Group #36470646 — subgroups

Subgroups — #36470646

Order 22784.

434 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 135 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	#26021863	C₂	1	1	yes	maximal, characteristic
3	#26018335	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2112	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₄	1	1	yes	characteristic
6	C₂×#15667971	C₄	1	1	yes	characteristic
7	C₂×#15667952	C₂²	1	2	yes	—
8	#19151553	—	2	2	no	—
9	#19151553	—	2	2	no	—
10	C₂×#15667922	C₂²	1	2	yes	—
11	C₂×C₈×Dic₈₉	C₄	1	2	yes	—
12	C₂²⋊#13083549	—	2	2	no	—
13	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×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	2	yes	—
22	C₂×C₂²⋊Dic₈₉	C₈	1	2	yes	—
23	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
24	#15667952	—	2	4	no	—
25	C₂×#13083549	—	2	2	no	—
26	#15667922	—	2	2	no	—
27	#15667922	—	2	2	no	—
28	C₈×Dic₈₉	—	2	4	no	—
29	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	—	2	4	no	—
31	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	D₄	1	2	yes	—
33	C₂²×C₈×C₈₉	D₄	1	2	yes	—
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	M₃₂×C₈₉	—	2	4	no	—
36	C₂×C₁₆×C₈₉	—	2	2	no	—
37	C₂²×Dic₈₉	C₂×C₈	1	4	yes	—
38	#13083558	—	2	8	no	—
39	C₂×#10623494	—	2	2	no	—
40	C₂×#10623494	—	2	2	no	—
41	C₂×#10623494	M₁₆	1	4	yes	—
42	C₄⋊Dic₈₉	—	2	4	no	—
43	C₂²⋊Dic₈₉	—	2	4	no	—
44	C₄×Dic₈₉	—	2	2	no	—
45	C₄×Dic₈₉	—	2	2	no	—
46	#13083549	—	4	2	no	—
47	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	—	2	2	no	—
50	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
53	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	—	2	4	no	—
56	C₂×C₈×C₈₉	—	2	4	no	—
57	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	#41	1	4	yes	—
61	C₁₆×C₈₉	—	4	2	no	—
62	C₂³×C₈₉	—	2	2	no	—
63	C₂³×C₈₉	—	4	2	no	—
64	C₂³×C₈₉	C₂²⋊C₈	1	2	yes	—
65	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	—	2	2	no	—
69	C₂×C₄×C₈₉	—	2	4	no	—
70	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#117	1	2	yes	—
73	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₈₉	—	2	4	no	—
76	C₂×Dic₈₉	—	4	4	no	—
77	C₈×C₈₉	—	2	2	no	—
78	C₈×C₈₉	#123	1	2	yes	—
79	C₈×C₈₉	#123	1	2	yes	—
80	#10623494	—	2	4	no	—
81	C₂²×C₈₉	—	2	2	no	—
82	C₂²×C₈₉	—	2	2	no	—
83	C₂²×C₈₉	—	2	4	no	—
84	C₂²×C₈₉	—	4	4	no	—
85	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
86	C₂²×C₈₉	#431	1	2	yes	—
87	C₄×C₈₉	—	2	2	no	—
88	C₄×C₈₉	#381	1	1	yes	characteristic
89	C₄×C₈₉	#407	1	1	yes	characteristic
90	C₄×C₈₉	#432	1	2	yes	—
91	Dic₈₉	—	4	4	no	—
92	#12847	—	89	1	no	maximal, Sylow
93	C₂×C₈₉	—	2	2	no	—
94	C₂×C₈₉	—	4	2	no	—
95	C₂×C₈₉	#1817	1	1	yes	characteristic
96	C₂×C₈₉	#2115	1	2	yes	—
97	C₂×#405	—	89	1	no	—
98	#2112	—	89	1	no	—
99	#2112	D₈₉	1	1	yes	characteristic
100	#1377	—	89	4	no	—
101	C₈₉	#12847	1	1	yes	characteristic, complemented, Sylow
102	C₂²×M₁₆	—	89	1	no	—
103	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
104	C₂×#117	—	89	1	no	—
105	C₂×M₃₂	—	89	2	no	—
106	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
107	#405	—	178	2	no	—
108	#405	—	178	2	no	—
109	C₂×#97	—	89	2	no	—
110	C₂×C₄×C₈	—	89	2	no	—
111	C₂²⋊C₁₆	—	2	2	no	—
112	C₂²⋊C₁₆	—	178	2	no	—
113	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
114	C₂×M₁₆	—	178	4	no	—
115	C₂×M₁₆	—	178	2	no	—
116	C₂²×C₈	—	2	4	no	—
117	C₂²×C₈	—	89	2	no	—
118	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
119	C₂²×C₈	D₃₅₆	1	2	yes	—
120	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
121	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
122	#117	—	178	2	no	—
123	#117	—	178	2	no	—
124	C₂×C₄⋊C₄	—	89	2	no	—
125	C₂×C₂²⋊C₄	—	89	2	no	—
126	C₂×C₄²	—	89	2	no	—
127	M₃₂	—	2	4	no	—
128	M₃₂	—	178	4	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	178	2	no	—
131	#97	—	178	2	no	—
132	#97	—	178	2	no	—
133	C₄×C₈	—	178	4	no	—
134	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	89	4	no	—
138	C₂²×C₄	#13083551	1	2	yes	—
139	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
140	C₂²×C₄	#13083557	1	2	yes	—
141	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
142	M₁₆	—	178	8	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	89	4	no	—
147	C₂×C₈	—	178	2	no	—
148	C₂×C₈	—	178	2	no	—
149	C₂×C₈	#13083551	1	2	yes	—
150	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
151	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
152	C₂×C₈	#13083573	1	4	yes	—
153	C₄⋊C₄	—	178	4	no	—
154	C₂²⋊C₄	—	178	4	no	—
155	C₄²	—	178	2	no	—
156	C₄²	—	178	2	no	—
157	C₁₆	—	4	2	no	—
158	C₁₆	—	356	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	4	2	no	—
161	C₂³	#15667924	1	2	yes	—
162	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	356	4	no	—
168	C₂×C₄	—	178	4	no	—
169	C₂×C₄	#15667924	1	2	yes	—
170	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
171	C₂×C₄	#15667968	1	2	yes	—
172	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
173	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	178	4	no	—
176	C₈	#15667993	1	2	yes	—
177	C₈	#15667996	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	2	4	no	—
181	C₂²	—	4	4	no	—
182	C₂²	C₂×#15667924	1	1	yes	characteristic
183	C₂²	#19151648	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	356	4	no	—
186	C₄	#19151459	1	1	yes	characteristic
187	C₄	#19151561	1	1	yes	characteristic
188	C₄	#19151652	1	2	yes	—
189	C₂	—	2	2	no	—
190	C₂	—	4	2	no	—
191	C₂	#26020318	1	1	yes	characteristic
192	C₂	#26021872	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151553

C₂

yes

maximal, characteristic

#26021863

C₂

yes

maximal, characteristic

#26018335

C₂

yes

maximal, complemented

C₈₉×#2112

C₂

yes

maximal, characteristic

C₂²×#13083558

C₄

yes

characteristic

C₂×#15667971

C₄

yes

characteristic

C₂×#15667952

C₂²

yes

—

#19151553

—

#19151553

—

C₂×#15667922

C₂²

yes

—

C₂×C₈×Dic₈₉

C₄

yes

—

C₂²⋊#13083549

—

C₂³×C₈×C₈₉

C₂²

yes

characteristic

C₂×M₃₂×C₈₉

C₂²

yes

—

C₂²⋊C₁₆×C₈₉

—

C₂×#13083558

—

C₂×#13083558

—

C₂²×#10623494

C₂×C₄

yes

—

#15667971

—

#15667971

—

C₂×C₄⋊Dic₈₉

C₈

yes

—

C₂×C₂²⋊Dic₈₉

C₈

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

—

—

—

—

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

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂²×Dic₈₉

C₂×C₈

yes

—

—

—

—

M₁₆

yes

—

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

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

—

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

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

#12847

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1817

yes

characteristic

C₂×C₈₉

#2115

yes

—

C₂×#405

—

#2112

—

#2112

D₈₉

yes

characteristic

100

#1377

—

101

C₈₉

#12847

yes

characteristic, complemented, Sylow

102

C₂²×M₁₆

—

103

C₂³×C₈

C₂×D₈₉

yes

characteristic

104

C₂×#117

—

105

C₂×M₃₂

—

106

C₂×M₃₂

C₂×D₈₉

yes

—

107

#405

—

178

—

108

#405

—

178

—

109

C₂×#97

—

110

C₂×C₄×C₈

—

111

C₂²⋊C₁₆

—

112

C₂²⋊C₁₆

—

178

—

113

C₂³×C₄

C₄×D₈₉

yes

characteristic

114

C₂×M₁₆

—

178

—

115

C₂×M₁₆

—

178

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂²×C₈

C₄×D₈₉

yes

characteristic

119

C₂²×C₈

D₃₅₆

yes

—

120

C₂²×C₈

C₂²⋊D₈₉

yes

—

121

C₂²×C₈

C₂²×D₈₉

yes

characteristic

122

#117

—

178

—

123

#117

—

178

—

124

C₂×C₄⋊C₄

—

125

C₂×C₂²⋊C₄

—

126

C₂×C₄²

—

127

M₃₂

—

128

M₃₂

—

178

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

178

—

131

#97

—

178

—

132

#97

—

178

—

133

C₄×C₈

—

178

—

134

C₂⁴

C₈×D₈₉

yes

characteristic

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

#13083551

yes

—

139

C₂²×C₄

C₈×D₈₉

yes

characteristic

140

C₂²×C₄

#13083557

yes

—

141

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

142

M₁₆

—

178

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

178

—

148

C₂×C₈

—

178

—

149

C₂×C₈

#13083551

yes

—

150

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

151

C₂×C₈

C₂×D₃₅₆

yes

characteristic

152

C₂×C₈

#13083573

yes

—

153

C₄⋊C₄

—

178

—

154

C₂²⋊C₄

—

178

—

155

C₄²

—

178

—

156

C₄²

—

178

—

157

C₁₆

—

158

C₁₆

—

356

—

159

C₂³

—

160

C₂³

—

161

C₂³

#15667924

yes

—

162

C₂³

C₂×C₈×D₈₉

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

356

—

168

C₂×C₄

—

178

—

169

C₂×C₄

#15667924

yes

—

170

C₂×C₄

C₂×#13083551

yes

characteristic

171

C₂×C₄

#15667968

yes

—

172

C₂×C₄

M₁₆×D₈₉

yes

—

173

C₂×C₄

C₂×#13083557

yes

characteristic

174

C₈

—

175

C₈

—

178

—

176

C₈

#15667993

yes

—

177

C₈

#15667996

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

C₂×#15667924

yes

characteristic

183

C₂²

#19151648

yes

—

184

C₄

—

185

C₄

—

356

—

186

C₄

#19151459

yes

characteristic

187

C₄

#19151561

yes

characteristic

188

C₄

#19151652

yes

—

189

C₂

—

190

C₂

—

191

C₂

#26020318

yes

characteristic

192

C₂

#26021872

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#13083551
commutator	C₂³×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2112	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₈₉	C₂×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2112	D₈₉
hypocenter	C₈₉	#12847
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ #2112

subgroup

G/H

center

C₂×C₄

C₂×#13083551

commutator

C₂³×C₈₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#2112

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₈₉

C₂×C₈

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ #2112

N	H ≅ G/N	type
3. #26018335	C₂	semidirect
101. C₈₉	#12847	semidirect

H ≅ G/N

type

3. #26018335

C₂

semidirect

101. C₈₉

#12847

semidirect

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

subgroup	index
C₂×#19151553	2
#26021863	2
#26018335	2
C₈₉×#2112	2
#12847	89

subgroup

index

C₈₉×#2112

Subgroups — #36470646

Conjugacy classes of subgroups

Complements

Maximal subgroups