Group #26049997 — subgroups

Subgroups — C₄₇×#10299 #26049997

Order 12032.

576 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 130 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	C₄₇×#3229	C₂	1	2	yes	maximal
2	C₄₇×#2868	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1994	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1546	C₂	1	2	yes	maximal
5	C₄₇×#1276	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#533	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
9	C₄₇×#498	—	2	2	no	—
10	C₄₇×#487	—	2	2	no	—
11	C₄₇×#483	—	2	4	no	—
12	C₄₇×#475	—	2	4	no	—
13	C₄₇×#431	C₂²	1	2	yes	—
14	C₄₇×#423	C₂²	1	1	yes	characteristic
15	C₄₇×#384	—	2	4	no	—
16	C₄³×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#339	—	2	2	no	—
18	C₄₇×#332	—	2	4	no	—
19	C₂²×Q₈×C₄₇	—	2	2	no	—
20	C₂×M₁₆×C₄₇	—	2	1	no	—
21	C₄⋊Q₈×C₄₇	—	2	2	no	—
22	C₄⋊Q₈×C₄₇	—	2	2	no	—
23	C₄⋊Q₈×C₄₇	—	2	2	no	—
24	C₄⋊Q₈×C₄₇	—	2	2	no	—
25	C₄⋊Q₈×C₄₇	—	4	4	no	—
26	C₄⋊Q₈×C₄₇	—	4	4	no	—
27	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
28	#125×C₄₇	—	2	1	no	—
29	C₂²⋊Q₈×C₄₇	—	2	2	no	—
30	C₄×Q₈×C₄₇	—	2	2	no	—
31	#117×C₄₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	D₄	1	4	yes	—
35	C₂×C₄²×C₄₇	—	2	2	no	—
36	C₂×C₄²×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	#106×C₄₇	—	2	8	no	—
39	C₄⋊C₈×C₄₇	—	2	2	no	—
40	C₄⋊C₈×C₄₇	D₄	1	4	yes	—
41	#103×C₄₇	—	2	8	no	—
42	C₂²⋊C₈×C₄₇	—	2	2	no	—
43	C₄×C₈×C₄₇	—	2	2	no	—
44	C₂×Q₈×C₄₇	—	2	2	no	—
45	C₂×Q₈×C₄₇	—	4	2	no	—
46	C₂×Q₈×C₄₇	—	4	2	no	—
47	C₂×Q₈×C₄₇	—	4	4	no	—
48	C₂×Q₈×C₄₇	—	4	4	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	M₁₆×C₄₇	—	4	2	no	—
54	C₂×C₈×C₄₇	—	2	4	no	—
55	C₂×C₈×C₄₇	—	4	1	no	—
56	C₄⋊C₄×C₄₇	—	2	4	no	—
57	C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₄⋊C₄×C₄₇	—	2	4	no	—
59	C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₄⋊C₄×C₄₇	—	4	16	no	—
61	C₄⋊C₄×C₄₇	—	4	2	no	—
62	C₄⋊C₄×C₄₇	—	4	2	no	—
63	C₄⋊C₄×C₄₇	—	4	2	no	—
64	C₂²⋊C₄×C₄₇	—	4	1	no	—
65	C₄²×C₄₇	—	2	8	no	—
66	C₄²×C₄₇	—	2	2	no	—
67	C₄²×C₄₇	—	4	1	no	—
68	C₄²×C₄₇	SD₁₆	1	4	yes	—
69	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×D₄	1	2	yes	—
71	C₄²×C₄₇	#41	1	1	yes	characteristic
72	C₂³×C₄₇	#120	1	1	yes	characteristic
73	Q₈×C₄₇	—	4	4	no	—
74	Q₈×C₄₇	—	8	4	no	—
75	Q₈×C₄₇	—	8	2	no	—
76	Q₈×C₄₇	—	8	2	no	—
77	C₂×C₄×C₄₇	—	2	8	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	—	4	8	no	—
83	C₂×C₄×C₄₇	#120	1	2	yes	—
84	C₂×C₄×C₄₇	#121	1	2	yes	—
85	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
89	C₈×C₄₇	—	4	4	no	—
90	C₈×C₄₇	—	8	1	no	—
91	#10299	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×C₄₇	—	2	2	no	—
93	C₂²×C₄₇	#392	1	1	yes	characteristic
94	C₂²×C₄₇	#450	1	2	yes	—
95	C₄×C₄₇	—	2	8	no	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	—	8	2	no	—
98	C₄×C₄₇	—	8	4	no	—
99	C₄×C₄₇	#456	1	4	yes	—
100	C₄×C₄₇	#460	1	2	yes	—
101	C₄×C₄₇	#492	1	1	yes	characteristic
102	C₄×C₄₇	#496	1	1	yes	characteristic
103	#3229	C₂×C₄₇	1	2	yes	—
104	#2868	C₂×C₄₇	1	1	yes	characteristic
105	#1994	C₂×C₄₇	1	1	yes	characteristic
106	#1546	C₂×C₄₇	1	2	yes	—
107	#1276	C₂×C₄₇	1	1	yes	characteristic
108	C₂×C₄₇	—	2	2	no	—
109	C₂×C₄₇	#2005	1	2	yes	—
110	C₂×C₄₇	#2014	1	1	yes	characteristic
111	#533	C₂²×C₄₇	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	2	2	no	—
113	C₂×C₄⋊Q₈	C₂²×C₄₇	1	2	yes	—
114	#498	—	2	2	no	—
115	#487	—	2	2	no	—
116	#483	—	2	4	no	—
117	#475	—	2	4	no	—
118	#431	C₂²×C₄₇	1	2	yes	—
119	#423	C₂²×C₄₇	1	1	yes	characteristic
120	#384	—	2	4	no	—
121	C₄³	C₂²×C₄₇	1	1	yes	characteristic
122	#339	—	2	2	no	—
123	#332	—	2	4	no	—
124	C₄₇	#10299	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×Q₈	—	2	2	no	—
126	C₂×M₁₆	—	2	1	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	4	4	no	—
132	C₄⋊Q₈	—	4	4	no	—
133	C₄⋊Q₈	D₄×C₄₇	1	2	yes	—
134	#125	—	2	1	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	#117	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	8	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	D₄×C₄₇	1	4	yes	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	D₄×C₄₇	1	2	yes	—
143	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
144	#106	—	2	8	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	D₄×C₄₇	1	4	yes	—
147	#103	—	2	8	no	—
148	C₂²⋊C₈	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₂×Q₈	—	2	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	4	4	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
158	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
159	M₁₆	—	4	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	4	1	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	4	16	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₂²⋊C₄	—	4	1	no	—
171	C₄²	—	2	8	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	4	1	no	—
174	C₄²	SD₁₆×C₄₇	1	4	yes	—
175	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
176	C₄²	C₂×D₄×C₄₇	1	2	yes	—
177	C₄²	#41×C₄₇	1	1	yes	characteristic
178	C₂³	#120×C₄₇	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	Q₈	—	8	4	no	—
181	Q₈	—	8	2	no	—
182	Q₈	—	8	2	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	#120×C₄₇	1	2	yes	—
190	C₂×C₄	#121×C₄₇	1	2	yes	—
191	C₂×C₄	#121×C₄₇	1	1	yes	characteristic
192	C₂×C₄	#127×C₄₇	1	1	yes	characteristic
193	C₂×C₄	C₂×SD₁₆×C₄₇	1	2	yes	—
194	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	8	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₄₇×#392	1	1	yes	characteristic
199	C₂²	C₄₇×#450	1	2	yes	—
200	C₄	—	2	8	no	—
201	C₄	—	2	2	no	—
202	C₄	—	8	4	no	—
203	C₄	—	8	2	no	—
204	C₄	C₄₇×#456	1	4	yes	—
205	C₄	C₄₇×#460	1	2	yes	—
206	C₄	C₄₇×#492	1	1	yes	characteristic
207	C₄	C₄₇×#496	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	C₄₇×#2005	1	2	yes	—
210	C₂	C₄₇×#2014	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₇×#3229

C₂

yes

maximal

C₄₇×#2868

C₂

yes

maximal, characteristic

C₄₇×#1994

C₂

yes

maximal, characteristic

C₄₇×#1546

C₂

yes

maximal

C₄₇×#1276

C₂

yes

maximal, characteristic

C₄₇×#533

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₄₇

—

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

—

C₄₇×#498

—

C₄₇×#487

—

C₄₇×#483

—

C₄₇×#475

—

C₄₇×#431

C₂²

yes

—

C₄₇×#423

C₂²

yes

characteristic

C₄₇×#384

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#339

—

C₄₇×#332

—

C₂²×Q₈×C₄₇

—

C₂×M₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

D₄

yes

—

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#106×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

D₄

yes

—

#103×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

yes

C₂³×C₄₇

yes

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

#10299

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#392

yes

characteristic

C₂²×C₄₇

#450

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#456

yes

—

100

C₄×C₄₇

#460

yes

—

101

C₄×C₄₇

#492

yes

characteristic

102

C₄×C₄₇

#496

yes

characteristic

103

#3229

C₂×C₄₇

yes

—

104

#2868

C₂×C₄₇

yes

characteristic

105

#1994

C₂×C₄₇

yes

characteristic

106

#1546

C₂×C₄₇

yes

—

107

#1276

C₂×C₄₇

yes

characteristic

108

C₂×C₄₇

—

109

C₂×C₄₇

#2005

yes

—

110

C₂×C₄₇

#2014

yes

characteristic

111

#533

C₂²×C₄₇

yes

characteristic

112

C₂×C₄⋊Q₈

—

113

C₂×C₄⋊Q₈

C₂²×C₄₇

yes

—

114

#498

—

115

#487

—

116

#483

—

117

#475

—

118

#431

C₂²×C₄₇

yes

—

119

#423

C₂²×C₄₇

yes

characteristic

120

#384

—

121

C₄³

C₂²×C₄₇

yes

characteristic

122

#339

—

123

#332

—

124

C₄₇

#10299

yes

characteristic, direct product factor, Sylow

125

C₂²×Q₈

—

126

C₂×M₁₆

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

133

C₄⋊Q₈

D₄×C₄₇

yes

—

134

#125

—

135

C₂²⋊Q₈

—

136

C₄×Q₈

—

137

#117

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

D₄×C₄₇

yes

—

141

C₂×C₄²

—

142

C₂×C₄²

D₄×C₄₇

yes

—

143

C₂×C₄²

C₂³×C₄₇

yes

characteristic

144

#106

—

145

C₄⋊C₈

—

146

C₄⋊C₈

D₄×C₄₇

yes

—

147

#103

—

148

C₂²⋊C₈

—

149

C₄×C₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

C₂×D₄×C₄₇

yes

—

158

C₂²×C₄

C₂×D₄×C₄₇

yes

characteristic

159

M₁₆

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₂²⋊C₄

—

171

C₄²

—

172

C₄²

—

173

C₄²

—

174

C₄²

SD₁₆×C₄₇

yes

—

175

C₄²

C₂×D₄×C₄₇

yes

characteristic

176

C₄²

C₂×D₄×C₄₇

yes

—

177

C₄²

#41×C₄₇

yes

characteristic

178

C₂³

#120×C₄₇

yes

characteristic

179

Q₈

—

180

Q₈

—

181

Q₈

—

182

Q₈

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

#120×C₄₇

yes

—

190

C₂×C₄

#121×C₄₇

yes

—

191

C₂×C₄

#121×C₄₇

yes

characteristic

192

C₂×C₄

#127×C₄₇

yes

characteristic

193

C₂×C₄

C₂×SD₁₆×C₄₇

yes

—

194

C₂×C₄

AGL(1,ℤ/8)×C₄₇

yes

—

195

C₈

—

196

C₈

—

197

C₂²

—

198

C₂²

C₄₇×#392

yes

characteristic

199

C₂²

C₄₇×#450

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

C₄₇×#456

yes

—

205

C₄

C₄₇×#460

yes

—

206

C₄

C₄₇×#492

yes

characteristic

207

C₄

C₄₇×#496

yes

characteristic

208

C₂

—

209

C₂

C₄₇×#2005

yes

—

210

C₂

C₄₇×#2014

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₄₇
prime radical	C₂³×C₄₇	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₄₇

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₄₇

prime radical

C₂³×C₄₇

#120

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

N	H ≅ G/N	type
91. #10299	C₄₇	direct (N × H)
124. C₄₇	#10299	direct (N × H)

H ≅ G/N

type

91. #10299

C₄₇

direct (N × H)

124. C₄₇

#10299

direct (N × H)

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

subgroup	index
C₄₇×#3229	2
C₄₇×#2868	2
C₄₇×#1994	2
C₄₇×#1546	2
C₄₇×#1276	2
#10299	47

subgroup

index

C₄₇×#3229

C₄₇×#2868

C₄₇×#1994

C₄₇×#1546

C₄₇×#1276

#10299

Subgroups — C₄₇×#10299 #26049997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47×#10299 #26049997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#10299 #26049997