Group #26230174 — subgroups

Subgroups — #26230174

Order 12032.

826 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 211 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	#20253834	C₂	1	1	yes	maximal, characteristic
2	#20252107	C₂	1	2	yes	maximal, complemented
3	#20244216	C₂	1	2	yes	maximal, complemented
4	#20243714	C₂	1	8	yes	maximal, complemented
5	#20243529	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#1745	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13094046	C₂²	1	4	yes	—
8	C₂×#13094030	C₂²	1	4	yes	—
9	#15689673	C₂²	1	2	yes	—
10	#15689517	C₄	1	4	yes	complemented
11	#15689512	C₄	1	4	yes	complemented
12	#15689196	—	2	8	no	—
13	#15689147	C₂²	1	2	yes	—
14	C₂×#13093952	C₂²	1	8	yes	—
15	#15689089	C₂²	1	4	yes	—
16	#15688888	C₂²	1	4	yes	—
17	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#336	C₂²	1	4	yes	—
21	C₂×#10625350	C₂³	1	4	yes	—
22	#13094046	—	4	8	no	—
23	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
24	C₂×#10625325	C₂×C₄	1	8	yes	—
25	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
26	#13094030	—	4	8	no	—
27	C₂×#10625329	C₂³	1	4	yes	—
28	C₂×#10625324	—	2	4	no	—
29	C₂×C₈×D₄₇	—	2	4	no	—
30	#13094024	C₂×C₄	1	2	yes	—
31	#13094021	C₂×C₄	1	2	yes	—
32	#13094012	C₂×C₄	1	4	yes	—
33	#13093998	—	2	4	no	—
34	C₄×D₁₈₈	—	2	4	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
36	#13093952	—	4	8	no	—
37	(C₂×D₄₇)⋊C₈	—	2	8	no	—
38	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	#97×C₄₇	C₂×C₄	1	4	yes	—
44	#97×C₄₇	D₄	1	4	yes	—
45	C₂²×C₄×C₄₇	—	2	2	no	—
46	C₂²×C₄×C₄₇	#41	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂×C₂²⋊D₄₇	—	2	4	no	—
49	#10625350	—	2	8	no	—
50	#10625350	—	2	8	no	—
51	C₂×D₁₈₈	—	2	4	no	—
52	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
53	C₂×C₄×D₄₇	—	2	4	no	—
54	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₄₇	—	2	4	no	—
56	C₂×C₈×C₄₇	—	2	4	no	—
57	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
59	C₄²×C₄₇	—	2	4	no	—
60	C₄²×C₄₇	—	4	4	no	—
61	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
62	C₄²×C₄₇	C₂×D₄	1	2	yes	—
63	(C₂×D₄₇)⋊C₄	—	2	8	no	—
64	C₄⋊Dic₄₇	—	2	4	no	—
65	#10625329	—	2	8	no	—
66	C₂×C₄₇⋊C₈	—	2	4	no	—
67	Dic₁₈₈	—	4	4	no	—
68	D₃₇₆	—	4	4	no	—
69	#10625325	—	4	8	no	—
70	#10625324	—	4	8	no	—
71	C₈×D₄₇	—	4	8	no	—
72	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
73	C₂²×D₄₇	—	2	4	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	4	2	no	—
76	C₂×C₄×C₄₇	—	4	2	no	—
77	C₂×C₄×C₄₇	—	4	2	no	—
78	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
79	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
82	C₂²⋊D₄₇	—	4	8	no	—
83	C₂×Dic₄₇	—	2	4	no	—
84	D₁₈₈	—	2	8	no	—
85	D₁₈₈	—	8	4	no	—
86	C₄×D₄₇	—	4	4	no	—
87	C₄×D₄₇	—	4	4	no	—
88	Dic₉₄	—	2	8	no	—
89	C₈×C₄₇	—	2	4	no	—
90	C₈×C₄₇	—	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	C₂×C₄×D₄	1	1	yes	characteristic
95	C₂²×C₄₇	#540	1	2	yes	—
96	C₂×D₄₇	—	4	4	no	—
97	C₂×D₄₇	—	4	8	no	—
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	—	47	1	no	—
104	#2932	—	47	2	no	—
105	#1962	—	47	2	no	—
106	#1876	—	47	8	no	—
107	#1838	—	47	1	no	—
108	#1745	D₄₇	1	1	yes	characteristic, complemented
109	C₂×C₄₇	—	2	2	no	—
110	C₂×C₄₇	#2328	1	1	yes	characteristic
111	C₂×C₄₇	#2956	1	2	yes	—
112	D₄₇	—	8	4	no	—
113	C₂×#135	—	47	4	no	—
114	C₂×#131	—	47	4	no	—
115	#532	—	47	2	no	—
116	#497	—	47	4	no	—
117	#496	—	47	4	no	—
118	#435	—	94	8	no	—
119	#423	—	47	2	no	—
120	C₂×#104	—	47	8	no	—
121	#408	—	47	4	no	—
122	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
123	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
124	C₄³	C₂×D₄₇	1	1	yes	characteristic
125	#341	—	47	4	no	—
126	#336	C₂×D₄₇	1	4	yes	—
127	C₄₇	#22283	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	—	47	4	no	—
129	#135	—	188	8	no	—
130	C₂×Q₁₆	—	47	4	no	—
131	C₂×SD₁₆	—	47	8	no	—
132	C₂×D₈	—	47	4	no	—
133	#131	—	188	8	no	—
134	C₂×M₁₆	—	47	4	no	—
135	C₂×M₁₆	—	94	4	no	—
136	C₂²×C₈	—	94	4	no	—
137	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
138	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
139	C₄⋊Q₈	—	47	2	no	—
140	#127	—	47	2	no	—
141	#124	—	47	4	no	—
142	#121	—	94	4	no	—
143	C₄×D₄	—	94	4	no	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
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	C₄×D₄₇	1	4	yes	—
151	#97	D₁₈₈	1	4	yes	—
152	#41	—	94	8	no	—
153	#41	—	94	8	no	—
154	C₂×Q₈	—	47	4	no	—
155	C₂×D₄	—	47	4	no	—
156	C₂×D₄	—	94	4	no	—
157	C₂×D₄	—	94	4	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	94	4	no	—
160	C₂²×C₄	#10625350	1	2	yes	—
161	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
162	Q₁₆	—	188	4	no	—
163	SD₁₆	—	188	8	no	—
164	D₈	—	188	4	no	—
165	M₁₆	—	94	8	no	—
166	M₁₆	—	188	8	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	94	4	no	—
170	C₂×C₈	—	188	8	no	—
171	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
172	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
173	C₄⋊C₄	—	94	4	no	—
174	C₂²⋊C₄	—	94	8	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	4	4	no	—
177	C₄²	C₂×C₄×D₄₇	1	2	yes	—
178	C₄²	C₂×D₁₈₈	1	2	yes	—
179	C₂³	—	94	4	no	—
180	C₂³	C₂×#10625350	1	1	yes	characteristic
181	Q₈	—	94	8	no	—
182	D₄	—	94	8	no	—
183	D₄	—	376	4	no	—
184	D₄	—	188	8	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	94	4	no	—
190	C₂×C₄	—	188	4	no	—
191	C₂×C₄	—	188	4	no	—
192	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
193	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
194	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
195	C₂×C₄	#13094070	1	4	yes	—
196	C₈	—	2	4	no	—
197	C₈	—	2	4	no	—
198	C₈	—	188	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	188	4	no	—
201	C₂²	—	188	8	no	—
202	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
203	C₂²	#15689728	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	4	4	no	—
206	C₄	—	188	4	no	—
207	C₄	#15689597	1	2	yes	—
208	C₄	#15689692	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	376	4	no	—
211	C₂	#20246113	1	1	yes	characteristic
212	C₂	#20252299	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20253834

C₂

yes

maximal, characteristic

#20252107

C₂

yes

maximal, complemented

#20244216

C₂

yes

maximal, complemented

#20243714

C₂

yes

maximal, complemented

#20243529

C₂

yes

maximal, characteristic, complemented

C₄₇×#1745

C₂

yes

maximal, characteristic, complemented

C₂×#13094046

C₂²

yes

—

C₂×#13094030

C₂²

yes

—

#15689673

C₂²

yes

—

#15689517

C₄

yes

complemented

#15689512

C₄

yes

complemented

#15689196

—

#15689147

C₂²

yes

—

C₂×#13093952

C₂²

yes

—

#15689089

C₂²

yes

—

#15688888

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₂×#97×C₄₇

C₂²

yes

characteristic

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

C₂²

yes

—

C₂×#10625350

C₂³

yes

—

#13094046

—

C₂×Dic₁₈₈

C₂×C₄

yes

—

C₂×#10625325

C₂×C₄

yes

—

C₂×D₃₇₆

C₂×C₄

yes

—

#13094030

—

C₂×#10625329

C₂³

yes

—

C₂×#10625324

—

C₂×C₈×D₄₇

—

#13094024

C₂×C₄

yes

—

#13094021

C₂×C₄

yes

—

#13094012

C₂×C₄

yes

—

#13093998

—

C₄×D₁₈₈

—

C₄⋊(C₄₇⋊C₈)

—

#13093952

—

(C₂×D₄₇)⋊C₈

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#97×C₄₇

C₂×C₄

yes

—

#97×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

#41

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂²×C₄

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂²×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

D₃₇₆

—

#10625325

—

#10625324

—

C₈×D₄₇

—

C₂³×C₄₇

C₂×#41

yes

characteristic

C₂²×D₄₇

—

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

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#22283

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₇

#540

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

100

C₄×C₄₇

#518

yes

—

101

C₄×C₄₇

#535

yes

—

102

Dic₄₇

—

103

#3150

—

104

#2932

—

105

#1962

—

106

#1876

—

107

#1838

—

108

#1745

D₄₇

yes

characteristic, complemented

109

C₂×C₄₇

—

110

C₂×C₄₇

#2328

yes

characteristic

111

C₂×C₄₇

#2956

yes

—

112

D₄₇

—

113

C₂×#135

—

114

C₂×#131

—

115

#532

—

116

#497

—

117

#496

—

118

#435

—

119

#423

—

120

C₂×#104

—

121

#408

—

122

C₂×#97

C₂×D₄₇

yes

characteristic

123

C₂×#97

C₂×D₄₇

yes

characteristic

124

C₄³

C₂×D₄₇

yes

characteristic

125

#341

—

126

#336

C₂×D₄₇

yes

—

127

C₄₇

#22283

yes

characteristic, complemented, Sylow

128

C₂×#41

—

129

#135

—

188

—

130

C₂×Q₁₆

—

131

C₂×SD₁₆

—

132

C₂×D₈

—

133

#131

—

188

—

134

C₂×M₁₆

—

135

C₂×M₁₆

—

136

C₂²×C₈

—

137

C₂²×C₈

C₂²×D₄₇

yes

—

138

C₂²×C₈

C₂²×D₄₇

yes

—

139

C₄⋊Q₈

—

140

#127

—

141

#124

—

142

#121

—

143

C₄×D₄

—

144

C₂×C₄²

—

145

C₂×C₄²

C₂²×D₄₇

yes

—

146

C₂×C₄²

C₂²×D₄₇

yes

characteristic

147

C₄⋊C₈

—

148

#104

—

188

—

149

C₂²⋊C₈

—

150

#97

C₄×D₄₇

yes

—

151

#97

D₁₈₈

yes

—

152

#41

—

153

#41

—

154

C₂×Q₈

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

#10625350

yes

—

161

C₂²×C₄

C₂³×D₄₇

yes

characteristic

162

Q₁₆

—

188

—

163

SD₁₆

—

188

—

164

D₈

—

188

—

165

M₁₆

—

166

M₁₆

—

188

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

188

—

171

C₂×C₈

C₂×C₄×D₄₇

yes

—

172

C₂×C₈

C₂×D₁₈₈

yes

—

173

C₄⋊C₄

—

174

C₂²⋊C₄

—

175

C₄²

—

176

C₄²

—

177

C₄²

C₂×C₄×D₄₇

yes

—

178

C₄²

C₂×D₁₈₈

yes

—

179

C₂³

—

180

C₂³

C₂×#10625350

yes

characteristic

181

Q₈

—

182

D₄

—

183

D₄

—

376

—

184

D₄

—

188

—

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

C₄×D₁₈₈

yes

—

193

C₂×C₄

C₂²×C₄×D₄₇

yes

characteristic

194

C₂×C₄

C₂²×D₁₈₈

yes

characteristic

195

C₂×C₄

#13094070

yes

—

196

C₈

—

197

C₈

—

198

C₈

—

188

—

199

C₂²

—

200

C₂²

—

188

—

201

C₂²

—

188

—

202

C₂²

C₂×C₄×D₁₈₈

yes

characteristic

203

C₂²

#15689728

yes

—

204

C₄

—

205

C₄

—

206

C₄

—

188

—

207

C₄

#15689597

yes

—

208

C₄

#15689692

yes

—

209

C₂

—

210

C₂

—

376

—

211

C₂

#20246113

yes

characteristic

212

C₂

#20252299

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1745	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#20253834	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1745	D₄₇
hypocenter	C₄₇	#22283
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1745

subgroup

G/H

center

C₂×C₄

C₂²×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₇

Fitting

C₄₇×#1745

C₂

socle

C₂²×C₄₇

C₂×C₄×D₄

prime radical

#20253834

C₂

cosocle

C₂²×C₄×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1745

N	H ≅ G/N	type
2. #20252107	C₂	semidirect
3. #20244216	C₂	semidirect
4. #20243714	C₂	semidirect
5. #20243529	C₂	semidirect
6. C₄₇×#1745	C₂	semidirect
10. #15689517	C₄	semidirect
11. #15689512	C₄	semidirect
108. #1745	D₄₇	semidirect
127. C₄₇	#22283	semidirect

H ≅ G/N

type

2. #20252107

C₂

semidirect

3. #20244216

C₂

semidirect

4. #20243714

C₂

semidirect

5. #20243529

C₂

semidirect

6. C₄₇×#1745

C₂

semidirect

10. #15689517

C₄

semidirect

11. #15689512

C₄

semidirect

108. #1745

D₄₇

semidirect

127. C₄₇

#22283

semidirect

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

subgroup	index
#20253834	2
#20252107	2
#20244216	2
#20243714	2
#20243529	2
C₄₇×#1745	2
#22283	47

subgroup

index

C₄₇×#1745

Subgroups — #26230174

Conjugacy classes of subgroups

Complements

Maximal subgroups