Group #26132466 — subgroups

Subgroups — #26132466

Order 12032.

362 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂×#15689541	C₂	1	1	yes	maximal, characteristic
2	#20245170	C₂	1	2	yes	maximal, complemented
3	#20243431	C₂	1	2	yes	maximal
4	#20241490	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13094030	C₂²	1	1	yes	characteristic
7	#15689541	—	2	2	no	—
8	#15689541	—	2	2	no	—
9	C₂×#13093963	C₄	1	2	yes	complemented
10	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₄₇	C₄	1	2	yes	complemented
12	#15689161	C₂²	1	2	yes	—
13	#15689096	—	2	2	no	—
14	#15688950	—	2	2	no	—
15	#15688948	—	2	2	no	—
16	#15688879	—	2	2	no	—
17	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#427	C₂²	1	2	yes	—
19	C₂×#10625350	D₄	1	1	yes	characteristic
20	#13094030	—	2	2	no	—
21	#13094030	—	2	2	no	—
22	C₂×#10625329	D₄	1	1	yes	characteristic
23	C₂×#10625324	Q₈	1	2	yes	—
24	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
25	#13093983	—	2	2	no	—
26	#13093963	—	2	4	no	—
27	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
30	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
31	C₁₆×D₄₇	—	2	2	no	—
32	C₁₆×D₄₇	—	2	2	no	—
33	#13093958	—	2	2	no	—
34	#13093956	—	2	2	no	—
35	#13093938	—	4	4	no	—
36	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
37	#117×C₄₇	—	2	2	no	—
38	M₃₂×C₄₇	—	2	2	no	—
39	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
40	#107×C₄₇	—	2	2	no	—
41	#106×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₄₇	D₈	1	2	yes	—
44	#10625350	—	2	2	no	—
45	#10625350	—	2	2	no	—
46	C₂×D₁₈₈	Q₁₆	1	1	yes	characteristic
47	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
48	C₂×Dic₉₄	Q₁₆	1	1	yes	characteristic
49	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₄₇	#41	1	2	yes	—
53	C₄⋊C₄×C₄₇	—	4	2	no	—
54	C₂²⋊C₄×C₄₇	—	4	2	no	—
55	C₄²×C₄₇	—	4	2	no	—
56	C₂²⋊Dic₄₇	—	4	2	no	—
57	C₄⋊Dic₄₇	—	4	2	no	—
58	C₄×Dic₄₇	—	4	2	no	—
59	#10625329	—	2	2	no	—
60	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
61	#10625324	—	2	4	no	—
62	C₈×D₄₇	—	2	2	no	—
63	C₈×D₄₇	—	2	2	no	—
64	C₁₆×C₄₇	—	2	2	no	—
65	C₄₇⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
66	C₄₇⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
67	C₄₇⋊C₁₆	C₂²⋊C₄	1	2	yes	—
68	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
69	C₂²×D₄₇	#107	1	2	yes	—
70	C₂×C₄×C₄₇	—	8	2	no	—
71	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#121	1	2	yes	—
73	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#135	1	2	yes	—
75	C₂²⋊D₄₇	—	2	4	no	—
76	C₂×Dic₄₇	—	8	2	no	—
77	C₂×Dic₄₇	#107	1	2	yes	—
78	D₁₈₈	—	2	2	no	—
79	C₄×D₄₇	—	2	2	no	—
80	C₄×D₄₇	—	2	2	no	—
81	Dic₉₄	—	2	2	no	—
82	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₈×C₄₇	#117	1	1	yes	characteristic
84	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
85	C₄₇⋊C₈	—	2	2	no	—
86	#12839	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	—	4	1	no	—
88	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
89	C₂²×C₄₇	#466	1	2	yes	—
90	C₂×D₄₇	—	2	2	no	—
91	C₂×D₄₇	—	4	2	no	—
92	C₄×C₄₇	—	16	2	no	—
93	C₄×C₄₇	#380	1	1	yes	characteristic
94	C₄×C₄₇	#427	1	1	yes	characteristic
95	C₄×C₄₇	#467	1	2	yes	—
96	Dic₄₇	—	2	2	no	—
97	Dic₄₇	—	16	2	no	—
98	C₂×#504	—	47	1	no	—
99	#2158	—	47	2	no	—
100	#1812	—	47	2	no	—
101	#1384	—	47	1	no	—
102	#1384	D₄₇	1	1	yes	characteristic, complemented
103	C₂×C₄₇	—	2	2	no	—
104	C₂×C₄₇	—	2	1	no	—
105	C₂×C₄₇	#1808	1	1	yes	characteristic
106	D₄₇	—	4	2	no	—
107	C₂×#131	—	47	1	no	—
108	#504	—	94	2	no	—
109	#504	—	94	2	no	—
110	C₂×M₃₂	—	47	2	no	—
111	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
112	C₂²×C₁₆	—	47	1	no	—
113	C₂²×C₁₆	—	47	2	no	—
114	#427	—	47	2	no	—
115	#427	C₂×D₄₇	1	2	yes	—
116	#410	—	94	2	no	—
117	#367	—	94	2	no	—
118	#366	—	94	2	no	—
119	H(3,ℤ/4)	—	94	2	no	—
120	C₄₇	#12839	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	—	47	1	no	—
122	#131	—	94	2	no	—
123	#131	—	94	2	no	—
124	C₂×M₁₆	—	47	1	no	—
125	C₂×M₁₆	—	47	2	no	—
126	C₂²×C₈	—	47	2	no	—
127	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
128	#117	—	2	2	no	—
129	#117	—	94	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	94	4	no	—
132	C₂×C₁₆	—	47	1	no	—
133	C₂×C₁₆	—	47	1	no	—
134	C₂×C₁₆	—	47	2	no	—
135	C₂×C₁₆	—	47	2	no	—
136	C₂×C₁₆	—	94	2	no	—
137	C₂×C₁₆	—	94	2	no	—
138	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
139	#107	—	2	2	no	—
140	#107	—	94	2	no	—
141	#106	—	2	2	no	—
142	#106	—	94	2	no	—
143	#99	—	188	4	no	—
144	#41	—	94	2	no	—
145	#41	—	94	2	no	—
146	C₂×Q₈	—	47	1	no	—
147	C₂×D₄	—	47	2	no	—
148	C₂×D₄	—	47	1	no	—
149	C₂²×C₄	—	47	2	no	—
150	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
151	M₁₆	—	94	2	no	—
152	M₁₆	—	94	4	no	—
153	C₂×C₈	—	47	2	no	—
154	C₂×C₈	—	94	2	no	—
155	C₂×C₈	—	94	2	no	—
156	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
157	C₂×C₈	#10625350	1	2	yes	—
158	C₂×C₈	D₄×D₄₇	1	2	yes	—
159	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	188	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	188	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	188	2	no	—
166	C₁₆	—	2	2	no	—
167	C₁₆	—	47	1	no	—
168	C₁₆	—	47	1	no	—
169	C₁₆	—	47	2	no	—
170	C₂³	—	47	2	no	—
171	C₂³	D₈×D₄₇	1	1	yes	characteristic
172	Q₈	—	94	2	no	—
173	D₄	—	94	4	no	—
174	D₄	—	94	2	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	47	2	no	—
177	C₂×C₄	—	94	2	no	—
178	C₂×C₄	—	94	2	no	—
179	C₂×C₄	—	376	2	no	—
180	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
181	C₂×C₄	#13093997	1	2	yes	—
182	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
183	C₂×C₄	#13094045	1	2	yes	—
184	C₈	—	94	2	no	—
185	C₈	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
186	C₈	#13093981	1	1	yes	characteristic
187	C₈	#13094002	1	2	yes	—
188	C₂²	—	4	1	no	—
189	C₂²	—	94	2	no	—
190	C₂²	—	188	2	no	—
191	C₂²	#107×D₄₇	1	1	yes	characteristic
192	C₂²	#15689358	1	2	yes	—
193	C₄	—	16	2	no	—
194	C₄	—	94	2	no	—
195	C₄	—	752	2	no	—
196	C₄	#15688982	1	1	yes	characteristic
197	C₄	#15689158	1	1	yes	characteristic
198	C₄	#15689363	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	188	2	no	—
202	C₂	#20243416	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689541

C₂

yes

maximal, characteristic

#20245170

C₂

yes

maximal, complemented

#20243431

C₂

yes

maximal

#20241490

C₂

yes

maximal, characteristic, complemented

C₄₇×#1384

C₂

yes

maximal, characteristic, complemented

C₂×#13094030

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₆×D₄₇

C₄

yes

complemented

#15689161

C₂²

yes

—

—

—

—

—

C₂×M₃₂×C₄₇

C₂²

yes

characteristic

C₄₇×#427

C₂²

yes

—

C₂×#10625350

D₄

yes

—

—

D₄

yes

characteristic

C₂×#10625324

Q₈

yes

—

C₂×C₈×D₄₇

C₂×C₄

yes

—

#13093983

—

#13093963

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₁₆×D₄₇

—

C₁₆×D₄₇

—

#13093958

—

#13093956

—

#13093938

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#117×C₄₇

—

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

#107×C₄₇

—

#106×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₄₇

D₈

yes

—

#10625350

—

#10625350

—

C₂×D₁₈₈

Q₁₆

yes

characteristic

C₂×C₄×D₄₇

C₄⋊C₄

yes

—

C₂×Dic₉₄

Q₁₆

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

#10625324

—

C₈×D₄₇

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₄₇⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₄₇⋊C₁₆

C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂×D₈

yes

characteristic

C₂²×D₄₇

#107

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#107

yes

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₂²⋊Q₈

yes

—

C₄₇⋊C₈

—

#12839

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#107

yes

characteristic

C₂²×C₄₇

#466

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

Dic₄₇

—

C₂×#504

—

#2158

—

100

#1812

—

101

#1384

—

102

#1384

D₄₇

yes

characteristic, complemented

103

C₂×C₄₇

—

104

C₂×C₄₇

—

105

C₂×C₄₇

#1808

yes

characteristic

106

D₄₇

—

107

C₂×#131

—

108

#504

—

109

#504

—

110

C₂×M₃₂

—

111

C₂×M₃₂

C₂×D₄₇

yes

characteristic

112

C₂²×C₁₆

—

113

C₂²×C₁₆

—

114

#427

—

115

#427

C₂×D₄₇

yes

—

116

#410

—

117

#367

—

118

#366

—

119

H(3,ℤ/4)

—

120

C₄₇

#12839

yes

characteristic, complemented, Sylow

121

C₂×#41

—

122

#131

—

123

#131

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂²×D₄₇

yes

characteristic

128

#117

—

129

#117

—

130

M₃₂

—

131

M₃₂

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

C₄×D₄₇

yes

—

139

#107

—

140

#107

—

141

#106

—

142

#106

—

143

#99

—

188

—

144

#41

—

145

#41

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₄₇

yes

characteristic

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

157

C₂×C₈

#10625350

yes

—

158

C₂×C₈

D₄×D₄₇

yes

—

159

C₂×C₈

Q₈×D₄₇

yes

characteristic

160

C₄⋊C₄

—

161

C₄⋊C₄

—

188

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

188

—

164

C₄²

—

165

C₄²

—

188

—

166

C₁₆

—

167

C₁₆

—

168

C₁₆

—

169

C₁₆

—

170

C₂³

—

171

C₂³

D₈×D₄₇

yes

characteristic

172

Q₈

—

173

D₄

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

376

—

180

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

181

C₂×C₄

#13093997

yes

—

182

C₂×C₄

Q₁₆×D₄₇

yes

characteristic

183

C₂×C₄

#13094045

yes

—

184

C₈

—

185

C₈

C₂²⋊C₄×D₄₇

yes

characteristic

186

C₈

#13093981

yes

characteristic

187

C₈

#13094002

yes

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

188

—

191

C₂²

#107×D₄₇

yes

characteristic

192

C₂²

#15689358

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

752

—

196

C₄

#15688982

yes

characteristic

197

C₄

#15689158

yes

characteristic

198

C₄

#15689363

yes

—

199

C₂

—

200

C₂

—

201

C₂

—

188

—

202

C₂

#20243416

yes

characteristic

	subgroup	G/H
center	C₄	#15689158
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1384	C₂
socle	C₂×C₄₇	#1808
prime radical	C₂×#10625350	D₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1384	D₄₇
hypocenter	C₄₇	#12839
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#15689158

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1384

C₂

C₂×C₄₇

D₄

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
2. #20245170	C₂	semidirect
4. #20241490	C₂	semidirect
5. C₄₇×#1384	C₂	semidirect
9. C₂×#13093963	C₄	semidirect
11. C₂×C₁₆×D₄₇	C₄	semidirect
102. #1384	D₄₇	semidirect
120. C₄₇	#12839	semidirect

H ≅ G/N

type

2. #20245170

C₂

semidirect

4. #20241490

C₂

semidirect

5. C₄₇×#1384

C₂

semidirect

9. C₂×#13093963

C₄

semidirect

11. C₂×C₁₆×D₄₇

C₄

semidirect

102. #1384

D₄₇

semidirect

120. C₄₇

#12839

semidirect

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

subgroup	index
C₂×#15689541	2
#20245170	2
#20243431	2
#20241490	2
C₄₇×#1384	2
#12839	47

subgroup

index

C₄₇×#1384

Subgroups — #26132466

Conjugacy classes of subgroups

Complements

Maximal subgroups