Group #36470640 — subgroups

Subgroups — #36470640

Order 22784.

454 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 151 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×#19151553	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#13083549	C₂	1	1	yes	maximal, characteristic
3	#26018329	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₄	1	1	yes	characteristic
6	C₂×#15667971	C₄	1	1	yes	characteristic
7	C₂²×#13083549	C₂²	1	2	yes	—
8	#19151553	—	2	2	no	—
9	#19151553	—	2	2	no	—
10	C₂×#15667922	C₄	1	2	yes	—
11	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
12	C₂²⋊#13083549	—	2	2	no	—
13	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂²⋊C₁₆×C₈₉	—	2	2	no	—
16	C₂×#13083558	—	2	4	no	—
17	C₂×#13083558	—	2	2	no	—
18	C₂²×#10623494	C₂×C₄	1	2	yes	—
19	#15667971	—	2	2	no	—
20	#15667971	—	2	2	no	—
21	C₂×C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
22	C₂×C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₈₉	C₈	1	2	yes	—
24	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
25	C₂×#13083549	—	2	2	no	—
26	C₂×#13083549	—	2	4	no	—
27	#15667922	—	2	4	no	—
28	C₈×Dic₈₉	—	2	2	no	—
29	C₈×Dic₈₉	—	2	2	no	—
30	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	—	2	4	no	—
32	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	D₄	1	2	yes	—
34	C₂²×C₈×C₈₉	D₄	1	2	yes	—
35	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×C₁₆×C₈₉	—	2	2	no	—
37	C₂×C₁₆×C₈₉	—	2	4	no	—
38	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
39	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
40	#13083558	—	2	8	no	—
41	C₂×#10623494	—	2	2	no	—
42	C₂×#10623494	—	2	2	no	—
43	C₂×#10623494	M₁₆	1	4	yes	—
44	C₄⋊Dic₈₉	—	2	2	no	—
45	C₄⋊Dic₈₉	—	2	2	no	—
46	C₂²⋊Dic₈₉	—	2	4	no	—
47	C₄×Dic₈₉	—	2	2	no	—
48	C₄×Dic₈₉	—	2	2	no	—
49	#13083549	—	4	2	no	—
50	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	—	2	2	no	—
52	C₂²×C₄×C₈₉	—	2	2	no	—
53	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₈₉	—	2	2	no	—
58	C₂×C₈×C₈₉	—	2	4	no	—
59	C₂×C₈×C₈₉	—	2	4	no	—
60	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	#41	1	4	yes	—
64	C₁₆×C₈₉	—	4	2	no	—
65	C₂³×C₈₉	—	2	2	no	—
66	C₂³×C₈₉	—	2	2	no	—
67	C₂³×C₈₉	—	2	2	no	—
68	C₂³×C₈₉	C₂²⋊C₈	1	2	yes	—
69	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	—	2	2	no	—
73	C₂×C₄×C₈₉	—	2	4	no	—
74	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#117	1	2	yes	—
77	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
79	C₂×Dic₈₉	—	2	2	no	—
80	C₂×Dic₈₉	—	2	2	no	—
81	C₂×Dic₈₉	—	2	4	no	—
82	C₂×Dic₈₉	—	2	4	no	—
83	C₈×C₈₉	—	2	2	no	—
84	C₈×C₈₉	#123	1	2	yes	—
85	C₈×C₈₉	#123	1	2	yes	—
86	#10623494	—	2	4	no	—
87	C₂²×C₈₉	—	2	2	no	—
88	C₂²×C₈₉	—	2	2	no	—
89	C₂²×C₈₉	—	2	2	no	—
90	C₂²×C₈₉	—	4	4	no	—
91	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
92	C₂²×C₈₉	#431	1	2	yes	—
93	C₂²×C₈₉	#504	1	2	yes	—
94	C₂²×C₈₉	#504	1	2	yes	—
95	C₄×C₈₉	—	2	2	no	—
96	C₄×C₈₉	#381	1	1	yes	characteristic
97	C₄×C₈₉	#407	1	1	yes	characteristic
98	C₄×C₈₉	#432	1	2	yes	—
99	Dic₈₉	—	4	2	no	—
100	Dic₈₉	—	4	2	no	—
101	#12845	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	—	4	2	no	—
103	C₂×C₈₉	#1817	1	1	yes	characteristic
104	C₂×C₈₉	#2113	1	1	yes	characteristic
105	C₂×C₈₉	#2113	1	1	yes	characteristic
106	C₂×C₈₉	#2164	1	4	yes	—
107	C₂×#405	—	89	1	no	—
108	C₂×C₂²⋊C₁₆	—	89	1	no	—
109	C₂×C₂²⋊C₁₆	D₈₉	1	1	yes	characteristic
110	#1374	—	89	4	no	—
111	C₈₉	#12845	1	1	yes	characteristic, complemented, Sylow
112	C₂²×M₁₆	—	89	1	no	—
113	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
114	C₂×#117	—	89	1	no	—
115	C₂²×C₁₆	—	89	2	no	—
116	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
117	#405	—	178	2	no	—
118	#405	—	178	2	no	—
119	C₂×#97	—	89	2	no	—
120	C₂×C₄×C₈	—	89	2	no	—
121	C₂²⋊C₁₆	—	2	2	no	—
122	C₂²⋊C₁₆	—	178	2	no	—
123	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
124	C₂×M₁₆	—	178	4	no	—
125	C₂×M₁₆	—	178	2	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	89	2	no	—
128	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
129	C₂²×C₈	D₃₅₆	1	2	yes	—
130	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
131	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
132	#117	—	178	2	no	—
133	#117	—	178	2	no	—
134	C₂×C₄⋊C₄	—	89	1	no	—
135	C₂×C₄⋊C₄	—	89	1	no	—
136	C₂×C₂²⋊C₄	—	89	2	no	—
137	C₂×C₄²	—	89	2	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	2	4	no	—
140	C₂×C₁₆	—	178	2	no	—
141	C₂×C₁₆	—	178	4	no	—
142	#97	—	178	4	no	—
143	C₄×C₈	—	178	2	no	—
144	C₄×C₈	—	178	2	no	—
145	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	89	2	no	—
149	C₂²×C₄	—	89	2	no	—
150	C₂²×C₄	#13083551	1	2	yes	—
151	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
152	C₂²×C₄	#13083557	1	2	yes	—
153	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
154	M₁₆	—	178	8	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	89	4	no	—
159	C₂×C₈	—	178	2	no	—
160	C₂×C₈	—	178	2	no	—
161	C₂×C₈	#13083551	1	2	yes	—
162	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
163	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
164	C₂×C₈	#13083573	1	4	yes	—
165	C₄⋊C₄	—	178	2	no	—
166	C₄⋊C₄	—	178	2	no	—
167	C₂²⋊C₄	—	178	4	no	—
168	C₄²	—	178	2	no	—
169	C₄²	—	178	2	no	—
170	C₁₆	—	4	2	no	—
171	C₁₆	—	356	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	#15667924	1	2	yes	—
176	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	178	4	no	—
182	C₂×C₄	—	178	4	no	—
183	C₂×C₄	—	178	2	no	—
184	C₂×C₄	—	178	2	no	—
185	C₂×C₄	#15667924	1	2	yes	—
186	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
187	C₂×C₄	#15667968	1	2	yes	—
188	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
189	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	—	178	4	no	—
192	C₈	#15667993	1	2	yes	—
193	C₈	#15667996	1	2	yes	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	C₂×#15667924	1	1	yes	characteristic
199	C₂²	#19151648	1	2	yes	—
200	C₂²	#19152014	1	2	yes	—
201	C₂²	#19152016	1	2	yes	—
202	C₄	—	2	2	no	—
203	C₄	—	356	2	no	—
204	C₄	—	356	2	no	—
205	C₄	#19151459	1	1	yes	characteristic
206	C₄	#19151561	1	1	yes	characteristic
207	C₄	#19151652	1	2	yes	—
208	C₂	—	4	2	no	—
209	C₂	#26020318	1	1	yes	characteristic
210	C₂	#26021865	1	1	yes	characteristic
211	C₂	#26021867	1	1	yes	characteristic
212	C₂	#26022056	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151553

C₂

yes

maximal, characteristic

C₂×C₂²⋊#13083549

C₂

yes

maximal, characteristic

#26018329

C₂

yes

maximal, complemented

C₂×C₂²⋊C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂²×#13083558

C₄

yes

characteristic

C₂×#15667971

C₄

yes

characteristic

C₂²×#13083549

C₂²

yes

—

#19151553

—

#19151553

—

C₂×#15667922

C₄

yes

—

C₂×C₈×Dic₈₉

C₂²

yes

—

C₂²⋊#13083549

—

C₂³×C₈×C₈₉

C₂²

yes

characteristic

C₂²×C₁₆×C₈₉

C₂²

yes

—

C₂²⋊C₁₆×C₈₉

—

C₂×#13083558

—

C₂×#13083558

—

C₂²×#10623494

C₂×C₄

yes

—

#15667971

—

#15667971

—

C₂×C₄⋊Dic₈₉

C₈

yes

characteristic

C₂×C₄⋊Dic₈₉

C₈

yes

characteristic

C₂×C₂²⋊Dic₈₉

C₈

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

C₂×#13083549

—

C₂×#13083549

—

#15667922

—

C₈×Dic₈₉

—

C₈×Dic₈₉

—

C₂³×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂²×Dic₈₉

C₂×C₈

yes

—

C₂²×Dic₈₉

C₂×C₈

yes

—

—

—

—

M₁₆

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

C₂⁴×C₈₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

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

#41

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²⋊C₈

yes

—

C₂³×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#123

yes

—

C₈×C₈₉

#123

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₈₉

#431

yes

—

C₂²×C₈₉

#504

yes

—

C₂²×C₈₉

#504

yes

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

100

Dic₈₉

—

101

#12845

—

maximal, Sylow

102

C₂×C₈₉

—

103

C₂×C₈₉

#1817

yes

characteristic

104

C₂×C₈₉

#2113

yes

characteristic

105

C₂×C₈₉

#2113

yes

characteristic

106

C₂×C₈₉

#2164

yes

—

107

C₂×#405

—

108

C₂×C₂²⋊C₁₆

—

109

C₂×C₂²⋊C₁₆

D₈₉

yes

characteristic

110

#1374

—

111

C₈₉

#12845

yes

characteristic, complemented, Sylow

112

C₂²×M₁₆

—

113

C₂³×C₈

C₂×D₈₉

yes

characteristic

114

C₂×#117

—

115

C₂²×C₁₆

—

116

C₂²×C₁₆

C₂×D₈₉

yes

—

117

#405

—

178

—

118

#405

—

178

—

119

C₂×#97

—

120

C₂×C₄×C₈

—

121

C₂²⋊C₁₆

—

122

C₂²⋊C₁₆

—

178

—

123

C₂³×C₄

C₄×D₈₉

yes

characteristic

124

C₂×M₁₆

—

178

—

125

C₂×M₁₆

—

178

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

C₄×D₈₉

yes

characteristic

129

C₂²×C₈

D₃₅₆

yes

—

130

C₂²×C₈

C₂²⋊D₈₉

yes

—

131

C₂²×C₈

C₂²×D₈₉

yes

characteristic

132

#117

—

178

—

133

#117

—

178

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₂²⋊C₄

—

137

C₂×C₄²

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

C₂×C₁₆

—

178

—

141

C₂×C₁₆

—

178

—

142

#97

—

178

—

143

C₄×C₈

—

178

—

144

C₄×C₈

—

178

—

145

C₂⁴

C₈×D₈₉

yes

characteristic

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

#13083551

yes

—

151

C₂²×C₄

C₈×D₈₉

yes

characteristic

152

C₂²×C₄

#13083557

yes

—

153

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

154

M₁₆

—

178

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

178

—

160

C₂×C₈

—

178

—

161

C₂×C₈

#13083551

yes

—

162

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

163

C₂×C₈

C₂×D₃₅₆

yes

characteristic

164

C₂×C₈

#13083573

yes

—

165

C₄⋊C₄

—

178

—

166

C₄⋊C₄

—

178

—

167

C₂²⋊C₄

—

178

—

168

C₄²

—

178

—

169

C₄²

—

178

—

170

C₁₆

—

171

C₁₆

—

356

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

#15667924

yes

—

176

C₂³

C₂×C₈×D₈₉

yes

characteristic

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

178

—

182

C₂×C₄

—

178

—

183

C₂×C₄

—

178

—

184

C₂×C₄

—

178

—

185

C₂×C₄

#15667924

yes

—

186

C₂×C₄

C₂×#13083551

yes

characteristic

187

C₂×C₄

#15667968

yes

—

188

C₂×C₄

M₁₆×D₈₉

yes

—

189

C₂×C₄

C₂×#13083557

yes

characteristic

190

C₈

—

191

C₈

—

178

—

192

C₈

#15667993

yes

—

193

C₈

#15667996

yes

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

C₂×#15667924

yes

characteristic

199

C₂²

#19151648

yes

—

200

C₂²

#19152014

yes

—

201

C₂²

#19152016

yes

—

202

C₄

—

203

C₄

—

356

—

204

C₄

—

356

—

205

C₄

#19151459

yes

characteristic

206

C₄

#19151561

yes

characteristic

207

C₄

#19151652

yes

—

208

C₂

—

209

C₂

#26020318

yes

characteristic

210

C₂

#26021865

yes

characteristic

211

C₂

#26021867

yes

characteristic

212

C₂

#26022056

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₉
commutator, socle	C₂³×C₈₉	C₂²×C₈
Fitting	C₂×C₂²⋊C₁₆×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₈₉
hypocenter	C₈₉	#12845
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₈₉

commutator, socle

C₂³×C₈₉

C₂²×C₈

Fitting

C₂×C₂²⋊C₁₆×C₈₉

C₂

prime radical

C₂⁴×C₈₉

C₂×C₈

cosocle

C₂²×C₈×C₈₉

C₂³

hypercenter

C₂×C₂²⋊C₁₆

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

N	H ≅ G/N	type
3. #26018329	C₂	semidirect
111. C₈₉	#12845	semidirect

H ≅ G/N

type

3. #26018329

C₂

semidirect

111. C₈₉

#12845

semidirect

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

subgroup	index
C₂×#19151553	2
C₂×C₂²⋊#13083549	2
#26018329	2
C₂×C₂²⋊C₁₆×C₈₉	2
#12845	89

subgroup

index

C₂×#19151553

C₂×C₂²⋊#13083549

#26018329

C₂×C₂²⋊C₁₆×C₈₉

#12845

Subgroups — #36470640

Conjugacy classes of subgroups

Complements

Maximal subgroups