Group #26122850 — subgroups

Subgroups — #26122850

Order 12032.

1318 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 1318 conjugacy classes of subgroups — too many to draw. Only its 267 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×#13093943	C₂	1	4	yes	maximal
2	#20243082	C₂	1	2	yes	maximal
3	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	C₂²	1	4	yes	—
5	C₂×#13093943	—	2	16	no	—
6	C₂×#13093943	—	2	8	no	—
7	#15688878	—	2	4	no	—
8	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₄₇	—	2	8	no	—
10	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
11	C₂²×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#10625329	—	2	4	no	—
14	C₂×#10625329	—	2	8	no	—
15	C₂×#10625329	—	2	8	no	—
16	C₂×#10625329	D₄	1	8	yes	—
17	C₂²×C₄₇⋊C₈	—	2	4	no	—
18	#13093943	—	2	32	no	—
19	C₂²×Q₈×C₄₇	—	2	4	no	—
20	C₂²×Q₈×C₄₇	—	2	4	no	—
21	C₂²×Q₈×C₄₇	—	2	8	no	—
22	C₂²×Q₈×C₄₇	—	2	8	no	—
23	C₂²×Q₈×C₄₇	D₄	1	8	yes	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₄₇	—	2	32	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
30	C₂×C₄²×C₄₇	—	2	4	no	—
31	C₂×C₄²×C₄₇	D₄	1	4	yes	—
32	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×Q₈×C₄₇	—	2	16	no	—
34	C₂×Q₈×C₄₇	—	2	8	no	—
35	C₂×Q₈×C₄₇	—	2	16	no	—
36	C₂×Q₈×C₄₇	—	2	8	no	—
37	C₂×Q₈×C₄₇	—	4	4	no	—
38	C₂×Q₈×C₄₇	—	4	8	no	—
39	C₂×Q₈×C₄₇	—	4	8	no	—
40	C₂×Q₈×C₄₇	—	4	4	no	—
41	C₂×Q₈×C₄₇	—	4	4	no	—
42	C₂×Q₈×C₄₇	—	4	4	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	2	8	no	—
45	C₂²×C₄×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	—	2	4	no	—
47	C₂²×C₄×C₄₇	—	2	16	no	—
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₄₇	#41	1	4	yes	—
54	C₄⋊C₄×C₄₇	—	2	32	no	—
55	C₄²×C₄₇	—	2	8	no	—
56	#10625329	—	2	16	no	—
57	#10625329	—	4	8	no	—
58	C₂×C₄₇⋊C₈	—	2	8	no	—
59	C₂×C₄₇⋊C₈	—	4	4	no	—
60	C₂×C₄₇⋊C₈	—	4	4	no	—
61	C₂³×C₄₇	—	2	4	no	—
62	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
63	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂³×C₄₇	C₄×D₄	1	4	yes	—
65	Q₈×C₄₇	—	4	16	no	—
66	Q₈×C₄₇	—	4	16	no	—
67	C₂×C₄×C₄₇	—	2	16	no	—
68	C₂×C₄×C₄₇	—	2	4	no	—
69	C₂×C₄×C₄₇	—	2	4	no	—
70	C₂×C₄×C₄₇	—	2	8	no	—
71	C₂×C₄×C₄₇	—	2	16	no	—
72	C₂×C₄×C₄₇	—	4	8	no	—
73	C₂×C₄×C₄₇	—	4	8	no	—
74	C₂×C₄×C₄₇	#101	1	8	yes	complemented
75	C₂×C₄×C₄₇	#120	1	4	yes	—
76	C₂×C₄×C₄₇	#120	1	4	yes	—
77	C₂×C₄×C₄₇	#121	1	4	yes	—
78	C₂×C₄×C₄₇	#121	1	8	yes	—
79	C₂×C₄×C₄₇	#123	1	4	yes	—
80	C₂×C₄×C₄₇	#124	1	2	yes	—
81	C₂×C₄×C₄₇	#127	1	2	yes	—
82	C₄₇⋊C₈	—	4	8	no	—
83	#10983	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	—	2	8	no	—
85	C₂²×C₄₇	—	2	4	no	—
86	C₂²×C₄₇	#379	1	1	yes	characteristic
87	C₂²×C₄₇	#386	1	4	yes	—
88	C₂²×C₄₇	#390	1	2	yes	—
89	C₂²×C₄₇	C₂×#101	1	4	yes	—
90	C₄×C₄₇	—	2	8	no	—
91	C₄×C₄₇	—	4	8	no	—
92	C₄×C₄₇	#392	1	4	yes	—
93	C₄×C₄₇	#394	1	4	yes	—
94	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
95	C₂²×#101	—	47	4	no	—
96	#1728	—	47	2	no	—
97	C₂×C₄₇	—	2	4	no	—
98	C₂×C₄₇	#1438	1	1	yes	characteristic
99	C₂×C₄₇	#1784	1	2	yes	—
100	C₂×C₄₇	#1966	1	4	yes	—
101	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
102	C₂²×M₁₆	—	47	4	no	—
103	C₂×C₄⋊Q₈	—	2	8	no	—
104	C₂×C₄⋊Q₈	—	2	4	no	—
105	C₂²×C₄⋊C₄	Dic₄₇	1	4	yes	—
106	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
107	C₂×#101	—	94	16	no	—
108	C₂×#101	—	94	8	no	—
109	#336	—	94	4	no	—
110	C₄₇	#10983	1	1	yes	characteristic, complemented, Sylow
111	C₂²×Q₈	—	2	8	no	—
112	C₂²×Q₈	—	2	8	no	—
113	C₂²×Q₈	—	2	4	no	—
114	C₂²×Q₈	—	2	4	no	—
115	C₂²×Q₈	C₂²⋊D₄₇	1	8	yes	—
116	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
117	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
118	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
119	C₂×M₁₆	—	47	8	no	—
120	C₂×M₁₆	—	94	4	no	—
121	C₂×M₁₆	—	94	8	no	—
122	C₂×M₁₆	—	94	8	no	—
123	C₂²×C₈	—	94	4	no	—
124	C₄⋊Q₈	—	2	32	no	—
125	C₂×C₄⋊C₄	—	2	16	no	—
126	C₂×C₄⋊C₄	—	2	8	no	—
127	C₂×C₄²	—	2	4	no	—
128	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
129	#101	—	94	32	no	—
130	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
131	C₂×Q₈	—	2	8	no	—
132	C₂×Q₈	—	2	16	no	—
133	C₂×Q₈	—	2	16	no	—
134	C₂×Q₈	—	2	8	no	—
135	C₂×Q₈	—	4	4	no	—
136	C₂×Q₈	—	4	8	no	—
137	C₂×Q₈	—	4	8	no	—
138	C₂×Q₈	—	4	4	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂×Q₈	—	4	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	16	no	—
146	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
147	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
148	C₂²×C₄	D₄×D₄₇	1	4	yes	—
149	C₂²×C₄	#10625352	1	4	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
151	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
152	M₁₆	—	94	16	no	—
153	M₁₆	—	188	8	no	—
154	C₂×C₈	—	94	8	no	—
155	C₂×C₈	—	188	4	no	—
156	C₂×C₈	—	188	4	no	—
157	C₄⋊C₄	—	2	32	no	—
158	C₄²	—	2	8	no	—
159	C₂³	—	2	4	no	—
160	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
161	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
162	C₂³	D₄×Dic₄₇	1	4	yes	—
163	Q₈	—	4	16	no	—
164	Q₈	—	4	16	no	—
165	C₂×C₄	—	2	16	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	—	4	8	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	#13093943	1	8	yes	complemented
173	C₂×C₄	#13093992	1	4	yes	—
174	C₂×C₄	#13093993	1	4	yes	—
175	C₂×C₄	#13093995	1	4	yes	—
176	C₂×C₄	#13093996	1	8	yes	—
177	C₂×C₄	#13094005	1	4	yes	—
178	C₂×C₄	#13094011	1	2	yes	—
179	C₂×C₄	#13094020	1	2	yes	—
180	C₈	—	188	8	no	—
181	C₂²	—	2	8	no	—
182	C₂²	—	2	4	no	—
183	C₂²	#15688979	1	1	yes	characteristic
184	C₂²	#15689003	1	4	yes	—
185	C₂²	#15689026	1	2	yes	—
186	C₂²	C₂×#13093943	1	4	yes	—
187	C₄	—	2	8	no	—
188	C₄	—	4	8	no	—
189	C₄	#15689031	1	4	yes	—
190	C₄	#15689037	1	4	yes	—
191	C₂	—	2	4	no	—
192	C₂	#20241629	1	1	yes	characteristic
193	C₂	#20243312	1	2	yes	—
194	C₂	#20244233	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13093943

C₂

yes

maximal

#20243082

C₂

yes

maximal

C₂²×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×#10625329

C₂²

yes

—

C₂×#13093943

—

C₂×#13093943

—

#15688878

—

C₂³×Q₈×C₄₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₄₇

—

C₂×C₄⋊Q₈×C₄₇

—

C₂²×C₄⋊C₄×C₄₇

C₄

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

—

—

—

D₄

yes

—

C₂²×C₄₇⋊C₈

—

#13093943

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

D₄

yes

—

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂⁴×C₄₇

C₂²×C₄

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₄×D₄

yes

—

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

#101

yes

complemented

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

#127

yes

—

C₄₇⋊C₈

—

#10983

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#379

yes

characteristic

C₂²×C₄₇

#386

yes

—

C₂²×C₄₇

#390

yes

—

C₂²×C₄₇

C₂×#101

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#392

yes

—

C₄×C₄₇

#394

yes

—

C₂²×C₄⋊Q₈

D₄₇

yes

characteristic

C₂²×#101

—

#1728

—

C₂×C₄₇

—

C₂×C₄₇

#1438

yes

characteristic

C₂×C₄₇

#1784

yes

—

100

C₂×C₄₇

#1966

yes

—

101

C₂³×Q₈

C₂×D₄₇

yes

—

102

C₂²×M₁₆

—

103

C₂×C₄⋊Q₈

—

104

C₂×C₄⋊Q₈

—

105

C₂²×C₄⋊C₄

Dic₄₇

yes

—

106

C₂²×C₄²

C₂×D₄₇

yes

characteristic

107

C₂×#101

—

108

C₂×#101

—

109

#336

—

110

C₄₇

#10983

yes

characteristic, complemented, Sylow

111

C₂²×Q₈

—

112

C₂²×Q₈

—

113

C₂²×Q₈

—

114

C₂²×Q₈

—

115

C₂²×Q₈

C₂²⋊D₄₇

yes

—

116

C₂³×C₄

C₂×Dic₄₇

yes

—

117

C₂³×C₄

C₂×Dic₄₇

yes

—

118

C₂³×C₄

C₂²×D₄₇

yes

characteristic

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂²×C₈

—

124

C₄⋊Q₈

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

—

128

C₂×C₄²

C₂²⋊D₄₇

yes

—

129

#101

—

130

C₂⁴

C₂²×Dic₄₇

yes

characteristic

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

147

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

148

C₂²×C₄

D₄×D₄₇

yes

—

149

C₂²×C₄

#10625352

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

151

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

152

M₁₆

—

153

M₁₆

—

188

—

154

C₂×C₈

—

155

C₂×C₈

—

188

—

156

C₂×C₈

—

188

—

157

C₄⋊C₄

—

158

C₄²

—

159

C₂³

—

160

C₂³

C₂×C₂²⋊Dic₄₇

yes

—

161

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

162

C₂³

D₄×Dic₄₇

yes

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

#13093943

yes

complemented

173

C₂×C₄

#13093992

yes

—

174

C₂×C₄

#13093993

yes

—

175

C₂×C₄

#13093995

yes

—

176

C₂×C₄

#13093996

yes

—

177

C₂×C₄

#13094005

yes

—

178

C₂×C₄

#13094011

yes

—

179

C₂×C₄

#13094020

yes

—

180

C₈

—

188

—

181

C₂²

—

182

C₂²

—

183

C₂²

#15688979

yes

characteristic

184

C₂²

#15689003

yes

—

185

C₂²

#15689026

yes

—

186

C₂²

C₂×#13093943

yes

—

187

C₄

—

188

C₄

—

189

C₄

#15689031

yes

—

190

C₄

#15689037

yes

—

191

C₂

—

192

C₂

#20241629

yes

characteristic

193

C₂

#20243312

yes

—

194

C₂

#20244233

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#10983
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₇

commutator, prime radical

C₂⁴×C₄₇

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₄₇

Fitting

C₂²×C₄⋊Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂³×C₄×C₄₇

C₂³

hypercenter

C₂²×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

#10983

LCS

G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

N	H ≅ G/N	type
74. C₂×C₄×C₄₇	#101	semidirect
110. C₄₇	#10983	semidirect
172. C₂×C₄	#13093943	semidirect

H ≅ G/N

type

74. C₂×C₄×C₄₇

#101

semidirect

110. C₄₇

#10983

semidirect

172. C₂×C₄

#13093943

semidirect

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

subgroup	index
C₂²×#13093943	2
#20243082	2
C₂²×C₄⋊Q₈×C₄₇	2
#10983	47

subgroup

index

C₂²×#13093943

#20243082

C₂²×C₄⋊Q₈×C₄₇

#10983

Subgroups — #26122850

Conjugacy classes of subgroups

Complements

Maximal subgroups