Group #20246259 — subgroups

Subgroups — #20246259

Order 6016.

604 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 213 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×#13093980	C₂	1	2	yes	maximal
2	C₂²×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15689019	C₂	1	2	yes	maximal, complemented
4	#15689006	C₂	1	4	yes	maximal, complemented
5	#15688984	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₄₇	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#380	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₄₇	C₂²	1	2	yes	—
10	#13093980	—	2	8	no	—
11	C₂×C₄⋊Dic₄₇	C₄	1	8	yes	complemented
12	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
16	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
19	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
20	C₄₇⋊#95	C₂²	1	2	yes	—
21	C₄₇⋊#95	C₂²	1	4	yes	—
22	C₄₇⋊#95	C₂²	1	4	yes	—
23	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
26	#95×C₄₇	C₂²	1	2	yes	—
27	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	—	2	4	no	—
29	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₄₇	—	2	4	no	—
34	C₂²×Dic₄₇	—	2	4	no	—
35	C₂²×Dic₄₇	C₂×C₄	1	8	yes	—
36	C₂²×Dic₄₇	C₂³	1	2	yes	—
37	C₂²×Dic₄₇	C₂³	1	2	yes	—
38	C₂²×Dic₄₇	C₂³	1	2	yes	—
39	C₂²×Dic₄₇	C₂³	1	2	yes	—
40	C₄⋊C₄×C₄₇	—	2	4	no	—
41	C₂²⋊C₄×C₄₇	—	2	4	no	—
42	C₂²⋊Dic₄₇	—	2	2	no	—
43	C₂²⋊Dic₄₇	D₄	1	4	yes	—
44	C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
45	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
46	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
47	Dic₄₇⋊C₄	—	2	8	no	—
48	C₄×Dic₄₇	—	2	4	no	—
49	C₄×Dic₄₇	—	2	8	no	—
50	C₂³×C₄₇	—	2	2	no	—
51	C₂³×C₄₇	—	2	2	no	—
52	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂³×C₄₇	C₂×D₄	1	2	yes	—
54	C₂³×C₄₇	#41	1	2	yes	—
55	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	8	no	—
58	C₂×C₄×C₄₇	—	2	4	no	—
59	C₂×C₄×C₄₇	—	2	4	no	—
60	C₂×C₄×C₄₇	—	2	4	no	—
61	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₄₇	#41	1	4	yes	—
64	C₂×Dic₄₇	—	2	16	no	—
65	C₂×Dic₄₇	—	2	2	no	—
66	C₂×Dic₄₇	—	2	2	no	—
67	C₂×Dic₄₇	—	2	2	no	—
68	C₂×Dic₄₇	—	2	2	no	—
69	C₂×Dic₄₇	—	2	2	no	—
70	C₂×Dic₄₇	—	2	2	no	—
71	C₂×Dic₄₇	—	2	2	no	—
72	C₂×Dic₄₇	—	2	2	no	—
73	C₂×Dic₄₇	—	2	2	no	—
74	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
75	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
76	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
77	C₂²×C₄₇	—	2	2	no	—
78	C₂²×C₄₇	—	2	2	no	—
79	C₂²×C₄₇	—	2	4	no	—
80	C₂²×C₄₇	—	2	4	no	—
81	C₂²×C₄₇	C₄×D₄	1	4	yes	—
82	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
83	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₄₇	C₂×#41	1	2	yes	—
85	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
86	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
87	C₄×C₄₇	—	2	2	no	—
88	C₄×C₄₇	—	4	4	no	—
89	C₄×C₄₇	#117	1	4	yes	complemented
90	Dic₄₇	—	2	4	no	—
91	Dic₄₇	—	2	4	no	—
92	Dic₄₇	—	2	4	no	—
93	Dic₄₇	—	4	2	no	—
94	#2344	—	47	1	no	maximal, Sylow
95	C₂×C₄₇	—	2	4	no	—
96	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
97	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
98	C₂×C₄₇	#519	1	1	yes	characteristic
99	C₂×C₄₇	#547	1	2	yes	—
100	C₂×C₄₇	#552	1	2	yes	—
101	C₂×#117	—	47	2	no	—
102	C₂²×C₄⋊C₄	—	47	1	no	—
103	#389	—	47	2	no	—
104	#387	—	47	4	no	—
105	#381	—	47	2	no	—
106	#380	D₄₇	1	1	yes	characteristic
107	C₄×C₄⋊C₄	—	47	2	no	—
108	C₄×C₂²⋊C₄	—	47	1	no	—
109	C₄₇	#2344	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₄	—	47	2	no	—
111	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
112	#117	—	94	8	no	—
113	C₂×C₄⋊C₄	—	47	8	no	—
114	C₂×C₄⋊C₄	—	47	1	no	—
115	C₂×C₄⋊C₄	—	47	2	no	—
116	C₂×C₄⋊C₄	—	47	1	no	—
117	C₂×C₄⋊C₄	—	47	4	no	—
118	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
119	C₂×C₂²⋊C₄	—	47	1	no	—
120	C₂×C₂²⋊C₄	—	47	1	no	—
121	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
122	C₂×C₄²	—	47	4	no	—
123	C₂×C₄²	—	47	2	no	—
124	#95	—	47	2	no	—
125	#95	—	47	4	no	—
126	#95	—	47	4	no	—
127	#95	C₂×D₄₇	1	2	yes	—
128	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	47	8	no	—
131	C₂²×C₄	—	47	2	no	—
132	C₂²×C₄	—	47	2	no	—
133	C₂²×C₄	—	47	2	no	—
134	C₂²×C₄	—	47	2	no	—
135	C₂²×C₄	—	94	4	no	—
136	C₂²×C₄	—	94	4	no	—
137	C₂²×C₄	C₄×D₄₇	1	4	yes	—
138	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
139	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	47	8	no	—
143	C₄⋊C₄	—	47	4	no	—
144	C₄⋊C₄	—	47	4	no	—
145	C₄⋊C₄	—	94	8	no	—
146	C₂²⋊C₄	—	2	4	no	—
147	C₂²⋊C₄	—	47	4	no	—
148	C₂²⋊C₄	—	94	2	no	—
149	C₄²	—	94	8	no	—
150	C₄²	—	94	4	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	2	2	no	—
153	C₂³	C₂×C₄×D₄₇	1	2	yes	—
154	C₂³	D₄×D₄₇	1	2	yes	—
155	C₂³	#10625354	1	2	yes	—
156	C₂³	C₂³×D₄₇	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	47	4	no	—
163	C₂×C₄	—	47	4	no	—
164	C₂×C₄	—	47	4	no	—
165	C₂×C₄	—	94	16	no	—
166	C₂×C₄	—	94	2	no	—
167	C₂×C₄	—	94	2	no	—
168	C₂×C₄	—	94	2	no	—
169	C₂×C₄	—	94	2	no	—
170	C₂×C₄	—	94	2	no	—
171	C₂×C₄	—	94	2	no	—
172	C₂×C₄	—	94	2	no	—
173	C₂×C₄	—	94	2	no	—
174	C₂×C₄	—	94	2	no	—
175	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
176	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
177	C₂×C₄	#10625352	1	4	yes	—
178	C₂²	—	2	4	no	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	#13093986	1	4	yes	—
183	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
184	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
185	C₂²	C₂×#10625352	1	2	yes	—
186	C₂²	C₂×#10625354	1	1	yes	characteristic
187	C₂²	#13094071	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	4	4	no	—
190	C₄	—	94	4	no	—
191	C₄	—	94	4	no	—
192	C₄	—	94	4	no	—
193	C₄	—	188	2	no	—
194	C₄	#13093980	1	4	yes	complemented
195	C₂	—	2	4	no	—
196	C₂	C₂×#13093980	1	1	yes	characteristic
197	C₂	C₂×#13093986	1	1	yes	characteristic
198	C₂	#15689600	1	1	yes	characteristic
199	C₂	#15689773	1	2	yes	—
200	C₂	#15689815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093980

C₂

yes

maximal

C₂²×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic

#15689019

C₂

yes

maximal, complemented

#15689006

C₂

yes

maximal, complemented

#15688984

C₂

yes

maximal

C₄⋊C₄×Dic₄₇

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₄₇

C₂

yes

maximal, characteristic

C₄₇×#380

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₂²

yes

—

#13093980

—

C₂×C₄⋊Dic₄₇

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₂²⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

#95×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

D₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

#41

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄×D₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#117

yes

complemented

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2344

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#117

yes

characteristic

C₂×C₄₇

C₂×C₄×D₄

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

100

C₂×C₄₇

#552

yes

—

101

C₂×#117

—

102

C₂²×C₄⋊C₄

—

103

#389

—

104

#387

—

105

#381

—

106

#380

D₄₇

yes

characteristic

107

C₄×C₄⋊C₄

—

108

C₄×C₂²⋊C₄

—

109

C₄₇

#2344

yes

characteristic, complemented, Sylow

110

C₂³×C₄

—

111

C₂³×C₄

C₂×D₄₇

yes

characteristic

112

#117

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

119

C₂×C₂²⋊C₄

—

120

C₂×C₂²⋊C₄

—

121

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

122

C₂×C₄²

—

123

C₂×C₄²

—

124

#95

—

125

#95

—

126

#95

—

127

#95

C₂×D₄₇

yes

—

128

C₂⁴

C₂²×D₄₇

yes

characteristic

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

C₄×D₄₇

yes

—

138

C₂²×C₄

C₂²×D₄₇

yes

—

139

C₂²×C₄

C₂²×D₄₇

yes

characteristic

140

C₂²×C₄

C₂²×D₄₇

yes

characteristic

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₂²⋊C₄

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₂³

—

152

C₂³

—

153

C₂³

C₂×C₄×D₄₇

yes

—

154

C₂³

D₄×D₄₇

yes

—

155

C₂³

#10625354

yes

—

156

C₂³

C₂³×D₄₇

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

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

C₂×C₄×D₄₇

yes

—

176

C₂×C₄

C₂×C₄×D₄₇

yes

—

177

C₂×C₄

#10625352

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

#13093986

yes

—

183

C₂²

C₂²×C₄×D₄₇

yes

characteristic

184

C₂²

C₂×D₄×D₄₇

yes

characteristic

185

C₂²

C₂×#10625352

yes

—

186

C₂²

C₂×#10625354

yes

characteristic

187

C₂²

#13094071

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

188

—

194

C₄

#13093980

yes

complemented

195

C₂

—

196

C₂

C₂×#13093980

yes

characteristic

197

C₂

C₂×#13093986

yes

characteristic

198

C₂

#15689600

yes

characteristic

199

C₂

#15689773

yes

—

200

C₂

#15689815

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#380	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	#380	D₄₇
hypocenter	C₄₇	#2344
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #380

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#380

C₂

socle

C₂³×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #380

N	H ≅ G/N	type
3. #15689019	C₂	semidirect
4. #15689006	C₂	semidirect
6. C₄⋊C₄×Dic₄₇	C₂	semidirect
11. C₂×C₄⋊Dic₄₇	C₄	semidirect
89. C₄×C₄₇	#117	semidirect
109. C₄₇	#2344	semidirect
194. C₄	#13093980	semidirect

H ≅ G/N

type

3. #15689019

C₂

semidirect

4. #15689006

C₂

semidirect

6. C₄⋊C₄×Dic₄₇

C₂

semidirect

11. C₂×C₄⋊Dic₄₇

C₄

semidirect

89. C₄×C₄₇

#117

semidirect

109. C₄₇

#2344

semidirect

194. C₄

#13093980

semidirect

Subgroups — #20246259

Conjugacy classes of subgroups

Complements