Group #20243369 — subgroups

Subgroups — #20243369

Order 6016.

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

normal subgroups only

This group has 312 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×#13093981	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×(C₂×D₄₇)⋊C₈	C₂	1	2	yes	maximal
4	#15688877	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#13093981	—	2	2	no	—
9	#13093981	—	2	2	no	—
10	C₂×Dic₄₇⋊C₄	C₄	1	1	yes	characteristic
11	C₂×Dic₄₇⋊C₄	C₄	1	1	yes	characteristic
12	C₂×(C₂×D₄₇)⋊C₄	C₄	1	1	yes	characteristic
13	C₂×(C₂×D₄₇)⋊C₄	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
15	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
16	(C₂×D₄₇)⋊C₈	—	2	4	no	—
17	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₄₇	—	2	2	no	—
20	C₂³×D₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₄₇	—	2	1	no	—
28	C₂×C₄×D₄₇	—	2	1	no	—
29	C₂×C₄×D₄₇	—	2	1	no	—
30	C₂×C₄×D₄₇	—	2	1	no	—
31	C₂×C₄×D₄₇	D₄	1	2	yes	—
32	C₂×C₄×D₄₇	Q₈	1	2	yes	—
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₄²×C₄₇	D₄	1	2	yes	—
37	C₄²×C₄₇	D₄	1	2	yes	—
38	(C₂×D₄₇)⋊C₄	—	2	2	no	—
39	(C₂×D₄₇)⋊C₄	—	2	2	no	—
40	Dic₄₇⋊C₄	—	2	2	no	—
41	Dic₄₇⋊C₄	—	2	2	no	—
42	C₄×Dic₄₇	—	2	1	no	—
43	C₄×Dic₄₇	—	2	1	no	—
44	C₂×C₄₇⋊C₈	—	2	2	no	—
45	C₂×C₄₇⋊C₈	—	2	2	no	—
46	C₂×C₄₇⋊C₈	—	2	2	no	—
47	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×D₄₇	—	2	1	no	—
49	C₂²×D₄₇	—	2	1	no	—
50	C₂²×D₄₇	—	2	2	no	—
51	C₂²×D₄₇	—	2	2	no	—
52	C₂²×D₄₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
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	2	no	—
68	C₂×Dic₄₇	—	2	2	no	—
69	C₂×Dic₄₇	C₄⋊C₄	1	2	yes	—
70	C₄×D₄₇	—	2	2	no	—
71	C₄×D₄₇	—	2	2	no	—
72	C₄×D₄₇	—	2	2	no	—
73	C₄×D₄₇	—	2	2	no	—
74	C₈×C₄₇	—	4	2	no	—
75	C₄₇⋊C₈	—	4	2	no	—
76	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄₇	#117	1	1	yes	characteristic
79	C₂²×C₄₇	#131	1	2	yes	—
80	C₂²×C₄₇	#131	1	2	yes	—
81	C₂×D₄₇	—	2	2	no	—
82	C₂×D₄₇	—	2	4	no	—
83	C₂×D₄₇	—	4	1	no	—
84	C₂×D₄₇	—	4	1	no	—
85	C₂×D₄₇	—	4	2	no	—
86	C₄×C₄₇	—	2	2	no	—
87	C₄×C₄₇	—	2	2	no	—
88	C₄×C₄₇	#121	1	1	yes	characteristic
89	C₄×C₄₇	#121	1	1	yes	characteristic
90	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
91	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
92	Dic₄₇	—	2	2	no	—
93	Dic₄₇	—	4	1	no	—
94	Dic₄₇	—	4	1	no	—
95	#1798	—	47	1	no	maximal, Sylow
96	C₂×C₄₇	#380	1	1	yes	characteristic
97	C₂×C₄₇	#408	1	1	yes	characteristic
98	C₂×C₄₇	#408	1	1	yes	characteristic
99	C₂×C₄₇	#424	1	2	yes	—
100	C₂×C₄₇	#433	1	2	yes	—
101	D₄₇	—	4	2	no	—
102	C₂×#117	—	47	1	no	—
103	C₂×C₄⋊C₈	—	47	1	no	—
104	C₂×C₄⋊C₈	D₄₇	1	1	yes	characteristic, complemented
105	C₂×C₂²⋊C₈	—	47	2	no	—
106	#336	—	47	2	no	—
107	C₄₇	#1798	1	1	yes	characteristic, complemented, Sylow
108	C₂³×C₄	—	47	1	no	—
109	C₂²×C₈	—	47	2	no	—
110	C₂²×C₈	C₂×D₄₇	1	2	yes	—
111	#117	—	94	2	no	—
112	#117	—	94	2	no	—
113	C₂×C₄⋊C₄	—	47	1	no	—
114	C₂×C₄⋊C₄	—	47	1	no	—
115	C₂×C₂²⋊C₄	—	47	1	no	—
116	C₂×C₂²⋊C₄	—	47	1	no	—
117	C₂×C₄²	—	47	1	no	—
118	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	94	2	no	—
121	C₂²⋊C₈	—	94	4	no	—
122	C₂⁴	—	47	1	no	—
123	C₂²×C₄	—	47	1	no	—
124	C₂²×C₄	—	47	2	no	—
125	C₂²×C₄	—	47	2	no	—
126	C₂²×C₄	—	47	1	no	—
127	C₂²×C₄	—	47	1	no	—
128	C₂²×C₄	—	94	1	no	—
129	C₂²×C₄	—	94	1	no	—
130	C₂²×C₄	—	94	1	no	—
131	C₂²×C₄	—	94	1	no	—
132	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	94	2	no	—
139	C₂×C₈	—	94	2	no	—
140	C₂×C₈	—	94	2	no	—
141	C₄⋊C₄	—	94	2	no	—
142	C₄⋊C₄	—	94	2	no	—
143	C₂²⋊C₄	—	94	2	no	—
144	C₂²⋊C₄	—	94	2	no	—
145	C₄²	—	94	1	no	—
146	C₄²	—	94	1	no	—
147	C₄²	D₁₈₈	1	2	yes	—
148	C₄²	C₂²⋊D₄₇	1	2	yes	—
149	C₂³	—	47	2	no	—
150	C₂³	—	94	1	no	—
151	C₂³	—	94	1	no	—
152	C₂³	—	94	2	no	—
153	C₂³	—	94	2	no	—
154	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	47	2	no	—
158	C₂×C₄	—	94	1	no	—
159	C₂×C₄	—	94	1	no	—
160	C₂×C₄	—	94	2	no	—
161	C₂×C₄	—	94	2	no	—
162	C₂×C₄	—	94	2	no	—
163	C₂×C₄	—	94	2	no	—
164	C₂×C₄	—	94	1	no	—
165	C₂×C₄	—	94	1	no	—
166	C₂×C₄	—	94	2	no	—
167	C₂×C₄	—	94	2	no	—
168	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
169	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
170	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
171	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
172	C₂×C₄	#10625352	1	1	yes	characteristic
173	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
174	C₂×C₄	#10625354	1	1	yes	characteristic
175	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
176	C₈	—	4	2	no	—
177	C₈	—	188	2	no	—
178	C₂²	—	94	2	no	—
179	C₂²	—	94	4	no	—
180	C₂²	—	188	1	no	—
181	C₂²	—	188	1	no	—
182	C₂²	—	188	2	no	—
183	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
184	C₂²	C₄⋊C₄×D₄₇	1	1	yes	characteristic
185	C₂²	#13093982	1	1	yes	characteristic
186	C₂²	#13094030	1	2	yes	—
187	C₂²	#13094032	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	2	no	—
190	C₄	—	94	2	no	—
191	C₄	—	188	1	no	—
192	C₄	—	188	1	no	—
193	C₄	#13093995	1	1	yes	characteristic
194	C₄	#13093999	1	1	yes	characteristic
195	C₄	#13094000	1	1	yes	characteristic
196	C₄	#13094004	1	1	yes	characteristic
197	C₂	—	188	2	no	—
198	C₂	#15688980	1	1	yes	characteristic
199	C₂	#15689089	1	1	yes	characteristic
200	C₂	#15689091	1	1	yes	characteristic
201	C₂	#15689148	1	2	yes	—
202	C₂	#15689183	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093981

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂×(C₂×D₄₇)⋊C₈

C₂

yes

maximal

#15688877

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13093981

—

#13093981

—

C₂×Dic₄₇⋊C₄

C₄

yes

characteristic

C₂×Dic₄₇⋊C₄

C₄

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

C₄

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

(C₂×D₄₇)⋊C₈

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

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

D₄

yes

—

C₂×C₄×D₄₇

Q₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

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

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

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

C₄⋊C₄

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×C₄⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#131

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

100

C₂×C₄₇

#433

yes

—

101

D₄₇

—

102

C₂×#117

—

103

C₂×C₄⋊C₈

—

104

C₂×C₄⋊C₈

D₄₇

yes

characteristic, complemented

105

C₂×C₂²⋊C₈

—

106

#336

—

107

C₄₇

#1798

yes

characteristic, complemented, Sylow

108

C₂³×C₄

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂×D₄₇

yes

—

111

#117

—

112

#117

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₂²⋊C₄

—

116

C₂×C₂²⋊C₄

—

117

C₂×C₄²

—

118

C₂×C₄²

C₂×D₄₇

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₂²⋊C₈

—

122

C₂⁴

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₄×D₄₇

yes

characteristic

133

C₂²×C₄

C₄×D₄₇

yes

characteristic

134

C₂²×C₄

C₂²×D₄₇

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

—

147

C₄²

D₁₈₈

yes

—

148

C₄²

C₂²⋊D₄₇

yes

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

C₂×C₄×D₄₇

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

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

(C₂×D₄₇)⋊C₄

yes

—

169

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

170

C₂×C₄

C₂×D₁₈₈

yes

characteristic

171

C₂×C₄

D₄×D₄₇

yes

characteristic

172

C₂×C₄

#10625352

yes

characteristic

173

C₂×C₄

Q₈×D₄₇

yes

characteristic

174

C₂×C₄

#10625354

yes

characteristic

175

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

176

C₈

—

177

C₈

—

188

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

188

—

181

C₂²

—

188

—

182

C₂²

—

188

—

183

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

184

C₂²

C₄⋊C₄×D₄₇

yes

characteristic

185

C₂²

#13093982

yes

characteristic

186

C₂²

#13094030

yes

—

187

C₂²

#13094032

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

188

—

192

C₄

—

188

—

193

C₄

#13093995

yes

characteristic

194

C₄

#13093999

yes

characteristic

195

C₄

#13094000

yes

characteristic

196

C₄

#13094004

yes

characteristic

197

C₂

—

188

—

198

C₂

#15688980

yes

characteristic

199

C₂

#15689089

yes

characteristic

200

C₂

#15689091

yes

characteristic

201

C₂

#15689148

yes

—

202

C₂

#15689183

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₄₇	C₂
hypercenter	C₂×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#1798
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₇

commutator, socle

C₂³×C₄₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₄₇

C₂

hypercenter

C₂×C₄⋊C₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. C₂×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
4. #15688877	C₂	semidirect
5. C₂×C₄⋊C₈×C₄₇	C₂	semidirect
104. C₂×C₄⋊C₈	D₄₇	semidirect
107. C₄₇	#1798	semidirect

H ≅ G/N

type

2. C₂×C₄⋊(C₄₇⋊C₈)

C₂

semidirect

4. #15688877

C₂

semidirect

5. C₂×C₄⋊C₈×C₄₇

C₂

semidirect

104. C₂×C₄⋊C₈

D₄₇

semidirect

107. C₄₇

#1798

semidirect

Subgroups — #20243369

Conjugacy classes of subgroups

Complements