Group #20243455 — subgroups

Subgroups — #20243455

Order 6016.

380 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 117 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×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₄₇)⋊C₈	C₂	1	2	yes	maximal
3	#15688878	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688877	C₂	1	2	yes	maximal, complemented
5	C₄₇×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₄×D₄₇	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×Dic₄₇	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₄₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₄²×D₄₇	—	2	4	no	—
14	(C₂×D₄₇)⋊C₈	—	2	4	no	—
15	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂³×D₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₄₇	—	2	2	no	—
25	C₂×C₄×D₄₇	—	2	2	no	—
26	C₂×C₄×D₄₇	—	2	2	no	—
27	C₂×C₄×D₄₇	—	2	2	no	—
28	C₂×C₄×D₄₇	—	2	2	no	—
29	C₂×C₄×D₄₇	—	2	1	no	—
30	C₂×C₄×D₄₇	—	2	1	no	—
31	C₂×C₄×D₄₇	—	2	2	no	—
32	C₂×C₄×D₄₇	C₈	1	4	yes	complemented
33	C₂×C₄×D₄₇	D₄	1	4	yes	—
34	C₂×C₈×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₄²×C₄₇	—	2	2	no	—
38	C₄×Dic₄₇	—	2	2	no	—
39	C₄×Dic₄₇	—	2	2	no	—
40	C₄×Dic₄₇	—	2	2	no	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂×C₄₇⋊C₈	—	2	2	no	—
43	C₂×C₄₇⋊C₈	—	2	2	no	—
44	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×D₄₇	—	2	1	no	—
46	C₂²×D₄₇	—	2	1	no	—
47	C₂²×D₄₇	—	2	1	no	—
48	C₂²×D₄₇	—	2	1	no	—
49	C₂²×D₄₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×D₄₇	C₂×C₈	1	2	yes	—
51	C₂²×D₄₇	M₁₆	1	2	yes	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
57	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
58	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₄₇	#41	1	2	yes	—
60	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂×Dic₄₇	—	2	1	no	—
63	C₂×Dic₄₇	—	2	1	no	—
64	C₂×Dic₄₇	—	2	1	no	—
65	C₂×Dic₄₇	—	2	1	no	—
66	C₂×Dic₄₇	—	2	1	no	—
67	C₂×Dic₄₇	—	2	1	no	—
68	C₂×Dic₄₇	C₂²⋊C₄	1	2	yes	—
69	C₂×Dic₄₇	C₂×C₈	1	2	yes	—
70	C₂×Dic₄₇	M₁₆	1	2	yes	—
71	C₄×D₄₇	—	2	8	no	—
72	C₄×D₄₇	—	2	8	no	—
73	C₄×D₄₇	—	2	4	no	—
74	C₄×D₄₇	—	2	4	no	—
75	C₈×C₄₇	—	4	2	no	—
76	C₄₇⋊C₈	—	4	2	no	—
77	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄₇	#117	1	1	yes	characteristic
79	C₂²×C₄₇	#117	1	1	yes	characteristic
80	C₂²×C₄₇	C₂²×C₈	1	1	yes	characteristic
81	C₂²×C₄₇	C₂×M₁₆	1	1	yes	characteristic
82	C₂²×C₄₇	C₂×M₁₆	1	1	yes	characteristic
83	C₂²×C₄₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂×D₄₇	—	2	2	no	—
85	C₂×D₄₇	—	2	2	no	—
86	C₂×D₄₇	—	2	2	no	—
87	C₂×D₄₇	—	2	2	no	—
88	C₂×D₄₇	—	2	2	no	—
89	C₂×D₄₇	—	2	2	no	—
90	C₂×D₄₇	C₂²⋊C₈	1	4	yes	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	#123	1	2	yes	—
94	C₄×C₄₇	#123	1	2	yes	—
95	Dic₄₇	—	2	2	no	—
96	Dic₄₇	—	2	2	no	—
97	Dic₄₇	—	2	2	no	—
98	#1817	—	47	1	no	maximal, Sylow
99	C₂×C₄₇	#381	1	1	yes	characteristic
100	C₂×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₄₇	#407	1	1	yes	characteristic
102	C₂×C₄₇	#431	1	1	yes	characteristic
103	C₂×C₄₇	#431	1	1	yes	characteristic
104	C₂×C₄₇	#432	1	1	yes	characteristic
105	C₂×C₄₇	#432	1	1	yes	characteristic
106	D₄₇	—	2	4	no	—
107	C₂²×C₄²	—	47	1	no	—
108	C₂×C₂²⋊C₈	—	47	2	no	—
109	#336	—	47	2	no	—
110	#336	—	47	1	no	—
111	#336	D₄₇	1	1	yes	characteristic, complemented
112	C₄₇	#1817	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₄	—	47	1	no	—
114	C₂³×C₄	—	47	1	no	—
115	C₂³×C₄	—	47	1	no	—
116	C₂²×C₈	—	47	2	no	—
117	C₂²×C₈	C₂×D₄₇	1	2	yes	—
118	C₂×C₄²	—	47	1	no	—
119	C₂×C₄²	—	47	1	no	—
120	C₂×C₄²	—	47	1	no	—
121	C₂×C₄²	—	94	4	no	—
122	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
123	C₂²⋊C₈	—	94	4	no	—
124	C₂⁴	—	47	1	no	—
125	C₂²×C₄	—	47	1	no	—
126	C₂²×C₄	—	47	1	no	—
127	C₂²×C₄	—	47	1	no	—
128	C₂²×C₄	—	47	4	no	—
129	C₂²×C₄	—	47	4	no	—
130	C₂²×C₄	—	94	2	no	—
131	C₂²×C₄	—	94	2	no	—
132	C₂²×C₄	—	94	2	no	—
133	C₂²×C₄	—	94	2	no	—
134	C₂²×C₄	—	94	2	no	—
135	C₂²×C₄	—	94	2	no	—
136	C₂²×C₄	—	94	1	no	—
137	C₂²×C₄	—	94	1	no	—
138	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
139	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	94	2	no	—
145	C₂×C₈	—	94	2	no	—
146	C₂×C₈	—	94	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	94	2	no	—
149	C₄²	—	94	2	no	—
150	C₄²	—	94	2	no	—
151	C₂³	—	47	2	no	—
152	C₂³	—	47	2	no	—
153	C₂³	—	47	2	no	—
154	C₂³	—	94	1	no	—
155	C₂³	—	94	1	no	—
156	C₂³	—	94	1	no	—
157	C₂³	—	94	1	no	—
158	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	47	2	no	—
164	C₂×C₄	—	47	2	no	—
165	C₂×C₄	—	47	2	no	—
166	C₂×C₄	—	94	8	no	—
167	C₂×C₄	—	94	8	no	—
168	C₂×C₄	—	94	1	no	—
169	C₂×C₄	—	94	1	no	—
170	C₂×C₄	—	94	1	no	—
171	C₂×C₄	—	94	1	no	—
172	C₂×C₄	—	94	1	no	—
173	C₂×C₄	—	94	1	no	—
174	C₂×C₄	—	94	4	no	—
175	C₂×C₄	—	94	4	no	—
176	C₂×C₄	C₈×D₄₇	1	2	yes	—
177	C₂×C₄	#10625324	1	2	yes	—
178	C₂×C₄	#10625350	1	2	yes	—
179	C₂×C₄	D₄×D₄₇	1	2	yes	—
180	C₂×C₄	#10625352	1	1	yes	characteristic
181	C₂×C₄	#10625354	1	1	yes	characteristic
182	C₈	—	4	2	no	—
183	C₈	—	188	2	no	—
184	C₂²	—	47	4	no	—
185	C₂²	—	94	2	no	—
186	C₂²	—	94	2	no	—
187	C₂²	—	94	2	no	—
188	C₂²	—	94	2	no	—
189	C₂²	—	94	2	no	—
190	C₂²	—	94	2	no	—
191	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
192	C₂²	#13093981	1	1	yes	characteristic
193	C₂²	#13093982	1	1	yes	characteristic
194	C₂²	C₂×C₈×D₄₇	1	1	yes	characteristic
195	C₂²	M₁₆×D₄₇	1	1	yes	characteristic
196	C₂²	M₁₆×D₄₇	1	1	yes	characteristic
197	C₂²	C₂×#10625324	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	94	2	no	—
201	C₄	—	94	2	no	—
202	C₄	—	94	2	no	—
203	C₄	#13094006	1	2	yes	—
204	C₄	#13094009	1	2	yes	—
205	C₂	—	94	4	no	—
206	C₂	#15688985	1	1	yes	characteristic
207	C₂	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
208	C₂	#15689085	1	1	yes	characteristic
209	C₂	#15689174	1	1	yes	characteristic
210	C₂	#15689175	1	1	yes	characteristic
211	C₂	#15689178	1	1	yes	characteristic
212	C₂	#15689179	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×D₄₇

C₂

yes

maximal, characteristic

C₂×(C₂×D₄₇)⋊C₈

C₂

yes

maximal

#15688878

C₂

yes

maximal, characteristic, complemented

#15688877

C₂

yes

maximal, complemented

C₄₇×#336

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₄

yes

characteristic

C₂²×C₄×D₄₇

C₄

yes

characteristic

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄×Dic₄₇

C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄²×D₄₇

—

(C₂×D₄₇)⋊C₈

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₈

yes

complemented

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂²⋊C₄

yes

—

C₂²×D₄₇

C₂×C₈

yes

—

C₂²×D₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²⋊C₄

yes

—

C₂×Dic₄₇

C₂×C₈

yes

—

C₂×Dic₄₇

M₁₆

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

C₂²×C₈

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂²⋊C₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#123

yes

—

C₄×C₄₇

#123

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

100

C₂×C₄₇

C₂×C₂²⋊C₈

yes

characteristic

101

C₂×C₄₇

#407

yes

characteristic

102

C₂×C₄₇

#431

yes

characteristic

103

C₂×C₄₇

#431

yes

characteristic

104

C₂×C₄₇

#432

yes

characteristic

105

C₂×C₄₇

#432

yes

characteristic

106

D₄₇

—

107

C₂²×C₄²

—

108

C₂×C₂²⋊C₈

—

109

#336

—

110

#336

—

111

#336

D₄₇

yes

characteristic, complemented

112

C₄₇

#1817

yes

characteristic, complemented, Sylow

113

C₂³×C₄

—

114

C₂³×C₄

—

115

C₂³×C₄

—

116

C₂²×C₈

—

117

C₂²×C₈

C₂×D₄₇

yes

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

C₂×C₄²

—

122

C₂×C₄²

C₂×D₄₇

yes

characteristic

123

C₂²⋊C₈

—

124

C₂⁴

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₄×D₄₇

yes

characteristic

139

C₂²×C₄

C₄×D₄₇

yes

characteristic

140

C₂²×C₄

C₂²×D₄₇

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

C₂×C₄×D₄₇

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

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

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

C₈×D₄₇

yes

—

177

C₂×C₄

#10625324

yes

—

178

C₂×C₄

#10625350

yes

—

179

C₂×C₄

D₄×D₄₇

yes

—

180

C₂×C₄

#10625352

yes

characteristic

181

C₂×C₄

#10625354

yes

characteristic

182

C₈

—

183

C₈

—

188

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

192

C₂²

#13093981

yes

characteristic

193

C₂²

#13093982

yes

characteristic

194

C₂²

C₂×C₈×D₄₇

yes

characteristic

195

C₂²

M₁₆×D₄₇

yes

characteristic

196

C₂²

M₁₆×D₄₇

yes

characteristic

197

C₂²

C₂×#10625324

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

#13094006

yes

—

204

C₄

#13094009

yes

—

205

C₂

—

206

C₂

#15688985

yes

characteristic

207

C₂

C₂²⋊C₈×D₄₇

yes

characteristic

208

C₂

#15689085

yes

characteristic

209

C₂

#15689174

yes

characteristic

210

C₂

#15689175

yes

characteristic

211

C₂

#15689178

yes

characteristic

212

C₂

#15689179

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₈
Fitting	C₄₇×#336	C₂
socle	C₂³×C₄₇	C₂²×C₄
hypercenter	#336	D₄₇
hypocenter	C₄₇	#1817
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂²×C₈

Fitting

C₄₇×#336

C₂

socle

C₂³×C₄₇

C₂²×C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
3. #15688878	C₂	semidirect
4. #15688877	C₂	semidirect
5. C₄₇×#336	C₂	semidirect
32. C₂×C₄×D₄₇	C₈	semidirect
111. #336	D₄₇	semidirect
112. C₄₇	#1817	semidirect

H ≅ G/N

type

3. #15688878

C₂

semidirect

4. #15688877

C₂

semidirect

5. C₄₇×#336

C₂

semidirect

32. C₂×C₄×D₄₇

C₈

semidirect

111. #336

D₄₇

semidirect

112. C₄₇

#1817

semidirect

Subgroups — #20243455

Conjugacy classes of subgroups

Complements