Group #26116014 — subgroups

Subgroups — #26116014

Order 12032.

526 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 207 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	#20244350	C₂	1	2	yes	maximal
2	#20244191	C₂	1	2	yes	maximal
3	#20243545	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093956	—	2	2	no	—
7	C₂×#13093948	—	2	4	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688993	C₂²	1	1	yes	characteristic
10	#15688993	C₂²	1	1	yes	characteristic
11	#15688885	—	2	4	no	—
12	C₈×C₄₇⋊C₈	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
14	C₄₇×#498	—	2	2	no	—
15	C₄₇×#498	C₄	1	4	yes	complemented
16	C₂×#106×C₄₇	—	2	2	no	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	2	1	no	—
22	C₂×C₄⋊Dic₄₇	—	2	1	no	—
23	#13093956	—	4	4	no	—
24	#13093948	—	4	8	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×Q₈×C₄₇	—	2	2	no	—
28	C₂²×C₈×C₄₇	D₄	1	2	yes	—
29	C₄⋊Q₈×C₄₇	—	2	2	no	—
30	C₄⋊Q₈×C₄₇	—	2	2	no	—
31	C₄⋊Q₈×C₄₇	—	2	2	no	—
32	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	#106×C₄₇	—	2	8	no	—
37	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₄₇	D₄	1	2	yes	—
39	C₂×Q₈×C₄₇	—	2	4	no	—
40	C₂×Q₈×C₄₇	—	4	2	no	—
41	C₂×Q₈×C₄₇	—	4	4	no	—
42	C₂×Q₈×C₄₇	—	4	4	no	—
43	C₂²×C₄×C₄₇	—	4	2	no	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
46	C₂²×Dic₄₇	—	4	1	no	—
47	C₂²×Dic₄₇	—	4	1	no	—
48	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
49	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
50	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
51	C₄⋊C₄×C₄₇	—	2	4	no	—
52	C₄⋊C₄×C₄₇	—	2	4	no	—
53	C₄⋊C₄×C₄₇	—	4	2	no	—
54	C₄⋊C₄×C₄₇	—	4	2	no	—
55	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₇	#41	1	2	yes	—
58	C₄⋊Dic₄₇	—	4	4	no	—
59	C₄⋊Dic₄₇	—	4	2	no	—
60	C₄⋊Dic₄₇	—	4	2	no	—
61	C₄⋊Dic₄₇	—	4	4	no	—
62	C₂×C₄₇⋊C₈	—	2	4	no	—
63	C₂×C₄₇⋊C₈	—	2	4	no	—
64	C₂×C₄₇⋊C₈	SD₁₆	1	8	yes	—
65	C₂³×C₄₇	#127	1	1	yes	characteristic
66	Q₈×C₄₇	—	4	8	no	—
67	Q₈×C₄₇	—	8	4	no	—
68	C₂×C₄×C₄₇	—	4	4	no	—
69	C₂×C₄×C₄₇	—	8	2	no	—
70	C₂×C₄×C₄₇	—	8	2	no	—
71	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₄₇	#121	1	2	yes	—
74	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
76	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₄₇	#135	1	4	yes	—
79	C₂×Dic₄₇	—	8	2	no	—
80	C₂×Dic₄₇	—	8	1	no	—
81	C₂×Dic₄₇	—	8	1	no	—
82	C₂×Dic₄₇	—	8	2	no	—
83	C₈×C₄₇	#103	1	8	yes	complemented
84	C₄₇⋊C₈	—	2	8	no	—
85	#9819	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	#390	1	1	yes	characteristic
87	C₂²×C₄₇	#492	1	2	yes	—
88	C₂²×C₄₇	#492	1	1	yes	characteristic
89	C₂²×C₄₇	#494	1	1	yes	characteristic
90	C₂²×C₄₇	#495	1	2	yes	—
91	C₄×C₄₇	—	8	4	no	—
92	C₄×C₄₇	C₂×#103	1	2	yes	—
93	C₄×C₄₇	C₄×SD₁₆	1	4	yes	—
94	C₄×C₄₇	#465	1	4	yes	—
95	C₄×C₄₇	#487	1	2	yes	—
96	Dic₄₇	—	16	1	no	—
97	Dic₄₇	—	16	1	no	—
98	C₂×#498	D₄₇	1	1	yes	characteristic
99	#1990	—	47	2	no	—
100	#1956	—	47	2	no	—
101	#1844	—	47	1	no	—
102	C₂×C₈²	—	47	1	no	—
103	C₂×C₄₇	#1705	1	1	yes	characteristic
104	C₂×C₄₇	#1707	1	2	yes	—
105	C₂×C₄₇	#1708	1	1	yes	characteristic
106	C₂×C₄₇	#1956	1	1	yes	characteristic
107	C₂×C₄₇	#1968	1	2	yes	—
108	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
109	#498	—	2	2	no	—
110	#498	Dic₄₇	1	4	yes	complemented
111	C₂×#106	—	2	2	no	—
112	C₂×#106	—	94	2	no	—
113	C₂×#103	—	94	4	no	—
114	C₂×C₄×C₈	—	47	2	no	—
115	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
116	#384	—	47	1	no	—
117	#384	—	47	1	no	—
118	#340	—	94	4	no	—
119	C₈²	—	94	2	no	—
120	C₄₇	#9819	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	2	2	no	—
122	C₂²×C₈	—	47	4	no	—
123	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	94	2	no	—
131	C₂×C₄⋊C₄	—	94	2	no	—
132	C₂×C₄⋊C₄	—	94	1	no	—
133	C₂×C₄⋊C₄	—	94	1	no	—
134	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
135	#106	—	2	8	no	—
136	#106	—	188	4	no	—
137	#103	—	188	8	no	—
138	C₄×C₈	—	94	2	no	—
139	C₄×C₈	—	94	2	no	—
140	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
141	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	188	1	no	—
148	C₂²×C₄	—	188	1	no	—
149	C₂²×C₄	D₄×D₄₇	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
151	C₂×C₈	—	47	8	no	—
152	C₂×C₈	—	94	4	no	—
153	C₂×C₈	—	94	4	no	—
154	C₂×C₈	#10625336	1	4	yes	—
155	C₂×C₈	#10625337	1	4	yes	—
156	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	188	4	no	—
162	C₄⋊C₄	—	188	2	no	—
163	C₄⋊C₄	—	188	2	no	—
164	C₄⋊C₄	—	188	4	no	—
165	C₄²	#10625352	1	2	yes	—
166	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
168	C₂³	#13094020	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	376	2	no	—
175	C₂×C₄	—	376	1	no	—
176	C₂×C₄	—	376	1	no	—
177	C₂×C₄	—	376	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
180	C₂×C₄	#13093995	1	2	yes	—
181	C₂×C₄	#13094011	1	1	yes	characteristic
182	C₂×C₄	SD₁₆×D₄₇	1	4	yes	—
183	C₂×C₄	C₂×#10625336	1	2	yes	—
184	C₂×C₄	C₂×#10625337	1	2	yes	—
185	C₂×C₄	#13094045	1	4	yes	—
186	C₈	—	94	8	no	—
187	C₈	#13093948	1	8	yes	complemented
188	C₂²	#15689026	1	1	yes	characteristic
189	C₂²	#15689494	1	2	yes	—
190	C₂²	#15689497	1	1	yes	characteristic
191	C₂²	#15689502	1	1	yes	characteristic
192	C₂²	#15689504	1	2	yes	—
193	C₄	—	8	4	no	—
194	C₄	—	752	1	no	—
195	C₄	—	752	1	no	—
196	C₄	C₂×#13093948	1	2	yes	—
197	C₄	SD₁₆×Dic₄₇	1	4	yes	—
198	C₄	#15689349	1	4	yes	—
199	C₄	#15689473	1	2	yes	—
200	C₂	#20242976	1	1	yes	characteristic
201	C₂	#20242983	1	2	yes	—
202	C₂	#20242990	1	1	yes	characteristic
203	C₂	#20244191	1	1	yes	characteristic
204	C₂	#20244244	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244350

C₂

yes

maximal

#20244191

C₂

yes

maximal

#20243545

C₂

yes

maximal, characteristic

C₂×C₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄₇×#498

C₂

yes

maximal, characteristic

C₂×#13093956

—

C₂×#13093948

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688993

C₂²

yes

characteristic

#15688993

C₂²

yes

characteristic

#15688885

—

C₈×C₄₇⋊C₈

—

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

—

C₄₇×#498

—

C₄₇×#498

C₄

yes

complemented

C₂×#106×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093956

—

#13093948

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×Q₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#106×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

SD₁₆

yes

—

C₂³×C₄₇

#127

yes

characteristic

Q₈×C₄₇

—

Q₈×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

#121

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

#103

yes

complemented

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#103

yes

—

C₄×C₄₇

C₄×SD₁₆

yes

—

C₄×C₄₇

#465

yes

—

C₄×C₄₇

#487

yes

—

Dic₄₇

—

Dic₄₇

—

C₂×#498

D₄₇

yes

characteristic

#1990

—

100

#1956

—

101

#1844

—

102

C₂×C₈²

—

103

C₂×C₄₇

#1705

yes

characteristic

104

C₂×C₄₇

#1707

yes

—

105

C₂×C₄₇

#1708

yes

characteristic

106

C₂×C₄₇

#1956

yes

characteristic

107

C₂×C₄₇

#1968

yes

—

108

C₂×C₄⋊Q₈

C₂×D₄₇

yes

—

109

#498

—

110

#498

Dic₄₇

yes

complemented

111

C₂×#106

—

112

C₂×#106

—

113

C₂×#103

—

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

116

#384

—

117

#384

—

118

#340

—

119

C₈²

—

120

C₄₇

#9819

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²⋊D₄₇

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

C₂×Dic₄₇

yes

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄²

C₂²×D₄₇

yes

characteristic

135

#106

—

136

#106

—

188

—

137

#103

—

188

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

C₂×Dic₄₇

yes

—

141

C₄×C₈

C₂²⋊D₄₇

yes

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂²×C₄

—

147

C₂²×C₄

—

188

—

148

C₂²×C₄

—

188

—

149

C₂²×C₄

D₄×D₄₇

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

#10625336

yes

—

155

C₂×C₈

#10625337

yes

—

156

C₂×C₈

C₂²⋊Dic₄₇

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

188

—

162

C₄⋊C₄

—

188

—

163

C₄⋊C₄

—

188

—

164

C₄⋊C₄

—

188

—

165

C₄²

#10625352

yes

—

166

C₄²

C₂²×Dic₄₇

yes

characteristic

167

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

168

C₂³

#13094020

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

376

—

175

C₂×C₄

—

376

—

176

C₂×C₄

—

376

—

177

C₂×C₄

—

376

—

178

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

179

C₂×C₄

D₄×Dic₄₇

yes

—

180

C₂×C₄

#13093995

yes

—

181

C₂×C₄

#13094011

yes

characteristic

182

C₂×C₄

SD₁₆×D₄₇

yes

—

183

C₂×C₄

C₂×#10625336

yes

—

184

C₂×C₄

C₂×#10625337

yes

—

185

C₂×C₄

#13094045

yes

—

186

C₈

—

187

C₈

#13093948

yes

complemented

188

C₂²

#15689026

yes

characteristic

189

C₂²

#15689494

yes

—

190

C₂²

#15689497

yes

characteristic

191

C₂²

#15689502

yes

characteristic

192

C₂²

#15689504

yes

—

193

C₄

—

194

C₄

—

752

—

195

C₄

—

752

—

196

C₄

C₂×#13093948

yes

—

197

C₄

SD₁₆×Dic₄₇

yes

—

198

C₄

#15689349

yes

—

199

C₄

#15689473

yes

—

200

C₂

#20242976

yes

characteristic

201

C₂

#20242983

yes

—

202

C₂

#20242990

yes

characteristic

203

C₂

#20244191

yes

characteristic

204

C₂

#20244244

yes

—

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#498	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#498	D₄₇
hypocenter	C₄₇	#9819
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#498

C₂

socle, prime radical

C₂³×C₄₇

#127

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
15. C₄₇×#498	C₄	semidirect
83. C₈×C₄₇	#103	semidirect
110. #498	Dic₄₇	semidirect
120. C₄₇	#9819	semidirect
187. C₈	#13093948	semidirect

H ≅ G/N

type

15. C₄₇×#498

C₄

semidirect

83. C₈×C₄₇

#103

semidirect

110. #498

Dic₄₇

semidirect

120. C₄₇

#9819

semidirect

187. C₈

#13093948

semidirect

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

subgroup	index
#20244350	2
#20244191	2
#20243545	2
C₂×C₈×C₄₇⋊C₈	2
C₂×C₄₇×#498	2
#9819	47

subgroup

index

#20244350

#20244191

#20243545

C₂×C₈×C₄₇⋊C₈

C₂×C₄₇×#498

#9819

Subgroups — #26116014

Conjugacy classes of subgroups

Complements

Maximal subgroups