Group #26125608 — subgroups

Subgroups — #26125608

Order 12032.

356 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20245191	C₂	1	2	yes	maximal, complemented
2	#20241314	C₂	1	4	yes	maximal, complemented
3	C₄₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
5	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	—
6	#15688899	—	2	2	no	—
7	C₂×#131×C₄₇	C₄	1	2	yes	—
8	C₄₇×#532	C₄	1	1	yes	characteristic
9	C₄₇×#436	—	2	2	no	—
10	C₈×D₄×C₄₇	—	2	2	no	—
11	C₄₇×#423	C₄	1	1	yes	characteristic
12	C₄₇×#408	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄₇⋊C₁₆	—	2	4	no	—
15	C₂×#41×C₄₇	C₂×C₄	1	2	yes	—
16	#131×C₄₇	—	4	4	no	—
17	C₂×M₁₆×C₄₇	—	2	2	no	—
18	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₄₇	—	2	2	no	—
20	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
22	C₄⋊Q₈×C₄₇	C₈	1	1	yes	characteristic
23	#127×C₄₇	C₈	1	1	yes	characteristic
24	#124×C₄₇	C₈	1	2	yes	—
25	#121×C₄₇	—	2	2	no	—
26	C₄×D₄×C₄₇	—	2	2	no	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₄₇	—	2	2	no	—
29	C₂²⋊C₈×C₄₇	—	2	4	no	—
30	C₄×C₈×C₄₇	D₄	1	2	yes	—
31	C₄×C₈×C₄₇	D₄	1	2	yes	—
32	#41×C₄₇	—	4	2	no	—
33	#41×C₄₇	—	4	2	no	—
34	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	—
35	C₂×D₄×C₄₇	—	2	2	no	—
36	C₂×D₄×C₄₇	—	2	2	no	—
37	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	M₁₆×C₄₇	—	4	4	no	—
42	M₁₆×C₄₇	—	4	2	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	—	4	1	no	—
46	C₂×C₈×C₄₇	—	4	4	no	—
47	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
50	C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂²⋊C₄×C₄₇	—	2	4	no	—
52	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
55	C₄₇⋊C₁₆	—	4	4	no	—
56	C₂³×C₄₇	—	2	2	no	—
57	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
58	Q₈×C₄₇	—	4	2	no	—
59	D₄×C₄₇	—	4	2	no	—
60	D₄×C₄₇	—	4	4	no	—
61	D₄×C₄₇	—	4	4	no	—
62	C₂×C₄×C₄₇	—	2	1	no	—
63	C₂×C₄×C₄₇	—	2	1	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	—	4	2	no	—
67	C₂×C₄×C₄₇	—	4	2	no	—
68	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
69	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
72	C₈×C₄₇	—	2	2	no	—
73	C₈×C₄₇	—	2	1	no	—
74	C₈×C₄₇	—	4	2	no	—
75	C₈×C₄₇	#100	1	2	yes	—
76	#11491	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	—	2	2	no	—
78	C₂²×C₄₇	—	2	1	no	—
79	C₂²×C₄₇	—	4	4	no	—
80	C₂²×C₄₇	—	4	2	no	—
81	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
82	C₄×C₄₇	—	2	2	no	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	—	2	1	no	—
85	C₄×C₄₇	—	4	2	no	—
86	C₄×C₄₇	#410	1	1	yes	characteristic
87	C₄×C₄₇	C₂×#100	1	1	yes	characteristic
88	#2931	D₄₇	1	1	yes	characteristic
89	#2163	—	47	2	no	—
90	#1315	—	47	4	no	—
91	C₂×C₄₇	—	4	1	no	—
92	C₂×C₄₇	—	8	2	no	—
93	C₂×C₄₇	#1460	1	1	yes	characteristic
94	C₂×C₄₇	#2117	1	2	yes	—
95	C₂×#131	Dic₄₇	1	2	yes	—
96	#532	Dic₄₇	1	1	yes	characteristic
97	#436	—	2	2	no	—
98	C₈×D₄	—	2	2	no	—
99	#423	Dic₄₇	1	1	yes	characteristic
100	#408	C₂×D₄₇	1	2	yes	—
101	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
102	C₄⋊C₁₆	—	94	2	no	—
103	C₂²⋊C₁₆	—	47	4	no	—
104	#346	—	94	2	no	—
105	C₄₇	#11491	1	1	yes	characteristic, complemented, Sylow
106	C₂×#41	C₂×Dic₄₇	1	2	yes	—
107	#131	—	4	4	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
113	C₄⋊Q₈	C₄₇⋊C₈	1	1	yes	characteristic
114	#127	C₄₇⋊C₈	1	1	yes	characteristic
115	#124	C₄₇⋊C₈	1	2	yes	—
116	#121	—	2	2	no	—
117	C₄×D₄	—	2	2	no	—
118	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
119	C₂×C₁₆	—	94	4	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	2	4	no	—
122	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
123	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
124	#41	—	4	2	no	—
125	#41	—	4	2	no	—
126	C₂×Q₈	C₂×C₄₇⋊C₈	1	2	yes	—
127	C₂×D₄	—	2	2	no	—
128	C₂×D₄	—	2	2	no	—
129	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	#10625329	1	2	yes	—
132	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
133	M₁₆	—	4	4	no	—
134	M₁₆	—	4	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	4	no	—
139	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
141	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
142	C₄⋊C₄	—	2	2	no	—
143	C₂²⋊C₄	—	2	4	no	—
144	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
145	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
146	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
147	C₁₆	—	188	4	no	—
148	C₂³	—	2	2	no	—
149	C₂³	C₂×#10625329	1	1	yes	characteristic
150	Q₈	—	4	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	4	no	—
153	D₄	—	4	4	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
161	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
163	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	1	no	—
166	C₈	—	4	2	no	—
167	C₈	#13093941	1	2	yes	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	4	4	no	—
171	C₂²	—	4	2	no	—
172	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	1	no	—
176	C₄	—	4	2	no	—
177	C₄	#15689097	1	1	yes	characteristic
178	C₄	C₂×#13093941	1	1	yes	characteristic
179	C₂	—	4	1	no	—
180	C₂	—	8	2	no	—
181	C₂	#20241709	1	1	yes	characteristic
182	C₂	#20245016	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245191

C₂

yes

maximal, complemented

#20241314

C₂

yes

maximal, complemented

C₄₇×#2931

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₁₆)

—

C₂²⋊(C₄₇⋊C₁₆)

C₂²

yes

—

#15688899

—

C₂×#131×C₄₇

C₄

yes

—

C₄₇×#532

C₄

yes

characteristic

C₄₇×#436

—

C₈×D₄×C₄₇

—

C₄₇×#423

C₄

yes

characteristic

C₄₇×#408

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄₇⋊C₁₆

—

C₂×#41×C₄₇

C₂×C₄

yes

—

#131×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊Q₈×C₄₇

C₈

yes

characteristic

#127×C₄₇

C₈

yes

characteristic

#124×C₄₇

C₈

yes

—

#121×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₈

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×M₁₆

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#100

yes

—

#11491

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

characteristic

#2163

—

#1315

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1460

yes

characteristic

C₂×C₄₇

#2117

yes

—

C₂×#131

Dic₄₇

yes

—

#532

Dic₄₇

yes

characteristic

#436

—

C₈×D₄

—

#423

Dic₄₇

yes

characteristic

100

#408

C₂×D₄₇

yes

—

101

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

102

C₄⋊C₁₆

—

103

C₂²⋊C₁₆

—

104

#346

—

105

C₄₇

#11491

yes

characteristic, complemented, Sylow

106

C₂×#41

C₂×Dic₄₇

yes

—

107

#131

—

108

C₂×M₁₆

—

109

C₂×M₁₆

C₂×Dic₄₇

yes

—

110

C₂²×C₈

—

111

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

112

C₂²×C₈

C₂²×D₄₇

yes

characteristic

113

C₄⋊Q₈

C₄₇⋊C₈

yes

characteristic

114

#127

C₄₇⋊C₈

yes

characteristic

115

#124

C₄₇⋊C₈

yes

—

116

#121

—

117

C₄×D₄

—

118

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

119

C₂×C₁₆

—

120

C₄⋊C₈

—

121

C₂²⋊C₈

—

122

C₄×C₈

C₂²⋊D₄₇

yes

—

123

C₄×C₈

C₂²⋊D₄₇

yes

—

124

#41

—

125

#41

—

126

C₂×Q₈

C₂×C₄₇⋊C₈

yes

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

C₂×C₄₇⋊C₈

yes

—

130

C₂²×C₄

—

131

C₂²×C₄

#10625329

yes

—

132

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

140

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

141

C₂×C₈

C₂×C₂²⋊D₄₇

yes

—

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₄²

C₂×C₄₇⋊C₈

yes

characteristic

145

C₄²

C₂×C₄₇⋊C₈

yes

characteristic

146

C₄²

C₂²⋊Dic₄₇

yes

—

147

C₁₆

—

188

—

148

C₂³

—

149

C₂³

C₂×#10625329

yes

characteristic

150

Q₈

—

151

D₄

—

152

D₄

—

153

D₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

161

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

162

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

163

C₂×C₄

C₂²×C₄₇⋊C₈

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

#13093941

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

C₂×C₂²⋊(C₄₇⋊C₈)

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#15689097

yes

characteristic

178

C₄

C₂×#13093941

yes

characteristic

179

C₂

—

180

C₂

—

181

C₂

#20241709

yes

characteristic

182

C₂

#20245016

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2931	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₄₇×#532	C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2931	D₄₇
hypocenter	C₄₇	#11491
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#2931

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₈

prime radical

C₄₇×#532

C₄

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

N	H ≅ G/N	type
1. #20245191	C₂	semidirect
2. #20241314	C₂	semidirect
105. C₄₇	#11491	semidirect

H ≅ G/N

type

1. #20245191

C₂

semidirect

2. #20241314

C₂

semidirect

105. C₄₇

#11491

semidirect

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

subgroup	index
#20245191	2
#20241314	2
C₄₇×#2931	2
#11491	47

subgroup

index

#20245191

#20241314

C₄₇×#2931

#11491

Subgroups — #26125608

Conjugacy classes of subgroups

Complements

Maximal subgroups