Group #72254212 — subgroups

Subgroups — #72254212

Order 4704.

434 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 91 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	#72254211	C₂	1	2	yes	maximal, complemented
2	C₇×#66627	C₂	1	2	yes	maximal, complemented
3	#72254195	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×C₄×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic
5	#13096284	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₇⋊D₇	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×C₇²	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₇⋊Dic₇	C₄	1	1	yes	characteristic
9	C₃×C₄×C₇⋊D₇	C₄	1	1	yes	characteristic
10	C₃×C₄×C₇⋊D₇	C₄	1	1	yes	characteristic
11	C₃×C₄×C₇⋊D₇	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₇⋊D₇	C₂²	1	1	yes	characteristic
13	C₇×C₃⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
14	C₂×C₄×C₇⋊D₇	D₃	1	1	yes	characteristic
15	C₇×#1136	—	3	2	no	—
16	#11716031	—	3	2	no	—
17	#11716029	—	3	2	no	—
18	D₇×#208	—	7	2	no	maximal
19	C₂²×C₃×C₇²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₇⋊D₇	—	2	2	no	—
21	C₂×C₃×C₇⋊D₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₇⋊D₇	C₂×C₄	1	1	yes	characteristic
23	C₃×C₄×C₇²	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄×C₇²	C₂³	1	1	yes	characteristic
25	C₃×C₇⋊Dic₇	C₂×C₄	1	1	yes	characteristic
26	C₃×C₇⋊Dic₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇⋊D₇	Dic₃	1	1	yes	characteristic
28	C₂×C₄×C₇²	C₂×D₃	1	1	yes	characteristic
29	C₂×C₇⋊Dic₇	Dic₃	1	1	yes	characteristic
30	C₄×C₇⋊D₇	Dic₃	1	1	yes	characteristic
31	C₄×C₇⋊D₇	Dic₃	1	1	yes	characteristic
32	C₄×C₇⋊D₇	C₂×D₃	1	1	yes	characteristic
33	C₄×C₇⋊D₇	C₂×D₃	1	1	yes	characteristic
34	C₇×C₇⋊C₈	—	3	4	no	—
35	C₂×C₃×C₄×D₇	—	7	2	no	—
36	C₂×C₃×C₄×D₇	—	14	3	no	—
37	#66627	D₇	1	2	yes	complemented
38	C₇×#208	—	7	2	no	—
39	#66557	—	7	4	no	—
40	D₇×C₃⋊C₈	—	7	4	no	—
41	C₂×C₃×C₇²	—	2	1	no	—
42	C₂×C₃×C₇²	C₂²×C₄	1	1	yes	characteristic
43	C₃×C₇⋊D₇	—	2	1	no	—
44	C₃×C₇⋊D₇	M₁₆	1	2	yes	complemented
45	D₇×M₁₆	—	21	2	no	—
46	C₂²×C₇²	C₂×Dic₃	1	1	yes	characteristic
47	C₂×C₇⋊D₇	—	2	2	no	—
48	C₂×C₇⋊D₇	C₂×Dic₃	1	1	yes	characteristic
49	C₂×C₇⋊D₇	C₂×Dic₃	1	1	yes	characteristic
50	C₄×C₇²	C₂×Dic₃	1	1	yes	characteristic
51	C₄×C₇²	C₂²×D₃	1	1	yes	characteristic
52	C₇⋊Dic₇	C₂×Dic₃	1	1	yes	characteristic
53	C₇⋊Dic₇	C₂×Dic₃	1	1	yes	characteristic
54	C₂²×C₃×D₇	—	7	2	no	—
55	C₂²×C₃×D₇	—	14	3	no	—
56	C₂×C₃×C₄×C₇	—	2	3	no	—
57	C₂×C₃×C₄×C₇	C₂×D₇	1	2	yes	—
58	C₂×C₃×Dic₇	—	7	2	no	—
59	C₂×C₃×Dic₇	—	14	3	no	—
60	C₃×C₄×D₇	—	7	2	no	—
61	C₃×C₄×D₇	—	7	2	no	—
62	C₃×C₄×D₇	—	7	2	no	—
63	C₃×C₄×D₇	—	7	2	no	—
64	C₃×C₄×D₇	—	14	3	no	—
65	C₃×C₄×D₇	—	14	3	no	—
66	C₃×C₄×D₇	—	14	3	no	—
67	C₃×C₄×D₇	—	14	3	no	—
68	C₃⋊(C₇⋊C₈)	C₂×D₇	1	4	yes	complemented
69	C₇×C₃⋊C₈	—	7	4	no	—
70	C₃×C₇²	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
71	C₂×C₄×D₇	—	7	2	no	—
72	C₂×C₄×D₇	—	14	3	no	—
73	C₇×M₁₆	—	21	2	no	—
74	#1136	—	3	2	no	—
75	#1131	—	21	4	no	—
76	C₈×D₇	—	21	4	no	—
77	C₂×C₇²	—	2	1	no	—
78	C₂×C₇²	C₂²×Dic₃	1	1	yes	characteristic
79	C₇⋊D₇	—	2	1	no	—
80	C₇⋊D₇	#208	1	2	yes	complemented
81	C₂×#208	—	49	1	no	Hall
82	C₂²×C₃×C₇	—	2	3	no	—
83	C₂²×C₃×C₇	C₄×D₇	1	2	yes	—
84	C₂×C₃×D₇	—	7	2	no	—
85	C₂×C₃×D₇	—	7	2	no	—
86	C₂×C₃×D₇	—	14	3	no	—
87	C₂×C₃×D₇	—	14	3	no	—
88	C₂×C₃×D₇	—	14	4	no	—
89	C₂×C₃×D₇	—	14	12	no	—
90	C₃×C₄×C₇	—	2	3	no	—
91	C₃×C₄×C₇	—	2	3	no	—
92	C₃×C₄×C₇	C₄×D₇	1	2	yes	—
93	C₃×C₄×C₇	C₂²×D₇	1	2	yes	—
94	C₃×Dic₇	—	7	2	no	—
95	C₃×Dic₇	—	7	2	no	—
96	C₃×Dic₇	—	14	3	no	—
97	C₃×Dic₇	—	14	3	no	—
98	C₂²×D₇	—	7	2	no	—
99	C₂²×D₇	—	14	3	no	—
100	C₂×C₄×C₇	—	2	3	no	—
101	C₂×C₄×C₇	D₃×D₇	1	2	yes	—
102	C₂×Dic₇	—	7	2	no	—
103	C₂×Dic₇	—	14	3	no	—
104	C₄×D₇	—	7	2	no	—
105	C₄×D₇	—	7	2	no	—
106	C₄×D₇	—	7	2	no	—
107	C₄×D₇	—	7	2	no	—
108	C₄×D₇	—	14	3	no	—
109	C₄×D₇	—	14	3	no	—
110	C₄×D₇	—	14	3	no	—
111	C₄×D₇	—	14	3	no	—
112	C₇×C₈	—	21	4	no	—
113	C₇⋊C₈	—	3	4	no	—
114	C₇²	C₂×#208	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₃×C₄	—	49	1	no	—
116	#208	—	7	2	no	—
117	#208	—	49	2	no	—
118	C₂×C₃⋊C₈	—	49	2	no	—
119	C₂×C₃×C₇	—	2	6	no	—
120	C₂×C₃×C₇	—	2	2	no	—
121	C₂×C₃×C₇	—	2	3	no	—
122	C₂×C₃×C₇	C₂×C₄×D₇	1	2	yes	—
123	C₃×D₇	—	7	4	no	—
124	C₃×D₇	—	14	6	no	—
125	C₃×D₇	—	14	6	no	—
126	C₃×D₇	—	14	2	no	—
127	C₂×M₁₆	—	147	1	no	Sylow
128	C₂²×C₇	—	2	3	no	—
129	C₂²×C₇	Dic₃×D₇	1	2	yes	—
130	C₂×D₇	—	7	2	no	—
131	C₂×D₇	—	7	2	no	—
132	C₂×D₇	—	14	3	no	—
133	C₂×D₇	—	14	3	no	—
134	C₂×D₇	—	14	4	no	—
135	C₂×D₇	—	14	12	no	—
136	C₄×C₇	—	2	3	no	—
137	C₄×C₇	—	2	3	no	—
138	C₄×C₇	Dic₃×D₇	1	2	yes	—
139	C₄×C₇	C₂×D₃×D₇	1	2	yes	—
140	Dic₇	—	7	2	no	—
141	Dic₇	—	7	2	no	—
142	Dic₇	—	14	3	no	—
143	Dic₇	—	14	3	no	—
144	C₂³×C₃	—	49	1	no	—
145	C₂×C₃×C₄	—	49	1	no	—
146	C₂×C₃×C₄	—	49	1	no	—
147	C₂×C₃×C₄	—	49	1	no	—
148	C₂×C₃×C₄	—	49	1	no	—
149	C₂×C₃×C₄	—	49	1	no	—
150	C₂×C₃×C₄	D₇²	1	1	yes	characteristic
151	C₃⋊C₈	—	7	4	no	—
152	C₃×C₇	—	2	3	no	—
153	C₃×C₇	D₇×M₁₆	1	2	yes	complemented
154	C₂²×C₄	—	49	1	no	—
155	M₁₆	—	21	2	no	—
156	M₁₆	—	147	2	no	—
157	C₂×C₈	—	147	2	no	—
158	C₂×C₇	—	2	6	no	—
159	C₂×C₇	—	2	2	no	—
160	C₂×C₇	—	2	3	no	—
161	C₂×C₇	C₂×Dic₃×D₇	1	2	yes	—
162	D₇	—	7	4	no	—
163	D₇	—	14	6	no	—
164	D₇	—	14	6	no	—
165	D₇	—	14	2	no	—
166	C₂²×C₃	—	49	1	no	—
167	C₂²×C₃	—	49	1	no	—
168	C₂²×C₃	—	98	2	no	—
169	C₂²×C₃	#87639	1	1	yes	characteristic
170	C₃×C₄	—	49	1	no	—
171	C₃×C₄	—	49	1	no	—
172	C₃×C₄	#87639	1	1	yes	characteristic
173	C₃×C₄	C₂×D₇²	1	1	yes	characteristic
174	C₂³	—	49	1	no	—
175	C₂×C₄	—	49	1	no	—
176	C₂×C₄	—	49	1	no	—
177	C₂×C₄	—	49	1	no	—
178	C₂×C₄	—	49	1	no	—
179	C₂×C₄	—	49	1	no	—
180	C₂×C₄	D₂₁⋊D₇	1	1	yes	characteristic
181	C₈	—	21	4	no	—
182	C₇	—	2	3	no	—
183	C₇	D₇×#208	1	2	yes	complemented
184	C₂×C₃	—	2	1	no	—
185	C₂×C₃	—	49	2	no	—
186	C₂×C₃	—	98	1	no	—
187	C₂×C₃	C₂×#87639	1	1	yes	characteristic
188	C₂²	—	49	1	no	—
189	C₂²	—	49	1	no	—
190	C₂²	—	98	2	no	—
191	C₂²	#11924252	1	1	yes	characteristic
192	C₄	—	49	1	no	—
193	C₄	—	49	1	no	—
194	C₄	#11924252	1	1	yes	characteristic
195	C₄	C₂×D₂₁⋊D₇	1	1	yes	characteristic
196	C₃	#13096284	1	1	yes	characteristic, complemented, Sylow
197	C₂	—	2	1	no	—
198	C₂	—	49	2	no	—
199	C₂	—	98	1	no	—
200	C₂	C₂×#11924252	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#72254211

C₂

yes

maximal, complemented

C₇×#66627

C₂

yes

maximal, complemented

#72254195

C₂

yes

maximal, complemented

C₂×C₃×C₄×C₇⋊D₇

C₂

yes

maximal, characteristic

#13096284

—

maximal, Hall

C₂²×C₃×C₇⋊D₇

C₄

yes

characteristic

C₂×C₃×C₄×C₇²

C₂²

yes

characteristic

C₂×C₃×C₇⋊Dic₇

C₄

yes

characteristic

C₃×C₄×C₇⋊D₇

C₄

yes

characteristic

C₃×C₄×C₇⋊D₇

C₄

yes

characteristic

C₃×C₄×C₇⋊D₇

C₂²

yes

characteristic

C₃×C₄×C₇⋊D₇

C₂²

yes

characteristic

C₇×C₃⋊(C₇⋊C₈)

C₂²

yes

complemented

C₂×C₄×C₇⋊D₇

D₃

yes

—

—

—

—

C₂²×C₃×C₇²

C₂×C₄

yes

characteristic

C₂×C₃×C₇⋊D₇

—

C₂×C₃×C₇⋊D₇

C₂×C₄

yes

characteristic

C₂×C₃×C₇⋊D₇

C₂×C₄

yes

characteristic

C₃×C₄×C₇²

C₂×C₄

yes

characteristic

C₃×C₄×C₇²

C₂³

yes

characteristic

C₃×C₇⋊Dic₇

C₂×C₄

yes

characteristic

C₃×C₇⋊Dic₇

C₂×C₄

yes

characteristic

C₂²×C₇⋊D₇

Dic₃

yes

characteristic

C₂×C₄×C₇²

C₂×D₃

yes

characteristic

C₂×C₇⋊Dic₇

Dic₃

yes

characteristic

C₄×C₇⋊D₇

Dic₃

yes

characteristic

C₄×C₇⋊D₇

Dic₃

yes

characteristic

C₄×C₇⋊D₇

C₂×D₃

yes

characteristic

C₄×C₇⋊D₇

C₂×D₃

yes

characteristic

C₇×C₇⋊C₈

—

C₂×C₃×C₄×D₇

—

C₂×C₃×C₄×D₇

—

#66627

D₇

yes

complemented

C₇×#208

—

#66557

—

D₇×C₃⋊C₈

—

C₂×C₃×C₇²

—

C₂×C₃×C₇²

C₂²×C₄

yes

characteristic

C₃×C₇⋊D₇

—

C₃×C₇⋊D₇

M₁₆

yes

complemented

D₇×M₁₆

—

C₂²×C₇²

C₂×Dic₃

yes

characteristic

C₂×C₇⋊D₇

—

C₂×C₇⋊D₇

C₂×Dic₃

yes

characteristic

C₂×C₇⋊D₇

C₂×Dic₃

yes

characteristic

C₄×C₇²

C₂×Dic₃

yes

characteristic

C₄×C₇²

C₂²×D₃

yes

characteristic

C₇⋊Dic₇

C₂×Dic₃

yes

characteristic

C₇⋊Dic₇

C₂×Dic₃

yes

characteristic

C₂²×C₃×D₇

—

C₂²×C₃×D₇

—

C₂×C₃×C₄×C₇

—

C₂×C₃×C₄×C₇

C₂×D₇

yes

—

C₂×C₃×Dic₇

—

C₂×C₃×Dic₇

—

C₃×C₄×D₇

—

C₃×C₄×D₇

—

C₃×C₄×D₇

—

C₃×C₄×D₇

—

C₃×C₄×D₇

—

C₃×C₄×D₇

—

C₃×C₄×D₇

—

C₃×C₄×D₇

—

C₃⋊(C₇⋊C₈)

C₂×D₇

yes

complemented

C₇×C₃⋊C₈

—

C₃×C₇²

C₂×M₁₆

yes

characteristic, complemented, Hall

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₇×M₁₆

—

#1136

—

#1131

—

C₈×D₇

—

C₂×C₇²

—

C₂×C₇²

C₂²×Dic₃

yes

characteristic

C₇⋊D₇

—

C₇⋊D₇

yes

—

C₂²×C₃×C₇

—

C₂²×C₃×C₇

C₄×D₇

yes

—

C₂×C₃×D₇

—

C₂×C₃×D₇

—

C₂×C₃×D₇

—

C₂×C₃×D₇

—

C₂×C₃×D₇

—

C₂×C₃×D₇

—

C₃×C₄×C₇

—

C₃×C₄×C₇

—

C₃×C₄×C₇

C₄×D₇

yes

—

C₃×C₄×C₇

C₂²×D₇

yes

—

C₃×Dic₇

—

C₃×Dic₇

—

C₃×Dic₇

—

C₃×Dic₇

—

C₂²×D₇

—

C₂²×D₇

—

100

C₂×C₄×C₇

—

101

C₂×C₄×C₇

D₃×D₇

yes

—

102

C₂×Dic₇

—

103

C₂×Dic₇

—

104

C₄×D₇

—

105

C₄×D₇

—

106

C₄×D₇

—

107

C₄×D₇

—

108

C₄×D₇

—

109

C₄×D₇

—

110

C₄×D₇

—

111

C₄×D₇

—

112

C₇×C₈

—

113

C₇⋊C₈

—

114

C₇²

C₂×#208

yes

characteristic, complemented, Sylow

115

C₂²×C₃×C₄

—

116

#208

—

117

#208

—

118

C₂×C₃⋊C₈

—

119

C₂×C₃×C₇

—

120

C₂×C₃×C₇

—

121

C₂×C₃×C₇

—

122

C₂×C₃×C₇

C₂×C₄×D₇

yes

—

123

C₃×D₇

—

124

C₃×D₇

—

125

C₃×D₇

—

126

C₃×D₇

—

127

C₂×M₁₆

—

147

Sylow

128

C₂²×C₇

—

129

C₂²×C₇

Dic₃×D₇

yes

—

130

C₂×D₇

—

131

C₂×D₇

—

132

C₂×D₇

—

133

C₂×D₇

—

134

C₂×D₇

—

135

C₂×D₇

—

136

C₄×C₇

—

137

C₄×C₇

—

138

C₄×C₇

Dic₃×D₇

yes

—

139

C₄×C₇

C₂×D₃×D₇

yes

—

140

Dic₇

—

141

Dic₇

—

142

Dic₇

—

143

Dic₇

—

144

C₂³×C₃

—

145

C₂×C₃×C₄

—

146

C₂×C₃×C₄

—

147

C₂×C₃×C₄

—

148

C₂×C₃×C₄

—

149

C₂×C₃×C₄

—

150

C₂×C₃×C₄

D₇²

yes

characteristic

151

C₃⋊C₈

—

152

C₃×C₇

—

153

C₃×C₇

D₇×M₁₆

yes

complemented

154

C₂²×C₄

—

155

M₁₆

—

156

M₁₆

—

147

—

157

C₂×C₈

—

147

—

158

C₂×C₇

—

159

C₂×C₇

—

160

C₂×C₇

—

161

C₂×C₇

C₂×Dic₃×D₇

yes

—

162

D₇

—

163

D₇

—

164

D₇

—

165

D₇

—

166

C₂²×C₃

—

167

C₂²×C₃

—

168

C₂²×C₃

—

169

C₂²×C₃

#87639

yes

characteristic

170

C₃×C₄

—

171

C₃×C₄

—

172

C₃×C₄

#87639

yes

characteristic

173

C₃×C₄

C₂×D₇²

yes

characteristic

174

C₂³

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

D₂₁⋊D₇

yes

characteristic

181

C₈

—

182

C₇

—

183

C₇

D₇×#208

yes

complemented

184

C₂×C₃

—

185

C₂×C₃

—

186

C₂×C₃

—

187

C₂×C₃

C₂×#87639

yes

characteristic

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

#11924252

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

#11924252

yes

characteristic

195

C₄

C₂×D₂₁⋊D₇

yes

characteristic

196

C₃

#13096284

yes

characteristic, complemented, Sylow

197

C₂

—

198

C₂

—

199

C₂

—

200

C₂

C₂×#11924252

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×D₂₁⋊D₇
commutator, socle	C₂×C₃×C₇²	C₂²×C₄
Fitting	C₂×C₃×C₄×C₇²	C₂²
prime radical	C₂²×C₃×C₇⋊D₇	C₄
cosocle	C₃×C₄×C₇²	C₂³
hypercenter	C₂×C₄	D₂₁⋊D₇
hypocenter	C₃×C₇²	C₂×M₁₆
LCS	G ▷ C₂×C₃×C₇² ▷ C₃×C₇²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

subgroup

G/H

center, Frattini

C₄

C₂×D₂₁⋊D₇

commutator, socle

C₂×C₃×C₇²

C₂²×C₄

Fitting

C₂×C₃×C₄×C₇²

C₂²

prime radical

C₂²×C₃×C₇⋊D₇

C₄

cosocle

C₃×C₄×C₇²

C₂³

hypercenter

C₂×C₄

D₂₁⋊D₇

hypocenter

C₃×C₇²

C₂×M₁₆

LCS

G ▷ C₂×C₃×C₇² ▷ C₃×C₇²

UCS

C₁ ◁ C₄ ◁ C₂×C₄

N	H ≅ G/N	type
1. #72254211	C₂	semidirect
2. C₇×#66627	C₂	semidirect
3. #72254195	C₂	semidirect
13. C₇×C₃⋊(C₇⋊C₈)	C₂²	semidirect
37. #66627	D₇	semidirect
44. C₃×C₇⋊D₇	M₁₆	semidirect
68. C₃⋊(C₇⋊C₈)	C₂×D₇	semidirect
70. C₃×C₇²	C₂×M₁₆	semidirect
80. C₇⋊D₇	#208	semidirect
114. C₇²	C₂×#208	semidirect
153. C₃×C₇	D₇×M₁₆	semidirect
183. C₇	D₇×#208	semidirect
196. C₃	#13096284	semidirect

H ≅ G/N

type

1. #72254211

C₂

semidirect

2. C₇×#66627

C₂

semidirect

3. #72254195

C₂

semidirect

13. C₇×C₃⋊(C₇⋊C₈)

C₂²

semidirect

37. #66627

D₇

semidirect

44. C₃×C₇⋊D₇

M₁₆

semidirect

68. C₃⋊(C₇⋊C₈)

C₂×D₇

semidirect

70. C₃×C₇²

C₂×M₁₆

semidirect

80. C₇⋊D₇

#208

semidirect

114. C₇²

C₂×#208

semidirect

153. C₃×C₇

D₇×M₁₆

semidirect

183. C₇

D₇×#208

semidirect

196. C₃

#13096284

semidirect

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

subgroup	index
#72254211	2
C₇×#66627	2
#72254195	2
C₂×C₃×C₄×C₇⋊D₇	2
#13096284	3
D₇×#208	7

subgroup

index

C₂×C₃×C₄×C₇⋊D₇

Subgroups — #72254212

Conjugacy classes of subgroups

Complements

Maximal subgroups