Group #36445003 — subgroups

Subgroups — #36445003

Order 22784.

430 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 161 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×Q₈×#10623494	C₂	1	1	yes	maximal, characteristic
2	#26018576	C₂	1	1	yes	maximal, characteristic
3	#26018539	C₂	1	2	yes	maximal
4	#26018103	C₂	1	2	yes	maximal
5	C₄×#101×C₈₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10623494	—	2	2	no	—
7	Q₈×#10623494	—	2	2	no	—
8	#19151651	C₂²	1	2	yes	—
9	C₂×C₄⋊#10623494	C₄	1	2	yes	—
10	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10623494	C₄	1	2	yes	—
12	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
13	#19151324	C₄	1	4	yes	—
14	#19151320	—	2	2	no	—
15	#19151320	—	2	2	no	—
16	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#101×C₈₉	C₄	1	2	yes	—
18	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
19	#343×C₈₉	C₄	1	2	yes	—
20	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
21	C₂²×#10623494	C₂×C₄	1	2	yes	—
22	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
23	C₄⋊#10623494	—	2	4	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₄⋊#10623494	—	2	2	no	—
26	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
27	C₄×#10623494	—	2	2	no	—
28	C₄×#10623494	—	2	2	no	—
29	C₄×#10623494	—	2	2	no	—
30	C₄×#10623494	D₄	1	2	yes	—
31	C₄×#10623494	D₄	1	2	yes	—
32	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
34	C₄×Q₈×C₈₉	—	2	2	no	—
35	C₄×Q₈×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	#101×C₈₉	—	2	4	no	—
41	#97×C₈₉	—	2	2	no	—
42	C₄×C₈×C₈₉	—	2	2	no	—
43	C₂×#10623494	—	2	2	no	—
44	C₂×#10623494	—	2	2	no	—
45	C₂×#10623494	—	2	4	no	—
46	C₂×#10623494	—	4	1	no	—
47	C₂×#10623494	—	4	2	no	—
48	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
50	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Q₈×C₈₉	—	2	2	no	—
52	C₂×Q₈×C₈₉	—	2	2	no	—
53	C₂×Q₈×C₈₉	M₁₆	1	4	yes	—
54	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
55	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₈₉	—	2	4	no	—
60	M₁₆×C₈₉	—	2	4	no	—
61	C₂×C₈×C₈₉	—	2	4	no	—
62	C₄⋊C₄×C₈₉	—	2	4	no	—
63	C₄⋊C₄×C₈₉	—	2	2	no	—
64	C₄²×C₈₉	—	2	2	no	—
65	C₄²×C₈₉	—	2	2	no	—
66	C₄²×C₈₉	C₂×D₄	1	2	yes	—
67	C₄²×C₈₉	#41	1	2	yes	—
68	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
69	Q₈×C₈₉	—	2	8	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	—	2	1	no	—
73	C₂×C₄×C₈₉	—	2	1	no	—
74	C₂×C₄×C₈₉	—	2	2	no	—
75	C₂×C₄×C₈₉	—	4	1	no	—
76	C₂×C₄×C₈₉	—	4	2	no	—
77	C₂×C₄×C₈₉	#97	1	4	yes	—
78	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₈₉	#131	1	2	yes	—
84	C₈×C₈₉	—	4	4	no	—
85	#10623494	—	2	1	no	—
86	#10623494	—	2	2	no	—
87	#10623494	—	4	4	no	—
88	#10623494	—	4	2	no	—
89	#10623494	#101	1	2	yes	complemented
90	C₂²×C₈₉	—	2	1	no	—
91	C₂²×C₈₉	—	2	1	no	—
92	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
94	C₂²×C₈₉	#405	1	1	yes	characteristic
95	C₄×C₈₉	—	2	4	no	—
96	C₄×C₈₉	—	2	1	no	—
97	C₄×C₈₉	—	2	2	no	—
98	C₄×C₈₉	—	4	2	no	—
99	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
100	C₄×C₈₉	#413	1	1	yes	characteristic
101	C₄×C₈₉	#435	1	4	yes	—
102	#8165	—	89	1	no	maximal, Sylow
103	C₂×C₈₉	—	2	1	no	—
104	C₂×C₈₉	—	2	1	no	—
105	C₂×C₈₉	#1752	1	1	yes	characteristic
106	C₂×C₈₉	C₄×#101	1	1	yes	characteristic
107	C₂×C₈₉	#1757	1	1	yes	characteristic
108	C₂×C₈×Q₈	—	89	1	no	—
109	C₄×#101	D₈₉	1	1	yes	characteristic
110	#1461	—	89	1	no	—
111	#1451	—	89	2	no	—
112	#1283	—	89	2	no	—
113	C₈₉	#8165	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
115	C₈×Q₈	—	178	2	no	—
116	C₈×Q₈	—	178	2	no	—
117	#431	—	89	2	no	—
118	C₂×C₄⋊C₈	—	89	2	no	—
119	C₂×C₄⋊C₈	—	89	1	no	—
120	C₂×#101	Dic₈₉	1	2	yes	—
121	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
122	C₂×C₄×C₈	—	89	1	no	—
123	C₂×C₄×C₈	—	89	2	no	—
124	#343	Dic₈₉	1	2	yes	—
125	#324	—	89	4	no	—
126	#322	—	178	2	no	—
127	#322	—	178	2	no	—
128	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
129	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
130	C₂²×C₈	—	89	1	no	—
131	C₂²×C₈	—	89	1	no	—
132	C₂²×C₈	—	89	2	no	—
133	C₄×Q₈	—	2	2	no	—
134	C₄×Q₈	—	2	2	no	—
135	C₂×C₄⋊C₄	C₄×D₈₉	1	2	yes	—
136	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
137	C₂×C₄²	C₄×D₈₉	1	2	yes	—
138	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
139	C₄⋊C₈	—	178	4	no	—
140	C₄⋊C₈	—	178	2	no	—
141	C₄⋊C₈	—	178	2	no	—
142	#101	—	2	4	no	—
143	C₂²⋊C₈	—	89	4	no	—
144	#97	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	89	2	no	—
147	C₄×C₈	—	89	2	no	—
148	C₄×C₈	—	178	2	no	—
149	C₄×C₈	—	178	2	no	—
150	C₄×C₈	—	178	2	no	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	#13083558	1	4	yes	—
154	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
155	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
156	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
159	M₁₆	—	2	4	no	—
160	M₁₆	—	2	4	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	89	1	no	—
163	C₂×C₈	—	89	1	no	—
164	C₂×C₈	—	89	2	no	—
165	C₂×C₈	—	356	2	no	—
166	C₂×C₈	—	356	1	no	—
167	C₂×C₈	—	178	2	no	—
168	C₂×C₈	—	178	4	no	—
169	C₂×C₈	—	178	2	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	D₄×D₈₉	1	2	yes	—
175	C₄²	#13083573	1	2	yes	—
176	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	#15667923	1	4	yes	—
186	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
187	C₂×C₄	#15667968	1	1	yes	characteristic
188	C₂×C₄	#15667972	1	2	yes	—
189	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
190	C₂×C₄	C₂×#13083558	1	2	yes	—
191	C₂×C₄	#15668019	1	2	yes	—
192	C₈	—	4	4	no	—
193	C₈	—	89	2	no	—
194	C₈	—	356	4	no	—
195	C₈	—	356	2	no	—
196	C₈	—	178	1	no	—
197	C₈	—	178	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₂²⋊C₄×Dic₈₉	1	1	yes	characteristic
201	C₂²	C₂×#15667923	1	1	yes	characteristic
202	C₂²	#19151554	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	#101×D₈₉	1	1	yes	characteristic
208	C₄	#19151581	1	1	yes	characteristic
209	C₄	#19151670	1	4	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#26020030	1	1	yes	characteristic
213	C₂	#101×Dic₈₉	1	1	yes	characteristic
214	C₂	#26020055	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×#10623494

C₂

yes

maximal, characteristic

#26018576

C₂

yes

maximal, characteristic

#26018539

C₂

yes

maximal

#26018103

C₂

yes

maximal

C₄×#101×C₈₉

C₂

yes

maximal, characteristic

Q₈×#10623494

—

Q₈×#10623494

—

#19151651

C₂²

yes

—

C₂×C₄⋊#10623494

C₄

yes

—

C₂×C₄⋊#10623494

C₂²

yes

characteristic

C₂×C₄×#10623494

C₄

yes

—

C₂×C₄×#10623494

C₂²

yes

characteristic

#19151324

C₄

yes

—

#19151320

—

#19151320

—

C₂×C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₂×#101×C₈₉

C₄

yes

—

C₄×M₁₆×C₈₉

C₂²

yes

—

#343×C₈₉

C₄

yes

—

C₂²×#10623494

C₂×C₄

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

—

C₂²×#10623494

C₂×C₄

yes

—

—

—

C₂×C₄

yes

—

—

—

—

D₄

yes

—

C₄×#10623494

D₄

yes

—

C₂²×Q₈×C₈₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₈₉

C₂×C₄

yes

—

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#101×C₈₉

—

#97×C₈₉

—

C₄×C₈×C₈₉

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10623494

C₂²⋊C₄

yes

—

C₂×#10623494

C₂²⋊C₄

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

C₄²

yes

—

C₂²×C₄×C₈₉

C₄²

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₄²

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#97

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#117

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₈×C₈₉

—

—

—

—

—

yes

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

C₂×#97

yes

characteristic

C₂²×C₈₉

#405

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#101

yes

characteristic

100

C₄×C₈₉

#413

yes

characteristic

101

C₄×C₈₉

#435

yes

—

102

#8165

—

maximal, Sylow

103

C₂×C₈₉

—

104

C₂×C₈₉

—

105

C₂×C₈₉

#1752

yes

characteristic

106

C₂×C₈₉

C₄×#101

yes

characteristic

107

C₂×C₈₉

#1757

yes

characteristic

108

C₂×C₈×Q₈

—

109

C₄×#101

D₈₉

yes

characteristic

110

#1461

—

111

#1451

—

112

#1283

—

113

C₈₉

#8165

yes

characteristic, complemented, Sylow

114

C₂×C₄×Q₈

C₂×D₈₉

yes

characteristic

115

C₈×Q₈

—

178

—

116

C₈×Q₈

—

178

—

117

#431

—

118

C₂×C₄⋊C₈

—

119

C₂×C₄⋊C₈

—

120

C₂×#101

Dic₈₉

yes

—

121

C₄×M₁₆

C₂×D₈₉

yes

—

122

C₂×C₄×C₈

—

123

C₂×C₄×C₈

—

124

#343

Dic₈₉

yes

—

125

#324

—

126

#322

—

178

—

127

#322

—

178

—

128

C₂²×Q₈

C₂×Dic₈₉

yes

characteristic

129

C₂×M₁₆

C₂×Dic₈₉

yes

—

130

C₂²×C₈

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

C₄×Q₈

—

134

C₄×Q₈

—

135

C₂×C₄⋊C₄

C₄×D₈₉

yes

—

136

C₂×C₄⋊C₄

C₂×Dic₈₉

yes

characteristic

137

C₂×C₄²

C₄×D₈₉

yes

—

138

C₂×C₄²

C₂²×D₈₉

yes

characteristic

139

C₄⋊C₈

—

178

—

140

C₄⋊C₈

—

178

—

141

C₄⋊C₈

—

178

—

142

#101

—

143

C₂²⋊C₈

—

144

#97

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

—

178

—

149

C₄×C₈

—

178

—

150

C₄×C₈

—

178

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

#13083558

yes

—

154

C₂²×C₄

C₄×Dic₈₉

yes

—

155

C₂²×C₄

C₄×Dic₈₉

yes

—

156

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

157

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

158

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

159

M₁₆

—

160

M₁₆

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

356

—

166

C₂×C₈

—

356

—

167

C₂×C₈

—

178

—

168

C₂×C₈

—

178

—

169

C₂×C₈

—

178

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄²

—

173

C₄²

—

174

C₄²

D₄×D₈₉

yes

—

175

C₄²

#13083573

yes

—

176

C₂³

C₂×C₄×Dic₈₉

yes

characteristic

177

Q₈

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

#15667923

yes

—

186

C₂×C₄

C₂²⋊C₄×D₈₉

yes

characteristic

187

C₂×C₄

#15667968

yes

characteristic

188

C₂×C₄

#15667972

yes

—

189

C₂×C₄

D₄×Dic₈₉

yes

—

190

C₂×C₄

C₂×#13083558

yes

—

191

C₂×C₄

#15668019

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

356

—

195

C₈

—

356

—

196

C₈

—

178

—

197

C₈

—

178

—

198

C₂²

—

199

C₂²

—

200

C₂²

C₂²⋊C₄×Dic₈₉

yes

characteristic

201

C₂²

C₂×#15667923

yes

characteristic

202

C₂²

#19151554

yes

characteristic

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

#101×D₈₉

yes

characteristic

208

C₄

#19151581

yes

characteristic

209

C₄

#19151670

yes

—

210

C₂

—

211

C₂

—

212

C₂

#26020030

yes

characteristic

213

C₂

#101×Dic₈₉

yes

characteristic

214

C₂

#26020055

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₈₉
commutator, prime radical	C₂³×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#101×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#101	D₈₉
hypocenter	C₈₉	#8165
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#101

subgroup

G/H

center

C₂×C₄

C₂²⋊C₄×D₈₉

commutator, prime radical

C₂³×C₈₉

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₄×#101×C₈₉

C₂

socle

C₂²×C₈₉

C₄×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#101

N	H ≅ G/N	type
89. #10623494	#101	semidirect
113. C₈₉	#8165	semidirect

H ≅ G/N

type

89. #10623494

#101

semidirect

113. C₈₉

#8165

semidirect

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

subgroup	index
C₂×Q₈×#10623494	2
#26018576	2
#26018539	2
#26018103	2
C₄×#101×C₈₉	2
#8165	89

subgroup

index

C₄×#101×C₈₉

Subgroups — #36445003

Conjugacy classes of subgroups

Complements

Maximal subgroups