Group #13187533 — subgroups

Subgroups — C₇×#10317 #13187533

Order 1792.

554 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 170 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×C₇×#498	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#487	C₂	1	2	yes	maximal
3	C₇×#1990	C₂	1	2	yes	maximal
4	C₇×#1956	C₂	1	1	yes	maximal, characteristic
5	C₇×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₇×#498	—	2	4	no	—
9	C₇×#487	—	2	4	no	—
10	C₇×#487	—	2	4	no	—
11	C₂×C₇×#106	—	2	2	no	—
12	C₂×C₇×#103	—	2	4	no	—
13	C₂×C₇×#103	—	2	2	no	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
16	C₇×#384	C₂²	1	1	yes	characteristic
17	C₇×#340	—	2	4	no	—
18	#10317	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₇×Q₈	—	2	1	no	—
20	C₂²×C₇×Q₈	—	2	2	no	—
21	C₂²×C₇×C₈	D₄	1	4	yes	—
22	C₂²×C₇×C₈	D₄	1	2	yes	—
23	C₇×C₄⋊Q₈	—	2	2	no	—
24	C₇×C₄⋊Q₈	—	2	2	no	—
25	C₇×C₄⋊Q₈	—	2	2	no	—
26	C₇×C₄⋊Q₈	—	2	2	no	—
27	C₇×C₄⋊Q₈	—	2	1	no	—
28	C₇×C₄⋊Q₈	D₄	1	2	yes	—
29	C₇×C₄⋊Q₈	D₄	1	4	yes	—
30	C₂×C₇×C₄⋊C₄	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	—	2	2	no	—
32	C₂×C₇×C₄⋊C₄	—	2	2	no	—
33	C₂×C₇×C₄⋊C₄	—	2	2	no	—
34	C₂×C₇×C₄⋊C₄	—	2	1	no	—
35	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
36	C₇×#106	—	2	8	no	—
37	C₇×#103	—	2	16	no	—
38	C₇×#103	—	4	4	no	—
39	C₄×C₇×C₈	—	2	2	no	—
40	C₄×C₇×C₈	—	2	2	no	—
41	C₄×C₇×C₈	—	2	2	no	—
42	C₂×#498	C₂×C₇	1	1	yes	characteristic
43	C₂×#487	C₂×C₇	1	2	yes	—
44	#1990	C₂×C₇	1	2	yes	—
45	#1956	C₂×C₇	1	1	yes	characteristic
46	#1273	C₂×C₇	1	1	yes	characteristic
47	C₂×C₇×Q₈	—	2	4	no	—
48	C₂×C₇×Q₈	—	2	2	no	—
49	C₂×C₇×Q₈	—	4	1	no	—
50	C₂×C₇×Q₈	—	4	2	no	—
51	C₂×C₇×Q₈	—	4	2	no	—
52	C₂×C₇×Q₈	—	4	4	no	—
53	C₂×C₇×Q₈	—	4	2	no	—
54	C₂×C₇×Q₈	—	4	4	no	—
55	C₂²×C₄×C₇	—	4	1	no	—
56	C₂²×C₄×C₇	—	4	2	no	—
57	C₂²×C₄×C₇	—	4	1	no	—
58	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
60	C₂×C₇×C₈	—	2	2	no	—
61	C₂×C₇×C₈	—	2	2	no	—
62	C₂×C₇×C₈	—	2	4	no	—
63	C₂×C₇×C₈	—	2	4	no	—
64	C₂×C₇×C₈	—	2	4	no	—
65	C₂×C₇×C₈	SD₁₆	1	4	yes	—
66	C₇×C₄⋊C₄	—	2	4	no	—
67	C₇×C₄⋊C₄	—	2	4	no	—
68	C₇×C₄⋊C₄	—	2	4	no	—
69	C₇×C₄⋊C₄	—	4	2	no	—
70	C₇×C₄⋊C₄	—	4	2	no	—
71	C₇×C₄⋊C₄	—	4	2	no	—
72	C₇×C₄⋊C₄	—	4	2	no	—
73	C₇×C₄⋊C₄	—	4	4	no	—
74	C₄²×C₇	C₂×D₄	1	2	yes	—
75	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇	#41	1	1	yes	characteristic
77	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
78	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
79	#498	—	2	4	no	—
80	#487	—	2	4	no	—
81	#487	—	2	4	no	—
82	C₂×#106	—	2	2	no	—
83	C₂×#103	—	2	4	no	—
84	C₂×#103	—	2	2	no	—
85	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
86	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
87	#384	C₂²×C₇	1	1	yes	characteristic
88	#340	—	2	4	no	—
89	C₂³×C₇	#127	1	1	yes	characteristic
90	C₇×Q₈	—	4	8	no	—
91	C₇×Q₈	—	4	4	no	—
92	C₇×Q₈	—	8	2	no	—
93	C₇×Q₈	—	8	4	no	—
94	C₂×C₄×C₇	—	4	2	no	—
95	C₂×C₄×C₇	—	4	4	no	—
96	C₂×C₄×C₇	—	8	1	no	—
97	C₂×C₄×C₇	—	8	2	no	—
98	C₂×C₄×C₇	—	8	1	no	—
99	C₂×C₄×C₇	—	8	2	no	—
100	C₂×C₄×C₇	—	8	1	no	—
101	C₂×C₄×C₇	—	8	2	no	—
102	C₂×C₄×C₇	#120	1	1	yes	characteristic
103	C₂×C₄×C₇	#120	1	2	yes	—
104	C₂×C₄×C₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₇	#121	1	2	yes	—
106	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
107	C₂×C₄×C₇	#135	1	4	yes	—
108	C₂×C₄×C₇	#137	1	2	yes	—
109	C₂×C₄×C₇	#137	1	4	yes	—
110	C₇×C₈	—	2	4	no	—
111	C₇×C₈	—	4	4	no	—
112	C₂²×Q₈	—	2	1	no	—
113	C₂²×Q₈	—	2	2	no	—
114	C₂²×C₈	C₇×D₄	1	4	yes	—
115	C₂²×C₈	C₇×D₄	1	2	yes	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	—	2	2	no	—
120	C₄⋊Q₈	—	2	1	no	—
121	C₄⋊Q₈	C₇×D₄	1	2	yes	—
122	C₄⋊Q₈	C₇×D₄	1	4	yes	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
129	#106	—	2	8	no	—
130	#103	—	2	16	no	—
131	#103	—	4	4	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₂²×C₇	#392	1	1	yes	characteristic
136	C₂²×C₇	#492	1	1	yes	characteristic
137	C₂²×C₇	#495	1	2	yes	—
138	C₂²×C₇	#497	1	2	yes	—
139	C₂²×C₇	#497	1	1	yes	characteristic
140	C₄×C₇	—	8	2	no	—
141	C₄×C₇	—	8	4	no	—
142	C₄×C₇	—	16	1	no	—
143	C₄×C₇	#448	1	2	yes	—
144	C₄×C₇	#452	1	4	yes	—
145	C₄×C₇	#461	1	2	yes	—
146	C₄×C₇	#464	1	4	yes	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	4	no	—
155	C₂²×C₄	—	4	1	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	—	4	1	no	—
158	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	C₇×SD₁₆	1	4	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄²	C₂×C₇×D₄	1	2	yes	—
175	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
176	C₄²	C₇×#41	1	1	yes	characteristic
177	C₂×C₇	#1708	1	1	yes	characteristic
178	C₂×C₇	#1716	1	1	yes	characteristic
179	C₂×C₇	#1718	1	2	yes	—
180	C₂×C₇	#2013	1	1	yes	characteristic
181	C₂×C₇	#2015	1	2	yes	—
182	C₂³	C₇×#127	1	1	yes	characteristic
183	Q₈	—	4	8	no	—
184	Q₈	—	4	4	no	—
185	Q₈	—	8	2	no	—
186	Q₈	—	8	4	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	8	1	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	1	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	C₇×#120	1	1	yes	characteristic
196	C₂×C₄	C₇×#120	1	2	yes	—
197	C₂×C₄	C₇×#121	1	1	yes	characteristic
198	C₂×C₄	C₇×#121	1	2	yes	—
199	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
200	C₂×C₄	C₇×#135	1	4	yes	—
201	C₂×C₄	C₇×#137	1	2	yes	—
202	C₂×C₄	C₇×#137	1	4	yes	—
203	C₈	—	2	4	no	—
204	C₈	—	4	4	no	—
205	C₇	#10317	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²	C₇×#392	1	1	yes	characteristic
207	C₂²	C₇×#492	1	1	yes	characteristic
208	C₂²	C₇×#495	1	2	yes	—
209	C₂²	C₇×#497	1	2	yes	—
210	C₂²	C₇×#497	1	1	yes	characteristic
211	C₄	—	8	2	no	—
212	C₄	—	8	4	no	—
213	C₄	—	16	1	no	—
214	C₄	C₇×#448	1	2	yes	—
215	C₄	C₇×#452	1	4	yes	—
216	C₄	C₇×#461	1	2	yes	—
217	C₄	C₇×#464	1	4	yes	—
218	C₂	C₇×#1708	1	1	yes	characteristic
219	C₂	C₇×#1716	1	1	yes	characteristic
220	C₂	C₇×#1718	1	2	yes	—
221	C₂	C₇×#2013	1	1	yes	characteristic
222	C₂	C₇×#2015	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×#498

C₂

yes

maximal, characteristic

C₂×C₇×#487

C₂

yes

maximal

C₇×#1990

C₂

yes

maximal

C₇×#1956

C₂

yes

maximal, characteristic

C₇×#1273

C₂

yes

maximal, characteristic

C₂×C₇×C₄⋊Q₈

C₂²

yes

—

C₂×C₇×C₄⋊Q₈

C₂²

yes

—

—

—

C₂×C₇×#106

—

C₂×C₇×#103

—

C₂×C₇×#103

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₂×C₄×C₇×C₈

C₂²

yes

—

C₇×#384

C₂²

yes

characteristic

C₇×#340

—

#10317

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₇×Q₈

—

C₂²×C₇×Q₈

—

C₂²×C₇×C₈

D₄

yes

—

C₂²×C₇×C₈

D₄

yes

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

D₄

yes

—

C₇×C₄⋊Q₈

D₄

yes

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

C₂³

yes

—

—

—

C₄×C₇×C₈

—

C₄×C₇×C₈

—

C₄×C₇×C₈

—

C₂×#498

C₂×C₇

yes

characteristic

C₂×#487

C₂×C₇

yes

—

#1990

C₂×C₇

yes

—

#1956

C₂×C₇

yes

characteristic

#1273

C₂×C₇

yes

characteristic

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

SD₁₆

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

#41

yes

characteristic

C₂×C₄⋊Q₈

C₂²×C₇

yes

characteristic

C₂×C₄⋊Q₈

C₂²×C₇

yes

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂²×C₇

yes

—

C₂×C₄×C₈

C₂²×C₇

yes

characteristic

#384

C₂²×C₇

yes

characteristic

#340

—

C₂³×C₇

#127

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

100

C₂×C₄×C₇

—

101

C₂×C₄×C₇

—

102

C₂×C₄×C₇

#120

yes

characteristic

103

C₂×C₄×C₇

#120

yes

—

104

C₂×C₄×C₇

#121

yes

characteristic

105

C₂×C₄×C₇

#121

yes

—

106

C₂×C₄×C₇

C₂×SD₁₆

yes

—

107

C₂×C₄×C₇

#135

yes

—

108

C₂×C₄×C₇

#137

yes

—

109

C₂×C₄×C₇

#137

yes

—

110

C₇×C₈

—

111

C₇×C₈

—

112

C₂²×Q₈

—

113

C₂²×Q₈

—

114

C₂²×C₈

C₇×D₄

yes

—

115

C₂²×C₈

C₇×D₄

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

C₇×D₄

yes

—

122

C₄⋊Q₈

C₇×D₄

yes

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂³×C₇

yes

characteristic

129

#106

—

130

#103

—

131

#103

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₂²×C₇

#392

yes

characteristic

136

C₂²×C₇

#492

yes

characteristic

137

C₂²×C₇

#495

yes

—

138

C₂²×C₇

#497

yes

—

139

C₂²×C₇

#497

yes

characteristic

140

C₄×C₇

—

141

C₄×C₇

—

142

C₄×C₇

—

143

C₄×C₇

#448

yes

—

144

C₄×C₇

#452

yes

—

145

C₄×C₇

#461

yes

—

146

C₄×C₇

#464

yes

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

C₂×C₇×D₄

yes

characteristic

159

C₂²×C₄

C₂×C₇×D₄

yes

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

C₇×SD₁₆

yes

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄²

C₂×C₇×D₄

yes

—

175

C₄²

C₂×C₇×D₄

yes

characteristic

176

C₄²

C₇×#41

yes

characteristic

177

C₂×C₇

#1708

yes

characteristic

178

C₂×C₇

#1716

yes

characteristic

179

C₂×C₇

#1718

yes

—

180

C₂×C₇

#2013

yes

characteristic

181

C₂×C₇

#2015

yes

—

182

C₂³

C₇×#127

yes

characteristic

183

Q₈

—

184

Q₈

—

185

Q₈

—

186

Q₈

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

C₇×#120

yes

characteristic

196

C₂×C₄

C₇×#120

yes

—

197

C₂×C₄

C₇×#121

yes

characteristic

198

C₂×C₄

C₇×#121

yes

—

199

C₂×C₄

C₂×C₇×SD₁₆

yes

—

200

C₂×C₄

C₇×#135

yes

—

201

C₂×C₄

C₇×#137

yes

—

202

C₂×C₄

C₇×#137

yes

—

203

C₈

—

204

C₈

—

205

C₇

#10317

yes

characteristic, direct product factor, Sylow

206

C₂²

C₇×#392

yes

characteristic

207

C₂²

C₇×#492

yes

characteristic

208

C₂²

C₇×#495

yes

—

209

C₂²

C₇×#497

yes

—

210

C₂²

C₇×#497

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

C₇×#448

yes

—

215

C₄

C₇×#452

yes

—

216

C₄

C₇×#461

yes

—

217

C₄

C₇×#464

yes

—

218

C₂

C₇×#1708

yes

characteristic

219

C₂

C₇×#1716

yes

characteristic

220

C₂

C₇×#1718

yes

—

221

C₂

C₇×#2013

yes

characteristic

222

C₂

C₇×#2015

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₇

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₇

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

N	H ≅ G/N	type
18. #10317	C₇	direct (N × H)
205. C₇	#10317	direct (N × H)

H ≅ G/N

type

18. #10317

C₇

direct (N × H)

205. C₇

#10317

direct (N × H)

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

subgroup	index
C₂×C₇×#498	2
C₂×C₇×#487	2
C₇×#1990	2
C₇×#1956	2
C₇×#1273	2
#10317	7

subgroup

index

C₂×C₇×#498

C₂×C₇×#487

Subgroups — C₇×#10317 #13187533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#10317 #13187533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10317 #13187533