Group #20251722 — subgroups

Subgroups — #20251722

Order 6016.

564 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 177 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	C₂²×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#15689030	C₂	1	2	yes	maximal, complemented
3	#15689019	C₂	1	2	yes	maximal, complemented
4	#15689004	C₂	1	4	yes	maximal, complemented
5	#15688995	C₂	1	2	yes	maximal
6	#15688984	C₂	1	2	yes	maximal
7	#15688983	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂²⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
13	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₄×Dic₄₇	—	2	4	no	—
15	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
17	C₄₇⋊#95	C₂²	1	2	yes	—
18	C₄₇⋊#95	C₂²	1	4	yes	—
19	C₄₇⋊#95	C₂²	1	4	yes	—
20	C₄₇⋊#95	C₂²	1	4	yes	—
21	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²⋊Q₈×C₄₇	—	2	4	no	—
24	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
27	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×Q₈×C₄₇	—	2	2	no	—
29	C₂×Q₈×C₄₇	—	2	4	no	—
30	C₂²×C₄×C₄₇	—	2	4	no	—
31	C₂²×C₄×C₄₇	D₄	1	4	yes	—
32	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×Dic₄₇	—	2	2	no	—
37	C₂²×Dic₄₇	—	2	2	no	—
38	C₂²×Dic₄₇	—	2	2	no	—
39	C₂²×Dic₄₇	—	2	4	no	—
40	C₂²×Dic₄₇	Q₈	1	4	yes	complemented
41	C₂²×Dic₄₇	C₂³	1	4	yes	—
42	C₂²×Dic₄₇	C₂³	1	2	yes	—
43	C₂²×Dic₄₇	C₂³	1	2	yes	—
44	C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₂²⋊C₄×C₄₇	—	2	4	no	—
47	C₂²⋊Dic₄₇	—	2	4	no	—
48	C₄⋊Dic₄₇	—	2	4	no	—
49	Dic₄₇⋊C₄	—	2	8	no	—
50	C₄×Dic₄₇	—	2	4	no	—
51	C₄×Dic₄₇	—	2	4	no	—
52	C₂³×C₄₇	—	2	2	no	—
53	C₂³×C₄₇	—	2	2	no	—
54	C₂³×C₄₇	C₂×D₄	1	2	yes	—
55	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
56	C₂³×C₄₇	#41	1	2	yes	—
57	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
58	Q₈×C₄₇	—	4	4	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	8	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	4	no	—
64	C₂×C₄×C₄₇	—	2	4	no	—
65	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₄₇	#41	1	4	yes	—
68	C₂×Dic₄₇	—	2	8	no	—
69	C₂×Dic₄₇	—	2	8	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	—	2	4	no	—
75	C₂×Dic₄₇	—	2	4	no	—
76	C₂×Dic₄₇	—	2	4	no	—
77	C₂×Dic₄₇	C₂×Q₈	1	4	yes	—
78	C₂×Dic₄₇	#41	1	4	yes	—
79	C₂²×C₄₇	—	2	2	no	—
80	C₂²×C₄₇	—	2	2	no	—
81	C₂²×C₄₇	—	2	4	no	—
82	C₂²×C₄₇	—	2	4	no	—
83	C₂²×C₄₇	C₂²⋊Q₈	1	4	yes	—
84	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₄₇	C₂×#41	1	2	yes	—
87	C₂²×C₄₇	C₂×#41	1	2	yes	—
88	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	4	2	no	—
91	C₄×C₄₇	—	4	2	no	—
92	C₄×C₄₇	#123	1	4	yes	—
93	Dic₄₇	—	2	4	no	—
94	Dic₄₇	—	2	4	no	—
95	Dic₄₇	—	4	4	no	—
96	#2855	—	47	1	no	maximal, Sylow
97	C₂×C₄₇	—	2	4	no	—
98	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄₇	C₂×#123	1	1	yes	characteristic
100	C₂×C₄₇	#525	1	1	yes	characteristic
101	C₂×C₄₇	#529	1	2	yes	—
102	C₂×C₄₇	#533	1	2	yes	—
103	C₂×C₂²⋊Q₈	D₄₇	1	1	yes	characteristic
104	C₂²×C₄²	—	47	1	no	—
105	#391	—	47	2	no	—
106	#389	—	47	2	no	—
107	#387	—	47	4	no	—
108	#385	—	47	2	no	—
109	#381	—	47	2	no	—
110	#380	—	47	1	no	—
111	C₄₇	#2855	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
113	C₂³×C₄	—	47	1	no	—
114	C₂³×C₄	—	47	1	no	—
115	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
116	C₂²⋊Q₈	—	2	4	no	—
117	C₂×C₄⋊C₄	—	47	4	no	—
118	C₂×C₄⋊C₄	—	47	2	no	—
119	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
120	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	—	47	2	no	—
122	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
123	C₂×C₄²	—	47	2	no	—
124	C₂×C₄²	—	47	2	no	—
125	C₂×C₄²	—	94	4	no	—
126	#95	—	47	2	no	—
127	#95	—	47	4	no	—
128	#95	—	47	4	no	—
129	#95	—	47	4	no	—
130	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	47	4	no	—
135	C₂²×C₄	—	47	2	no	—
136	C₂²×C₄	—	47	2	no	—
137	C₂²×C₄	—	47	4	no	—
138	C₂²×C₄	—	94	4	no	—
139	C₂²×C₄	—	94	2	no	—
140	C₂²×C₄	—	94	2	no	—
141	C₂²×C₄	—	94	2	no	—
142	C₂²×C₄	C₂²⋊D₄₇	1	4	yes	—
143	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
144	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
145	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
146	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	94	8	no	—
150	C₄⋊C₄	—	94	4	no	—
151	C₂²⋊C₄	—	2	4	no	—
152	C₂²⋊C₄	—	94	4	no	—
153	C₄²	—	94	4	no	—
154	C₄²	—	94	4	no	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	Q₈×D₄₇	1	2	yes	—
158	C₂³	#10625354	1	2	yes	—
159	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
160	C₂³	C₂³×D₄₇	1	1	yes	characteristic
161	Q₈	—	4	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	47	4	no	—
169	C₂×C₄	—	47	4	no	—
170	C₂×C₄	—	94	8	no	—
171	C₂×C₄	—	94	8	no	—
172	C₂×C₄	—	94	2	no	—
173	C₂×C₄	—	94	2	no	—
174	C₂×C₄	—	94	2	no	—
175	C₂×C₄	—	94	2	no	—
176	C₂×C₄	—	94	4	no	—
177	C₂×C₄	—	94	4	no	—
178	C₂×C₄	—	94	4	no	—
179	C₂×C₄	#10625352	1	4	yes	—
180	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
181	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	4	no	—
185	C₂²	—	2	4	no	—
186	C₂²	#13094000	1	4	yes	—
187	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
189	C₂²	#41×D₄₇	1	2	yes	—
190	C₂²	C₂×#10625352	1	2	yes	—
191	C₂²	C₂×#10625354	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	94	4	no	—
196	C₄	—	94	4	no	—
197	C₄	—	188	4	no	—
198	C₄	#13094005	1	4	yes	—
199	C₂	—	2	4	no	—
200	C₂	C₂×#13094000	1	1	yes	characteristic
201	C₂	C₂×#13094005	1	1	yes	characteristic
202	C₂	#15689630	1	1	yes	characteristic
203	C₂	#15689659	1	2	yes	—
204	C₂	#15689675	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×Dic₄₇

C₂

yes

maximal, characteristic

#15689030

C₂

yes

maximal, complemented

#15689019

C₂

yes

maximal, complemented

#15689004

C₂

yes

maximal, complemented

#15688995

C₂

yes

maximal

#15688984

C₂

yes

maximal

#15688983

C₂

yes

maximal, characteristic

C₂×C₂²⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₂²

yes

characteristic

C₂³×Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂²⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

Q₈

yes

complemented

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

#41

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

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×Q₈

yes

—

C₂×Dic₄₇

#41

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²⋊Q₈

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#123

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2855

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₄₇

C₂×#123

yes

characteristic

100

C₂×C₄₇

#525

yes

characteristic

101

C₂×C₄₇

#529

yes

—

102

C₂×C₄₇

#533

yes

—

103

C₂×C₂²⋊Q₈

D₄₇

yes

characteristic

104

C₂²×C₄²

—

105

#391

—

106

#389

—

107

#387

—

108

#385

—

109

#381

—

110

#380

—

111

C₄₇

#2855

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

C₂×D₄₇

yes

characteristic

113

C₂³×C₄

—

114

C₂³×C₄

—

115

C₂³×C₄

C₂×D₄₇

yes

characteristic

116

C₂²⋊Q₈

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

120

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

121

C₂×C₂²⋊C₄

—

122

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

123

C₂×C₄²

—

124

C₂×C₄²

—

125

C₂×C₄²

—

126

#95

—

127

#95

—

128

#95

—

129

#95

—

130

C₂⁴

C₂²×D₄₇

yes

characteristic

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

C₂²⋊D₄₇

yes

—

143

C₂²×C₄

C₂²×D₄₇

yes

—

144

C₂²×C₄

C₂²×D₄₇

yes

—

145

C₂²×C₄

C₂²×D₄₇

yes

characteristic

146

C₂²×C₄

C₂²×D₄₇

yes

characteristic

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₂²⋊C₄

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₂³

—

156

C₂³

—

157

C₂³

Q₈×D₄₇

yes

—

158

C₂³

#10625354

yes

—

159

C₂³

C₂×C₂²⋊D₄₇

yes

—

160

C₂³

C₂³×D₄₇

yes

characteristic

161

Q₈

—

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

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#10625352

yes

—

180

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

181

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

#13094000

yes

—

187

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

188

C₂²

C₂×Q₈×D₄₇

yes

characteristic

189

C₂²

#41×D₄₇

yes

—

190

C₂²

C₂×#10625352

yes

—

191

C₂²

C₂×#10625354

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

188

—

198

C₄

#13094005

yes

—

199

C₂

—

200

C₂

C₂×#13094000

yes

characteristic

201

C₂

C₂×#13094005

yes

characteristic

202

C₂

#15689630

yes

characteristic

203

C₂

#15689659

yes

—

204

C₂

#15689675

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₂²⋊Q₈×C₄₇	C₂
hypercenter	C₂×C₂²⋊Q₈	D₄₇
hypocenter	C₄₇	#2855
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×C₂²⋊Q₈×C₄₇

C₂

hypercenter

C₂×C₂²⋊Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

N	H ≅ G/N	type
2. #15689030	C₂	semidirect
3. #15689019	C₂	semidirect
4. #15689004	C₂	semidirect
40. C₂²×Dic₄₇	Q₈	semidirect
111. C₄₇	#2855	semidirect

H ≅ G/N

type

2. #15689030

C₂

semidirect

3. #15689019

C₂

semidirect

4. #15689004

C₂

semidirect

40. C₂²×Dic₄₇

Q₈

semidirect

111. C₄₇

#2855

semidirect

Subgroups — #20251722

Conjugacy classes of subgroups

Complements