Group #36472596 — subgroups

Subgroups — #36472596

Order 22784.

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

normal subgroups only

This group has 442 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	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151382	C₂	1	4	yes	maximal, complemented
3	C₂×#19151374	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2931	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667952	C₂²	1	2	yes	—
6	C₂²×#13083549	C₂²	1	2	yes	—
7	C₄⋊#13083549	—	2	2	no	—
8	#19151382	—	2	8	no	—
9	#19151374	—	2	2	no	—
10	C₂×#131×C₈₉	C₂²	1	2	yes	—
11	#532×C₈₉	C₄	1	1	yes	characteristic
12	#436×C₈₉	—	2	2	no	—
13	C₈×D₄×C₈₉	—	2	2	no	—
14	#423×C₈₉	C₄	1	1	yes	characteristic
15	#408×C₈₉	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#15667952	—	2	4	no	—
18	C₂×#13083549	—	2	2	no	—
19	C₂×#13083549	—	2	2	no	—
20	C₂×#13083549	—	2	2	no	—
21	C₂×#13083549	D₄	1	4	yes	complemented
22	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
23	#131×C₈₉	—	2	4	no	—
24	#131×C₈₉	—	2	4	no	—
25	C₂×M₁₆×C₈₉	—	2	2	no	—
26	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₈₉	—	2	2	no	—
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₈₉	C₈	1	1	yes	characteristic
31	#127×C₈₉	C₈	1	1	yes	characteristic
32	#124×C₈₉	C₈	1	2	yes	—
33	#121×C₈₉	—	2	2	no	—
34	C₄×D₄×C₈₉	—	2	2	no	—
35	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₈₉	—	2	2	no	—
37	C₂²⋊C₈×C₈₉	—	2	4	no	—
38	C₄×C₈×C₈₉	D₄	1	2	yes	—
39	C₄×C₈×C₈₉	D₄	1	2	yes	—
40	#13083549	—	2	4	no	—
41	#13083549	—	4	2	no	—
42	#41×C₈₉	—	2	4	no	—
43	#41×C₈₉	—	2	4	no	—
44	C₂×Q₈×C₈₉	C₂×C₈	1	2	yes	—
45	C₂×D₄×C₈₉	—	2	2	no	—
46	C₂×D₄×C₈₉	—	2	2	no	—
47	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
50	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	M₁₆×C₈₉	—	2	4	no	—
52	M₁₆×C₈₉	—	4	4	no	—
53	C₂×C₈×C₈₉	—	2	2	no	—
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	—	4	2	no	—
56	C₂×C₈×C₈₉	—	4	2	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	2	yes	—
60	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	#41	1	2	yes	—
62	C₄⋊C₄×C₈₉	—	2	2	no	—
63	C₂²⋊C₄×C₈₉	—	2	4	no	—
64	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
67	C₂³×C₈₉	—	2	2	no	—
68	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
69	Q₈×C₈₉	—	2	4	no	—
70	D₄×C₈₉	—	2	4	no	—
71	D₄×C₈₉	—	4	4	no	—
72	D₄×C₈₉	—	4	4	no	—
73	C₂×C₄×C₈₉	—	2	2	no	—
74	C₂×C₄×C₈₉	—	4	2	no	—
75	C₂×C₄×C₈₉	—	4	2	no	—
76	C₂×C₄×C₈₉	—	4	2	no	—
77	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#131	1	2	yes	—
83	C₈×C₈₉	—	2	2	no	—
84	C₈×C₈₉	—	4	2	no	—
85	C₈×C₈₉	#121	1	2	yes	—
86	C₈×C₈₉	#124	1	1	yes	characteristic
87	C₈×C₈₉	#127	1	1	yes	characteristic
88	C₂²×C₈₉	—	2	2	no	—
89	C₂²×C₈₉	—	4	4	no	—
90	C₂²×C₈₉	—	4	2	no	—
91	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
92	C₂²×C₈₉	#436	1	2	yes	—
93	C₄×C₈₉	—	2	2	no	—
94	C₄×C₈₉	—	2	2	no	—
95	C₄×C₈₉	—	4	2	no	—
96	C₄×C₈₉	#390	1	1	yes	characteristic
97	C₄×C₈₉	#408	1	1	yes	characteristic
98	C₄×C₈₉	C₈×D₄	1	2	yes	—
99	#13192	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	—	2	2	no	—
101	C₂×C₈₉	—	8	2	no	—
102	C₂×C₈₉	#1954	1	1	yes	characteristic
103	C₂×C₈₉	#2117	1	2	yes	—
104	#2931	D₈₉	1	1	yes	characteristic
105	#2150	—	89	1	no	—
106	C₂×#350	—	89	4	no	—
107	C₂×#346	—	89	1	no	—
108	C₈₉	#13192	1	1	yes	characteristic, complemented, Sylow
109	C₂×#131	C₂×D₈₉	1	2	yes	—
110	#532	Dic₈₉	1	1	yes	characteristic
111	C₂×M₃₂	—	89	2	no	—
112	C₂²×C₁₆	—	89	2	no	—
113	#436	—	2	2	no	—
114	C₈×D₄	—	2	2	no	—
115	#423	Dic₈₉	1	1	yes	characteristic
116	#408	Dic₈₉	1	2	yes	—
117	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
118	C₄⋊C₁₆	—	178	2	no	—
119	#350	—	178	8	no	—
120	#346	—	178	2	no	—
121	C₂×#41	C₂×Dic₈₉	1	2	yes	—
122	#131	—	2	4	no	—
123	#131	—	2	4	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
129	C₄⋊Q₈	#10623494	1	1	yes	characteristic
130	#127	#10623494	1	1	yes	characteristic
131	#124	#10623494	1	2	yes	—
132	#121	—	2	2	no	—
133	C₄×D₄	—	2	2	no	—
134	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
135	M₃₂	—	178	4	no	—
136	C₂×C₁₆	—	89	4	no	—
137	C₂×C₁₆	—	178	2	no	—
138	C₂×C₁₆	—	178	2	no	—
139	C₂×C₁₆	—	178	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	4	no	—
142	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
143	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
144	#41	—	2	4	no	—
145	#41	—	2	4	no	—
146	C₂×Q₈	C₂×#10623494	1	2	yes	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	C₂×#10623494	1	2	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	#13083558	1	2	yes	—
152	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
153	M₁₆	—	2	4	no	—
154	M₁₆	—	4	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
160	C₂×C₈	D₄×D₈₉	1	2	yes	—
161	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
162	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
163	C₂×C₈	#13083573	1	2	yes	—
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	4	no	—
166	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
167	C₄²	C₂×#10623494	1	1	yes	characteristic
168	C₄²	C₂×#10623494	1	1	yes	characteristic
169	C₁₆	—	356	2	no	—
170	C₁₆	—	178	4	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₂×#13083558	1	1	yes	characteristic
173	Q₈	—	2	4	no	—
174	D₄	—	2	4	no	—
175	D₄	—	4	4	no	—
176	D₄	—	4	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂²⋊#10623494	1	2	yes	—
182	C₂×C₄	C₂²⋊#10623494	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
184	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
185	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
186	C₂×C₄	#15668019	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	4	2	no	—
189	C₈	#15667983	1	2	yes	—
190	C₈	#15667999	1	1	yes	characteristic
191	C₈	#15668008	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	—	4	4	no	—
195	C₂²	C₂×C₂²⋊#10623494	1	1	yes	characteristic
196	C₂²	#19151676	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	2	no	—
200	C₄	#19151501	1	1	yes	characteristic
201	C₄	#19151565	1	1	yes	characteristic
202	C₄	D₄×#10623494	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	8	2	no	—
205	C₂	#26021047	1	1	yes	characteristic
206	C₂	#26021880	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022008

C₂

yes

maximal, characteristic, complemented

C₂×#19151382

C₂

yes

maximal, complemented

C₂×#19151374

C₂

yes

maximal, characteristic, complemented

C₈₉×#2931

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

—

C₂²×#13083549

C₂²

yes

—

C₄⋊#13083549

—

#19151382

—

#19151374

—

C₂×#131×C₈₉

C₂²

yes

—

#532×C₈₉

C₄

yes

characteristic

#436×C₈₉

—

C₈×D₄×C₈₉

—

#423×C₈₉

C₄

yes

characteristic

#408×C₈₉

C₄

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

—

—

—

D₄

yes

complemented

C₂×#41×C₈₉

C₂×C₄

yes

—

#131×C₈₉

—

#131×C₈₉

—

C₂×M₁₆×C₈₉

—

C₂×M₁₆×C₈₉

C₂×C₄

yes

—

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊Q₈×C₈₉

C₈

yes

characteristic

#127×C₈₉

C₈

yes

characteristic

#124×C₈₉

C₈

yes

—

#121×C₈₉

—

C₄×D₄×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#13083549

—

#13083549

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₂×C₈

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂×C₈

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

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

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

C₂×M₁₆

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

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×D₄

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

#131

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#121

yes

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₈₉

#436

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₈×D₄

yes

—

#13192

—

maximal, Sylow

100

C₂×C₈₉

—

101

C₂×C₈₉

—

102

C₂×C₈₉

#1954

yes

characteristic

103

C₂×C₈₉

#2117

yes

—

104

#2931

D₈₉

yes

characteristic

105

#2150

—

106

C₂×#350

—

107

C₂×#346

—

108

C₈₉

#13192

yes

characteristic, complemented, Sylow

109

C₂×#131

C₂×D₈₉

yes

—

110

#532

Dic₈₉

yes

characteristic

111

C₂×M₃₂

—

112

C₂²×C₁₆

—

113

#436

—

114

C₈×D₄

—

115

#423

Dic₈₉

yes

characteristic

116

#408

Dic₈₉

yes

—

117

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

118

C₄⋊C₁₆

—

178

—

119

#350

—

178

—

120

#346

—

178

—

121

C₂×#41

C₂×Dic₈₉

yes

—

122

#131

—

123

#131

—

124

C₂×M₁₆

—

125

C₂×M₁₆

C₂×Dic₈₉

yes

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂×Dic₈₉

yes

characteristic

128

C₂²×C₈

C₂²×D₈₉

yes

characteristic

129

C₄⋊Q₈

yes

130

yes

131

yes

—

132

#121

—

133

C₄×D₄

—

134

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

135

M₃₂

—

178

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

178

—

138

C₂×C₁₆

—

178

—

139

C₂×C₁₆

—

178

—

140

C₄⋊C₈

—

141

C₂²⋊C₈

—

142

C₄×C₈

C₂²⋊D₈₉

yes

—

143

C₄×C₈

C₂²⋊D₈₉

yes

—

144

#41

—

145

#41

—

146

C₂×Q₈

C₂×#10623494

yes

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

C₂×#10623494

yes

—

150

C₂²×C₄

—

151

C₂²×C₄

#13083558

yes

—

152

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

153

M₁₆

—

154

M₁₆

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

C₂²⋊Dic₈₉

yes

—

160

C₂×C₈

D₄×D₈₉

yes

—

161

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

162

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

163

C₂×C₈

#13083573

yes

—

164

C₄⋊C₄

—

165

C₂²⋊C₄

—

166

C₄²

C₂²⋊Dic₈₉

yes

—

167

C₄²

C₂×#10623494

yes

characteristic

168

C₄²

C₂×#10623494

yes

characteristic

169

C₁₆

—

356

—

170

C₁₆

—

178

—

171

C₂³

—

172

C₂³

C₂×#13083558

yes

characteristic

173

Q₈

—

174

D₄

—

175

D₄

—

176

D₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

C₂²⋊#10623494

yes

—

182

C₂×C₄

C₂²⋊#10623494

yes

—

183

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

184

C₂×C₄

D₄×Dic₈₉

yes

—

185

C₂×C₄

C₂²×#10623494

yes

characteristic

186

C₂×C₄

#15668019

yes

—

187

C₈

—

188

C₈

—

189

C₈

#15667983

yes

—

190

C₈

#15667999

yes

characteristic

191

C₈

#15668008

yes

characteristic

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

C₂×C₂²⋊#10623494

yes

characteristic

196

C₂²

#19151676

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

#19151501

yes

characteristic

201

C₄

#19151565

yes

characteristic

202

C₄

D₄×#10623494

yes

—

203

C₂

—

204

C₂

—

205

C₂

#26021047

yes

characteristic

206

C₂

#26021880

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2931	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	#532×C₈₉	C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2931	D₈₉
hypocenter	C₈₉	#13192
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#2931

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₈

prime radical

#532×C₈₉

C₄

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

N	H ≅ G/N	type
1. #26022008	C₂	semidirect
2. C₂×#19151382	C₂	semidirect
3. C₂×#19151374	C₂	semidirect
21. C₂×#13083549	D₄	semidirect
108. C₈₉	#13192	semidirect

H ≅ G/N

type

1. #26022008

C₂

semidirect

2. C₂×#19151382

C₂

semidirect

3. C₂×#19151374

C₂

semidirect

21. C₂×#13083549

D₄

semidirect

108. C₈₉

#13192

semidirect

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

subgroup	index
#26022008	2
C₂×#19151382	2
C₂×#19151374	2
C₈₉×#2931	2
#13192	89

subgroup

index

C₈₉×#2931

Subgroups — #36472596

Conjugacy classes of subgroups

Complements

Maximal subgroups