Group #26107195 — subgroups

Subgroups — #26107195

Order 12032.

532 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 141 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	#20245943	C₂	1	1	yes	maximal, characteristic
2	#97×Dic₄₇	C₂	1	2	yes	maximal
3	C₂×#15688864	C₂	1	2	yes	maximal
4	#20241269	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1295	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094016	C₂²	1	1	yes	characteristic
7	C₂×#13093935	C₂²	1	2	yes	—
8	C₂×#13093934	—	2	2	no	—
9	C₂×C₈×Dic₄₇	—	2	2	no	—
10	#15689019	C₄	1	2	yes	—
11	#15689005	—	2	2	no	—
12	C₄⋊C₄×Dic₄₇	C₄	1	2	yes	—
13	#15688974	—	2	2	no	—
14	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
15	#15688864	C₄	1	8	yes	complemented
16	C₂×#97×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#382	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	#13094016	—	2	1	no	—
20	#13094016	—	2	1	no	—
21	#13094016	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
23	C₂×Dic₄₇⋊C₄	—	2	2	no	—
24	C₂×Dic₄₇⋊C₄	—	2	2	no	—
25	C₂×C₄×Dic₄₇	—	2	2	no	—
26	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
29	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
30	#13093935	—	2	2	no	—
31	#13093935	C₂×C₄	1	4	yes	—
32	#13093934	—	2	8	no	—
33	C₈×Dic₄₇	—	2	8	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	#97×C₄₇	—	2	2	no	—
40	#97×C₄₇	C₂×C₄	1	4	yes	—
41	#95×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂²×Dic₄₇	—	2	2	no	—
46	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₄₇	—	2	4	no	—
50	C₂×C₈×C₄₇	—	2	4	no	—
51	C₂×C₈×C₄₇	—	2	4	no	—
52	C₄⋊C₄×C₄₇	—	4	2	no	—
53	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
54	C₄²×C₄₇	#41	1	1	yes	characteristic
55	C₄²×C₄₇	#41	1	1	yes	characteristic
56	C₄⋊Dic₄₇	—	2	2	no	—
57	C₄⋊Dic₄₇	C₄²	1	4	yes	—
58	Dic₄₇⋊C₄	—	2	4	no	—
59	Dic₄₇⋊C₄	—	2	4	no	—
60	Dic₄₇⋊C₄	—	4	4	no	—
61	C₄×Dic₄₇	—	2	2	no	—
62	C₄×Dic₄₇	—	2	4	no	—
63	C₄×Dic₄₇	—	2	4	no	—
64	C₄×Dic₄₇	—	4	1	no	—
65	C₄×Dic₄₇	—	4	4	no	—
66	C₂×C₄₇⋊C₈	—	2	4	no	—
67	C₂×C₄₇⋊C₈	—	2	4	no	—
68	C₂×C₄₇⋊C₈	—	2	4	no	—
69	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	1	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	4	2	no	—
75	C₂×C₄×C₄₇	—	4	2	no	—
76	C₂×C₄×C₄₇	—	4	2	no	—
77	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
81	C₂×Dic₄₇	—	2	2	no	—
82	C₂×Dic₄₇	—	2	2	no	—
83	C₂×Dic₄₇	—	2	4	no	—
84	C₂×Dic₄₇	—	2	1	no	—
85	C₂×Dic₄₇	—	2	2	no	—
86	C₂×Dic₄₇	—	4	2	no	—
87	C₂×Dic₄₇	—	4	2	no	—
88	C₂×Dic₄₇	#101	1	2	yes	—
89	C₂×Dic₄₇	#104	1	4	yes	—
90	C₂×Dic₄₇	#104	1	4	yes	—
91	C₈×C₄₇	—	2	8	no	—
92	C₄₇⋊C₈	—	2	8	no	—
93	C₄×#330	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
96	C₂²×C₄₇	#413	1	1	yes	characteristic
97	C₂²×C₄₇	C₂×#104	1	2	yes	—
98	C₂²×C₄₇	C₂×#104	1	2	yes	—
99	C₄×C₄₇	—	2	1	no	—
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	2	2	no	—
102	C₄×C₄₇	—	2	2	no	—
103	C₄×C₄₇	—	8	2	no	—
104	Dic₄₇	—	2	4	no	—
105	Dic₄₇	—	4	4	no	—
106	Dic₄₇	—	4	1	no	—
107	Dic₄₇	#330	1	4	yes	complemented
108	C₄×#125	—	47	1	no	—
109	C₄×#97	—	47	2	no	—
110	C₂×#330	—	47	2	no	—
111	#1295	—	47	1	no	—
112	#1295	D₄₇	1	1	yes	characteristic
113	C₂×C₄₇	C₂×#330	1	1	yes	characteristic
114	C₂×C₄₇	#1524	1	2	yes	—
115	C₂×C₄₇	#1526	1	1	yes	characteristic
116	C₂×C₄₇	C₄×#101	1	1	yes	characteristic
117	C₂×C₄₇	C₄×#104	1	2	yes	—
118	C₂×#125	—	47	1	no	—
119	C₂×#97	—	47	2	no	—
120	C₂×#97	—	94	2	no	—
121	C₂×#97	C₂×D₄₇	1	2	yes	—
122	C₂×C₄×C₈	—	94	2	no	—
123	#389	—	47	2	no	—
124	#387	—	94	2	no	—
125	#382	C₂×D₄₇	1	1	yes	characteristic
126	C₄×C₄⋊C₄	—	47	2	no	—
127	C₄×C₄⋊C₄	—	94	2	no	—
128	C₄³	—	47	1	no	—
129	#330	—	47	8	no	—
130	C₄₇	C₄×#330	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	94	2	no	—
133	#125	—	47	4	no	—
134	#125	—	94	1	no	—
135	#125	—	94	1	no	—
136	C₂×C₄⋊C₄	—	47	2	no	—
137	C₂×C₄⋊C₄	—	94	2	no	—
138	C₂×C₄⋊C₄	—	94	2	no	—
139	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
140	C₂×C₄²	—	47	1	no	—
141	C₂×C₄²	—	47	2	no	—
142	C₂×C₄²	—	47	2	no	—
143	C₂×C₄²	—	47	1	no	—
144	C₂×C₄²	—	94	2	no	—
145	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
146	#97	—	2	2	no	—
147	#97	—	47	4	no	—
148	#97	—	94	8	no	—
149	#97	—	94	2	no	—
150	#97	C₄×D₄₇	1	4	yes	—
151	C₄×C₈	—	94	8	no	—
152	#95	—	2	2	no	—
153	#95	—	94	2	no	—
154	#95	—	94	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	47	1	no	—
157	C₂²×C₄	—	47	2	no	—
158	C₂²×C₄	—	47	1	no	—
159	C₂²×C₄	—	94	2	no	—
160	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
161	C₂²×C₄	D₄×D₄₇	1	2	yes	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	94	4	no	—
166	C₂×C₈	—	94	4	no	—
167	C₂×C₈	—	94	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	47	4	no	—
170	C₄⋊C₄	—	94	2	no	—
171	C₄⋊C₄	—	94	4	no	—
172	C₄⋊C₄	—	94	4	no	—
173	C₄⋊C₄	—	188	4	no	—
174	C₄²	—	94	4	no	—
175	C₄²	—	94	4	no	—
176	C₄²	—	94	2	no	—
177	C₄²	—	188	1	no	—
178	C₄²	—	188	4	no	—
179	C₄²	C₂×C₄×D₄₇	1	2	yes	—
180	C₄²	#10625352	1	1	yes	characteristic
181	C₄²	#10625354	1	1	yes	characteristic
182	C₂³	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	47	2	no	—
191	C₂×C₄	—	47	4	no	—
192	C₂×C₄	—	47	4	no	—
193	C₂×C₄	—	94	2	no	—
194	C₂×C₄	—	94	2	no	—
195	C₂×C₄	—	94	4	no	—
196	C₂×C₄	—	94	1	no	—
197	C₂×C₄	—	94	2	no	—
198	C₂×C₄	—	188	2	no	—
199	C₂×C₄	—	188	2	no	—
200	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
201	C₂×C₄	#13093982	1	1	yes	characteristic
202	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
203	C₂×C₄	#13093986	1	2	yes	—
204	C₈	—	2	8	no	—
205	C₈	—	94	8	no	—
206	C₂²	#15688970	1	1	yes	characteristic
207	C₂²	#101×D₄₇	1	1	yes	characteristic
208	C₂²	#15689104	1	1	yes	characteristic
209	C₂²	#104×D₄₇	1	2	yes	—
210	C₂²	#104×D₄₇	1	2	yes	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	8	2	no	—
216	C₄	—	47	4	no	—
217	C₄	—	94	4	no	—
218	C₄	—	188	1	no	—
219	C₄	—	188	4	no	—
220	C₂	#330×D₄₇	1	1	yes	characteristic
221	C₂	#20241973	1	2	yes	—
222	C₂	#20241981	1	1	yes	characteristic
223	C₂	#20243183	1	1	yes	characteristic
224	C₂	#20243192	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245943

C₂

yes

maximal, characteristic

#97×Dic₄₇

C₂

yes

maximal

C₂×#15688864

C₂

yes

maximal

#20241269

C₂

yes

maximal, characteristic

C₄₇×#1295

C₂

yes

maximal, characteristic

C₂×#13094016

C₂²

yes

characteristic

C₂×#13093935

C₂²

yes

—

C₂×#13093934

—

C₂×C₈×Dic₄₇

—

#15689019

C₄

yes

—

#15689005

—

C₄⋊C₄×Dic₄₇

C₄

yes

—

#15688974

—

C₄²×Dic₄₇

C₂²

yes

characteristic

#15688864

C₄

yes

complemented

C₂×#97×C₄₇

C₂²

yes

—

C₄₇×#382

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

#13094016

—

#13094016

—

#13094016

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×Dic₄₇⋊C₄

—

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₄₇

D₄

yes

—

C₂×C₄×Dic₄₇

D₄

yes

—

#13093935

—

#13093935

C₂×C₄

yes

—

#13093934

—

C₈×Dic₄₇

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#97×C₄₇

—

#97×C₄₇

C₂×C₄

yes

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂²×Dic₄₇

C₂²⋊C₄

yes

—

C₂²×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

yes

C₄²×C₄₇

yes

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₄²

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

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×C₄²

yes

characteristic

C₂×C₄×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#101

yes

—

C₂×Dic₄₇

#104

yes

—

C₂×Dic₄₇

#104

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄×#330

—

maximal, Sylow

C₂²×C₄₇

C₄×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×#104

yes

—

C₄×C₄₇

—

100

C₄×C₄₇

—

101

C₄×C₄₇

—

102

C₄×C₄₇

—

103

C₄×C₄₇

—

104

Dic₄₇

—

105

Dic₄₇

—

106

Dic₄₇

—

107

Dic₄₇

#330

yes

complemented

108

C₄×#125

—

109

C₄×#97

—

110

C₂×#330

—

111

#1295

—

112

#1295

D₄₇

yes

characteristic

113

C₂×C₄₇

C₂×#330

yes

characteristic

114

C₂×C₄₇

#1524

yes

—

115

C₂×C₄₇

#1526

yes

characteristic

116

C₂×C₄₇

C₄×#101

yes

characteristic

117

C₂×C₄₇

C₄×#104

yes

—

118

C₂×#125

—

119

C₂×#97

—

120

C₂×#97

—

121

C₂×#97

C₂×D₄₇

yes

—

122

C₂×C₄×C₈

—

123

#389

—

124

#387

—

125

#382

C₂×D₄₇

yes

characteristic

126

C₄×C₄⋊C₄

—

127

C₄×C₄⋊C₄

—

128

C₄³

—

129

#330

—

130

C₄₇

C₄×#330

yes

characteristic, complemented, Sylow

131

C₂²×C₈

—

132

C₂²×C₈

—

133

#125

—

134

#125

—

135

#125

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

C₄×D₄₇

yes

—

140

C₂×C₄²

—

141

C₂×C₄²

—

142

C₂×C₄²

—

143

C₂×C₄²

—

144

C₂×C₄²

—

145

C₂×C₄²

C₂²×D₄₇

yes

characteristic

146

#97

—

147

#97

—

148

#97

—

149

#97

—

150

#97

C₄×D₄₇

yes

—

151

C₄×C₈

—

152

#95

—

153

#95

—

154

#95

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

161

C₂²×C₄

D₄×D₄₇

yes

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

188

—

174

C₄²

—

175

C₄²

—

176

C₄²

—

177

C₄²

—

188

—

178

C₄²

—

188

—

179

C₄²

C₂×C₄×D₄₇

yes

—

180

C₄²

#10625352

yes

characteristic

181

C₄²

#10625354

yes

characteristic

182

C₂³

C₂²⋊C₄×D₄₇

yes

characteristic

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

188

—

199

C₂×C₄

—

188

—

200

C₂×C₄

C₄²×D₄₇

yes

characteristic

201

C₂×C₄

#13093982

yes

characteristic

202

C₂×C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

203

C₂×C₄

#13093986

yes

—

204

C₈

—

205

C₈

—

206

C₂²

#15688970

yes

characteristic

207

C₂²

#101×D₄₇

yes

characteristic

208

C₂²

#15689104

yes

characteristic

209

C₂²

#104×D₄₇

yes

—

210

C₂²

#104×D₄₇

yes

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

188

—

219

C₄

—

188

—

220

C₂

#330×D₄₇

yes

characteristic

221

C₂

#20241973

yes

—

222

C₂

#20241981

yes

characteristic

223

C₂

#20243183

yes

characteristic

224

C₂

#20243192

yes

—

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1295	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1295	D₄₇
hypocenter	C₄₇	C₄×#330
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1295

subgroup

G/H

center

C₂³

C₂²⋊C₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1295

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1295

N	H ≅ G/N	type
15. #15688864	C₄	semidirect
107. Dic₄₇	#330	semidirect
130. C₄₇	C₄×#330	semidirect

H ≅ G/N

type

15. #15688864

C₄

semidirect

107. Dic₄₇

#330

semidirect

130. C₄₇

C₄×#330

semidirect

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

subgroup	index
#20245943	2
#97×Dic₄₇	2
C₂×#15688864	2
#20241269	2
C₄₇×#1295	2
C₄×#330	47

subgroup

index

#97×Dic₄₇

C₄₇×#1295

Subgroups — #26107195

Conjugacy classes of subgroups

Complements

Maximal subgroups