Group #26225165 — subgroups

Subgroups — #26225165

Order 12032.

662 conjugacy classes of subgroups; only the 295 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 295 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	#20252323	C₂	1	2	yes	maximal
2	#20244370	C₂	1	2	yes	maximal
3	#20244137	C₂	1	4	yes	maximal
4	#20243563	C₂	1	1	yes	maximal, characteristic
5	#20243274	C₂	1	4	yes	maximal
6	#20243076	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#3162	C₂	1	1	yes	maximal, characteristic
8	#15689254	—	2	4	no	—
9	Q₈×C₄₇⋊C₈	—	2	4	no	—
10	C₂×#13093960	C₂²	1	4	yes	—
11	#15689150	C₂²	1	4	yes	—
12	#15689147	C₂²	1	2	yes	—
13	#15689079	C₂²	1	4	yes	—
14	C₂×#13093934	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
17	#15688969	C₂²	1	1	yes	characteristic
18	#15688882	C₂²	1	8	yes	—
19	C₄₇×#533	C₂²	1	2	yes	—
20	C₄₇×#501	C₄	1	8	yes	complemented
21	C₄₇×#427	C₂²	1	4	yes	—
22	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#10625329	C₂³	1	4	yes	—
24	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
26	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
27	#13093960	—	2	8	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
29	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
30	#13093935	—	2	4	no	—
31	#13093934	Q₈	1	8	yes	complemented
32	C₄×C₄₇⋊C₈	—	2	4	no	—
33	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
34	C₈×Dic₄₇	—	2	4	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
38	#125×C₄₇	C₂×C₄	1	4	yes	—
39	C₂²⋊Q₈×C₄₇	—	2	4	no	—
40	C₄×Q₈×C₄₇	—	2	4	no	—
41	#117×C₄₇	C₂³	1	4	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	#107×C₄₇	C₂×C₄	1	8	yes	—
44	#106×C₄₇	C₂×C₄	1	8	yes	—
45	#97×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×Q₈×C₄₇	—	2	4	no	—
47	C₂²×C₄×C₄₇	#41	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
49	C₂²×Dic₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
51	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
52	C₂×C₈×C₄₇	#41	1	4	yes	—
53	C₄⋊C₄×C₄₇	—	2	8	no	—
54	C₄⋊C₄×C₄₇	—	2	4	no	—
55	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
56	C₂²⋊C₄×C₄₇	—	2	4	no	—
57	C₄²×C₄₇	—	2	4	no	—
58	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
59	C₄²×C₄₇	C₂×D₄	1	2	yes	—
60	C₄×Dic₄₇	—	2	2	no	—
61	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
62	#10625329	—	2	8	no	—
63	C₂×C₄₇⋊C₈	—	2	2	no	—
64	C₂×C₄₇⋊C₈	—	2	4	no	—
65	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
66	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
67	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
68	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
69	Q₈×C₄₇	—	4	8	no	—
70	C₂×C₄×C₄₇	—	2	8	no	—
71	C₂×C₄×C₄₇	—	4	2	no	—
72	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
76	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
77	C₂×Dic₄₇	—	2	4	no	—
78	C₂×Dic₄₇	—	4	4	no	—
79	C₈×C₄₇	C₄×Q₈	1	8	yes	—
80	C₄₇⋊C₈	—	2	2	no	—
81	C₄₇⋊C₈	—	4	4	no	—
82	C₄₇⋊C₈	C₄⋊Q₈	1	4	yes	—
83	#21737	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	—	2	2	no	—
85	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
86	C₂²×C₄₇	#533	1	2	yes	—
87	C₄×C₄₇	—	2	4	no	—
88	C₄×C₄₇	—	4	8	no	—
89	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
90	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
91	C₄×C₄₇	#532	1	1	yes	characteristic
92	Dic₄₇	—	4	4	no	—
93	#3162	D₄₇	1	1	yes	characteristic
94	#2960	—	47	2	no	—
95	#1995	—	47	2	no	—
96	#1945	—	47	4	no	—
97	#1849	—	47	1	no	—
98	#1773	—	47	4	no	—
99	#1726	—	47	1	no	—
100	C₂×C₄₇	—	2	2	no	—
101	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄₇	#2955	1	2	yes	—
103	#533	C₂×D₄₇	1	2	yes	—
104	#501	Dic₄₇	1	8	yes	complemented
105	#446	—	94	4	no	—
106	C₈×Q₈	—	94	4	no	—
107	C₂×#108	—	47	4	no	—
108	#427	C₂×D₄₇	1	4	yes	—
109	#424	—	47	4	no	—
110	#423	—	47	2	no	—
111	#405	—	47	4	no	—
112	C₂×#97	—	47	2	no	—
113	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
114	C₂×C₄×C₈	—	47	1	no	—
115	C₂×C₄×C₈	—	47	2	no	—
116	#376	—	47	1	no	—
117	#339	—	47	8	no	—
118	C₄₇	#21737	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	47	4	no	—
120	C₂²×C₈	—	47	1	no	—
121	C₂²×C₈	—	47	1	no	—
122	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
123	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
124	#125	C₂×Dic₄₇	1	4	yes	—
125	C₂²⋊Q₈	—	2	4	no	—
126	C₄×Q₈	—	2	4	no	—
127	#117	C₂²×D₄₇	1	4	yes	—
128	C₂×C₄²	—	47	2	no	—
129	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
130	#108	—	94	8	no	—
131	#107	C₂×Dic₄₇	1	8	yes	—
132	#106	C₂×Dic₄₇	1	8	yes	—
133	C₄⋊C₈	—	94	8	no	—
134	C₄⋊C₈	—	94	4	no	—
135	#97	—	47	8	no	—
136	#97	—	94	4	no	—
137	#97	C₂×Dic₄₇	1	4	yes	—
138	C₄×C₈	—	47	4	no	—
139	C₄×C₈	—	94	4	no	—
140	C₄×C₈	—	94	4	no	—
141	#95	—	94	2	no	—
142	C₂×Q₈	—	2	4	no	—
143	C₂²×C₄	—	94	2	no	—
144	C₂²×C₄	#10625352	1	2	yes	—
145	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
146	M₁₆	—	94	8	no	—
147	C₂×C₈	—	47	2	no	—
148	C₂×C₈	—	47	4	no	—
149	C₂×C₈	—	47	2	no	—
150	C₂×C₈	—	94	2	no	—
151	C₂×C₈	—	94	4	no	—
152	C₂×C₈	Q₈×D₄₇	1	4	yes	—
153	C₂×C₈	#10625354	1	4	yes	—
154	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
155	C₄⋊C₄	—	2	8	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	47	4	no	—
161	C₄²	—	94	2	no	—
162	C₄²	D₄×D₄₇	1	2	yes	—
163	C₄²	C₂²×Dic₄₇	1	2	yes	—
164	C₂³	C₂×#10625352	1	1	yes	characteristic
165	Q₈	—	4	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	94	4	no	—
169	C₂×C₄	—	188	4	no	—
170	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
171	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
172	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
174	C₂×C₄	C₂×#10625354	1	2	yes	—
175	C₈	—	47	4	no	—
176	C₈	—	94	2	no	—
177	C₈	—	188	4	no	—
178	C₈	Q₈×Dic₄₇	1	8	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
181	C₂²	#15689678	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	4	8	no	—
184	C₄	—	188	4	no	—
185	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
186	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
187	C₄	#15689674	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	C₄⋊Q₈×Dic₄₇	1	1	yes	characteristic
190	C₂	#20252291	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20252323

C₂

yes

maximal

#20244370

C₂

yes

maximal

#20244137

C₂

yes

maximal

#20243563

C₂

yes

maximal, characteristic

#20243274

C₂

yes

maximal

#20243076

C₂

yes

maximal, characteristic

C₄₇×#3162

C₂

yes

maximal, characteristic

#15689254

—

Q₈×C₄₇⋊C₈

—

C₂×#13093960

C₂²

yes

—

#15689150

C₂²

yes

—

#15689147

C₂²

yes

—

#15689079

C₂²

yes

—

C₂×#13093934

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₄₇

C₂²

yes

—

#15688969

C₂²

yes

characteristic

#15688882

C₂²

yes

—

C₄₇×#533

C₂²

yes

—

C₄₇×#501

C₄

yes

complemented

C₄₇×#427

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₂×#10625329

C₂³

yes

—

C₂²×C₄₇⋊C₈

C₂³

yes

characteristic

C₂²×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄×Dic₄₇

C₂³

yes

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

#13093935

—

#13093934

Q₈

yes

complemented

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₈×Dic₄₇

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

—

#125×C₄₇

C₂×C₄

yes

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#107×C₄₇

C₂×C₄

yes

—

#106×C₄₇

C₂×C₄

yes

—

#97×C₄₇

C₂×C₄

yes

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

#41

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂²×C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×Q₈

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

C₂×C₄₇⋊C₈

C₂×Q₈

yes

—

C₂³×C₄₇

C₂×#41

yes

characteristic

Q₈×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

C₂²×Q₈

yes

—

C₂×C₄×C₄₇

C₂×#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

C₄×Q₈

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₄⋊Q₈

yes

—

#21737

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₇

#533

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄×Q₈

yes

—

C₄×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₄₇

#532

yes

characteristic

Dic₄₇

—

#3162

D₄₇

yes

—

—

—

—

—

—

100

C₂×C₄₇

—

101

C₂×C₄₇

C₄×C₄⋊Q₈

yes

characteristic

102

C₂×C₄₇

#2955

yes

—

103

#533

C₂×D₄₇

yes

—

104

#501

Dic₄₇

yes

complemented

105

#446

—

106

C₈×Q₈

—

107

C₂×#108

—

108

#427

C₂×D₄₇

yes

—

109

#424

—

110

#423

—

111

#405

—

112

C₂×#97

—

113

C₂×#97

C₂×D₄₇

yes

characteristic

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

—

116

#376

—

117

#339

—

118

C₄₇

#21737

yes

characteristic, complemented, Sylow

119

C₂×M₁₆

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂²×D₄₇

yes

—

123

C₄⋊Q₈

C₂×Dic₄₇

yes

—

124

#125

C₂×Dic₄₇

yes

—

125

C₂²⋊Q₈

—

126

C₄×Q₈

—

127

#117

C₂²×D₄₇

yes

—

128

C₂×C₄²

—

129

C₂×C₄²

C₂²×D₄₇

yes

characteristic

130

#108

—

131

#107

C₂×Dic₄₇

yes

—

132

#106

C₂×Dic₄₇

yes

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#97

—

136

#97

—

137

#97

C₂×Dic₄₇

yes

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

#95

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

#10625352

yes

—

145

C₂²×C₄

C₂³×D₄₇

yes

characteristic

146

M₁₆

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

Q₈×D₄₇

yes

—

153

C₂×C₈

#10625354

yes

—

154

C₂×C₈

C₂²×Dic₄₇

yes

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

C₂²×Dic₄₇

yes

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

D₄×D₄₇

yes

—

163

C₄²

C₂²×Dic₄₇

yes

—

164

C₂³

C₂×#10625352

yes

characteristic

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

188

—

170

C₂×C₄

D₄×Dic₄₇

yes

—

171

C₂×C₄

C₂³×Dic₄₇

yes

characteristic

172

C₂×C₄

C₂×D₄×D₄₇

yes

characteristic

173

C₂×C₄

C₂×Q₈×D₄₇

yes

—

174

C₂×C₄

C₂×#10625354

yes

—

175

C₈

—

176

C₈

—

177

C₈

—

188

—

178

C₈

Q₈×Dic₄₇

yes

—

179

C₂²

—

180

C₂²

C₂×D₄×Dic₄₇

yes

characteristic

181

C₂²

#15689678

yes

—

182

C₄

—

183

C₄

—

184

C₄

—

188

—

185

C₄

C₂×Q₈×Dic₄₇

yes

—

186

C₄

C₄⋊Q₈×D₄₇

yes

characteristic

187

C₄

#15689674

yes

characteristic

188

C₂

—

189

C₂

C₄⋊Q₈×Dic₄₇

yes

characteristic

190

C₂

#20252291

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#3162	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#3162	D₄₇
hypocenter	C₄₇	#21737
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₇

Fitting

C₄₇×#3162

C₂

socle

C₂²×C₄₇

C₂×C₄×D₄

prime radical

C₂³×C₄₇

C₂×#41

cosocle

C₂²×C₄×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

N	H ≅ G/N	type
20. C₄₇×#501	C₄	semidirect
31. #13093934	Q₈	semidirect
104. #501	Dic₄₇	semidirect
118. C₄₇	#21737	semidirect

H ≅ G/N

type

20. C₄₇×#501

C₄

semidirect

31. #13093934

Q₈

semidirect

104. #501

Dic₄₇

semidirect

118. C₄₇

#21737

semidirect

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

subgroup	index
#20252323	2
#20244370	2
#20244137	2
#20243563	2
#20243274	2
#20243076	2
C₄₇×#3162	2
#21737	47

subgroup

index

C₄₇×#3162

Subgroups — #26225165

Conjugacy classes of subgroups

Complements

Maximal subgroups