Group #26109922 — subgroups

Subgroups — #26109922

Order 12032.

310 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20243075	C₂	1	1	yes	maximal, characteristic
2	#20242134	C₂	1	2	yes	maximal
3	#20241232	C₂	1	2	yes	maximal
4	#20241217	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1274	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×#13093935	C₄	1	1	yes	characteristic
8	C₂×#13093935	C₄	1	1	yes	characteristic
9	C₂×#13093935	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	#15688875	—	2	2	no	—
14	#15688873	—	2	2	no	—
15	C₄₇×#431	C₂²	1	2	yes	—
16	C₄₇×#376	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
23	#13093935	—	2	2	no	—
24	#13093935	—	2	2	no	—
25	#13093935	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₄×C₄₇⋊C₈	—	2	2	no	—
28	C₄×C₄₇⋊C₈	C₈	1	4	yes	complemented
29	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
30	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
31	C₄×C₄₇⋊C₈	Q₈	1	1	yes	characteristic
32	C₄×C₄₇⋊C₈	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₄₇	—	2	2	no	—
37	C₂²⋊C₈×C₄₇	—	2	2	no	—
38	C₄×C₈×C₄₇	—	2	2	no	—
39	#95×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	2	1	no	—
41	C₂²×C₄×C₄₇	—	2	1	no	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	#41	1	2	yes	—
44	C₂×C₈×C₄₇	—	4	2	no	—
45	C₄²×C₄₇	—	2	1	no	—
46	C₄²×C₄₇	—	2	1	no	—
47	C₄²×C₄₇	C₂×C₈	1	2	yes	—
48	C₄²×C₄₇	M₁₆	1	2	yes	—
49	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₄₇	#41	1	2	yes	—
52	C₂×C₄₇⋊C₈	—	2	1	no	—
53	C₂×C₄₇⋊C₈	—	2	1	no	—
54	C₂×C₄₇⋊C₈	—	2	1	no	—
55	C₂×C₄₇⋊C₈	—	2	1	no	—
56	C₂×C₄₇⋊C₈	—	2	1	no	—
57	C₂×C₄₇⋊C₈	—	2	1	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
64	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
65	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
66	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
67	C₂³×C₄₇	#117	1	1	yes	characteristic
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	4	2	no	—
71	C₂×C₄×C₄₇	—	4	2	no	—
72	C₂×C₄×C₄₇	—	4	2	no	—
73	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#124	1	2	yes	—
76	C₂×C₄×C₄₇	#125	1	2	yes	—
77	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
81	C₈×C₄₇	—	8	2	no	—
82	C₄₇⋊C₈	—	2	1	no	—
83	C₄₇⋊C₈	—	2	1	no	—
84	C₄₇⋊C₈	—	2	2	no	—
85	C₄₇⋊C₈	—	2	2	no	—
86	C₄₇⋊C₈	—	8	2	no	—
87	C₄₇⋊C₈	C₄⋊C₈	1	2	yes	—
88	C₄₇⋊C₈	C₄⋊C₈	1	2	yes	—
89	#8780	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	—	2	2	no	—
91	C₂²×C₄₇	#382	1	1	yes	characteristic
92	C₂²×C₄₇	#431	1	1	yes	characteristic
93	C₂²×C₄₇	#432	1	1	yes	characteristic
94	C₄×C₄₇	—	2	2	no	—
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	—	4	2	no	—
97	C₄×C₄₇	—	4	2	no	—
98	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₄₇	#423	1	1	yes	characteristic
100	C₄×C₄₇	#431	1	1	yes	characteristic
101	C₄×C₄₇	#432	1	1	yes	characteristic
102	#1726	—	47	1	no	—
103	#1564	—	47	2	no	—
104	#1281	—	47	2	no	—
105	#1274	—	47	1	no	—
106	#1274	D₄₇	1	1	yes	characteristic
107	C₂×C₄₇	—	2	1	no	—
108	C₂×C₄₇	—	2	1	no	—
109	C₂×C₄₇	#1831	1	1	yes	characteristic
110	C₂×C₄₇	#1833	1	2	yes	—
111	#431	—	47	2	no	—
112	#431	C₂×D₄₇	1	2	yes	—
113	C₂×#97	—	47	1	no	—
114	C₂×#97	—	47	1	no	—
115	C₂×#97	—	47	1	no	—
116	C₂×C₄×C₈	—	47	1	no	—
117	C₂×C₄×C₈	—	47	1	no	—
118	C₂×C₄×C₈	—	47	1	no	—
119	#376	C₂×D₄₇	1	1	yes	characteristic
120	#335	—	94	2	no	—
121	#334	—	94	2	no	—
122	C₄₇	#8780	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	—	47	1	no	—
124	C₂²×C₈	—	47	1	no	—
125	C₂²×C₈	—	47	1	no	—
126	C₂²×C₈	—	47	1	no	—
127	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
128	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
129	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	94	2	no	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	94	2	no	—
134	#97	—	94	2	no	—
135	#97	—	94	2	no	—
136	#97	—	94	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	47	4	no	—
139	C₄×C₈	—	47	1	no	—
140	C₄×C₈	—	47	1	no	—
141	C₄×C₈	—	47	1	no	—
142	C₄×C₈	—	47	1	no	—
143	C₄×C₈	—	94	2	no	—
144	C₄×C₈	—	94	2	no	—
145	#95	—	2	2	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
149	C₂²×C₄	#10625352	1	2	yes	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	47	1	no	—
152	C₂×C₈	—	47	1	no	—
153	C₂×C₈	—	47	2	no	—
154	C₂×C₈	—	47	2	no	—
155	C₂×C₈	—	47	1	no	—
156	C₂×C₈	—	47	1	no	—
157	C₂×C₈	—	47	2	no	—
158	C₂×C₈	—	47	2	no	—
159	C₂×C₈	—	94	1	no	—
160	C₂×C₈	—	94	1	no	—
161	C₂×C₈	—	94	1	no	—
162	C₂×C₈	—	94	1	no	—
163	C₂×C₈	—	94	1	no	—
164	C₂×C₈	—	94	1	no	—
165	C₂×C₈	—	188	2	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₈×D₄₇	1	2	yes	—
169	C₄²	#10625324	1	2	yes	—
170	C₄²	#10625350	1	2	yes	—
171	C₄²	D₄×D₄₇	1	1	yes	characteristic
172	C₄²	Q₈×D₄₇	1	1	yes	characteristic
173	C₂³	#13093980	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
180	C₂×C₄	#13093981	1	1	yes	characteristic
181	C₂×C₄	#13094010	1	2	yes	—
182	C₂×C₄	#13094014	1	2	yes	—
183	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
184	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
185	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
186	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
187	C₈	—	8	2	no	—
188	C₈	—	47	2	no	—
189	C₈	—	47	2	no	—
190	C₈	—	94	1	no	—
191	C₈	—	94	1	no	—
192	C₈	—	94	2	no	—
193	C₈	—	94	2	no	—
194	C₈	—	376	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#15688987	1	1	yes	characteristic
197	C₂²	#15689173	1	1	yes	characteristic
198	C₂²	#15689177	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	2	no	—
203	C₄	C₄⋊C₈×D₄₇	1	1	yes	characteristic
204	C₄	#15689145	1	1	yes	characteristic
205	C₄	#15689174	1	1	yes	characteristic
206	C₄	#15689178	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	#20243503	1	1	yes	characteristic
210	C₂	#20243510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243075

C₂

yes

maximal, characteristic

#20242134

C₂

yes

maximal

#20241232

C₂

yes

maximal

#20241217

C₂

yes

maximal, characteristic

C₄₇×#1274

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₂×#13093935

C₄

yes

characteristic

C₂×#13093935

C₄

yes

characteristic

C₂×#13093935

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₄

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₄

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#15688875

—

#15688873

—

C₄₇×#431

C₂²

yes

—

C₄₇×#376

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

#13093935

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₈

yes

complemented

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

Q₈

yes

characteristic

C₄×C₄₇⋊C₈

Q₈

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₈

yes

—

C₂×C₄₇⋊C₈

C₂×C₈

yes

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₂³×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#125

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₄⋊C₈

yes

—

C₄₇⋊C₈

C₄⋊C₈

yes

—

#8780

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₈

yes

characteristic

C₄×C₄₇

#423

yes

characteristic

100

C₄×C₄₇

#431

yes

characteristic

101

C₄×C₄₇

#432

yes

characteristic

102

#1726

—

103

#1564

—

104

#1281

—

105

#1274

—

106

#1274

D₄₇

yes

characteristic

107

C₂×C₄₇

—

108

C₂×C₄₇

—

109

C₂×C₄₇

#1831

yes

characteristic

110

C₂×C₄₇

#1833

yes

—

111

#431

—

112

#431

C₂×D₄₇

yes

—

113

C₂×#97

—

114

C₂×#97

—

115

C₂×#97

—

116

C₂×C₄×C₈

—

117

C₂×C₄×C₈

—

118

C₂×C₄×C₈

—

119

#376

C₂×D₄₇

yes

characteristic

120

#335

—

121

#334

—

122

C₄₇

#8780

yes

characteristic, complemented, Sylow

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₂×C₄²

C₄×D₄₇

yes

characteristic

128

C₂×C₄²

C₄×D₄₇

yes

characteristic

129

C₂×C₄²

C₂²×D₄₇

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₂²⋊C₈

—

133

C₂²⋊C₈

—

134

#97

—

135

#97

—

136

#97

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

#95

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

149

C₂²×C₄

#10625352

yes

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

188

—

166

C₄²

—

167

C₄²

—

168

C₄²

C₈×D₄₇

yes

—

169

C₄²

#10625324

yes

—

170

C₄²

#10625350

yes

—

171

C₄²

D₄×D₄₇

yes

characteristic

172

C₄²

Q₈×D₄₇

yes

characteristic

173

C₂³

#13093980

yes

characteristic

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

180

C₂×C₄

#13093981

yes

characteristic

181

C₂×C₄

#13094010

yes

—

182

C₂×C₄

#13094014

yes

—

183

C₂×C₄

C₂×C₈×D₄₇

yes

characteristic

184

C₂×C₄

M₁₆×D₄₇

yes

characteristic

185

C₂×C₄

M₁₆×D₄₇

yes

characteristic

186

C₂×C₄

C₂×#10625324

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₈

—

192

C₈

—

193

C₈

—

194

C₈

—

376

—

195

C₂²

—

196

C₂²

#15688987

yes

characteristic

197

C₂²

#15689173

yes

characteristic

198

C₂²

#15689177

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

C₄⋊C₈×D₄₇

yes

characteristic

204

C₄

#15689145

yes

characteristic

205

C₄

#15689174

yes

characteristic

206

C₄

#15689178

yes

characteristic

207

C₂

—

208

C₂

—

209

C₂

#20243503

yes

characteristic

210

C₂

#20243510

yes

—

	subgroup	G/H
center	C₂×C₄	#13093981
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1274	C₂
socle	C₂²×C₄₇	#382
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1274	D₄₇
hypocenter	C₄₇	#8780
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

subgroup

G/H

center

C₂×C₄

#13093981

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1274

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

N	H ≅ G/N	type
28. C₄×C₄₇⋊C₈	C₈	semidirect
122. C₄₇	#8780	semidirect

H ≅ G/N

type

28. C₄×C₄₇⋊C₈

C₈

semidirect

122. C₄₇

#8780

semidirect

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

subgroup	index
#20243075	2
#20242134	2
#20241232	2
#20241217	2
C₄₇×#1274	2
#8780	47

subgroup

index

C₄₇×#1274

Subgroups — #26109922

Conjugacy classes of subgroups

Complements

Maximal subgroups