Group #13265365 — subgroups

Subgroups — #13265365

Order 1792.

270 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 115 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	#11727038	C₂	1	2	yes	maximal
2	C₈²×D₇	C₂	1	1	yes	maximal, characteristic
3	C₇×#1370	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725361	C₂	1	2	yes	maximal, complemented
5	#11725246	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#6372	—	2	2	no	—
7	C₄×C₈×D₇	C₄	1	1	yes	characteristic
8	C₄×C₈×D₇	C₄	1	1	yes	characteristic
9	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
11	C₇×C₈²	C₂²	1	1	yes	characteristic
12	Dic₇⋊C₁₆	—	2	2	no	—
13	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
14	C₈×C₇⋊C₈	C₄	1	1	yes	characteristic
15	C₈×C₇⋊C₈	C₄	1	1	yes	characteristic
16	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
17	#12004	—	7	1	no	maximal, Sylow
18	C₂×C₈×D₇	—	2	1	no	—
19	C₂×C₈×D₇	C₈	1	1	yes	characteristic
20	C₂×C₈×D₇	C₈	1	1	yes	characteristic
21	C₂×C₈×D₇	D₄	1	1	yes	characteristic
22	C₂×C₈×D₇	Q₈	1	1	yes	characteristic
23	C₄²×D₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₇×C₁₆	—	2	2	no	—
25	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₈×Dic₇	—	2	1	no	—
29	C₈×Dic₇	C₈	1	1	yes	characteristic
30	C₈×Dic₇	C₈	1	1	yes	characteristic
31	C₈×Dic₇	D₄	1	1	yes	characteristic
32	C₈×Dic₇	Q₈	1	1	yes	characteristic
33	C₂×C₇⋊C₁₆	—	2	2	no	—
34	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
37	#6372	—	4	4	no	—
38	#2150	—	7	2	no	—
39	C₂×C₈²	—	7	1	no	—
40	#1370	—	7	2	no	—
41	#1370	—	7	1	no	—
42	#1370	D₇	1	1	yes	characteristic, complemented
43	C₂×C₄×D₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×D₇	M₁₆	1	1	yes	characteristic
46	C₂×C₇×C₈	—	2	1	no	—
47	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
48	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
52	C₄×Dic₇	C₄⋊C₄	1	1	yes	characteristic
53	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
54	C₄×Dic₇	M₁₆	1	1	yes	characteristic
55	C₂×C₇⋊C₈	—	2	1	no	—
56	C₂×C₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₇⋊C₈	C₂×C₈	1	1	yes	characteristic
59	C₂×C₇⋊C₈	C₂×C₈	1	1	yes	characteristic
60	C₈×D₇	—	2	4	no	—
61	C₈×D₇	—	2	2	no	—
62	C₈×D₇	—	2	2	no	—
63	C₈×D₇	—	2	2	no	—
64	C₈×D₇	D₈	1	2	yes	—
65	C₈×D₇	Q₁₆	1	2	yes	—
66	C₇×C₁₆	—	4	2	no	—
67	C₇⋊C₁₆	—	4	2	no	—
68	C₂×M₃₂	—	14	2	no	—
69	C₂×C₄×C₈	—	7	1	no	—
70	C₂×C₄×C₈	—	7	1	no	—
71	C₂×C₄×C₈	—	7	1	no	—
72	C₄⋊C₁₆	—	7	2	no	—
73	C₄⋊C₁₆	—	14	2	no	—
74	C₄⋊C₁₆	C₂×D₇	1	2	yes	—
75	C₈²	—	7	1	no	—
76	C₈²	—	7	1	no	—
77	C₈²	—	7	1	no	—
78	C₈²	C₂×D₇	1	1	yes	characteristic
79	C₂²×D₇	C₄⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₇	C₄⋊C₈	1	1	yes	characteristic
84	C₂×Dic₇	C₄⋊C₈	1	1	yes	characteristic
85	C₂×Dic₇	C₄⋊C₈	1	1	yes	characteristic
86	C₄×D₇	—	2	1	no	—
87	C₄×D₇	—	2	1	no	—
88	C₄×D₇	—	2	2	no	—
89	C₄×D₇	#107	1	2	yes	—
90	C₄×D₇	#108	1	2	yes	—
91	C₇×C₈	—	2	2	no	—
92	C₇×C₈	—	2	1	no	—
93	C₇×C₈	—	2	1	no	—
94	C₇×C₈	M₃₂	1	2	yes	complemented
95	C₇×C₈	C₂×D₈	1	1	yes	characteristic
96	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
97	C₇⋊C₈	—	2	2	no	—
98	C₇⋊C₈	—	2	1	no	—
99	C₇⋊C₈	—	2	1	no	—
100	C₇⋊C₈	#107	1	2	yes	—
101	C₇⋊C₈	M₃₂	1	2	yes	complemented
102	C₂²×C₈	—	7	1	no	—
103	C₂²×C₈	—	7	1	no	—
104	C₂²×C₈	—	7	1	no	—
105	C₂²×C₈	—	7	1	no	—
106	C₂²×C₈	—	14	1	no	—
107	C₂×C₄²	—	7	1	no	—
108	M₃₂	—	28	4	no	—
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	14	2	no	—
111	C₄×C₈	—	7	1	no	—
112	C₄×C₈	—	7	1	no	—
113	C₄×C₈	—	7	1	no	—
114	C₄×C₈	—	7	1	no	—
115	C₄×C₈	—	7	1	no	—
116	C₄×C₈	—	7	1	no	—
117	C₄×C₈	—	7	1	no	—
118	C₄×C₈	—	14	1	no	—
119	C₄×C₈	C₄×D₇	1	1	yes	characteristic
120	C₄×C₈	C₄×D₇	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
122	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
123	C₂×D₇	—	2	1	no	—
124	C₂×D₇	—	2	1	no	—
125	C₂×D₇	#335	1	2	yes	—
126	C₄×C₇	—	2	1	no	—
127	C₄×C₇	C₂×#107	1	1	yes	characteristic
128	C₄×C₇	C₂×#108	1	1	yes	characteristic
129	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
130	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
131	Dic₇	—	2	1	no	—
132	Dic₇	#335	1	2	yes	—
133	Dic₇	#364	1	2	yes	—
134	C₂²×C₄	—	7	1	no	—
135	C₂²×C₄	—	7	1	no	—
136	C₂²×C₄	—	7	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	7	1	no	—
139	C₂×C₈	—	7	1	no	—
140	C₂×C₈	—	7	2	no	—
141	C₂×C₈	—	7	2	no	—
142	C₂×C₈	—	7	1	no	—
143	C₂×C₈	—	7	1	no	—
144	C₂×C₈	—	14	2	no	—
145	C₂×C₈	—	14	2	no	—
146	C₂×C₈	—	14	4	no	—
147	C₂×C₈	—	14	1	no	—
148	C₂×C₈	—	14	2	no	—
149	C₂×C₈	C₈×D₇	1	1	yes	characteristic
150	C₂×C₈	C₈×D₇	1	1	yes	characteristic
151	C₂×C₈	D₄×D₇	1	1	yes	characteristic
152	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
153	C₄²	—	7	1	no	—
154	C₄²	—	7	1	no	—
155	C₄²	—	7	1	no	—
156	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
157	C₁₆	—	4	2	no	—
158	C₁₆	—	28	2	no	—
159	C₂×C₇	C₂×#335	1	1	yes	characteristic
160	C₂×C₇	#2150	1	1	yes	characteristic
161	C₂×C₇	C₂×#364	1	1	yes	characteristic
162	D₇	—	2	2	no	—
163	C₂³	—	7	1	no	—
164	C₂×C₄	—	7	1	no	—
165	C₂×C₄	—	7	1	no	—
166	C₂×C₄	—	7	2	no	—
167	C₂×C₄	—	7	2	no	—
168	C₂×C₄	—	7	1	no	—
169	C₂×C₄	—	14	2	no	—
170	C₂×C₄	—	14	1	no	—
171	C₂×C₄	—	14	1	no	—
172	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
174	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	7	2	no	—
179	C₈	—	7	2	no	—
180	C₈	—	14	2	no	—
181	C₈	—	14	1	no	—
182	C₈	—	14	1	no	—
183	C₈	#6372	1	2	yes	complemented
184	C₈	D₇×D₈	1	1	yes	characteristic
185	C₈	D₇×Q₁₆	1	1	yes	characteristic
186	C₇	#12004	1	1	yes	characteristic, complemented, Sylow
187	C₂²	—	7	2	no	—
188	C₂²	—	14	1	no	—
189	C₂²	—	14	1	no	—
190	C₂²	D₇×C₄⋊C₈	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	7	2	no	—
193	C₄	—	7	2	no	—
194	C₄	—	14	1	no	—
195	C₄	D₇×#107	1	1	yes	characteristic
196	C₄	D₇×#108	1	1	yes	characteristic
197	C₄	C₂×#6372	1	1	yes	characteristic
198	C₄	D₇×M₃₂	1	1	yes	characteristic
199	C₂	—	14	2	no	—
200	C₂	D₇×#335	1	1	yes	characteristic
201	C₂	#11727038	1	1	yes	characteristic
202	C₂	D₇×#364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11727038

C₂

yes

maximal

C₈²×D₇

C₂

yes

maximal, characteristic

C₇×#1370

C₂

yes

maximal, characteristic, complemented

#11725361

C₂

yes

maximal, complemented

#11725246

C₂

yes

maximal, characteristic, complemented

C₂×#6372

—

C₄×C₈×D₇

C₄

yes

characteristic

C₄×C₈×D₇

C₄

yes

characteristic

C₄×C₈×D₇

C₂²

yes

characteristic

C₇×C₄⋊C₁₆

C₂²

yes

—

C₇×C₈²

C₂²

yes

characteristic

Dic₇⋊C₁₆

—

C₄⋊(C₇⋊C₁₆)

C₂²

yes

—

C₈×C₇⋊C₈

C₄

yes

characteristic

C₈×C₇⋊C₈

C₄

yes

characteristic

C₈×C₇⋊C₈

C₂²

yes

characteristic

#12004

—

maximal, Sylow

C₂×C₈×D₇

—

C₂×C₈×D₇

C₈

yes

characteristic

C₂×C₈×D₇

C₈

yes

characteristic

C₂×C₈×D₇

D₄

yes

characteristic

C₂×C₈×D₇

Q₈

yes

characteristic

C₄²×D₇

C₂×C₄

yes

characteristic

C₂×C₇×C₁₆

—

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

characteristic

C₈×Dic₇

—

C₈×Dic₇

C₈

yes

characteristic

C₈×Dic₇

C₈

yes

characteristic

C₈×Dic₇

D₄

yes

characteristic

C₈×Dic₇

Q₈

yes

characteristic

C₂×C₇⋊C₁₆

—

C₄×C₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇⋊C₈

C₂×C₄

yes

characteristic

#6372

—

#2150

—

C₂×C₈²

—

#1370

—

#1370

—

#1370

D₇

yes

characteristic, complemented

C₂×C₄×D₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇

C₂×C₈

yes

characteristic

C₂×C₄×D₇

M₁₆

yes

characteristic

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂×C₈

yes

characteristic

C₂×C₇×C₈

C₂×C₈

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×Q₈

yes

characteristic

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄×Dic₇

C₄⋊C₄

yes

characteristic

C₄×Dic₇

C₂×C₈

yes

characteristic

C₄×Dic₇

M₁₆

yes

characteristic

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇⋊C₈

C₂×C₈

yes

characteristic

C₂×C₇⋊C₈

C₂×C₈

yes

characteristic

C₈×D₇

—

C₈×D₇

—

C₈×D₇

—

C₈×D₇

—

C₈×D₇

D₈

yes

—

C₈×D₇

Q₁₆

yes

—

C₇×C₁₆

—

C₇⋊C₁₆

—

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₇

yes

—

C₈²

—

C₈²

—

C₈²

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₇

C₂×M₁₆

yes

characteristic

C₂×Dic₇

C₄⋊C₈

yes

characteristic

C₂×Dic₇

C₄⋊C₈

yes

characteristic

C₂×Dic₇

C₄⋊C₈

yes

characteristic

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

#107

yes

—

C₄×D₇

#108

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

M₃₂

yes

complemented

C₇×C₈

C₂×D₈

yes

characteristic

C₇×C₈

C₂×Q₁₆

yes

characteristic

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

100

C₇⋊C₈

#107

yes

—

101

C₇⋊C₈

M₃₂

yes

complemented

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄²

—

108

M₃₂

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

C₄×D₇

yes

characteristic

120

C₄×C₈

C₄×D₇

yes

characteristic

121

C₄×C₈

C₂²×D₇

yes

characteristic

122

C₂²×C₇

C₂×C₄⋊C₈

yes

characteristic

123

C₂×D₇

—

124

C₂×D₇

—

125

C₂×D₇

#335

yes

—

126

C₄×C₇

—

127

C₄×C₇

C₂×#107

yes

characteristic

128

C₄×C₇

C₂×#108

yes

characteristic

129

C₄×C₇

C₂×M₃₂

yes

characteristic

130

C₄×C₇

C₂×M₃₂

yes

characteristic

131

Dic₇

—

132

Dic₇

#335

yes

—

133

Dic₇

#364

yes

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

C₈×D₇

yes

characteristic

150

C₂×C₈

C₈×D₇

yes

characteristic

151

C₂×C₈

D₄×D₇

yes

characteristic

152

C₂×C₈

Q₈×D₇

yes

characteristic

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

C₂×C₄×D₇

yes

characteristic

157

C₁₆

—

158

C₁₆

—

159

C₂×C₇

C₂×#335

yes

characteristic

160

C₂×C₇

#2150

yes

characteristic

161

C₂×C₇

C₂×#364

yes

characteristic

162

D₇

—

163

C₂³

—

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

D₇×C₄⋊C₄

yes

characteristic

173

C₂×C₄

C₂×C₈×D₇

yes

characteristic

174

C₂×C₄

D₇×M₁₆

yes

characteristic

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

#6372

yes

complemented

184

C₈

D₇×D₈

yes

characteristic

185

C₈

D₇×Q₁₆

yes

characteristic

186

C₇

#12004

yes

characteristic, complemented, Sylow

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

D₇×C₄⋊C₈

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

D₇×#107

yes

characteristic

196

C₄

D₇×#108

yes

characteristic

197

C₄

C₂×#6372

yes

characteristic

198

C₄

D₇×M₃₂

yes

characteristic

199

C₂

—

200

C₂

D₇×#335

yes

characteristic

201

C₂

#11727038

yes

characteristic

202

C₂

D₇×#364

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1370	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂²×D₇	C₄⋊C₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#1370	D₇
hypocenter	C₇	#12004
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

subgroup

G/H

center

C₂×C₈

D₄×D₇

commutator

C₂×C₄×C₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇

Fitting

C₇×#1370

C₂

socle

C₂²×C₇

C₂×C₄⋊C₈

prime radical

C₂²×D₇

C₄⋊C₈

cosocle

C₄×C₇×C₈

C₂³

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

N	H ≅ G/N	type
3. C₇×#1370	C₂	semidirect
4. #11725361	C₂	semidirect
5. #11725246	C₂	semidirect
42. #1370	D₇	semidirect
94. C₇×C₈	M₃₂	semidirect
101. C₇⋊C₈	M₃₂	semidirect
183. C₈	#6372	semidirect
186. C₇	#12004	semidirect

H ≅ G/N

type

3. C₇×#1370

C₂

semidirect

4. #11725361

C₂

semidirect

5. #11725246

C₂

semidirect

42. #1370

D₇

semidirect

94. C₇×C₈

M₃₂

semidirect

101. C₇⋊C₈

M₃₂

semidirect

183. C₈

#6372

semidirect

186. C₇

#12004

semidirect

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

subgroup	index
#11727038	2
C₈²×D₇	2
C₇×#1370	2
#11725361	2
#11725246	2
#12004	7

subgroup

index

C₈²×D₇

Subgroups — #13265365

Conjugacy classes of subgroups

Complements

Maximal subgroups