Group #26113796 — subgroups

Subgroups — #26113796

Order 12032.

488 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 103 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	#20252459	C₂	1	1	yes	maximal, characteristic
2	#20242105	C₂	1	2	yes	maximal, complemented
3	#20242023	C₂	1	1	yes	maximal, characteristic
4	#20241260	C₂	1	2	yes	maximal, complemented
5	C₄₇×#3152	C₂	1	1	yes	maximal, characteristic
6	#15689195	—	2	4	no	—
7	#15689180	C₂²	1	2	yes	—
8	#15689180	C₂²	1	2	yes	—
9	#15689090	—	2	2	no	—
10	#15688869	—	2	4	no	—
11	#15688863	—	2	2	no	—
12	C₂×#137×C₄₇	—	2	2	no	—
13	C₄₇×#532	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#497	—	2	2	no	—
16	C₄₇×#494	C₄	1	2	yes	—
17	C₄₇×#492	C₄	1	2	yes	—
18	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#10625329	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
26	#13093935	—	2	2	no	—
27	#13093935	—	2	2	no	—
28	C₂×#41×C₄₇	D₄	1	2	yes	—
29	C₂²×Q₈×C₄₇	—	2	1	no	—
30	#137×C₄₇	—	4	4	no	—
31	#137×C₄₇	—	4	4	no	—
32	C₂×Q₁₆×C₄₇	—	2	4	no	—
33	C₂×SD₁₆×C₄₇	—	2	4	no	—
34	C₂×M₁₆×C₄₇	—	2	1	no	—
35	C₄⋊Q₈×C₄₇	—	2	2	no	—
36	C₄⋊Q₈×C₄₇	—	2	1	no	—
37	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
38	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	#127×C₄₇	C₂×C₄	1	1	yes	characteristic
40	#124×C₄₇	D₄	1	2	yes	—
41	#121×C₄₇	—	2	2	no	—
42	C₄×D₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	#97×C₄₇	—	2	1	no	—
46	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	#41×C₄₇	—	4	2	no	—
48	#41×C₄₇	—	4	2	no	—
49	C₂×Q₈×C₄₇	—	2	2	no	—
50	C₂×Q₈×C₄₇	—	4	2	no	—
51	C₂×Q₈×C₄₇	—	4	2	no	—
52	C₂×Q₈×C₄₇	—	4	1	no	—
53	C₂×Q₈×C₄₇	C₂²⋊C₄	1	2	yes	—
54	C₂×D₄×C₄₇	—	2	2	no	—
55	C₂×D₄×C₄₇	—	2	2	no	—
56	C₂×D₄×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	4	1	no	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	Q₁₆×C₄₇	—	4	8	no	—
62	SD₁₆×C₄₇	—	4	8	no	—
63	M₁₆×C₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	—	2	2	no	—
65	C₄⋊C₄×C₄₇	—	2	2	no	—
66	C₄⋊C₄×C₄₇	—	2	4	no	—
67	C₄⋊C₄×C₄₇	—	4	2	no	—
68	C₂²⋊C₄×C₄₇	—	2	4	no	—
69	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₄₇	C₂×D₄	1	2	yes	—
72	#10625329	—	8	2	no	—
73	C₂×C₄₇⋊C₈	—	2	4	no	—
74	C₂×C₄₇⋊C₈	—	4	2	no	—
75	C₂×C₄₇⋊C₈	—	8	2	no	—
76	C₂³×C₄₇	—	2	2	no	—
77	C₂³×C₄₇	#123	1	1	yes	characteristic
78	Q₈×C₄₇	—	4	2	no	—
79	Q₈×C₄₇	—	4	4	no	—
80	Q₈×C₄₇	—	8	2	no	—
81	D₄×C₄₇	—	4	2	no	—
82	D₄×C₄₇	—	8	2	no	—
83	D₄×C₄₇	—	8	2	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	4	2	no	—
87	C₂×C₄×C₄₇	—	4	2	no	—
88	C₂×C₄×C₄₇	—	4	2	no	—
89	C₂×C₄×C₄₇	—	4	2	no	—
90	C₂×C₄×C₄₇	—	8	1	no	—
91	C₂×C₄×C₄₇	—	8	1	no	—
92	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#121	1	2	yes	—
96	C₈×C₄₇	—	2	4	no	—
97	C₄₇⋊C₈	—	4	4	no	—
98	C₄₇⋊C₈	—	8	2	no	—
99	#9421	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	2	1	no	—
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	4	2	no	—
103	C₂²×C₄₇	—	4	4	no	—
104	C₂²×C₄₇	#386	1	1	yes	characteristic
105	C₄×C₄₇	—	2	1	no	—
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	—	4	2	no	—
109	C₄×C₄₇	—	8	2	no	—
110	C₄×C₄₇	#453	1	2	yes	—
111	#3152	D₄₇	1	1	yes	characteristic
112	#2980	—	47	1	no	—
113	#1556	—	47	2	no	—
114	#1536	—	47	1	no	—
115	#1291	—	47	2	no	—
116	C₂×C₄₇	—	4	1	no	—
117	C₂×C₄₇	—	8	2	no	—
118	C₂×C₄₇	#1887	1	1	yes	characteristic
119	C₂×C₄₇	#1902	1	2	yes	—
120	C₂×#137	—	2	2	no	—
121	#532	C₂×D₄₇	1	1	yes	characteristic
122	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
123	#497	—	2	2	no	—
124	#494	Dic₄₇	1	2	yes	—
125	#492	Dic₄₇	1	2	yes	—
126	#435	—	94	4	no	—
127	#432	—	47	2	no	—
128	#432	—	47	2	no	—
129	#408	—	94	2	no	—
130	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
131	#332	—	94	4	no	—
132	#329	—	94	2	no	—
133	C₄₇	#9421	1	1	yes	characteristic, complemented, Sylow
134	C₂×#41	C₂²⋊D₄₇	1	2	yes	—
135	C₂²×Q₈	—	2	1	no	—
136	#137	—	4	4	no	—
137	#137	—	4	4	no	—
138	C₂×Q₁₆	—	2	4	no	—
139	C₂×SD₁₆	—	2	4	no	—
140	C₂×M₁₆	—	2	1	no	—
141	C₂×M₁₆	—	94	2	no	—
142	C₂²×C₈	—	94	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	1	no	—
145	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
146	C₄⋊Q₈	C₂×Dic₄₇	1	1	yes	characteristic
147	#127	C₂×Dic₄₇	1	1	yes	characteristic
148	#124	C₂²⋊D₄₇	1	2	yes	—
149	#121	—	2	2	no	—
150	C₄×D₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
153	C₄⋊C₈	—	47	4	no	—
154	C₄⋊C₈	—	94	2	no	—
155	C₂²⋊C₈	—	94	4	no	—
156	C₂²⋊C₈	—	94	2	no	—
157	C₂²⋊C₈	—	94	2	no	—
158	#97	—	2	1	no	—
159	#97	—	94	2	no	—
160	#97	—	94	2	no	—
161	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
162	#41	—	4	2	no	—
163	#41	—	4	2	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂×Q₈	C₂²⋊Dic₄₇	1	2	yes	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	C₂²⋊Dic₄₇	1	2	yes	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	4	1	no	—
174	C₂²×C₄	#10625352	1	2	yes	—
175	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
176	Q₁₆	—	4	8	no	—
177	SD₁₆	—	4	8	no	—
178	M₁₆	—	2	4	no	—
179	M₁₆	—	376	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	94	4	no	—
182	C₂×C₈	—	376	2	no	—
183	C₂×C₈	—	188	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	2	4	no	—
188	C₄²	D₄×D₄₇	1	2	yes	—
189	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
190	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂³	—	2	2	no	—
192	C₂³	#13094005	1	1	yes	characteristic
193	Q₈	—	4	2	no	—
194	Q₈	—	4	4	no	—
195	Q₈	—	8	2	no	—
196	D₄	—	4	2	no	—
197	D₄	—	8	2	no	—
198	D₄	—	8	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	8	1	no	—
206	C₂×C₄	—	8	1	no	—
207	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
208	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
209	C₂×C₄	#13093992	1	1	yes	characteristic
210	C₂×C₄	#13093996	1	2	yes	—
211	C₈	—	2	4	no	—
212	C₈	—	376	2	no	—
213	C₈	—	188	4	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	#15689003	1	1	yes	characteristic
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	8	2	no	—
224	C₄	#15689290	1	2	yes	—
225	C₂	—	4	1	no	—
226	C₂	—	8	2	no	—
227	C₂	#20243787	1	1	yes	characteristic
228	C₂	#20243863	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20252459

C₂

yes

maximal, characteristic

#20242105

C₂

yes

maximal, complemented

#20242023

C₂

yes

maximal, characteristic

#20241260

C₂

yes

maximal, complemented

C₄₇×#3152

C₂

yes

maximal, characteristic

#15689195

—

#15689180

C₂²

yes

—

#15689180

C₂²

yes

—

#15689090

—

#15688869

—

#15688863

—

C₂×#137×C₄₇

—

C₄₇×#532

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄₇×#497

—

C₄₇×#494

C₄

yes

—

C₄₇×#492

C₄

yes

—

C₄×M₁₆×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

#13093935

—

C₂×#41×C₄₇

D₄

yes

—

C₂²×Q₈×C₄₇

—

#137×C₄₇

—

#137×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×SD₁₆×C₄₇

—

C₂×M₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

characteristic

#127×C₄₇

C₂×C₄

yes

characteristic

#124×C₄₇

D₄

yes

—

#121×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#97×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂²⋊C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

Q₁₆×C₄₇

—

SD₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

#123

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

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×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9421

—

maximal, Sylow

100

C₂²×C₄₇

—

101

C₂²×C₄₇

—

102

C₂²×C₄₇

—

103

C₂²×C₄₇

—

104

C₂²×C₄₇

#386

yes

characteristic

105

C₄×C₄₇

—

106

C₄×C₄₇

—

107

C₄×C₄₇

—

108

C₄×C₄₇

—

109

C₄×C₄₇

—

110

C₄×C₄₇

#453

yes

—

111

#3152

D₄₇

yes

characteristic

112

#2980

—

113

#1556

—

114

#1536

—

115

#1291

—

116

C₂×C₄₇

—

117

C₂×C₄₇

—

118

C₂×C₄₇

#1887

yes

characteristic

119

C₂×C₄₇

#1902

yes

—

120

C₂×#137

—

121

#532

C₂×D₄₇

yes

characteristic

122

C₂×C₄⋊Q₈

C₂×D₄₇

yes

characteristic

123

#497

—

124

#494

Dic₄₇

yes

—

125

#492

Dic₄₇

yes

—

126

#435

—

127

#432

—

128

#432

—

129

#408

—

130

C₄×M₁₆

C₂×D₄₇

yes

characteristic

131

#332

—

132

#329

—

133

C₄₇

#9421

yes

characteristic, complemented, Sylow

134

C₂×#41

C₂²⋊D₄₇

yes

—

135

C₂²×Q₈

—

136

#137

—

137

#137

—

138

C₂×Q₁₆

—

139

C₂×SD₁₆

—

140

C₂×M₁₆

—

141

C₂×M₁₆

—

142

C₂²×C₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

C₂×Dic₄₇

yes

—

146

C₄⋊Q₈

C₂×Dic₄₇

yes

characteristic

147

#127

C₂×Dic₄₇

yes

characteristic

148

#124

C₂²⋊D₄₇

yes

—

149

#121

—

150

C₄×D₄

—

151

C₂×C₄⋊C₄

—

152

C₂×C₄²

C₂²×D₄₇

yes

characteristic

153

C₄⋊C₈

—

154

C₄⋊C₈

—

155

C₂²⋊C₈

—

156

C₂²⋊C₈

—

157

C₂²⋊C₈

—

158

#97

—

159

#97

—

160

#97

—

161

C₄×C₈

C₂×Dic₄₇

yes

—

162

#41

—

163

#41

—

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂×Q₈

C₂²⋊Dic₄₇

yes

—

169

C₂×D₄

—

170

C₂×D₄

—

171

C₂×D₄

C₂²⋊Dic₄₇

yes

—

172

C₂²×C₄

—

173

C₂²×C₄

—

174

C₂²×C₄

#10625352

yes

—

175

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

176

Q₁₆

—

177

SD₁₆

—

178

M₁₆

—

179

M₁₆

—

376

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₂×C₈

—

376

—

183

C₂×C₈

—

188

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₂²⋊C₄

—

188

C₄²

D₄×D₄₇

yes

—

189

C₄²

C₂²×Dic₄₇

yes

characteristic

190

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

191

C₂³

—

192

C₂³

#13094005

yes

characteristic

193

Q₈

—

194

Q₈

—

195

Q₈

—

196

D₄

—

197

D₄

—

198

D₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

208

C₂×C₄

D₄×Dic₄₇

yes

—

209

C₂×C₄

#13093992

yes

characteristic

210

C₂×C₄

#13093996

yes

—

211

C₈

—

212

C₈

—

376

—

213

C₈

—

188

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

—

218

C₂²

#15689003

yes

characteristic

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

#15689290

yes

—

225

C₂

—

226

C₂

—

227

C₂

#20243787

yes

characteristic

228

C₂

#20243863

yes

—

	subgroup	G/H
center	C₂²	#15689003
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#3152	C₂
socle	C₂²×C₄₇	#386
prime radical	C₄₇×#532	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#3152	D₄₇
hypocenter	C₄₇	#9421
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3152

subgroup

G/H

center

C₂²

#15689003

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#3152

C₂

socle

C₂²×C₄₇

#386

prime radical

C₄₇×#532

C₂²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3152

N	H ≅ G/N	type
2. #20242105	C₂	semidirect
4. #20241260	C₂	semidirect
133. C₄₇	#9421	semidirect

H ≅ G/N

type

2. #20242105

C₂

semidirect

4. #20241260

C₂

semidirect

133. C₄₇

#9421

semidirect

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

subgroup	index
#20252459	2
#20242105	2
#20242023	2
#20241260	2
C₄₇×#3152	2
#9421	47

subgroup

index

C₄₇×#3152

Subgroups — #26113796

Conjugacy classes of subgroups

Complements

Maximal subgroups