Group #26109794 — subgroups

Subgroups — #26109794

Order 12032.

754 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20243309	C₂	1	2	yes	maximal
2	#20241296	C₂	1	4	yes	maximal, complemented
3	C₄₇×#2308	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093941	—	2	4	no	—
5	#15689087	C₂²	1	4	yes	—
6	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
7	C₂²×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
8	C₄₇×#386	—	2	4	no	—
9	C₄₇×#381	C₂²	1	2	yes	—
10	C₄₇×#379	C₄	1	2	yes	—
11	C₂×#10625329	—	2	4	no	—
12	#13093941	—	4	8	no	—
13	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
14	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×D₄×C₄₇	—	2	4	no	—
17	C₂²×D₄×C₄₇	—	4	4	no	—
18	C₂²×D₄×C₄₇	D₄	1	4	yes	—
19	C₂³×C₄×C₄₇	—	2	2	no	—
20	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
22	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
23	C₂×C₂²⋊C₄×C₄₇	—	4	4	no	—
24	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
25	#95×C₄₇	—	2	4	no	—
26	C₂⁴×C₄₇	—	2	4	no	—
27	C₂⁴×C₄₇	—	2	2	no	—
28	C₂⁴×C₄₇	—	2	2	no	—
29	C₂⁴×C₄₇	—	4	2	no	—
30	C₂⁴×C₄₇	—	8	2	no	—
31	C₂⁴×C₄₇	—	8	2	no	—
32	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
33	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
34	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×D₄×C₄₇	—	2	8	no	—
36	C₂×D₄×C₄₇	—	2	8	no	—
37	C₂×D₄×C₄₇	—	4	4	no	—
38	C₂×D₄×C₄₇	—	4	16	no	—
39	C₂²×C₄×C₄₇	—	2	4	no	—
40	C₂²×C₄×C₄₇	—	4	2	no	—
41	C₂²×C₄×C₄₇	—	4	4	no	—
42	C₂²×C₄×C₄₇	—	8	4	no	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₄₇	#41	1	4	yes	—
45	C₂²⋊C₄×C₄₇	—	4	4	no	—
46	C₂²⋊C₄×C₄₇	—	4	4	no	—
47	C₂²⋊C₄×C₄₇	—	8	4	no	—
48	#10625329	—	4	8	no	—
49	C₂×C₄₇⋊C₈	—	4	4	no	—
50	C₂³×C₄₇	—	2	8	no	—
51	C₂³×C₄₇	—	2	2	no	—
52	C₂³×C₄₇	—	2	2	no	—
53	C₂³×C₄₇	—	2	2	no	—
54	C₂³×C₄₇	—	2	2	no	—
55	C₂³×C₄₇	—	2	4	no	—
56	C₂³×C₄₇	—	4	2	no	—
57	C₂³×C₄₇	—	4	4	no	—
58	C₂³×C₄₇	—	4	4	no	—
59	C₂³×C₄₇	—	4	4	no	—
60	C₂³×C₄₇	—	8	4	no	—
61	C₂³×C₄₇	—	8	4	no	—
62	C₂³×C₄₇	—	8	8	no	—
63	C₂³×C₄₇	—	8	8	no	—
64	C₂³×C₄₇	#99	1	4	yes	—
65	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂³×C₄₇	#117	1	2	yes	—
67	D₄×C₄₇	—	4	16	no	—
68	C₂×C₄×C₄₇	—	2	4	no	—
69	C₂×C₄×C₄₇	—	4	4	no	—
70	C₂×C₄×C₄₇	—	4	8	no	—
71	C₂×C₄×C₄₇	—	4	4	no	—
72	C₂×C₄×C₄₇	—	8	8	no	—
73	C₂×C₄×C₄₇	#123	1	4	yes	—
74	C₄₇⋊C₈	—	8	4	no	—
75	#8751	—	47	1	no	maximal, Sylow
76	C₂²×C₄₇	—	2	2	no	—
77	C₂²×C₄₇	—	2	2	no	—
78	C₂²×C₄₇	—	2	4	no	—
79	C₂²×C₄₇	—	2	4	no	—
80	C₂²×C₄₇	—	4	4	no	—
81	C₂²×C₄₇	—	4	4	no	—
82	C₂²×C₄₇	—	4	4	no	—
83	C₂²×C₄₇	—	4	4	no	—
84	C₂²×C₄₇	—	4	2	no	—
85	C₂²×C₄₇	—	4	4	no	—
86	C₂²×C₄₇	—	8	4	no	—
87	C₂²×C₄₇	—	8	8	no	—
88	C₂²×C₄₇	—	8	8	no	—
89	C₂²×C₄₇	#381	1	1	yes	characteristic
90	C₂²×C₄₇	C₂×#99	1	2	yes	—
91	C₄×C₄₇	—	2	4	no	—
92	C₄×C₄₇	—	8	4	no	—
93	#2308	D₄₇	1	1	yes	characteristic
94	#1783	—	47	2	no	—
95	#1306	—	47	4	no	—
96	C₂×C₄₇	—	2	2	no	—
97	C₂×C₄₇	—	4	2	no	—
98	C₂×C₄₇	—	4	4	no	—
99	C₂×C₄₇	—	8	2	no	—
100	C₂×C₄₇	#1819	1	1	yes	characteristic
101	C₂×C₄₇	#1829	1	2	yes	—
102	C₂³×D₄	C₂×D₄₇	1	1	yes	characteristic
103	C₂²×C₂²⋊C₄	Dic₄₇	1	2	yes	—
104	C₂×#100	—	94	4	no	—
105	#407	—	47	4	no	—
106	#386	—	2	4	no	—
107	#381	C₂×D₄₇	1	2	yes	—
108	#379	Dic₄₇	1	2	yes	—
109	C₄₇	#8751	1	1	yes	characteristic, complemented, Sylow
110	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
111	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
112	C₂²×D₄	—	2	4	no	—
113	C₂²×D₄	—	4	4	no	—
114	C₂²×D₄	C₂²⋊D₄₇	1	4	yes	—
115	C₂³×C₄	—	2	2	no	—
116	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
117	C₂×M₁₆	—	94	4	no	—
118	C₂×C₂²⋊C₄	—	2	2	no	—
119	C₂×C₂²⋊C₄	—	2	4	no	—
120	C₂×C₂²⋊C₄	—	4	4	no	—
121	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
122	#100	—	188	8	no	—
123	C₂²⋊C₈	—	94	8	no	—
124	#95	—	2	4	no	—
125	C₂⁴	—	2	4	no	—
126	C₂⁴	—	2	2	no	—
127	C₂⁴	—	2	2	no	—
128	C₂⁴	—	4	2	no	—
129	C₂⁴	—	8	2	no	—
130	C₂⁴	—	8	2	no	—
131	C₂⁴	C₂²⋊Dic₄₇	1	2	yes	—
132	C₂⁴	C₂²⋊Dic₄₇	1	2	yes	—
133	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
134	C₂×D₄	—	2	8	no	—
135	C₂×D₄	—	2	8	no	—
136	C₂×D₄	—	4	4	no	—
137	C₂×D₄	—	4	16	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	4	4	no	—
141	C₂²×C₄	—	8	4	no	—
142	C₂²×C₄	#10625350	1	4	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
144	M₁₆	—	188	8	no	—
145	C₂×C₈	—	188	4	no	—
146	C₂²⋊C₄	—	4	4	no	—
147	C₂²⋊C₄	—	4	4	no	—
148	C₂²⋊C₄	—	8	4	no	—
149	C₂³	—	2	8	no	—
150	C₂³	—	2	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	2	2	no	—
153	C₂³	—	2	2	no	—
154	C₂³	—	2	4	no	—
155	C₂³	—	4	2	no	—
156	C₂³	—	4	4	no	—
157	C₂³	—	4	4	no	—
158	C₂³	—	4	4	no	—
159	C₂³	—	8	4	no	—
160	C₂³	—	8	4	no	—
161	C₂³	—	8	8	no	—
162	C₂³	—	8	8	no	—
163	C₂³	#13093939	1	4	yes	—
164	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
165	C₂³	#13093983	1	2	yes	—
166	D₄	—	4	16	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	4	8	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	8	8	no	—
172	C₂×C₄	#13094007	1	4	yes	—
173	C₈	—	376	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	4	no	—
178	C₂²	—	4	4	no	—
179	C₂²	—	4	4	no	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	4	no	—
184	C₂²	—	8	4	no	—
185	C₂²	—	8	8	no	—
186	C₂²	—	8	8	no	—
187	C₂²	#15688986	1	1	yes	characteristic
188	C₂²	C₂×#13093939	1	2	yes	—
189	C₄	—	2	4	no	—
190	C₄	—	8	4	no	—
191	C₂	—	2	2	no	—
192	C₂	—	4	2	no	—
193	C₂	—	4	4	no	—
194	C₂	—	8	2	no	—
195	C₂	#20243462	1	1	yes	characteristic
196	C₂	#20243499	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243309

C₂

yes

maximal

#20241296

C₂

yes

maximal, complemented

C₄₇×#2308

C₂

yes

maximal, characteristic

C₂×#13093941

—

#15689087

C₂²

yes

—

C₂³×D₄×C₄₇

C₂²

yes

characteristic

C₂²×C₂²⋊C₄×C₄₇

C₄

yes

—

C₄₇×#386

—

C₄₇×#381

C₂²

yes

—

C₄₇×#379

C₄

yes

—

C₂×#10625329

—

#13093941

—

C₂²⋊(C₄₇⋊C₈)

—

C₂⁵×C₄₇

C₂×C₄

yes

characteristic

C₂⁵×C₄₇

C₂×C₄

yes

characteristic

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

D₄

yes

—

C₂³×C₄×C₄₇

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

#95×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

C₂²⋊C₄

yes

—

C₂⁴×C₄₇

C₂²⋊C₄

yes

—

C₂⁴×C₄₇

C₂²×C₄

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#99

yes

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

#117

yes

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#123

yes

—

C₄₇⋊C₈

—

#8751

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#381

yes

characteristic

C₂²×C₄₇

C₂×#99

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

#2308

D₄₇

yes

characteristic

#1783

—

#1306

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

100

C₂×C₄₇

#1819

yes

characteristic

101

C₂×C₄₇

#1829

yes

—

102

C₂³×D₄

C₂×D₄₇

yes

characteristic

103

C₂²×C₂²⋊C₄

Dic₄₇

yes

—

104

C₂×#100

—

105

#407

—

106

#386

—

107

#381

C₂×D₄₇

yes

—

108

#379

Dic₄₇

yes

—

109

C₄₇

#8751

yes

characteristic, complemented, Sylow

110

C₂⁵

C₂×Dic₄₇

yes

characteristic

111

C₂⁵

C₂×Dic₄₇

yes

characteristic

112

C₂²×D₄

—

113

C₂²×D₄

—

114

C₂²×D₄

C₂²⋊D₄₇

yes

—

115

C₂³×C₄

—

116

C₂³×C₄

C₂²×D₄₇

yes

characteristic

117

C₂×M₁₆

—

118

C₂×C₂²⋊C₄

—

119

C₂×C₂²⋊C₄

—

120

C₂×C₂²⋊C₄

—

121

C₂×C₂²⋊C₄

C₂×Dic₄₇

yes

—

122

#100

—

188

—

123

C₂²⋊C₈

—

124

#95

—

125

C₂⁴

—

126

C₂⁴

—

127

C₂⁴

—

128

C₂⁴

—

129

C₂⁴

—

130

C₂⁴

—

131

C₂⁴

C₂²⋊Dic₄₇

yes

—

132

C₂⁴

C₂²⋊Dic₄₇

yes

—

133

C₂⁴

C₂²×Dic₄₇

yes

characteristic

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

#10625350

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

144

M₁₆

—

188

—

145

C₂×C₈

—

188

—

146

C₂²⋊C₄

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

#13093939

yes

—

164

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

165

C₂³

#13093983

yes

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

#13094007

yes

—

173

C₈

—

376

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#15688986

yes

characteristic

188

C₂²

C₂×#13093939

yes

—

189

C₄

—

190

C₄

—

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

—

195

C₂

#20243462

yes

characteristic

196

C₂

#20243499

yes

—

	subgroup	G/H
center	C₂²	#15688986
commutator	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2308	C₂
socle	C₂²×C₄₇	#381
prime radical	C₂³×D₄×C₄₇	C₂²
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2308	D₄₇
hypocenter	C₄₇	#8751
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2308

subgroup

G/H

center

C₂²

#15688986

commutator

C₂⁴×C₄₇

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₄₇

Fitting

C₄₇×#2308

C₂

socle

C₂²×C₄₇

#381

prime radical

C₂³×D₄×C₄₇

C₂²

cosocle

C₂³×C₄×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂⁴×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2308

N	H ≅ G/N	type
2. #20241296	C₂	semidirect
109. C₄₇	#8751	semidirect

H ≅ G/N

type

2. #20241296

C₂

semidirect

109. C₄₇

#8751

semidirect

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

subgroup	index
#20243309	2
#20241296	2
C₄₇×#2308	2
#8751	47

subgroup

index

#20243309

#20241296

C₄₇×#2308

#8751

Subgroups — #26109794

Conjugacy classes of subgroups

Complements

Maximal subgroups