Group #26114905 — subgroups

Subgroups — #26114905

Order 12032.

294 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20242140	C₂	1	2	yes	maximal
2	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
3	#20241680	C₂	1	1	yes	maximal, characteristic
4	#20241239	C₂	1	2	yes	maximal
5	C₄×#108×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	1	yes	characteristic
7	#15689176	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	#15688876	—	2	2	no	—
10	#15688876	—	2	2	no	—
11	#15688874	—	2	2	no	—
12	#15688845	—	2	2	no	—
13	C₂×#108×C₄₇	C₄	1	2	yes	—
14	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#341	C₄	1	1	yes	characteristic
17	C₄₇×#341	C₄	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
23	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
24	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
25	C₂²⋊(C₄₇⋊C₈)	Q₈	1	2	yes	—
26	C₄×C₄₇⋊C₈	—	2	1	no	—
27	C₄×C₄₇⋊C₈	—	2	1	no	—
28	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	#108×C₄₇	—	2	4	no	—
34	#97×C₄₇	—	2	2	no	—
35	C₄×C₈×C₄₇	—	2	2	no	—
36	C₄×C₈×C₄₇	D₄	1	2	yes	—
37	C₄×C₈×C₄₇	Q₈	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
41	M₁₆×C₄₇	—	2	2	no	—
42	M₁₆×C₄₇	—	2	2	no	—
43	M₁₆×C₄₇	—	2	2	no	—
44	M₁₆×C₄₇	M₁₆	1	4	yes	—
45	C₂×C₈×C₄₇	—	2	2	no	—
46	C₂×C₈×C₄₇	—	2	2	no	—
47	C₂×C₈×C₄₇	—	2	2	no	—
48	C₂×C₈×C₄₇	—	2	2	no	—
49	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
51	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₄₇	#41	1	1	yes	characteristic
55	C₂×C₄₇⋊C₈	—	2	1	no	—
56	C₂×C₄₇⋊C₈	—	2	1	no	—
57	C₂×C₄₇⋊C₈	—	2	2	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂×C₄₇⋊C₈	—	8	2	no	—
60	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	—	2	1	no	—
63	C₂×C₄×C₄₇	—	4	1	no	—
64	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₄₇	#131	1	2	yes	—
72	C₈×C₄₇	—	2	2	no	—
73	C₈×C₄₇	—	2	2	no	—
74	C₈×C₄₇	—	2	4	no	—
75	C₈×C₄₇	—	4	2	no	—
76	C₄₇⋊C₈	—	4	2	no	—
77	C₄₇⋊C₈	—	4	1	no	—
78	C₄₇⋊C₈	—	4	1	no	—
79	C₄₇⋊C₈	—	8	2	no	—
80	#9610	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	—	2	1	no	—
82	C₂²×C₄₇	—	2	1	no	—
83	C₂²×C₄₇	#389	1	1	yes	characteristic
84	C₂²×C₄₇	#446	1	2	yes	—
85	C₄×C₄₇	—	2	1	no	—
86	C₄×C₄₇	—	2	1	no	—
87	C₄×C₄₇	—	2	1	no	—
88	C₄×C₄₇	#423	1	1	yes	characteristic
89	C₄×C₄₇	#424	1	1	yes	characteristic
90	C₄×C₄₇	#435	1	2	yes	—
91	C₄×C₄₇	#472	1	1	yes	characteristic
92	C₄×C₄₇	#473	1	1	yes	characteristic
93	C₄×#108	D₄₇	1	1	yes	characteristic
94	#1565	—	47	2	no	—
95	C₂×#335	—	47	1	no	—
96	#1452	—	47	1	no	—
97	#1283	—	47	2	no	—
98	C₂×C₄₇	—	2	2	no	—
99	C₂×C₄₇	#1940	1	1	yes	characteristic
100	C₂×C₄₇	#1952	1	1	yes	characteristic
101	C₂×C₄₇	#1953	1	1	yes	characteristic
102	#431	—	47	1	no	—
103	#431	—	47	1	no	—
104	C₂×#108	Dic₄₇	1	2	yes	—
105	C₂×C₄⋊C₈	—	47	2	no	—
106	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
107	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
108	#341	Dic₄₇	1	1	yes	characteristic
109	#341	Dic₄₇	1	1	yes	characteristic
110	#335	—	94	2	no	—
111	#335	—	94	2	no	—
112	#334	—	94	2	no	—
113	#322	—	94	2	no	—
114	C₄₇	#9610	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
116	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
117	C₂²×C₈	—	94	2	no	—
118	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
119	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
121	#108	—	2	4	no	—
122	C₄⋊C₈	—	47	1	no	—
123	C₄⋊C₈	—	47	1	no	—
124	C₄⋊C₈	—	94	2	no	—
125	C₄⋊C₈	—	94	2	no	—
126	C₄⋊C₈	—	94	1	no	—
127	C₂²⋊C₈	—	47	2	no	—
128	C₂²⋊C₈	—	94	1	no	—
129	#97	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	94	1	no	—
132	C₄×C₈	—	94	1	no	—
133	C₄×C₈	Dic₉₄	1	2	yes	—
134	C₄×C₈	D₁₈₈	1	2	yes	—
135	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
136	C₂²×C₄	#10625354	1	1	yes	characteristic
137	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
138	M₁₆	—	2	2	no	—
139	M₁₆	—	2	2	no	—
140	M₁₆	—	2	2	no	—
141	M₁₆	#10625329	1	4	yes	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	94	1	no	—
147	C₂×C₈	—	94	1	no	—
148	C₂×C₈	—	94	2	no	—
149	C₂×C₈	—	376	2	no	—
150	C₂×C₈	—	188	2	no	—
151	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
152	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
153	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
154	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
155	C₄²	D₄×D₄₇	1	1	yes	characteristic
156	C₄²	#10625352	1	1	yes	characteristic
157	C₂³	Q₈×Dic₄₇	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
163	C₂×C₄	#13093999	1	1	yes	characteristic
164	C₂×C₄	#13094004	1	1	yes	characteristic
165	C₂×C₄	#13094015	1	1	yes	characteristic
166	C₂×C₄	#13094023	1	1	yes	characteristic
167	C₂×C₄	C₂×#10625329	1	2	yes	—
168	C₂×C₄	#13094031	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	4	no	—
172	C₈	—	4	2	no	—
173	C₈	—	376	2	no	—
174	C₈	—	188	2	no	—
175	C₈	—	188	1	no	—
176	C₈	—	188	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#15689022	1	1	yes	characteristic
180	C₂²	#15689254	1	2	yes	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	#15689147	1	1	yes	characteristic
185	C₄	#15689150	1	1	yes	characteristic
186	C₄	#15689195	1	2	yes	—
187	C₄	#15689394	1	1	yes	characteristic
188	C₄	#15689399	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	#20244112	1	1	yes	characteristic
191	C₂	#20244174	1	1	yes	characteristic
192	C₂	#20244179	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20242140

C₂

yes

maximal

C₂×#15688876

C₂

yes

maximal, characteristic

#20241680

C₂

yes

maximal, characteristic

#20241239

C₂

yes

maximal

C₄×#108×C₄₇

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

characteristic

#15689176

C₂²

yes

characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

—

—

C₂×#108×C₄₇

C₄

yes

—

C₄×M₁₆×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#341

C₄

yes

characteristic

C₄₇×#341

C₄

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

D₄

yes

characteristic

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

Q₈

yes

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#108×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

M₁₆

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₄×Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

#125

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9610

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#389

yes

characteristic

C₂²×C₄₇

#446

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

—

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#1940

yes

characteristic

100

C₂×C₄₇

#1952

yes

characteristic

101

C₂×C₄₇

#1953

yes

characteristic

102

#431

—

103

#431

—

104

C₂×#108

Dic₄₇

yes

—

105

C₂×C₄⋊C₈

—

106

C₄×M₁₆

C₂×D₄₇

yes

—

107

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

108

#341

Dic₄₇

yes

characteristic

109

#341

Dic₄₇

yes

characteristic

110

#335

—

111

#335

—

112

#334

—

113

#322

—

114

C₄₇

#9610

yes

characteristic, complemented, Sylow

115

C₂×M₁₆

C₂×Dic₄₇

yes

—

116

C₂×M₁₆

C₂×Dic₄₇

yes

—

117

C₂²×C₈

—

118

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

119

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

120

C₂×C₄²

C₂²×D₄₇

yes

characteristic

121

#108

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₂²⋊C₈

—

128

C₂²⋊C₈

—

129

#97

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

Dic₉₄

yes

—

134

C₄×C₈

D₁₈₈

yes

—

135

C₂²×C₄

Q₈×D₄₇

yes

characteristic

136

C₂²×C₄

#10625354

yes

characteristic

137

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

138

M₁₆

—

139

M₁₆

—

140

M₁₆

—

141

M₁₆

#10625329

yes

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

376

—

150

C₂×C₈

—

188

—

151

C₂×C₈

C₄⋊Dic₄₇

yes

—

152

C₂×C₈

C₄⋊Dic₄₇

yes

—

153

C₄²

C₂×Dic₉₄

yes

characteristic

154

C₄²

C₂×D₁₈₈

yes

characteristic

155

C₄²

D₄×D₄₇

yes

characteristic

156

C₄²

#10625352

yes

characteristic

157

C₂³

Q₈×Dic₄₇

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

162

C₂×C₄

D₄×Dic₄₇

yes

characteristic

163

C₂×C₄

#13093999

yes

characteristic

164

C₂×C₄

#13094004

yes

characteristic

165

C₂×C₄

#13094015

yes

characteristic

166

C₂×C₄

#13094023

yes

characteristic

167

C₂×C₄

C₂×#10625329

yes

—

168

C₂×C₄

#13094031

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

376

—

174

C₈

—

188

—

175

C₈

—

188

—

176

C₈

—

188

—

177

C₂²

—

178

C₂²

—

179

C₂²

#15689022

yes

characteristic

180

C₂²

#15689254

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#15689147

yes

characteristic

185

C₄

#15689150

yes

characteristic

186

C₄

#15689195

yes

—

187

C₄

#15689394

yes

characteristic

188

C₄

#15689399

yes

characteristic

189

C₂

—

190

C₂

#20244112

yes

characteristic

191

C₂

#20244174

yes

characteristic

192

C₂

#20244179

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093999
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#108×C₄₇	C₂
socle	C₂²×C₄₇	#389
prime radical	C₂³×C₄₇	C₄×Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#108	D₄₇
hypocenter	C₄₇	#9610
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

subgroup

G/H

center

C₂×C₄

#13093999

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄×#108×C₄₇

C₂

socle

C₂²×C₄₇

#389

prime radical

C₂³×C₄₇

C₄×Q₈

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

N	H ≅ G/N	type
114. C₄₇	#9610	semidirect

H ≅ G/N

type

114. C₄₇

#9610

semidirect

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

subgroup	index
#20242140	2
C₂×#15688876	2
#20241680	2
#20241239	2
C₄×#108×C₄₇	2
#9610	47

subgroup

index

C₄×#108×C₄₇

Subgroups — #26114905

Conjugacy classes of subgroups

Complements

Maximal subgroups