Group #20243401 — subgroups

Subgroups — #20243401

Order 6016.

292 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 109 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×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093948	C₂	1	1	yes	maximal, characteristic
4	#15688988	C₂	1	1	yes	maximal, characteristic
5	#15688886	C₂	1	1	yes	maximal, characteristic
6	#15688885	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	Q₈×Dic₄₇	—	2	2	no	—
10	Q₈×Dic₄₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₄₇	—	2	1	no	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₄₇	—	2	1	no	—
15	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
16	Dic₄₇⋊C₈	—	2	2	no	—
17	#13093948	—	2	2	no	—
18	C₄₇⋊#95	—	2	1	no	—
19	C₄₇⋊#95	—	2	1	no	—
20	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
23	#103×C₄₇	—	2	2	no	—
24	C₂×Q₈×C₄₇	—	2	1	no	—
25	C₂×Q₈×C₄₇	—	2	1	no	—
26	C₂×Q₈×C₄₇	—	2	1	no	—
27	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
28	C₂×Q₈×C₄₇	D₄	1	2	yes	—
29	C₂×Q₈×C₄₇	Q₈	1	2	yes	—
30	C₂²×C₄×C₄₇	—	2	1	no	—
31	C₂²×C₄×C₄₇	—	2	1	no	—
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₄₇	—	2	1	no	—
34	C₂²×Dic₄₇	—	2	1	no	—
35	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
36	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	—	2	1	no	—
38	C₂×C₈×C₄₇	—	2	1	no	—
39	C₂×C₈×C₄₇	—	2	1	no	—
40	C₄⋊C₄×C₄₇	—	2	1	no	—
41	C₄⋊C₄×C₄₇	—	2	1	no	—
42	C₄⋊Dic₄₇	—	2	1	no	—
43	C₄⋊Dic₄₇	—	2	4	no	—
44	C₄⋊Dic₄₇	—	2	1	no	—
45	C₄⋊Dic₄₇	—	2	1	no	—
46	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
47	C₄×Dic₄₇	—	2	4	no	—
48	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
49	C₄×Dic₄₇	D₄	1	2	yes	—
50	C₂×C₄₇⋊C₈	—	2	1	no	—
51	C₂×C₄₇⋊C₈	—	2	1	no	—
52	C₂×C₄₇⋊C₈	—	2	1	no	—
53	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
54	Q₈×C₄₇	—	2	2	no	—
55	Q₈×C₄₇	—	2	2	no	—
56	Q₈×C₄₇	—	2	2	no	—
57	Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	4	1	no	—
62	C₂×C₄×C₄₇	—	4	1	no	—
63	C₂×C₄×C₄₇	—	4	1	no	—
64	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
70	C₂×Dic₄₇	—	2	1	no	—
71	C₂×Dic₄₇	—	2	1	no	—
72	C₂×Dic₄₇	—	2	1	no	—
73	C₂×Dic₄₇	—	2	1	no	—
74	C₂×Dic₄₇	—	2	2	no	—
75	C₂×Dic₄₇	—	4	1	no	—
76	C₂×Dic₄₇	—	4	1	no	—
77	C₂×Dic₄₇	—	4	1	no	—
78	C₂×Dic₄₇	—	4	1	no	—
79	C₂×Dic₄₇	—	4	1	no	—
80	C₂×Dic₄₇	C₂²⋊C₄	1	2	yes	—
81	C₂×Dic₄₇	C₂²⋊C₄	1	2	yes	—
82	C₈×C₄₇	—	4	1	no	—
83	C₄₇⋊C₈	—	4	1	no	—
84	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄₇	#121	1	1	yes	characteristic
86	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂²×C₄₇	#135	1	1	yes	characteristic
88	C₂²×C₄₇	#135	1	1	yes	characteristic
89	C₂²×C₄₇	#137	1	1	yes	characteristic
90	C₂²×C₄₇	#137	1	1	yes	characteristic
91	C₄×C₄₇	—	2	4	no	—
92	C₄×C₄₇	—	8	1	no	—
93	C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₄₇	#117	1	1	yes	characteristic
95	C₄×C₄₇	#121	1	1	yes	characteristic
96	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
97	Dic₄₇	—	2	2	no	—
98	Dic₄₇	—	2	2	no	—
99	Dic₄₇	—	4	2	no	—
100	Dic₄₇	—	8	1	no	—
101	#1805	—	47	1	no	maximal, Sylow
102	C₂×C₄₇	#380	1	1	yes	characteristic
103	C₂×C₄₇	#416	1	1	yes	characteristic
104	C₂×C₄₇	#419	1	1	yes	characteristic
105	C₂×C₄₇	#464	1	1	yes	characteristic
106	C₂×C₄₇	#464	1	1	yes	characteristic
107	C₂×C₄₇	#479	1	1	yes	characteristic
108	C₂×C₄₇	#479	1	1	yes	characteristic
109	C₂×C₄×Q₈	—	47	1	no	—
110	C₂×C₄⋊C₈	—	47	1	no	—
111	C₂×#103	—	47	1	no	—
112	C₂×#103	D₄₇	1	1	yes	characteristic
113	#382	—	47	1	no	—
114	#340	—	47	1	no	—
115	#340	—	47	1	no	—
116	C₄₇	#1805	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
118	C₂²×C₈	—	47	1	no	—
119	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
120	C₄×Q₈	—	47	4	no	—
121	C₄×Q₈	—	94	2	no	—
122	C₂×C₄⋊C₄	—	47	1	no	—
123	C₂×C₄⋊C₄	—	47	1	no	—
124	C₂×C₄⋊C₄	—	94	1	no	—
125	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
126	C₂×C₄²	—	47	1	no	—
127	C₂×C₄²	—	94	1	no	—
128	C₄⋊C₈	—	94	2	no	—
129	#103	—	2	2	no	—
130	#103	—	94	2	no	—
131	#95	—	94	1	no	—
132	#95	—	94	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	Dic₉₄	1	2	yes	—
137	C₂×Q₈	C₄×D₄₇	1	2	yes	—
138	C₂×Q₈	C₂²⋊D₄₇	1	2	yes	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	47	1	no	—
142	C₂²×C₄	—	47	1	no	—
143	C₂²×C₄	—	94	1	no	—
144	C₂²×C₄	—	94	1	no	—
145	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	94	1	no	—
150	C₂×C₈	—	94	1	no	—
151	C₂×C₈	—	94	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	47	2	no	—
155	C₄⋊C₄	—	94	4	no	—
156	C₄⋊C₄	—	94	1	no	—
157	C₄⋊C₄	—	94	1	no	—
158	C₄⋊C₄	—	94	1	no	—
159	C₄²	—	47	2	no	—
160	C₄²	—	47	2	no	—
161	C₄²	—	94	4	no	—
162	C₂³	D₄×D₄₇	1	1	yes	characteristic
163	Q₈	—	2	2	no	—
164	Q₈	—	2	2	no	—
165	Q₈	—	2	2	no	—
166	Q₈	Dic₄₇⋊C₄	1	4	yes	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	47	2	no	—
174	C₂×C₄	—	47	2	no	—
175	C₂×C₄	—	94	1	no	—
176	C₂×C₄	—	94	1	no	—
177	C₂×C₄	—	94	1	no	—
178	C₂×C₄	—	94	1	no	—
179	C₂×C₄	—	94	2	no	—
180	C₂×C₄	—	188	1	no	—
181	C₂×C₄	—	188	1	no	—
182	C₂×C₄	—	188	1	no	—
183	C₂×C₄	—	188	1	no	—
184	C₂×C₄	—	188	1	no	—
185	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
187	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
188	C₂×C₄	#10625352	1	1	yes	characteristic
189	C₂×C₄	#10625352	1	1	yes	characteristic
190	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
191	C₈	—	4	1	no	—
192	C₈	—	188	1	no	—
193	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
194	C₂²	#13093996	1	1	yes	characteristic
195	C₂²	#13094001	1	1	yes	characteristic
196	C₂²	#13094045	1	1	yes	characteristic
197	C₂²	#13094047	1	1	yes	characteristic
198	C₂²	#13094053	1	1	yes	characteristic
199	C₂²	#13094056	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	—	8	1	no	—
202	C₄	—	94	2	no	—
203	C₄	—	94	2	no	—
204	C₄	—	376	1	no	—
205	C₄	—	188	2	no	—
206	C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
207	C₄	#13093980	1	1	yes	characteristic
208	C₄	#13093996	1	1	yes	characteristic
209	C₄	#13094001	1	1	yes	characteristic
210	C₂	#15688981	1	1	yes	characteristic
211	C₂	#15689118	1	1	yes	characteristic
212	C₂	#15689130	1	1	yes	characteristic
213	C₂	#15689343	1	1	yes	characteristic
214	C₂	#15689347	1	1	yes	characteristic
215	C₂	#15689428	1	1	yes	characteristic
216	C₂	#15689432	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₄₇

C₂

yes

maximal, characteristic

C₂×Dic₄₇⋊C₈

C₂

yes

maximal, characteristic

C₂×#13093948

C₂

yes

maximal, characteristic

#15688988

C₂

yes

maximal, characteristic

#15688886

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal, characteristic

C₂×#103×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

Q₈×Dic₄₇

—

Q₈×Dic₄₇

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

#13093948

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

#103×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

D₄

yes

characteristic

C₂²×Dic₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

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

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

characteristic

C₂²×C₄₇

C₂²⋊Q₈

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

100

Dic₄₇

—

101

#1805

—

maximal, Sylow

102

C₂×C₄₇

#380

yes

characteristic

103

C₂×C₄₇

#416

yes

characteristic

104

C₂×C₄₇

#419

yes

characteristic

105

C₂×C₄₇

#464

yes

characteristic

106

C₂×C₄₇

#464

yes

characteristic

107

C₂×C₄₇

#479

yes

characteristic

108

C₂×C₄₇

#479

yes

characteristic

109

C₂×C₄×Q₈

—

110

C₂×C₄⋊C₈

—

111

C₂×#103

—

112

C₂×#103

D₄₇

yes

characteristic

113

#382

—

114

#340

—

115

#340

—

116

C₄₇

#1805

yes

characteristic, complemented, Sylow

117

C₂²×Q₈

C₂×D₄₇

yes

characteristic

118

C₂²×C₈

—

119

C₂²×C₈

C₂×D₄₇

yes

characteristic

120

C₄×Q₈

—

121

C₄×Q₈

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

126

C₂×C₄²

—

127

C₂×C₄²

—

128

C₄⋊C₈

—

129

#103

—

130

#103

—

131

#95

—

132

#95

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

Dic₉₄

yes

—

137

C₂×Q₈

C₄×D₄₇

yes

—

138

C₂×Q₈

C₂²⋊D₄₇

yes

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂²×D₄₇

yes

characteristic

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₂³

D₄×D₄₇

yes

characteristic

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

Dic₄₇⋊C₄

yes

—

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

—

180

C₂×C₄

—

188

—

181

C₂×C₄

—

188

—

182

C₂×C₄

—

188

—

183

C₂×C₄

—

188

—

184

C₂×C₄

—

188

—

185

C₂×C₄

C₂×Dic₉₄

yes

characteristic

186

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

187

C₂×C₄

D₄×D₄₇

yes

characteristic

188

C₂×C₄

#10625352

yes

characteristic

189

C₂×C₄

#10625352

yes

characteristic

190

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

191

C₈

—

192

C₈

—

188

—

193

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

194

C₂²

#13093996

yes

characteristic

195

C₂²

#13094001

yes

characteristic

196

C₂²

#13094045

yes

characteristic

197

C₂²

#13094047

yes

characteristic

198

C₂²

#13094053

yes

characteristic

199

C₂²

#13094056

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

376

—

205

C₄

—

188

—

206

C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

207

C₄

#13093980

yes

characteristic

208

C₄

#13093996

yes

characteristic

209

C₄

#13094001

yes

characteristic

210

C₂

#15688981

yes

characteristic

211

C₂

#15689118

yes

characteristic

212

C₂

#15689130

yes

characteristic

213

C₂

#15689343

yes

characteristic

214

C₂

#15689347

yes

characteristic

215

C₂

#15689428

yes

characteristic

216

C₂

#15689432

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#103×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#103	D₄₇
hypocenter	C₄₇	#1805
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#103×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
10. Q₈×Dic₄₇	C₄	semidirect
116. C₄₇	#1805	semidirect

H ≅ G/N

type

10. Q₈×Dic₄₇

C₄

semidirect

116. C₄₇

#1805

semidirect

Subgroups — #20243401

Conjugacy classes of subgroups

Complements