

157

C₄²

#1159

yes

—

158

C₂×C₇

—

159

C₂×C₇

#1801

yes

characteristic

160

C₂×C₇

#1810

yes

characteristic

161

C₂×C₇

#1812

yes

characteristic

162

C₂³

C₂×Dic₇⋊C₄

yes

characteristic

163

Q₈

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

#6396

yes

—

173

C₂×C₄

#6440

yes

characteristic

174

C₂×C₄

#6441

yes

—

175

C₂×C₄

D₇×C₂²⋊C₄

yes

characteristic

176

C₂×C₄

#6502

yes

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₇

#8623

yes

characteristic, complemented, Sylow

184

C₂²

—

185

C₂²

—

186

C₂²

#89714

yes

characteristic

187

C₂²

C₂×#6396

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

D₇×#101

yes

characteristic

194

C₄

#89523

yes

characteristic

195

C₂

—

196

C₂

#11727736

yes

characteristic

197

C₂

#11727795

yes

characteristic

198

C₂

#11727851

yes

characteristic

	subgroup	G/H
center	C₂²	#89714
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1467	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1467	D₇
hypocenter	C₇	#8623
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

subgroup

G/H

center

C₂²

#89714

commutator

C₂²×C₄×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

N	H ≅ G/N	type
183. C₇	#8623	semidirect

H ≅ G/N

type

183. C₇

#8623

semidirect

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

subgroup	index
C₂×Q₈×C₇⋊C₈	2
C₇×#1467	2
#11729379	2
#11728548	2
#11725561	2
#8623	7

subgroup

index

C₂×Q₈×C₇⋊C₈

Subgroups — #13246752

Conjugacy classes of subgroups

Complements

Maximal subgroups