Group #20239502 — subgroups

Subgroups — C₄₇×#1892 #20239502

Order 6016.

380 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 144 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	C₂³×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#103×C₄₇	C₂	1	2	yes	maximal, complemented
4	C₄₇×#380	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#340	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₄₇	—	2	2	no	—
9	C₂²×C₈×C₄₇	—	2	4	no	—
10	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
11	C₂²⋊Q₈×C₄₇	—	2	2	no	—
12	C₂²⋊Q₈×C₄₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
17	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
18	#103×C₄₇	—	2	4	no	—
19	#95×C₄₇	—	2	1	no	—
20	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
21	C₂×Q₈×C₄₇	—	2	1	no	—
22	C₂×Q₈×C₄₇	—	2	2	no	—
23	C₂×Q₈×C₄₇	—	2	2	no	—
24	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₄₇	—	2	2	no	—
26	C₂²×C₄×C₄₇	—	2	2	no	—
27	C₂²×C₄×C₄₇	—	2	1	no	—
28	C₂²×C₄×C₄₇	—	2	1	no	—
29	C₂²×C₄×C₄₇	—	2	2	no	—
30	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	D₄	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₄₇	—	2	8	no	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	D₄	1	4	yes	—
38	C₄⋊C₄×C₄₇	—	2	4	no	—
39	C₄⋊C₄×C₄₇	—	2	1	no	—
40	C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂²⋊C₄×C₄₇	—	2	4	no	—
44	C₂²⋊C₄×C₄₇	—	4	2	no	—
45	C₂³×C₄₇	—	2	2	no	—
46	C₂³×C₄₇	—	2	2	no	—
47	C₂³×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₄₇	SD₁₆	1	2	yes	—
49	C₂³×C₄₇	Q₁₆	1	2	yes	—
50	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₄₇	—	2	4	no	—
52	Q₈×C₄₇	—	4	1	no	—
53	Q₈×C₄₇	—	4	1	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	4	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	4	2	no	—
61	C₂×C₄×C₄₇	—	4	2	no	—
62	C₂×C₄×C₄₇	—	4	2	no	—
63	C₂×C₄×C₄₇	—	4	2	no	—
64	C₂×C₄×C₄₇	—	4	2	no	—
65	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#41	1	2	yes	—
70	C₈×C₄₇	—	2	4	no	—
71	C₂²×C₄₇	—	2	2	no	—
72	C₂²×C₄₇	—	2	2	no	—
73	C₂²×C₄₇	—	2	4	no	—
74	C₂²×C₄₇	—	2	4	no	—
75	C₂²×C₄₇	#103	1	4	yes	complemented
76	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄₇	#121	1	2	yes	—
78	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂²×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
80	C₂²×C₄₇	#135	1	2	yes	—
81	C₄×C₄₇	—	2	2	no	—
82	C₄×C₄₇	—	4	2	no	—
83	C₄×C₄₇	—	4	2	no	—
84	C₄×C₄₇	—	8	2	no	—
85	C₄×C₄₇	C₄×D₄	1	2	yes	—
86	C₄×C₄₇	#120	1	1	yes	characteristic
87	C₄×C₄₇	#123	1	1	yes	characteristic
88	#1892	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₄₇	—	2	4	no	—
90	C₂×C₄₇	#386	1	1	yes	characteristic
91	C₂×C₄₇	C₂×#103	1	1	yes	characteristic
92	C₂×C₄₇	#416	1	1	yes	characteristic
93	C₂×C₄₇	#465	1	2	yes	—
94	C₂×C₄₇	#467	1	2	yes	—
95	C₂³×C₈	C₂×C₄₇	1	1	yes	characteristic
96	C₂×C₂²⋊Q₈	C₂×C₄₇	1	1	yes	characteristic
97	C₂×#103	C₂×C₄₇	1	2	yes	complemented
98	#380	C₂×C₄₇	1	1	yes	characteristic
99	#340	C₂×C₄₇	1	2	yes	complemented
100	C₄₇	#1892	1	1	yes	characteristic, direct product factor, Sylow
101	C₂²×Q₈	C₂²×C₄₇	1	1	yes	characteristic
102	C₂³×C₄	C₂²×C₄₇	1	1	yes	characteristic
103	C₂²×C₈	—	2	2	no	—
104	C₂²×C₈	—	2	4	no	—
105	C₂²×C₈	C₂²×C₄₇	1	2	yes	—
106	C₂²⋊Q₈	—	2	2	no	—
107	C₂²⋊Q₈	C₄×C₄₇	1	4	yes	complemented
108	C₂×C₄⋊C₄	—	2	1	no	—
109	C₂×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₂²⋊C₄	—	2	1	no	—
113	#103	—	2	4	no	—
114	#95	—	2	1	no	—
115	C₂⁴	D₄×C₄₇	1	1	yes	characteristic
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	C₂×C₄×C₄₇	1	2	yes	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	C₂×C₄×C₄₇	1	2	yes	—
126	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
127	C₂²×C₄	D₄×C₄₇	1	2	yes	—
128	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	D₄×C₄₇	1	4	yes	—
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
138	C₂²⋊C₄	—	2	4	no	—
139	C₂²⋊C₄	—	4	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	2	2	no	—
142	C₂³	C₂²⋊C₄×C₄₇	1	2	yes	—
143	C₂³	SD₁₆×C₄₇	1	2	yes	—
144	C₂³	Q₁₆×C₄₇	1	2	yes	—
145	C₂³	C₂×D₄×C₄₇	1	1	yes	characteristic
146	Q₈	—	2	4	no	—
147	Q₈	—	4	1	no	—
148	Q₈	—	4	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
161	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
162	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
163	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
164	C₂×C₄	#41×C₄₇	1	2	yes	—
165	C₈	—	2	4	no	—
166	C₂²	—	2	4	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	#103×C₄₇	1	4	yes	complemented
171	C₂²	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
172	C₂²	#121×C₄₇	1	2	yes	—
173	C₂²	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
174	C₂²	C₂×Q₁₆×C₄₇	1	1	yes	characteristic
175	C₂²	#135×C₄₇	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	2	no	—
180	C₄	C₄×D₄×C₄₇	1	2	yes	—
181	C₄	#120×C₄₇	1	1	yes	characteristic
182	C₄	#123×C₄₇	1	1	yes	characteristic
183	C₂	—	2	4	no	—
184	C₂	C₄₇×#386	1	1	yes	characteristic
185	C₂	C₂×#103×C₄₇	1	1	yes	characteristic
186	C₂	C₄₇×#416	1	1	yes	characteristic
187	C₂	C₄₇×#465	1	2	yes	—
188	C₂	C₄₇×#467	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₈×C₄₇

C₂

yes

maximal, characteristic

C₂×C₂²⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂×#103×C₄₇

C₂

yes

maximal, complemented

C₄₇×#380

C₂

yes

maximal, characteristic

C₄₇×#340

C₂

yes

maximal, complemented

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂²⋊Q₈×C₄₇

—

C₂²⋊Q₈×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

—

#103×C₄₇

—

#95×C₄₇

—

C₂⁴×C₄₇

D₄

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

—

C₂³×C₄₇

SD₁₆

yes

—

C₂³×C₄₇

Q₁₆

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

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

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#103

yes

complemented

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

C₂×SD₁₆

yes

characteristic

C₂²×C₄₇

C₂×Q₁₆

yes

characteristic

C₂²×C₄₇

#135

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#467

yes

—

C₂³×C₈

C₂×C₄₇

yes

characteristic

C₂×C₂²⋊Q₈

C₂×C₄₇

yes

characteristic

C₂×#103

C₂×C₄₇

yes

complemented

#380

C₂×C₄₇

yes

characteristic

#340

C₂×C₄₇

yes

complemented

100

C₄₇

#1892

yes

characteristic, direct product factor, Sylow

101

C₂²×Q₈

C₂²×C₄₇

yes

characteristic

102

C₂³×C₄

C₂²×C₄₇

yes

characteristic

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

C₂²×C₄₇

yes

—

106

C₂²⋊Q₈

—

107

C₂²⋊Q₈

C₄×C₄₇

yes

complemented

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂²×C₄₇

yes

—

110

C₂×C₄⋊C₄

C₂²×C₄₇

yes

characteristic

111

C₂×C₂²⋊C₄

—

112

C₂×C₂²⋊C₄

—

113

#103

—

114

#95

—

115

C₂⁴

D₄×C₄₇

yes

characteristic

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

C₂×C₄×C₄₇

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂×C₄×C₄₇

yes

—

126

C₂²×C₄

D₄×C₄₇

yes

characteristic

127

C₂²×C₄

D₄×C₄₇

yes

—

128

C₂²×C₄

C₂³×C₄₇

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

D₄×C₄₇

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

C₂×C₄×C₄₇

yes

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₂³

—

141

C₂³

—

142

C₂³

C₂²⋊C₄×C₄₇

yes

—

143

C₂³

SD₁₆×C₄₇

yes

—

144

C₂³

Q₁₆×C₄₇

yes

—

145

C₂³

C₂×D₄×C₄₇

yes

characteristic

146

Q₈

—

147

Q₈

—

148

Q₈

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

C₂²⋊C₄×C₄₇

yes

—

161

C₂×C₄

C₂²×C₄×C₄₇

yes

characteristic

162

C₂×C₄

C₂×D₄×C₄₇

yes

—

163

C₂×C₄

C₂×D₄×C₄₇

yes

characteristic

164

C₂×C₄

#41×C₄₇

yes

—

165

C₈

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

#103×C₄₇

yes

complemented

171

C₂²

C₂×C₂²⋊C₄×C₄₇

yes

characteristic

172

C₂²

#121×C₄₇

yes

—

173

C₂²

C₂×SD₁₆×C₄₇

yes

characteristic

174

C₂²

C₂×Q₁₆×C₄₇

yes

characteristic

175

C₂²

#135×C₄₇

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

C₄×D₄×C₄₇

yes

—

181

C₄

#120×C₄₇

yes

characteristic

182

C₄

#123×C₄₇

yes

characteristic

183

C₂

—

184

C₂

C₄₇×#386

yes

characteristic

185

C₂

C₂×#103×C₄₇

yes

characteristic

186

C₂

C₄₇×#416

yes

characteristic

187

C₂

C₄₇×#465

yes

—

188

C₂

C₄₇×#467

yes

—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂²×C₄	C₂³×C₄₇
prime radical	C₂⁴×C₄₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂³×C₄×C₄₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₄₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₄₇

Frattini, cosocle

C₂²×C₄

C₂³×C₄₇

prime radical

C₂⁴×C₄₇

D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₄₇ ◁ C₂³×C₄×C₄₇ ◁ G

N	H ≅ G/N	type
3. C₂×#103×C₄₇	C₂	semidirect
5. C₄₇×#340	C₂	semidirect
12. C₂²⋊Q₈×C₄₇	C₄	semidirect
75. C₂²×C₄₇	#103	semidirect
88. #1892	C₄₇	direct (N × H)
97. C₂×#103	C₂×C₄₇	semidirect
99. #340	C₂×C₄₇	semidirect
100. C₄₇	#1892	direct (N × H)
107. C₂²⋊Q₈	C₄×C₄₇	semidirect
170. C₂²	#103×C₄₇	semidirect

H ≅ G/N

type

3. C₂×#103×C₄₇

C₂

semidirect

5. C₄₇×#340

C₂

semidirect

12. C₂²⋊Q₈×C₄₇

C₄

semidirect

75. C₂²×C₄₇

#103

semidirect

88. #1892

C₄₇

direct (N × H)

97. C₂×#103

C₂×C₄₇

semidirect

99. #340

C₂×C₄₇

semidirect

100. C₄₇

#1892

direct (N × H)

107. C₂²⋊Q₈

C₄×C₄₇

semidirect

170. C₂²

#103×C₄₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₈×C₄₇	2
C₂×C₂²⋊Q₈×C₄₇	2
C₂×#103×C₄₇	2
C₄₇×#380	2
C₄₇×#340	2
#1892	47

subgroup

index

C₂³×C₈×C₄₇

C₂×C₂²⋊Q₈×C₄₇

C₂×#103×C₄₇

C₄₇×#380

C₄₇×#340

#1892

Subgroups — C₄₇×#1892 #20239502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47×#1892 #20239502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#1892 #20239502