Group #13271963 — subgroups

Subgroups — #13271963

Order 1792.

442 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 139 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	#11735903	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#89303	C₂	1	4	yes	maximal, complemented
4	#11727497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#6464	C₂²	1	2	yes	—
6	#90168	C₄	1	1	yes	characteristic
7	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	#89882	C₄	1	2	yes	—
10	C₂×#6386	C₂²	1	2	yes	—
11	#89696	C₄	1	1	yes	characteristic
12	#89460	—	2	2	no	—
13	C₈×D₂₈	—	2	2	no	—
14	C₇×#346	—	2	2	no	—
15	#89303	—	2	8	no	—
16	C₄⋊(C₇⋊C₁₆)	—	2	2	no	—
17	#13192	—	7	1	no	maximal, Sylow
18	C₂×#1157	C₂×C₄	1	2	yes	—
19	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
22	#6493	—	2	2	no	—
23	C₂×#1136	C₂×C₄	1	2	yes	—
24	#6464	—	2	4	no	—
25	#6464	—	2	4	no	—
26	C₂×#1131	—	2	2	no	—
27	C₂×C₈×D₇	—	2	2	no	—
28	#6438	C₈	1	2	yes	—
29	#6437	C₈	1	1	yes	characteristic
30	C₄×D₂₈	—	2	2	no	—
31	#6432	C₈	1	1	yes	characteristic
32	C₂×C₇×C₁₆	—	2	2	no	—
33	C₂×C₇×C₁₆	—	2	2	no	—
34	C₂×C₇×C₁₆	D₄	1	4	yes	complemented
35	C₄×C₇×C₈	D₄	1	2	yes	—
36	C₄×C₇×C₈	D₄	1	2	yes	—
37	(C₂×D₇)⋊C₈	—	2	4	no	—
38	#6386	—	2	4	no	—
39	C₂×C₇⋊C₁₆	—	2	2	no	—
40	C₄⋊(C₇⋊C₈)	—	2	2	no	—
41	#2931	—	7	1	no	—
42	#2150	—	7	1	no	—
43	C₂×#350	—	7	4	no	—
44	C₂×#346	D₇	1	1	yes	characteristic, complemented
45	C₂²×C₄×C₇	M₁₆	1	2	yes	—
46	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₇	—	2	2	no	—
48	#1157	—	2	4	no	—
49	#1157	—	2	4	no	—
50	C₂×D₂₈	—	2	2	no	—
51	C₂×D₂₈	C₂×C₈	1	2	yes	—
52	C₂×C₄×D₇	—	2	2	no	—
53	C₂×Dic₁₄	C₂×C₈	1	2	yes	—
54	C₂×C₇×C₈	—	2	2	no	—
55	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
58	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₇×C₈	#41	1	2	yes	—
60	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
63	(C₂×D₇)⋊C₄	—	2	4	no	—
64	C₄⋊Dic₇	—	2	2	no	—
65	#1136	—	2	4	no	—
66	C₂×C₇⋊C₈	—	2	2	no	—
67	#1131	—	4	4	no	—
68	C₈×D₇	—	4	2	no	—
69	C₈×D₇	—	4	2	no	—
70	C₇×C₁₆	—	2	4	no	—
71	C₇⋊C₁₆	—	4	2	no	—
72	C₂×#131	—	7	2	no	—
73	#532	—	7	1	no	—
74	C₂×M₃₂	—	7	2	no	—
75	C₂²×C₁₆	C₂×D₇	1	2	yes	—
76	#436	—	14	2	no	—
77	C₈×D₄	—	14	2	no	—
78	#423	—	7	1	no	—
79	#408	—	7	2	no	—
80	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
81	C₄⋊C₁₆	—	14	2	no	—
82	#350	—	14	8	no	—
83	#346	—	2	2	no	—
84	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂²×D₇	—	2	2	no	—
86	C₂×C₄×C₇	—	4	2	no	—
87	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
88	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
89	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₇	#131	1	2	yes	—
93	C₂²⋊D₇	—	4	4	no	—
94	C₂×Dic₇	—	2	2	no	—
95	D₂₈	—	2	4	no	—
96	D₂₈	—	4	4	no	—
97	C₄×D₇	—	4	2	no	—
98	C₄×D₇	—	4	2	no	—
99	Dic₁₄	—	2	4	no	—
100	C₇×C₈	—	2	2	no	—
101	C₇×C₈	#121	1	2	yes	—
102	C₇×C₈	#124	1	1	yes	characteristic
103	C₇×C₈	#127	1	1	yes	characteristic
104	C₇⋊C₈	—	4	2	no	—
105	C₂×#41	—	7	2	no	—
106	#131	—	14	4	no	—
107	#131	—	14	4	no	—
108	C₂×M₁₆	—	7	2	no	—
109	C₂×M₁₆	—	14	2	no	—
110	C₂²×C₈	—	14	2	no	—
111	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
113	C₄⋊Q₈	—	7	1	no	—
114	#127	—	7	1	no	—
115	#124	—	7	2	no	—
116	#121	—	14	2	no	—
117	C₄×D₄	—	14	2	no	—
118	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
119	M₃₂	—	14	4	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	14	2	no	—
123	C₂×C₁₆	D₂₈	1	4	yes	complemented
124	C₄⋊C₈	—	14	2	no	—
125	C₂²⋊C₈	—	14	4	no	—
126	C₄×C₈	D₂₈	1	2	yes	—
127	C₄×C₈	C₂²⋊D₇	1	2	yes	—
128	C₂²×C₇	—	2	2	no	—
129	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₇	#436	1	2	yes	—
131	C₂×D₇	—	4	4	no	—
132	C₂×D₇	—	4	2	no	—
133	C₄×C₇	—	2	2	no	—
134	C₄×C₇	—	2	2	no	—
135	C₄×C₇	#390	1	1	yes	characteristic
136	C₄×C₇	#408	1	1	yes	characteristic
137	C₄×C₇	C₈×D₄	1	2	yes	—
138	Dic₇	—	4	2	no	—
139	#41	—	14	4	no	—
140	#41	—	14	4	no	—
141	C₂×Q₈	—	7	2	no	—
142	C₂×D₄	—	7	2	no	—
143	C₂×D₄	—	14	2	no	—
144	C₂×D₄	—	14	2	no	—
145	C₂²×C₄	—	14	2	no	—
146	C₂²×C₄	#1131	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
148	M₁₆	—	14	4	no	—
149	M₁₆	—	28	4	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	14	2	no	—
152	C₂×C₈	—	28	2	no	—
153	C₂×C₈	—	28	2	no	—
154	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
155	C₂×C₈	C₂×D₂₈	1	2	yes	—
156	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
157	C₂×C₈	#1157	1	2	yes	—
158	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
159	C₄⋊C₄	—	14	2	no	—
160	C₂²⋊C₄	—	14	4	no	—
161	C₄²	C₈×D₇	1	1	yes	characteristic
162	C₄²	C₈×D₇	1	1	yes	characteristic
163	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
164	C₁₆	—	2	4	no	—
165	C₁₆	—	28	2	no	—
166	C₂×C₇	—	2	2	no	—
167	C₂×C₇	#1954	1	1	yes	characteristic
168	C₂×C₇	#2117	1	2	yes	—
169	D₇	—	8	2	no	—
170	C₂³	—	14	2	no	—
171	C₂³	C₂×#1131	1	1	yes	characteristic
172	Q₈	—	14	4	no	—
173	D₄	—	14	4	no	—
174	D₄	—	28	4	no	—
175	D₄	—	28	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	14	2	no	—
178	C₂×C₄	—	28	2	no	—
179	C₂×C₄	—	28	2	no	—
180	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
181	C₂×C₄	(C₂×D₇)⋊C₈	1	2	yes	—
182	C₂×C₄	C₄×D₂₈	1	2	yes	—
183	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
184	C₂×C₄	#6464	1	2	yes	—
185	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	28	2	no	—
188	C₈	#6437	1	1	yes	characteristic
189	C₈	#6438	1	1	yes	characteristic
190	C₈	#6493	1	2	yes	—
191	C₇	#13192	1	1	yes	characteristic, complemented, Sylow
192	C₂²	—	2	2	no	—
193	C₂²	—	28	4	no	—
194	C₂²	—	28	2	no	—
195	C₂²	#89460	1	2	yes	—
196	C₂²	C₂×(C₂×D₇)⋊C₈	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	28	2	no	—
200	C₄	C₈×D₂₈	1	2	yes	—
201	C₄	#89711	1	1	yes	characteristic
202	C₄	#89882	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	56	2	no	—
205	C₂	#11727541	1	1	yes	characteristic
206	C₂	#11729191	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11735903

C₂

yes

maximal, characteristic

C₂×C₇×#346

C₂

yes

maximal, characteristic, complemented

C₂×#89303

C₂

yes

maximal, complemented

#11727497

C₂

yes

maximal, characteristic, complemented

C₂×#6464

C₂²

yes

—

#90168

C₄

yes

characteristic

C₂²×C₇×C₁₆

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

#89882

C₄

yes

—

C₂×#6386

C₂²

yes

—

#89696

C₄

yes

characteristic

#89460

—

C₈×D₂₈

—

C₇×#346

—

#89303

—

C₄⋊(C₇⋊C₁₆)

—

#13192

—

maximal, Sylow

C₂×#1157

C₂×C₄

yes

—

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂²×C₇×C₈

C₂³

yes

characteristic

C₂×C₄²×C₇

C₂×C₄

yes

characteristic

#6493

—

C₂×#1136

C₂×C₄

yes

—

#6464

—

#6464

—

C₂×#1131

—

C₂×C₈×D₇

—

#6438

C₈

yes

—

#6437

C₈

yes

characteristic

C₄×D₂₈

—

#6432

C₈

yes

characteristic

C₂×C₇×C₁₆

—

C₂×C₇×C₁₆

—

C₂×C₇×C₁₆

D₄

yes

complemented

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

D₄

yes

—

(C₂×D₇)⋊C₈

—

#6386

—

C₂×C₇⋊C₁₆

—

C₄⋊(C₇⋊C₈)

—

—

—

—

D₇

yes

characteristic, complemented

C₂²×C₄×C₇

M₁₆

yes

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×C₂²⋊D₇

—

#1157

—

#1157

—

C₂×D₂₈

—

C₂×D₂₈

C₂×C₈

yes

—

C₂×C₄×D₇

—

C₂×Dic₁₄

C₂×C₈

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂²⋊C₄

yes

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

#41

yes

—

C₄²×C₇

C₂²⋊C₄

yes

—

C₄²×C₇

C₂×C₈

yes

characteristic

C₄²×C₇

C₂×C₈

yes

characteristic

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

#1136

—

C₂×C₇⋊C₈

—

#1131

—

C₈×D₇

—

C₈×D₇

—

C₇×C₁₆

—

C₇⋊C₁₆

—

C₂×#131

—

#532

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₇

yes

—

#436

—

C₈×D₄

—

#423

—

#408

—

C₂×C₄×C₈

C₂×D₇

yes

characteristic

C₄⋊C₁₆

—

#350

—

#346

—

C₂³×C₇

C₂×M₁₆

yes

characteristic

C₂²×D₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²⋊C₈

yes

—

C₂×C₄×C₇

C₂²⋊C₈

yes

—

C₂×C₄×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇

C₄×D₄

yes

—

C₂×C₄×C₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₇

#131

yes

—

C₂²⋊D₇

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

100

C₇×C₈

—

101

C₇×C₈

#121

yes

—

102

C₇×C₈

#124

yes

characteristic

103

C₇×C₈

#127

yes

characteristic

104

C₇⋊C₈

—

105

C₂×#41

—

106

#131

—

107

#131

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

110

C₂²×C₈

—

111

C₂²×C₈

C₄×D₇

yes

characteristic

112

C₂²×C₈

C₂²×D₇

yes

characteristic

113

C₄⋊Q₈

—

114

#127

—

115

#124

—

116

#121

—

117

C₄×D₄

—

118

C₂×C₄²

C₄×D₇

yes

characteristic

119

M₃₂

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

—

123

C₂×C₁₆

D₂₈

yes

complemented

124

C₄⋊C₈

—

125

C₂²⋊C₈

—

126

C₄×C₈

D₂₈

yes

—

127

C₄×C₈

C₂²⋊D₇

yes

—

128

C₂²×C₇

—

129

C₂²×C₇

C₂×C₂²⋊C₈

yes

characteristic

130

C₂²×C₇

#436

yes

—

131

C₂×D₇

—

132

C₂×D₇

—

133

C₄×C₇

—

134

C₄×C₇

—

135

C₄×C₇

#390

yes

characteristic

136

C₄×C₇

#408

yes

characteristic

137

C₄×C₇

C₈×D₄

yes

—

138

Dic₇

—

139

#41

—

140

#41

—

141

C₂×Q₈

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

—

146

C₂²×C₄

#1131

yes

—

147

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

148

M₁₆

—

149

M₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

(C₂×D₇)⋊C₄

yes

—

155

C₂×C₈

C₂×D₂₈

yes

—

156

C₂×C₈

C₂×D₂₈

yes

characteristic

157

C₂×C₈

#1157

yes

—

158

C₂×C₈

C₂×C₂²⋊D₇

yes

characteristic

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₈×D₇

yes

characteristic

162

C₄²

C₈×D₇

yes

characteristic

163

C₄²

(C₂×D₇)⋊C₄

yes

—

164

C₁₆

—

165

C₁₆

—

166

C₂×C₇

—

167

C₂×C₇

#1954

yes

characteristic

168

C₂×C₇

#2117

yes

—

169

D₇

—

170

C₂³

—

171

C₂³

C₂×#1131

yes

characteristic

172

Q₈

—

173

D₄

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

(C₂×D₇)⋊C₈

yes

—

181

C₂×C₄

(C₂×D₇)⋊C₈

yes

—

182

C₂×C₄

C₄×D₂₈

yes

—

183

C₂×C₄

C₂×C₈×D₇

yes

characteristic

184

C₂×C₄

#6464

yes

—

185

C₂×C₄

C₂×(C₂×D₇)⋊C₄

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

#6437

yes

characteristic

189

C₈

#6438

yes

characteristic

190

C₈

#6493

yes

—

191

C₇

#13192

yes

characteristic, complemented, Sylow

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

#89460

yes

—

196

C₂²

C₂×(C₂×D₇)⋊C₈

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

C₈×D₂₈

yes

—

201

C₄

#89711

yes

characteristic

202

C₄

#89882

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

#11727541

yes

characteristic

206

C₂

#11729191

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#346	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	#90168	C₄
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#346	D₇
hypocenter	C₇	#13192
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₈

C₂×D₂₈

commutator

C₂×C₄×C₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇

Fitting

C₂×C₇×#346

C₂

socle

C₂²×C₇

C₂×C₂²⋊C₈

prime radical

#90168

C₄

cosocle

C₂²×C₇×C₈

C₂³

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. C₂×C₇×#346	C₂	semidirect
3. C₂×#89303	C₂	semidirect
4. #11727497	C₂	semidirect
34. C₂×C₇×C₁₆	D₄	semidirect
44. C₂×#346	D₇	semidirect
123. C₂×C₁₆	D₂₈	semidirect
191. C₇	#13192	semidirect

H ≅ G/N

type

2. C₂×C₇×#346

C₂

semidirect

3. C₂×#89303

C₂

semidirect

4. #11727497

C₂

semidirect

34. C₂×C₇×C₁₆

D₄

semidirect

44. C₂×#346

D₇

semidirect

123. C₂×C₁₆

D₂₈

semidirect

191. C₇

#13192

semidirect

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

subgroup	index
#11735903	2
C₂×C₇×#346	2
C₂×#89303	2
#11727497	2
#13192	7

subgroup

index

C₂×C₇×#346

Subgroups — #13271963

Conjugacy classes of subgroups

Complements

Maximal subgroups