Group #36472950 — subgroups

Subgroups — #36472950

Order 22784.

386 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₂×C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151374	C₂	1	1	yes	maximal, characteristic
3	#26018329	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
6	C₂²×#13083549	C₂²	1	2	yes	—
7	C₈×Dic₁₇₈	—	2	2	no	—
8	C₈×Dic₁₇₈	—	2	2	no	—
9	C₂×C₄⋊#10623494	C₄	1	1	yes	characteristic
10	C₂×#15667941	C₄	1	2	yes	—
11	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
12	#19151374	—	2	2	no	—
13	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₈₉	—	2	2	no	—
16	C₂²×Dic₁₇₈	C₈	1	1	yes	characteristic
17	C₂²×#10623494	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₇₈	—	2	2	no	—
19	C₄×Dic₁₇₈	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
21	C₂×#13083553	C₈	1	2	yes	—
22	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
23	C₂×#13083549	—	2	2	no	—
24	C₂×#13083549	—	2	2	no	—
25	C₂×#13083549	Q₈	1	4	yes	complemented
26	C₄⋊#10623494	—	2	2	no	—
27	#15667941	—	2	4	no	—
28	C₈×Dic₈₉	—	2	2	no	—
29	C₈×Dic₈₉	—	2	2	no	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₈₉	—	2	2	no	—
34	C₂×C₁₆×C₈₉	—	2	4	no	—
35	C₄×C₈×C₈₉	D₄	1	2	yes	—
36	C₄×C₈×C₈₉	D₄	1	2	yes	—
37	C₂×Dic₁₇₈	—	2	2	no	—
38	C₂×Dic₁₇₈	—	2	4	no	—
39	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
40	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
41	C₂×#10623494	—	2	2	no	—
42	C₂×#10623494	—	2	4	no	—
43	C₄⋊Dic₈₉	—	2	2	no	—
44	#13083553	—	2	4	no	—
45	C₄×Dic₈₉	—	2	2	no	—
46	C₄×Dic₈₉	—	2	2	no	—
47	#13083549	—	2	4	no	—
48	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	—	2	2	no	—
52	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₈₉	#41	1	2	yes	—
57	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₈₉	M₁₆	1	2	yes	—
59	C₁₆×C₈₉	—	4	2	no	—
60	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₈₉	#131	1	2	yes	—
69	C₂×Dic₈₉	—	2	2	no	—
70	C₂×Dic₈₉	—	2	2	no	—
71	C₂×Dic₈₉	—	2	2	no	—
72	C₂×Dic₈₉	—	2	4	no	—
73	C₂×Dic₈₉	M₃₂	1	4	yes	—
74	Dic₁₇₈	—	2	8	no	—
75	C₈×C₈₉	—	2	2	no	—
76	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₈₉	#124	1	1	yes	characteristic
78	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
79	#10623494	—	4	2	no	—
80	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₂²×C₈₉	C₈×Q₈	1	2	yes	—
82	C₂²×C₈₉	C₂×M₃₂	1	2	yes	—
83	C₂²×C₈₉	#504	1	2	yes	—
84	C₄×C₈₉	—	2	2	no	—
85	C₄×C₈₉	—	2	2	no	—
86	C₄×C₈₉	#391	1	1	yes	characteristic
87	C₄×C₈₉	#408	1	1	yes	characteristic
88	C₄×C₈₉	#446	1	2	yes	—
89	Dic₈₉	—	2	4	no	—
90	Dic₈₉	—	4	2	no	—
91	#13264	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	#1984	1	1	yes	characteristic
93	C₂×C₈₉	#2112	1	1	yes	characteristic
94	C₂×C₈₉	#2113	1	1	yes	characteristic
95	C₂×C₈₉	#2183	1	4	yes	—
96	C₂×C₈×Q₈	—	89	1	no	—
97	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
98	C₂×#346	—	89	1	no	—
99	#1374	—	89	4	no	—
100	C₈₉	#13264	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄×Q₈	—	89	1	no	—
102	C₂²×C₁₆	—	89	2	no	—
103	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
104	C₈×Q₈	—	178	2	no	—
105	C₈×Q₈	—	178	2	no	—
106	C₂×C₄⋊C₈	—	89	2	no	—
107	C₂×C₄⋊C₈	—	89	1	no	—
108	C₂×C₄×C₈	—	89	2	no	—
109	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
110	C₄⋊C₁₆	—	2	2	no	—
111	#346	—	178	2	no	—
112	C₂²×Q₈	—	89	1	no	—
113	C₂²×C₈	—	89	2	no	—
114	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
116	C₄×Q₈	—	178	2	no	—
117	C₄×Q₈	—	178	2	no	—
118	C₂×C₄⋊C₄	—	89	2	no	—
119	C₂×C₄⋊C₄	—	89	1	no	—
120	C₂×C₄²	—	89	2	no	—
121	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	4	no	—
124	C₂×C₁₆	—	89	4	no	—
125	C₂×C₁₆	—	178	2	no	—
126	C₂×C₁₆	—	178	2	no	—
127	C₄⋊C₈	—	178	4	no	—
128	C₄⋊C₈	—	178	2	no	—
129	C₄×C₈	—	178	2	no	—
130	C₄×C₈	—	178	2	no	—
131	C₄×C₈	D₃₅₆	1	2	yes	—
132	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
133	C₂×Q₈	—	178	4	no	—
134	C₂×Q₈	—	178	2	no	—
135	C₂²×C₄	—	89	2	no	—
136	C₂²×C₄	—	89	2	no	—
137	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
138	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	178	4	no	—
142	C₂×C₈	—	178	2	no	—
143	C₂×C₈	#13083551	1	2	yes	—
144	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
145	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₈₉	1	2	yes	—
147	C₂×C₈	#13083573	1	2	yes	—
148	C₄⋊C₄	—	178	4	no	—
149	C₄⋊C₄	—	178	2	no	—
150	C₄²	—	178	2	no	—
151	C₄²	—	178	2	no	—
152	C₄²	#13083551	1	2	yes	—
153	C₄²	#13083557	1	2	yes	—
154	C₁₆	—	4	2	no	—
155	C₁₆	—	178	4	no	—
156	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
157	Q₈	—	178	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	89	4	no	—
161	C₂×C₄	—	178	4	no	—
162	C₂×C₄	—	178	2	no	—
163	C₂×C₄	—	178	2	no	—
164	C₂×C₄	—	178	2	no	—
165	C₂×C₄	#15667924	1	2	yes	—
166	C₂×C₄	#15667924	1	2	yes	—
167	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
168	C₂×C₄	#15667977	1	2	yes	—
169	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
170	C₂×C₄	#15668020	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	356	2	no	—
173	C₈	#15667992	1	2	yes	—
174	C₈	#15667999	1	1	yes	characteristic
175	C₈	#15668010	1	1	yes	characteristic
176	C₂²	C₂×#15667924	1	1	yes	characteristic
177	C₂²	#19151722	1	2	yes	—
178	C₂²	M₃₂×D₈₉	1	2	yes	—
179	C₂²	#19152014	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	356	2	no	—
183	C₄	—	178	4	no	—
184	C₄	#19151503	1	1	yes	characteristic
185	C₄	#19151566	1	1	yes	characteristic
186	C₄	#19151725	1	2	yes	—
187	C₂	#26021187	1	1	yes	characteristic
188	C₂	#26021862	1	1	yes	characteristic
189	C₂	#26021865	1	1	yes	characteristic
190	C₂	#26022159	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₇₈

C₂

yes

maximal, characteristic

C₂×#19151374

C₂

yes

maximal, characteristic

#26018329

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₇₈

C₄

yes

characteristic

C₂²×#13083549

C₂²

yes

—

C₈×Dic₁₇₈

—

C₈×Dic₁₇₈

—

C₂×C₄⋊#10623494

C₄

yes

characteristic

C₂×#15667941

C₄

yes

—

C₂×C₈×Dic₈₉

C₂²

yes

—

#19151374

—

C₂²×C₁₆×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₂²×Dic₁₇₈

C₈

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

—

C₄×Dic₁₇₈

—

C₄×Dic₁₇₈

—

C₂×C₄⋊Dic₈₉

C₈

yes

characteristic

C₂×#13083553

C₈

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

C₂×#13083549

—

C₂×#13083549

—

C₂×#13083549

Q₈

yes

complemented

C₄⋊#10623494

—

#15667941

—

C₈×Dic₈₉

—

C₈×Dic₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂×C₈

yes

—

C₂²×Dic₈₉

C₂×C₈

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

M₁₆

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×Q₈

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₉

#131

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

M₃₂

yes

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂²⋊Q₈

yes

—

C₈×C₈₉

#124

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

characteristic

#10623494

—

C₂²×C₈₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₈₉

C₈×Q₈

yes

—

C₂²×C₈₉

C₂×M₃₂

yes

—

C₂²×C₈₉

#504

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈×Q₈

—

C₂×C₄⋊C₁₆

D₈₉

yes

characteristic

C₂×#346

—

#1374

—

100

C₈₉

#13264

yes

characteristic, complemented, Sylow

101

C₂×C₄×Q₈

—

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₈₉

yes

—

104

C₈×Q₈

—

178

—

105

C₈×Q₈

—

178

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

110

C₄⋊C₁₆

—

111

#346

—

178

—

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₈₉

yes

characteristic

115

C₂²×C₈

C₂²×D₈₉

yes

characteristic

116

C₄×Q₈

—

178

—

117

C₄×Q₈

—

178

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

—

121

C₂×C₄²

C₄×D₈₉

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

178

—

126

C₂×C₁₆

—

178

—

127

C₄⋊C₈

—

178

—

128

C₄⋊C₈

—

178

—

129

C₄×C₈

—

178

—

130

C₄×C₈

—

178

—

131

C₄×C₈

D₃₅₆

yes

—

132

C₄×C₈

C₂²⋊D₈₉

yes

—

133

C₂×Q₈

—

178

—

134

C₂×Q₈

—

178

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₈×D₈₉

yes

characteristic

138

C₂²×C₄

C₈×D₈₉

yes

characteristic

139

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

178

—

142

C₂×C₈

—

178

—

143

C₂×C₈

#13083551

yes

—

144

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

145

C₂×C₈

C₂×D₃₅₆

yes

characteristic

146

C₂×C₈

Q₈×D₈₉

yes

—

147

C₂×C₈

#13083573

yes

—

148

C₄⋊C₄

—

178

—

149

C₄⋊C₄

—

178

—

150

C₄²

—

178

—

151

C₄²

—

178

—

152

C₄²

#13083551

yes

—

153

C₄²

#13083557

yes

—

154

C₁₆

—

155

C₁₆

—

178

—

156

C₂³

C₂×C₈×D₈₉

yes

characteristic

157

Q₈

—

178

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

178

—

162

C₂×C₄

—

178

—

163

C₂×C₄

—

178

—

164

C₂×C₄

—

178

—

165

C₂×C₄

#15667924

yes

—

166

C₂×C₄

#15667924

yes

—

167

C₂×C₄

C₂×#13083551

yes

characteristic

168

C₂×C₄

#15667977

yes

—

169

C₂×C₄

C₂×#13083557

yes

characteristic

170

C₂×C₄

#15668020

yes

—

171

C₈

—

172

C₈

—

356

—

173

C₈

#15667992

yes

—

174

C₈

#15667999

yes

characteristic

175

C₈

#15668010

yes

characteristic

176

C₂²

C₂×#15667924

yes

characteristic

177

C₂²

#19151722

yes

—

178

C₂²

M₃₂×D₈₉

yes

—

179

C₂²

#19152014

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

356

—

183

C₄

—

178

—

184

C₄

#19151503

yes

characteristic

185

C₄

#19151566

yes

characteristic

186

C₄

#19151725

yes

—

187

C₂

#26021187

yes

characteristic

188

C₂

#26021862

yes

characteristic

189

C₂

#26021865

yes

characteristic

190

C₂

#26022159

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂²×C₈
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#13264
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₈₉

commutator, socle, prime radical

C₂³×C₈₉

C₂²×C₈

Fitting

C₂×C₄⋊C₁₆×C₈₉

C₂

cosocle

C₂²×C₈×C₈₉

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
25. C₂×#13083549	Q₈	semidirect
100. C₈₉	#13264	semidirect

H ≅ G/N

type

25. C₂×#13083549

Q₈

semidirect

100. C₈₉

#13264

semidirect

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

subgroup	index
C₂×C₈×Dic₁₇₈	2
C₂×#19151374	2
#26018329	2
C₂×C₄⋊C₁₆×C₈₉	2
#13264	89

subgroup

index

C₂×C₈×Dic₁₇₈

C₂×#19151374

#26018329

C₂×C₄⋊C₁₆×C₈₉

#13264

Subgroups — #36472950

Conjugacy classes of subgroups

Complements

Maximal subgroups