Group #36448307 — subgroups

Subgroups — #36448307

Order 22784.

386 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 131 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	#26029337	C₂	1	1	yes	maximal, characteristic
2	#26021189	C₂	1	1	yes	maximal, characteristic
3	#26019004	C₂	1	2	yes	maximal
4	#26018090	C₂	1	2	yes	maximal
5	C₄×#101×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151729	—	2	2	no	—
7	#19151729	—	2	2	no	—
8	#19151655	—	2	2	no	—
9	#19151651	C₂²	1	2	yes	—
10	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
12	#19151353	—	2	2	no	—
13	#19151351	—	2	2	no	—
14	#19151349	—	2	2	no	—
15	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#101×C₈₉	C₄	1	2	yes	—
17	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
18	#343×C₈₉	C₄	1	1	yes	characteristic
19	#343×C₈₉	C₄	1	1	yes	characteristic
20	C₂²×#10623494	—	2	1	no	—
21	C₂²×#10623494	—	2	1	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₄⋊#10623494	D₄	1	2	yes	—
26	C₄⋊#10623494	D₄	1	2	yes	—
27	C₂²⋊#10623494	—	2	2	no	—
28	#15667923	—	2	4	no	—
29	C₄×#10623494	—	2	2	no	—
30	C₄×#10623494	—	2	2	no	—
31	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₈₉	—	2	2	no	—
35	C₄×Q₈×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₈₉	Q₈	1	2	yes	—
38	C₂×C₄²×C₈₉	D₄	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	2	no	—
46	C₂×#10623494	—	4	1	no	—
47	C₂×#10623494	—	4	1	no	—
48	C₂×#10623494	—	4	2	no	—
49	C₂×#10623494	—	4	1	no	—
50	C₂×Q₈×C₈₉	—	2	2	no	—
51	C₂×Q₈×C₈₉	—	2	2	no	—
52	C₂×Q₈×C₈₉	—	2	1	no	—
53	C₂×Q₈×C₈₉	—	2	1	no	—
54	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	M₁₆×C₈₉	—	2	2	no	—
60	M₁₆×C₈₉	—	2	2	no	—
61	M₁₆×C₈₉	—	2	2	no	—
62	M₁₆×C₈₉	M₁₆	1	4	yes	—
63	C₂×C₈×C₈₉	—	2	2	no	—
64	C₂×C₈×C₈₉	—	2	2	no	—
65	C₄⋊C₄×C₈₉	—	2	2	no	—
66	C₄⋊C₄×C₈₉	—	2	4	no	—
67	C₄²×C₈₉	—	2	2	no	—
68	C₄²×C₈₉	—	2	2	no	—
69	C₄²×C₈₉	C₂×D₄	1	2	yes	—
70	C₄²×C₈₉	#41	1	2	yes	—
71	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	Q₈×C₈₉	—	4	2	no	—
73	Q₈×C₈₉	—	4	2	no	—
74	C₂×C₄×C₈₉	—	2	1	no	—
75	C₂×C₄×C₈₉	—	2	1	no	—
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	—	2	2	no	—
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	—	2	2	no	—
80	C₂×C₄×C₈₉	—	4	1	no	—
81	C₂×C₄×C₈₉	—	4	2	no	—
82	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
89	C₈×C₈₉	—	2	4	no	—
90	C₈×C₈₉	—	4	2	no	—
91	#10623494	—	4	2	no	—
92	#10623494	—	4	2	no	—
93	#10623494	—	4	2	no	—
94	#10623494	—	8	1	no	—
95	C₂²×C₈₉	—	2	1	no	—
96	C₂²×C₈₉	—	2	1	no	—
97	C₂²×C₈₉	#385	1	1	yes	characteristic
98	C₂²×C₈₉	#423	1	1	yes	characteristic
99	C₂²×C₈₉	#424	1	1	yes	characteristic
100	C₄×C₈₉	—	2	1	no	—
101	C₄×C₈₉	—	2	2	no	—
102	C₄×C₈₉	—	4	2	no	—
103	C₄×C₈₉	—	4	2	no	—
104	C₄×C₈₉	#435	1	2	yes	—
105	C₄×C₈₉	#436	1	2	yes	—
106	C₄×C₈₉	#454	1	1	yes	characteristic
107	C₄×C₈₉	#456	1	1	yes	characteristic
108	#8838	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	—	2	1	no	—
110	C₂×C₈₉	—	2	1	no	—
111	C₂×C₈₉	#1851	1	1	yes	characteristic
112	C₂×C₈₉	#1857	1	1	yes	characteristic
113	C₂×C₈₉	#1858	1	1	yes	characteristic
114	#2983	—	89	1	no	—
115	#1984	—	89	1	no	—
116	C₄×#101	D₈₉	1	1	yes	characteristic
117	#1565	—	89	2	no	—
118	#1278	—	89	2	no	—
119	C₈₉	#8838	1	1	yes	characteristic, complemented, Sylow
120	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
121	#446	—	178	2	no	—
122	#446	—	178	2	no	—
123	#432	—	178	2	no	—
124	#431	—	89	2	no	—
125	C₂×C₄⋊C₈	—	89	1	no	—
126	C₂×#101	Dic₈₉	1	2	yes	—
127	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
128	C₂×C₄×C₈	—	89	1	no	—
129	#343	Dic₈₉	1	1	yes	characteristic
130	#343	Dic₈₉	1	1	yes	characteristic
131	#336	—	178	2	no	—
132	#335	—	178	2	no	—
133	#334	—	178	2	no	—
134	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
135	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
136	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
137	C₂²×C₈	—	178	1	no	—
138	C₂²×C₈	—	178	1	no	—
139	C₄×Q₈	—	2	2	no	—
140	C₄×Q₈	—	2	2	no	—
141	C₂×C₄⋊C₄	Dic₁₇₈	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
143	C₂×C₄²	D₃₅₆	1	2	yes	—
144	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
145	C₄⋊C₈	—	89	2	no	—
146	C₄⋊C₈	—	89	2	no	—
147	C₄⋊C₈	—	178	2	no	—
148	C₄⋊C₈	—	178	2	no	—
149	C₄⋊C₈	—	178	2	no	—
150	#101	—	2	4	no	—
151	C₂²⋊C₈	—	178	2	no	—
152	#97	—	2	2	no	—
153	#97	—	178	4	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	178	2	no	—
156	C₄×C₈	—	178	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
162	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
163	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
165	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
166	M₁₆	—	2	2	no	—
167	M₁₆	—	2	2	no	—
168	M₁₆	—	2	2	no	—
169	M₁₆	#13083558	1	4	yes	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	356	1	no	—
173	C₂×C₈	—	356	1	no	—
174	C₂×C₈	—	356	2	no	—
175	C₂×C₈	—	356	1	no	—
176	C₂×C₈	—	178	2	no	—
177	C₂×C₈	—	178	2	no	—
178	C₂×C₈	—	178	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	D₄×D₈₉	1	2	yes	—
184	C₄²	#13083573	1	2	yes	—
185	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
186	Q₈	—	4	2	no	—
187	Q₈	—	4	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
197	C₂×C₄	#15667982	1	1	yes	characteristic
198	C₂×C₄	#15667989	1	2	yes	—
199	C₂×C₄	#15667996	1	1	yes	characteristic
200	C₂×C₄	C₂×#13083558	1	2	yes	—
201	C₂×C₄	#15668019	1	1	yes	characteristic
202	C₂×C₄	#15668019	1	1	yes	characteristic
203	C₈	—	2	4	no	—
204	C₈	—	4	2	no	—
205	C₈	—	356	2	no	—
206	C₈	—	356	2	no	—
207	C₈	—	356	2	no	—
208	C₈	—	712	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	#19151473	1	1	yes	characteristic
212	C₂²	#19151622	1	1	yes	characteristic
213	C₂²	#19151625	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	#19151670	1	2	yes	—
219	C₄	#19151676	1	2	yes	—
220	C₄	#19151773	1	1	yes	characteristic
221	C₄	#19151783	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#26020437	1	1	yes	characteristic
225	C₂	#26020469	1	1	yes	characteristic
226	C₂	#26020474	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26029337

C₂

yes

maximal, characteristic

#26021189

C₂

yes

maximal, characteristic

#26019004

C₂

yes

maximal

#26018090

C₂

yes

maximal

C₄×#101×C₈₉

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂×C₄⋊#10623494

C₂²

yes

characteristic

C₂×C₄×#10623494

C₂²

yes

—

—

—

C₂×C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₂×#101×C₈₉

C₄

yes

—

C₄×M₁₆×C₈₉

C₂²

yes

—

#343×C₈₉

C₄

yes

characteristic

#343×C₈₉

C₄

yes

—

—

—

—

—

D₄

yes

—

C₄⋊#10623494

D₄

yes

—

—

—

—

—

C₂²×Q₈×C₈₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₈₉

C₂×C₄

yes

—

C₂×M₁₆×C₈₉

C₂×C₄

yes

—

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₈₉

Q₈

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#101×C₈₉

—

#97×C₈₉

—

C₄×C₈×C₈₉

—

—

—

—

—

—

—

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

M₁₆

yes

—

C₂×C₈×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×C₄⋊C₄

yes

characteristic

Q₈×C₈₉

—

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

—

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#123

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₈×C₈₉

—

C₈×C₈₉

—

—

—

—

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

100

C₄×C₈₉

—

101

C₄×C₈₉

—

102

C₄×C₈₉

—

103

C₄×C₈₉

—

104

C₄×C₈₉

#435

yes

—

105

C₄×C₈₉

#436

yes

—

106

C₄×C₈₉

#454

yes

characteristic

107

C₄×C₈₉

yes

108

—

109

C₂×C₈₉

—

110

C₂×C₈₉

—

111

C₂×C₈₉

#1851

yes

characteristic

112

C₂×C₈₉

#1857

yes

characteristic

113

C₂×C₈₉

#1858

yes

characteristic

114

#2983

—

115

#1984

—

116

C₄×#101

D₈₉

yes

characteristic

117

#1565

—

118

#1278

—

119

C₈₉

#8838

yes

characteristic, complemented, Sylow

120

C₂×C₄×Q₈

C₂×D₈₉

yes

characteristic

121

#446

—

178

—

122

#446

—

178

—

123

#432

—

178

—

124

#431

—

125

C₂×C₄⋊C₈

—

126

C₂×#101

Dic₈₉

yes

—

127

C₄×M₁₆

C₂×D₈₉

yes

—

128

C₂×C₄×C₈

—

129

#343

Dic₈₉

yes

characteristic

130

#343

Dic₈₉

yes

characteristic

131

#336

—

178

—

132

#335

—

178

—

133

#334

—

178

—

134

C₂²×Q₈

C₂×Dic₈₉

yes

characteristic

135

C₂×M₁₆

C₂×Dic₈₉

yes

—

136

C₂×M₁₆

C₂×Dic₈₉

yes

—

137

C₂²×C₈

—

178

—

138

C₂²×C₈

—

178

—

139

C₄×Q₈

—

140

C₄×Q₈

—

141

C₂×C₄⋊C₄

Dic₁₇₈

yes

—

142

C₂×C₄⋊C₄

C₂×Dic₈₉

yes

characteristic

143

C₂×C₄²

D₃₅₆

yes

—

144

C₂×C₄²

C₂²×D₈₉

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

178

—

148

C₄⋊C₈

—

178

—

149

C₄⋊C₈

—

178

—

150

#101

—

151

C₂²⋊C₈

—

178

—

152

#97

—

153

#97

—

178

—

154

C₄×C₈

—

155

C₄×C₈

—

178

—

156

C₄×C₈

—

178

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂²×C₄

C₄⋊Dic₈₉

yes

—

162

C₂²×C₄

C₄⋊Dic₈₉

yes

—

163

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

164

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

165

C₂²×C₄

C₂×Dic₁₇₈

yes

characteristic

166

M₁₆

—

167

M₁₆

—

168

M₁₆

—

169

M₁₆

#13083558

yes

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

356

—

173

C₂×C₈

—

356

—

174

C₂×C₈

—

356

—

175

C₂×C₈

—

356

—

176

C₂×C₈

—

178

—

177

C₂×C₈

—

178

—

178

C₂×C₈

—

178

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄²

—

182

C₄²

—

183

C₄²

D₄×D₈₉

yes

—

184

C₄²

#13083573

yes

—

185

C₂³

C₂×C₄⋊Dic₈₉

yes

characteristic

186

Q₈

—

187

Q₈

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

D₄×Dic₈₉

yes

—

197

C₂×C₄

#15667982

yes

characteristic

198

C₂×C₄

#15667989

yes

—

199

C₂×C₄

#15667996

yes

characteristic

200

C₂×C₄

C₂×#13083558

yes

—

201

C₂×C₄

#15668019

yes

characteristic

202

C₂×C₄

#15668019

yes

characteristic

203

C₈

—

204

C₈

—

205

C₈

—

356

—

206

C₈

—

356

—

207

C₈

—

356

—

208

C₈

—

712

—

209

C₂²

—

210

C₂²

—

211

C₂²

#19151473

yes

characteristic

212

C₂²

#19151622

yes

characteristic

213

C₂²

#19151625

yes

characteristic

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

#19151670

yes

—

219

C₄

#19151676

yes

—

220

C₄

#19151773

yes

characteristic

221

C₄

#19151783

yes

characteristic

222

C₂

—

223

C₂

—

224

C₂

#26020437

yes

characteristic

225

C₂

#26020469

yes

characteristic

226

C₂

#26020474

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#15667982
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#101×C₈₉	C₂
socle	C₂²×C₈₉	#385
prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#101	D₈₉
hypocenter	C₈₉	#8838
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#101

subgroup

G/H

center

C₂×C₄

#15667982

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₄×#101×C₈₉

C₂

socle

C₂²×C₈₉

#385

prime radical

C₂³×C₈₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#101

N	H ≅ G/N	type
119. C₈₉	#8838	semidirect

H ≅ G/N

type

119. C₈₉

#8838

semidirect

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

subgroup	index
#26029337	2
#26021189	2
#26019004	2
#26018090	2
C₄×#101×C₈₉	2
#8838	89

subgroup

index

C₄×#101×C₈₉

Subgroups — #36448307

Conjugacy classes of subgroups

Complements

Maximal subgroups