Group #26230175 — subgroups

Subgroups — #26230175

Order 12032.

826 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 243 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	#20252106	C₂	1	2	yes	maximal, complemented
2	#20244217	C₂	1	2	yes	maximal, complemented
3	#20243717	C₂	1	8	yes	maximal, complemented
4	#20243529	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243136	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#3150	C₂	1	1	yes	maximal, characteristic
7	C₂×#13094031	C₂²	1	4	yes	—
8	#15689195	—	2	8	no	—
9	#15689147	C₂²	1	2	yes	—
10	C₂×#13093951	C₂²	1	8	yes	—
11	#15689090	C₂²	1	4	yes	—
12	C₂×#13093935	C₂²	1	1	yes	characteristic
13	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
14	#15688888	C₂²	1	4	yes	—
15	#15688877	C₂²	1	4	yes	—
16	C₂×#135×C₄₇	C₂²	1	4	yes	—
17	C₄₇×#532	C₂²	1	2	yes	—
18	C₄₇×#497	C₄	1	4	yes	complemented
19	C₄₇×#496	C₄	1	4	yes	complemented
20	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
21	#13094031	—	4	8	no	—
22	C₂×#10625329	—	2	4	no	—
23	C₂×#10625329	C₂³	1	4	yes	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
28	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
29	#13093951	—	4	8	no	—
30	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
31	#13093935	D₄	1	4	yes	—
32	C₂×#41×C₄₇	C₂³	1	4	yes	—
33	#135×C₄₇	—	4	8	no	—
34	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
35	C₂×SD₁₆×C₄₇	C₂×C₄	1	8	yes	—
36	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
39	#127×C₄₇	C₂×C₄	1	2	yes	—
40	#124×C₄₇	C₂×C₄	1	4	yes	—
41	#121×C₄₇	—	2	4	no	—
42	C₄×D₄×C₄₇	—	2	4	no	—
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	#97×C₄₇	C₂×C₄	1	4	yes	—
45	#41×C₄₇	—	2	8	no	—
46	#41×C₄₇	—	2	8	no	—
47	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
48	C₂×D₄×C₄₇	—	2	4	no	—
49	C₂×D₄×C₄₇	—	2	4	no	—
50	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	#41	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	2	2	no	—
55	Q₁₆×C₄₇	—	4	4	no	—
56	SD₁₆×C₄₇	—	4	8	no	—
57	D₈×C₄₇	—	4	4	no	—
58	C₂×C₈×C₄₇	—	2	4	no	—
59	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
60	C₄⋊C₄×C₄₇	—	2	4	no	—
61	C₂²⋊C₄×C₄₇	—	2	8	no	—
62	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
63	C₄²×C₄₇	C₂×D₄	1	2	yes	—
64	C₄×Dic₄₇	—	2	4	no	—
65	C₄×Dic₄₇	—	4	4	no	—
66	#10625329	—	2	8	no	—
67	#10625329	—	4	8	no	—
68	C₂×C₄₇⋊C₈	—	2	4	no	—
69	C₂×C₄₇⋊C₈	—	2	4	no	—
70	C₂×C₄₇⋊C₈	—	4	8	no	—
71	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
72	C₂³×C₄₇	—	2	4	no	—
73	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
74	Q₈×C₄₇	—	2	8	no	—
75	D₄×C₄₇	—	2	8	no	—
76	D₄×C₄₇	—	4	8	no	—
77	D₄×C₄₇	—	8	4	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	—	4	4	no	—
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
83	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
86	C₂×Dic₄₇	—	2	4	no	—
87	C₂×Dic₄₇	—	4	2	no	—
88	C₂×Dic₄₇	—	4	2	no	—
89	C₈×C₄₇	—	2	4	no	—
90	C₄₇⋊C₈	—	2	4	no	—
91	C₄₇⋊C₈	—	4	4	no	—
92	#22283	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	—	2	2	no	—
94	C₂²×C₄₇	—	4	4	no	—
95	C₂²×C₄₇	—	4	8	no	—
96	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂²×C₄₇	#540	1	2	yes	—
98	C₄×C₄₇	—	2	4	no	—
99	C₄×C₄₇	—	4	4	no	—
100	C₄×C₄₇	#518	1	2	yes	—
101	C₄×C₄₇	#535	1	2	yes	—
102	Dic₄₇	—	4	4	no	—
103	#3150	D₄₇	1	1	yes	characteristic
104	#2932	—	47	2	no	—
105	#1962	—	47	2	no	—
106	#1876	—	47	8	no	—
107	#1838	—	47	1	no	—
108	#1745	—	47	1	no	—
109	C₂×C₄₇	—	2	2	no	—
110	C₂×C₄₇	—	8	4	no	—
111	C₂×C₄₇	#2328	1	1	yes	characteristic
112	C₂×C₄₇	#2956	1	2	yes	—
113	C₂×#135	C₂×D₄₇	1	4	yes	—
114	C₂×#131	—	47	4	no	—
115	#532	C₂×D₄₇	1	2	yes	—
116	#497	Dic₄₇	1	4	yes	complemented
117	#496	Dic₄₇	1	4	yes	complemented
118	#435	—	94	8	no	—
119	#423	—	47	2	no	—
120	C₂×#104	—	47	8	no	—
121	#408	—	47	4	no	—
122	C₂×#97	—	47	1	no	—
123	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
124	C₄³	—	47	1	no	—
125	#341	—	47	4	no	—
126	#336	—	47	4	no	—
127	C₄₇	#22283	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	C₂²×D₄₇	1	4	yes	—
129	#135	—	4	8	no	—
130	C₂×Q₁₆	C₂×Dic₄₇	1	4	yes	—
131	C₂×SD₁₆	C₂×Dic₄₇	1	8	yes	—
132	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
133	#131	—	188	8	no	—
134	C₂×M₁₆	—	47	4	no	—
135	C₂×M₁₆	—	94	4	no	—
136	C₂²×C₈	—	47	2	no	—
137	C₂²×C₈	—	94	4	no	—
138	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
139	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
140	#127	C₂×Dic₄₇	1	2	yes	—
141	#124	C₂×Dic₄₇	1	4	yes	—
142	#121	—	2	4	no	—
143	C₄×D₄	—	2	4	no	—
144	C₂×C₄²	—	47	2	no	—
145	C₂×C₄²	—	94	2	no	—
146	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
147	C₄⋊C₈	—	94	4	no	—
148	#104	—	188	8	no	—
149	C₂²⋊C₈	—	94	8	no	—
150	#97	—	47	4	no	—
151	#97	C₂×Dic₄₇	1	4	yes	—
152	#41	—	2	8	no	—
153	#41	—	2	8	no	—
154	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
155	C₂×D₄	—	2	4	no	—
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	94	2	no	—
160	C₂²×C₄	#10625352	1	2	yes	—
161	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
162	Q₁₆	—	4	4	no	—
163	SD₁₆	—	4	8	no	—
164	D₈	—	4	4	no	—
165	M₁₆	—	94	8	no	—
166	M₁₆	—	188	8	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	47	4	no	—
169	C₂×C₈	—	94	4	no	—
170	C₂×C₈	—	94	4	no	—
171	C₂×C₈	—	188	8	no	—
172	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
173	C₄⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	8	no	—
175	C₄²	—	94	4	no	—
176	C₄²	—	188	4	no	—
177	C₄²	D₄×D₄₇	1	2	yes	—
178	C₄²	C₂²×Dic₄₇	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×#10625352	1	1	yes	characteristic
181	Q₈	—	2	8	no	—
182	D₄	—	2	8	no	—
183	D₄	—	4	8	no	—
184	D₄	—	8	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	94	4	no	—
190	C₂×C₄	—	188	2	no	—
191	C₂×C₄	—	188	2	no	—
192	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
193	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
194	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
195	C₂×C₄	#13094069	1	4	yes	—
196	C₈	—	2	4	no	—
197	C₈	—	94	4	no	—
198	C₈	—	188	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	4	no	—
201	C₂²	—	4	8	no	—
202	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
203	C₂²	#15689727	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	4	4	no	—
206	C₄	—	188	4	no	—
207	C₄	#15689598	1	2	yes	—
208	C₄	#15689686	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	8	4	no	—
211	C₂	#20246114	1	1	yes	characteristic
212	C₂	#20252298	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20252106

C₂

yes

maximal, complemented

#20244217

C₂

yes

maximal, complemented

#20243717

C₂

yes

maximal, complemented

#20243529

C₂

yes

maximal, characteristic, complemented

#20243136

C₂

yes

maximal, characteristic, complemented

C₄₇×#3150

C₂

yes

maximal, characteristic

C₂×#13094031

C₂²

yes

—

#15689195

—

#15689147

C₂²

yes

—

C₂×#13093951

C₂²

yes

—

#15689090

C₂²

yes

—

C₂×#13093935

C₂²

yes

characteristic

C₄²×Dic₄₇

C₂²

yes

characteristic

#15688888

C₂²

yes

—

#15688877

C₂²

yes

—

C₂×#135×C₄₇

C₂²

yes

—

C₄₇×#532

C₂²

yes

—

C₄₇×#497

C₄

yes

complemented

C₄₇×#496

C₄

yes

complemented

C₂×#97×C₄₇

C₂²

yes

—

—

C₂³

yes

—

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

C₂³

yes

—

C₄⋊(C₄₇⋊C₈)

—

#13093951

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

D₄

yes

—

C₂×#41×C₄₇

C₂³

yes

—

#135×C₄₇

—

C₂×Q₁₆×C₄₇

C₂×C₄

yes

—

C₂×SD₁₆×C₄₇

C₂×C₄

yes

—

C₂×D₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

—

#127×C₄₇

C₂×C₄

yes

—

#124×C₄₇

C₂×C₄

yes

—

#121×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#97×C₄₇

C₂×C₄

yes

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

C₂²×C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

#41

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

—

Q₁₆×C₄₇

—

SD₁₆×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×#41

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₇

H(5,𝔽₂)

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#22283

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₇

#540

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

100

C₄×C₄₇

#518

yes

—

101

C₄×C₄₇

#535

yes

—

102

Dic₄₇

—

103

#3150

D₄₇

yes

characteristic

104

#2932

—

105

#1962

—

106

#1876

—

107

#1838

—

108

#1745

—

109

C₂×C₄₇

—

110

C₂×C₄₇

—

111

C₂×C₄₇

#2328

yes

characteristic

112

C₂×C₄₇

#2956

yes

—

113

C₂×#135

C₂×D₄₇

yes

—

114

C₂×#131

—

115

#532

C₂×D₄₇

yes

—

116

#497

Dic₄₇

yes

complemented

117

#496

Dic₄₇

yes

complemented

118

#435

—

119

#423

—

120

C₂×#104

—

121

#408

—

122

C₂×#97

—

123

C₂×#97

C₂×D₄₇

yes

characteristic

124

C₄³

—

125

#341

—

126

#336

—

127

C₄₇

#22283

yes

characteristic, complemented, Sylow

128

C₂×#41

C₂²×D₄₇

yes

—

129

#135

—

130

C₂×Q₁₆

C₂×Dic₄₇

yes

—

131

C₂×SD₁₆

C₂×Dic₄₇

yes

—

132

C₂×D₈

C₂×Dic₄₇

yes

—

133

#131

—

188

—

134

C₂×M₁₆

—

135

C₂×M₁₆

—

136

C₂²×C₈

—

137

C₂²×C₈

—

138

C₂²×C₈

C₂²×D₄₇

yes

—

139

C₄⋊Q₈

C₂×Dic₄₇

yes

—

140

#127

C₂×Dic₄₇

yes

—

141

#124

C₂×Dic₄₇

yes

—

142

#121

—

143

C₄×D₄

—

144

C₂×C₄²

—

145

C₂×C₄²

—

146

C₂×C₄²

C₂²×D₄₇

yes

characteristic

147

C₄⋊C₈

—

148

#104

—

188

—

149

C₂²⋊C₈

—

150

#97

—

151

#97

C₂×Dic₄₇

yes

—

152

#41

—

153

#41

—

154

C₂×Q₈

C₂²×Dic₄₇

yes

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

C₂²×Dic₄₇

yes

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

#10625352

yes

—

161

C₂²×C₄

C₂³×D₄₇

yes

characteristic

162

Q₁₆

—

163

SD₁₆

—

164

D₈

—

165

M₁₆

—

166

M₁₆

—

188

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

188

—

172

C₂×C₈

C₂²×Dic₄₇

yes

—

173

C₄⋊C₄

—

174

C₂²⋊C₄

—

175

C₄²

—

176

C₄²

—

188

—

177

C₄²

D₄×D₄₇

yes

—

178

C₄²

C₂²×Dic₄₇

yes

—

179

C₂³

—

180

C₂³

C₂×#10625352

yes

characteristic

181

Q₈

—

182

D₄

—

183

D₄

—

184

D₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

188

—

191

C₂×C₄

—

188

—

192

C₂×C₄

D₄×Dic₄₇

yes

—

193

C₂×C₄

C₂³×Dic₄₇

yes

characteristic

194

C₂×C₄

C₂×D₄×D₄₇

yes

characteristic

195

C₂×C₄

#13094069

yes

—

196

C₈

—

197

C₈

—

198

C₈

—

188

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

C₂×D₄×Dic₄₇

yes

characteristic

203

C₂²

#15689727

yes

—

204

C₄

—

205

C₄

—

206

C₄

—

188

—

207

C₄

#15689598

yes

—

208

C₄

#15689686

yes

—

209

C₂

—

210

C₂

—

211

C₂

#20246114

yes

characteristic

212

C₂

#20252298

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3150	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#3150	D₄₇
hypocenter	C₄₇	#22283
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3150

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₇

Fitting, prime radical

C₄₇×#3150

C₂

socle

C₂²×C₄₇

C₂×C₄×D₄

cosocle

C₂²×C₄×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3150

N	H ≅ G/N	type
1. #20252106	C₂	semidirect
2. #20244217	C₂	semidirect
3. #20243717	C₂	semidirect
4. #20243529	C₂	semidirect
5. #20243136	C₂	semidirect
18. C₄₇×#497	C₄	semidirect
19. C₄₇×#496	C₄	semidirect
116. #497	Dic₄₇	semidirect
117. #496	Dic₄₇	semidirect
127. C₄₇	#22283	semidirect

H ≅ G/N

type

1. #20252106

C₂

semidirect

2. #20244217

C₂

semidirect

3. #20243717

C₂

semidirect

4. #20243529

C₂

semidirect

5. #20243136

C₂

semidirect

18. C₄₇×#497

C₄

semidirect

19. C₄₇×#496

C₄

semidirect

116. #497

Dic₄₇

semidirect

117. #496

Dic₄₇

semidirect

127. C₄₇

#22283

semidirect

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

subgroup	index
#20252106	2
#20244217	2
#20243717	2
#20243529	2
#20243136	2
C₄₇×#3150	2
#22283	47

subgroup

index

C₄₇×#3150

Subgroups — #26230175

Conjugacy classes of subgroups

Complements

Maximal subgroups