Group #20243822 — subgroups

Subgroups — #20243822

Order 6016.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093948	C₂	1	2	yes	maximal, complemented
3	#15688983	C₂	1	1	yes	maximal, characteristic
4	#15688885	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625329	—	2	2	no	—
7	C₂×#10625329	—	2	4	no	—
8	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
11	#13093948	—	2	4	no	—
12	C₄₇⋊#95	—	2	1	no	—
13	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²⋊Q₈×C₄₇	—	2	2	no	—
16	C₂²⋊Q₈×C₄₇	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
18	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×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	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	D₄	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₄₇	—	2	2	no	—
34	C₄⋊C₄×C₄₇	—	2	4	no	—
35	C₄⋊C₄×C₄₇	—	2	1	no	—
36	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂²⋊C₄×C₄₇	—	2	4	no	—
38	C₂²⋊Dic₄₇	—	4	2	no	—
39	C₄⋊Dic₄₇	—	2	2	no	—
40	C₄⋊Dic₄₇	—	2	2	no	—
41	#10625329	—	2	8	no	—
42	C₂×C₄₇⋊C₈	—	2	2	no	—
43	C₂×C₄₇⋊C₈	—	2	2	no	—
44	C₂×C₄₇⋊C₈	D₄	1	4	yes	—
45	C₂³×C₄₇	—	2	2	no	—
46	C₂³×C₄₇	—	2	1	no	—
47	C₂³×C₄₇	—	2	1	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	C₂²⋊C₄	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	1	no	—
62	C₂×C₄×C₄₇	—	4	1	no	—
63	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#41	1	2	yes	—
68	C₂×Dic₄₇	—	4	2	no	—
69	C₂×Dic₄₇	—	4	2	no	—
70	C₂×Dic₄₇	—	4	2	no	—
71	C₄₇⋊C₈	—	2	4	no	—
72	C₂²×C₄₇	—	2	2	no	—
73	C₂²×C₄₇	—	2	2	no	—
74	C₂²×C₄₇	—	2	2	no	—
75	C₂²×C₄₇	—	4	2	no	—
76	C₂²×C₄₇	—	4	2	no	—
77	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄₇	#121	1	2	yes	—
79	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂²×C₄₇	#137	1	1	yes	characteristic
81	C₂²×C₄₇	#137	1	2	yes	—
82	C₄×C₄₇	—	2	2	no	—
83	C₄×C₄₇	—	4	2	no	—
84	C₄×C₄₇	—	4	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	Dic₄₇	—	8	2	no	—
89	#1894	—	47	1	no	maximal, Sylow
90	C₂×C₄₇	—	4	2	no	—
91	C₂×C₄₇	#386	1	1	yes	characteristic
92	C₂×C₄₇	#417	1	1	yes	characteristic
93	C₂×C₄₇	#419	1	1	yes	characteristic
94	C₂×C₄₇	#468	1	2	yes	—
95	C₂×C₄₇	#470	1	2	yes	—
96	C₂²×M₁₆	—	47	1	no	—
97	C₂×C₂²⋊Q₈	D₄₇	1	1	yes	characteristic
98	C₂×#103	—	47	2	no	—
99	#380	—	47	1	no	—
100	#340	—	47	2	no	—
101	C₄₇	#1894	1	1	yes	characteristic, complemented, Sylow
102	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
103	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
104	C₂×M₁₆	—	94	4	no	—
105	C₂×M₁₆	—	94	2	no	—
106	C₂²×C₈	—	47	2	no	—
107	C₂²⋊Q₈	—	2	2	no	—
108	C₂²⋊Q₈	Dic₄₇	1	4	yes	complemented
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₄⋊C₄	—	47	2	no	—
111	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
112	C₂×C₂²⋊C₄	—	2	1	no	—
113	C₂×C₂²⋊C₄	—	94	1	no	—
114	#103	—	94	4	no	—
115	#95	—	94	1	no	—
116	C₂⁴	C₂²⋊D₄₇	1	1	yes	characteristic
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	94	2	no	—
126	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
127	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
129	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
130	M₁₆	—	94	8	no	—
131	C₂×C₈	—	47	4	no	—
132	C₂×C₈	—	94	2	no	—
133	C₂×C₈	—	94	2	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	94	2	no	—
137	C₄⋊C₄	—	94	2	no	—
138	C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
139	C₂²⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	188	2	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	1	no	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	2	no	—
145	C₂³	C₂²⋊Dic₄₇	1	2	yes	—
146	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
147	Q₈	—	2	4	no	—
148	Q₈	—	4	1	no	—
149	Q₈	—	4	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	188	2	no	—
160	C₂×C₄	—	188	2	no	—
161	C₂×C₄	—	188	2	no	—
162	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
163	C₂×C₄	D₄×D₄₇	1	2	yes	—
164	C₂×C₄	#10625352	1	2	yes	—
165	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
166	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
167	C₈	—	94	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	2	no	—
173	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
174	C₂²	#13093996	1	2	yes	—
175	C₂²	#13094049	1	1	yes	characteristic
176	C₂²	#13094054	1	1	yes	characteristic
177	C₂²	#13094055	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	376	2	no	—
182	C₄	D₄×Dic₄₇	1	2	yes	—
183	C₄	#13093992	1	1	yes	characteristic
184	C₄	#13094005	1	1	yes	characteristic
185	C₂	—	4	2	no	—
186	C₂	#15689003	1	1	yes	characteristic
187	C₂	#15689124	1	1	yes	characteristic
188	C₂	#15689132	1	1	yes	characteristic
189	C₂	#15689371	1	2	yes	—
190	C₂	#15689379	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10625329

C₂

yes

maximal, characteristic

C₂×#13093948

C₂

yes

maximal, complemented

#15688983

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal, complemented

C₂×C₂²⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625329

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

—

#13093948

—

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂²⋊Q₈×C₄₇

—

C₂²⋊Q₈×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×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

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

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

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

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1894

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#470

yes

—

C₂²×M₁₆

—

C₂×C₂²⋊Q₈

D₄₇

yes

characteristic

C₂×#103

—

#380

—

100

#340

—

101

C₄₇

#1894

yes

characteristic, complemented, Sylow

102

C₂²×Q₈

C₂×D₄₇

yes

characteristic

103

C₂³×C₄

C₂×D₄₇

yes

characteristic

104

C₂×M₁₆

—

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

C₂²⋊Q₈

—

108

C₂²⋊Q₈

Dic₄₇

yes

complemented

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

112

C₂×C₂²⋊C₄

—

113

C₂×C₂²⋊C₄

—

114

#103

—

115

#95

—

116

C₂⁴

C₂²⋊D₄₇

yes

characteristic

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

C₂×Dic₄₇

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂×Dic₄₇

yes

—

127

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

128

C₂²×C₄

C₂²⋊D₄₇

yes

—

129

C₂²×C₄

C₂²×D₄₇

yes

characteristic

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

C₂×Dic₄₇

yes

—

139

C₂²⋊C₄

—

140

C₂²⋊C₄

—

188

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₂²⋊Dic₄₇

yes

—

146

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

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

—

188

—

160

C₂×C₄

—

188

—

161

C₂×C₄

—

188

—

162

C₂×C₄

C₂²⋊Dic₄₇

yes

—

163

C₂×C₄

D₄×D₄₇

yes

—

164

C₂×C₄

#10625352

yes

—

165

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

166

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

167

C₈

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

174

C₂²

#13093996

yes

—

175

C₂²

#13094049

yes

characteristic

176

C₂²

#13094054

yes

characteristic

177

C₂²

#13094055

yes

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

376

—

182

C₄

D₄×Dic₄₇

yes

—

183

C₄

#13093992

yes

characteristic

184

C₄

#13094005

yes

characteristic

185

C₂

—

186

C₂

#15689003

yes

characteristic

187

C₂

#15689124

yes

characteristic

188

C₂

#15689132

yes

characteristic

189

C₂

#15689371

yes

—

190

C₂

#15689379

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×C₂²⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₂²⋊Q₈	D₄₇
hypocenter	C₄₇	#1894
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×C₂²⋊Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₂²⋊Q₈

D₄₇

hypocenter

C₄₇

#1894

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

N	H ≅ G/N	type
2. C₂×#13093948	C₂	semidirect
4. #15688885	C₂	semidirect
16. C₂²⋊Q₈×C₄₇	C₄	semidirect
101. C₄₇	#1894	semidirect
108. C₂²⋊Q₈	Dic₄₇	semidirect

H ≅ G/N

type

2. C₂×#13093948

C₂

semidirect

4. #15688885

C₂

semidirect

16. C₂²⋊Q₈×C₄₇

C₄

semidirect

101. C₄₇

#1894

semidirect

108. C₂²⋊Q₈

Dic₄₇

semidirect

Subgroups — #20243822

Conjugacy classes of subgroups

Complements