Group #26107281 — subgroups

Subgroups — #26107281

Order 12032.

452 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 143 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	#20242050	C₂	1	1	yes	maximal, characteristic
2	#20241980	C₂	1	1	yes	maximal, characteristic
3	C₈×#10625329	C₂	1	2	yes	maximal, complemented
4	#20241277	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2931	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₄×#10625329	C₂²	1	2	yes	—
8	#15688872	C₄	1	2	yes	—
9	#15688867	C₄	1	2	yes	—
10	#15688863	C₄	1	4	yes	—
11	#15688845	—	2	2	no	—
12	C₈×C₄₇⋊C₈	—	2	2	no	—
13	C₂×#131×C₄₇	C₄	1	2	yes	—
14	C₄₇×#532	C₂²	1	1	yes	characteristic
15	C₄₇×#436	—	2	2	no	—
16	C₈×D₄×C₄₇	—	2	2	no	—
17	C₄₇×#423	C₂²	1	1	yes	characteristic
18	C₄₇×#408	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#10625329	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
23	#13093935	C₂×C₄	1	4	yes	—
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₂×#41×C₄₇	C₂×C₄	1	2	yes	—
27	#131×C₄₇	—	4	4	no	—
28	C₂×M₁₆×C₄₇	—	2	2	no	—
29	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₄₇	—	2	2	no	—
31	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	#127×C₄₇	C₂×C₄	1	1	yes	characteristic
35	#124×C₄₇	C₂×C₄	1	2	yes	—
36	#121×C₄₇	—	2	2	no	—
37	C₄×D₄×C₄₇	—	2	2	no	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₄₇	—	2	2	no	—
40	C₂²⋊C₈×C₄₇	—	2	4	no	—
41	C₄×C₈×C₄₇	D₄	1	2	yes	—
42	C₄×C₈×C₄₇	D₄	1	2	yes	—
43	#41×C₄₇	—	4	2	no	—
44	#41×C₄₇	—	4	2	no	—
45	C₂×Q₈×C₄₇	C₄²	1	2	yes	—
46	C₂×D₄×C₄₇	—	2	2	no	—
47	C₂×D₄×C₄₇	—	2	2	no	—
48	C₂×D₄×C₄₇	C₄²	1	2	yes	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	#41	1	2	yes	—
52	M₁₆×C₄₇	—	4	4	no	—
53	M₁₆×C₄₇	—	4	2	no	—
54	C₂×C₈×C₄₇	—	2	2	no	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	—	4	1	no	—
57	C₂×C₈×C₄₇	—	4	4	no	—
58	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₂²⋊C₄×C₄₇	—	2	4	no	—
63	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₄₇	C₂×D₄	1	2	yes	—
66	#10625329	—	2	8	no	—
67	C₂×C₄₇⋊C₈	—	2	4	no	—
68	C₂×C₄₇⋊C₈	—	2	4	no	—
69	C₂³×C₄₇	—	2	2	no	—
70	C₂³×C₄₇	#117	1	1	yes	characteristic
71	Q₈×C₄₇	—	4	2	no	—
72	D₄×C₄₇	—	4	2	no	—
73	D₄×C₄₇	—	4	4	no	—
74	D₄×C₄₇	—	4	4	no	—
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	—	2	1	no	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
85	C₈×C₄₇	—	2	1	no	—
86	C₈×C₄₇	—	2	2	no	—
87	C₈×C₄₇	—	4	2	no	—
88	C₈×C₄₇	#100	1	2	yes	—
89	C₄₇⋊C₈	—	2	8	no	—
90	C₄₇⋊C₈	—	4	4	no	—
91	#8241	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	—	2	1	no	—
93	C₂²×C₄₇	—	2	2	no	—
94	C₂²×C₄₇	—	4	2	no	—
95	C₂²×C₄₇	—	4	4	no	—
96	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₄₇	—	2	1	no	—
98	C₄×C₄₇	—	2	2	no	—
99	C₄×C₄₇	—	4	2	no	—
100	C₄×C₄₇	C₂×#100	1	1	yes	characteristic
101	C₄×C₄₇	#413	1	1	yes	characteristic
102	C₄×C₄₇	#443	1	4	yes	—
103	#2931	D₄₇	1	1	yes	characteristic
104	#1543	—	47	1	no	—
105	#1525	—	47	1	no	—
106	C₈×M₁₆	—	47	2	no	—
107	#1299	—	47	2	no	—
108	C₂×C₄₇	—	4	1	no	—
109	C₂×C₄₇	—	8	2	no	—
110	C₂×C₄₇	C₄×#100	1	1	yes	characteristic
111	C₂×C₄₇	#1764	1	2	yes	—
112	C₂×#131	Dic₄₇	1	2	yes	—
113	#532	C₂×D₄₇	1	1	yes	characteristic
114	#436	—	2	2	no	—
115	C₈×D₄	—	2	2	no	—
116	#431	—	47	2	no	—
117	#423	C₂×D₄₇	1	1	yes	characteristic
118	#408	Dic₄₇	1	2	yes	—
119	C₄×M₁₆	—	47	2	no	—
120	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
121	#333	—	47	2	no	—
122	#331	—	47	2	no	—
123	#329	—	47	4	no	—
124	#322	—	94	2	no	—
125	C₈²	—	94	2	no	—
126	C₄₇	#8241	1	1	yes	characteristic, complemented, Sylow
127	C₂×#41	C₂×Dic₄₇	1	2	yes	—
128	#131	—	4	4	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	94	2	no	—
131	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
134	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
135	C₄⋊Q₈	C₄×D₄₇	1	1	yes	characteristic
136	#127	C₄×D₄₇	1	1	yes	characteristic
137	#124	C₄×D₄₇	1	2	yes	—
138	#121	—	2	2	no	—
139	C₄×D₄	—	2	2	no	—
140	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	47	4	no	—
143	C₂²⋊C₈	—	2	4	no	—
144	C₂²⋊C₈	—	94	2	no	—
145	#97	—	47	4	no	—
146	C₄×C₈	—	94	2	no	—
147	C₄×C₈	—	94	2	no	—
148	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
149	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
150	#41	—	4	2	no	—
151	#41	—	4	2	no	—
152	C₂×Q₈	C₄×Dic₄₇	1	2	yes	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	C₄×Dic₄₇	1	2	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	#10625350	1	2	yes	—
158	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
159	M₁₆	—	4	4	no	—
160	M₁₆	—	4	2	no	—
161	M₁₆	—	94	8	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	4	no	—
166	C₂×C₈	—	94	4	no	—
167	C₂×C₈	—	94	4	no	—
168	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
169	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
174	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
175	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	#13093983	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
190	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
191	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
192	C₈	—	2	1	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	94	8	no	—
196	C₈	—	188	4	no	—
197	C₈	#13093941	1	2	yes	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	4	4	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	1	no	—
205	C₄	—	4	2	no	—
206	C₄	C₂×#13093941	1	1	yes	characteristic
207	C₄	#15689107	1	1	yes	characteristic
208	C₄	#15689238	1	4	yes	—
209	C₂	—	4	1	no	—
210	C₂	—	8	2	no	—
211	C₂	C₄×#13093941	1	1	yes	characteristic
212	C₂	#20243223	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20242050

C₂

yes

maximal, characteristic

#20241980

C₂

yes

maximal, characteristic

C₈×#10625329

C₂

yes

maximal, complemented

#20241277

C₂

yes

maximal, complemented

C₄₇×#2931

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₄×#10625329

C₂²

yes

—

#15688872

C₄

yes

—

#15688867

C₄

yes

—

#15688863

C₄

yes

—

#15688845

—

C₈×C₄₇⋊C₈

—

C₂×#131×C₄₇

C₄

yes

—

C₄₇×#532

C₂²

yes

characteristic

C₄₇×#436

—

C₈×D₄×C₄₇

—

C₄₇×#423

C₂²

yes

characteristic

C₄₇×#408

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

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

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₄₇

C₂×C₄

yes

characteristic

#127×C₄₇

C₂×C₄

yes

characteristic

#124×C₄₇

C₂×C₄

yes

—

#121×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

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

C₄²

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₄²

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

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

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

#117

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

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#100

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8241

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

100

C₄×C₄₇

C₂×#100

yes

characteristic

101

C₄×C₄₇

#413

yes

characteristic

102

C₄×C₄₇

#443

yes

—

103

#2931

D₄₇

yes

characteristic

104

#1543

—

105

#1525

—

106

C₈×M₁₆

—

107

#1299

—

108

C₂×C₄₇

—

109

C₂×C₄₇

—

110

C₂×C₄₇

C₄×#100

yes

characteristic

111

C₂×C₄₇

#1764

yes

—

112

C₂×#131

Dic₄₇

yes

—

113

#532

C₂×D₄₇

yes

characteristic

114

#436

—

115

C₈×D₄

—

116

#431

—

117

#423

C₂×D₄₇

yes

characteristic

118

#408

Dic₄₇

yes

—

119

C₄×M₁₆

—

120

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

121

#333

—

122

#331

—

123

#329

—

124

#322

—

125

C₈²

—

126

C₄₇

#8241

yes

characteristic, complemented, Sylow

127

C₂×#41

C₂×Dic₄₇

yes

—

128

#131

—

129

C₂×M₁₆

—

130

C₂×M₁₆

—

131

C₂×M₁₆

C₂×Dic₄₇

yes

—

132

C₂²×C₈

—

133

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

134

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

135

C₄⋊Q₈

C₄×D₄₇

yes

characteristic

136

#127

C₄×D₄₇

yes

characteristic

137

#124

C₄×D₄₇

yes

—

138

#121

—

139

C₄×D₄

—

140

C₂×C₄²

C₂²×D₄₇

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₂²⋊C₈

—

144

C₂²⋊C₈

—

145

#97

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

C₂²⋊D₄₇

yes

—

149

C₄×C₈

C₂²⋊D₄₇

yes

—

150

#41

—

151

#41

—

152

C₂×Q₈

C₄×Dic₄₇

yes

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

C₄×Dic₄₇

yes

—

156

C₂²×C₄

—

157

C₂²×C₄

#10625350

yes

—

158

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

159

M₁₆

—

160

M₁₆

—

161

M₁₆

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

C₂²⋊Dic₄₇

yes

—

169

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

170

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

171

C₄⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

C₂×C₄×D₄₇

yes

characteristic

174

C₄²

C₂×C₄×D₄₇

yes

characteristic

175

C₄²

C₂×C₂²⋊D₄₇

yes

—

176

C₂³

—

177

C₂³

#13093983

yes

characteristic

178

Q₈

—

179

D₄

—

180

D₄

—

181

D₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

C₂×C₄×Dic₄₇

yes

characteristic

189

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

190

C₂×C₄

C₄×C₂²⋊D₄₇

yes

—

191

C₂×C₄

C₄×C₂²⋊D₄₇

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

188

—

197

C₈

#13093941

yes

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

C₄×C₂²⋊Dic₄₇

yes

characteristic

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

C₂×#13093941

yes

characteristic

207

C₄

#15689107

yes

characteristic

208

C₄

#15689238

yes

—

209

C₂

—

210

C₂

—

211

C₂

C₄×#13093941

yes

characteristic

212

C₂

#20243223

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2931	C₂
socle	C₂²×C₄₇	C₄×C₂²⋊C₄
prime radical	C₄₇×#532	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2931	D₄₇
hypocenter	C₄₇	#8241
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#2931

C₂

socle

C₂²×C₄₇

C₄×C₂²⋊C₄

prime radical

C₄₇×#532

C₂²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

N	H ≅ G/N	type
3. C₈×#10625329	C₂	semidirect
4. #20241277	C₂	semidirect
126. C₄₇	#8241	semidirect

H ≅ G/N

type

3. C₈×#10625329

C₂

semidirect

4. #20241277

C₂

semidirect

126. C₄₇

#8241

semidirect

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

subgroup	index
#20242050	2
#20241980	2
C₈×#10625329	2
#20241277	2
C₄₇×#2931	2
#8241	47

subgroup

index

C₄₇×#2931

Subgroups — #26107281

Conjugacy classes of subgroups

Complements

Maximal subgroups