Group #36445568 — subgroups

Subgroups — #36445568

Order 22784.

470 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 199 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	C₄×#15667941	C₂	1	2	yes	maximal
2	C₄×#15667922	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151320	C₂	1	2	yes	maximal
4	#26018079	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667941	C₄	1	4	yes	—
7	C₂×#15667922	C₄	1	2	yes	—
8	C₂×#15667922	C₄	1	2	yes	—
9	C₂×C₄×#10623494	C₂²	1	2	yes	—
10	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
11	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
12	#19151352	C₄	1	4	yes	—
13	#19151320	—	2	4	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×#10623494	C₂×C₄	1	2	yes	—
17	C₂²×#10623494	C₂×C₄	1	4	yes	—
18	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
22	#15667941	—	2	8	no	—
23	#15667922	—	2	4	no	—
24	#15667922	—	2	4	no	—
25	C₄×#10623494	—	2	2	no	—
26	C₄×#10623494	—	2	2	no	—
27	C₄×#10623494	D₄	1	2	yes	—
28	C₄×#10623494	Q₈	1	2	yes	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₈₉	—	2	2	no	—
33	C₄×C₈×C₈₉	—	2	2	no	—
34	C₄×C₈×C₈₉	—	2	2	no	—
35	C₂²×Dic₈₉	C₄²	1	1	yes	characteristic
36	C₂²×Dic₈₉	C₄²	1	1	yes	characteristic
37	C₂²×Dic₈₉	C₄²	1	2	yes	—
38	C₂×#10623494	—	2	4	no	—
39	C₂×#10623494	—	2	2	no	—
40	C₂×#10623494	—	2	2	no	—
41	C₂×#10623494	—	2	4	no	—
42	C₂×#10623494	C₄⋊C₄	1	4	yes	—
43	C₂×#10623494	M₁₆	1	8	yes	—
44	C₄×Dic₈₉	—	2	2	no	—
45	C₄×Dic₈₉	—	2	4	no	—
46	C₄×Dic₈₉	—	2	4	no	—
47	C₄×Dic₈₉	M₁₆	1	4	yes	—
48	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₈₉	—	2	2	no	—
51	C₂×C₈×C₈₉	—	2	2	no	—
52	C₂×C₈×C₈₉	—	2	4	no	—
53	C₂×C₈×C₈₉	—	2	4	no	—
54	C₂×C₈×C₈₉	—	2	4	no	—
55	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
56	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₈₉	#41	1	2	yes	—
59	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
66	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₈₉	#131	1	4	yes	—
68	C₂×Dic₈₉	—	2	2	no	—
69	C₂×Dic₈₉	—	2	2	no	—
70	C₂×Dic₈₉	—	2	1	no	—
71	C₂×Dic₈₉	—	2	1	no	—
72	C₂×Dic₈₉	—	2	2	no	—
73	C₂×Dic₈₉	—	2	2	no	—
74	C₂×Dic₈₉	#97	1	4	yes	—
75	C₈×C₈₉	—	2	4	no	—
76	C₈×C₈₉	—	4	4	no	—
77	#10623494	—	2	4	no	—
78	#10623494	—	2	8	no	—
79	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
81	C₂²×C₈₉	C₄×M₁₆	1	2	yes	—
82	C₂²×C₈₉	C₄×M₁₆	1	1	yes	characteristic
83	C₂²×C₈₉	#405	1	2	yes	—
84	C₄×C₈₉	#423	1	2	yes	—
85	C₄×C₈₉	#432	1	2	yes	—
86	C₄×C₈₉	#435	1	4	yes	—
87	C₄×C₈₉	#446	1	4	yes	—
88	Dic₈₉	—	2	4	no	—
89	Dic₈₉	—	4	1	no	—
90	Dic₈₉	—	4	1	no	—
91	#8263	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	#1451	1	2	yes	—
93	C₂×C₈₉	#1453	1	1	yes	characteristic
94	C₂×C₈₉	#1455	1	1	yes	characteristic
95	C₂×C₈₉	#1767	1	1	yes	characteristic
96	C₂×C₈₉	#1771	1	2	yes	—
97	C₄×C₄⋊C₈	—	89	2	no	—
98	C₄×#97	—	89	1	no	—
99	C₂×#322	—	89	2	no	—
100	#1273	—	89	1	no	—
101	#1273	D₈₉	1	1	yes	characteristic
102	C₈₉	#8263	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊C₈	—	89	4	no	—
104	C₂×#97	—	89	2	no	—
105	C₂×#97	—	89	2	no	—
106	C₂×C₄×C₈	—	89	1	no	—
107	C₂×C₄×C₈	—	89	2	no	—
108	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
110	C₄³	—	89	1	no	—
111	#336	—	89	4	no	—
112	#322	—	178	4	no	—
113	C₂²×C₈	—	89	4	no	—
114	C₂²×C₈	—	89	2	no	—
115	C₂²×C₈	C₄×D₈₉	1	4	yes	—
116	C₂²×C₈	C₄×D₈₉	1	2	yes	—
117	C₂×C₄²	—	89	1	no	—
118	C₂×C₄²	—	89	2	no	—
119	C₂×C₄²	—	89	1	no	—
120	C₂×C₄²	—	89	2	no	—
121	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
122	C₄⋊C₈	—	178	8	no	—
123	#97	—	178	4	no	—
124	#97	—	178	4	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	89	2	no	—
129	C₄×C₈	—	89	2	no	—
130	C₄×C₈	—	178	2	no	—
131	C₄×C₈	—	178	2	no	—
132	C₂²×C₄	—	89	2	no	—
133	C₂²×C₄	—	89	1	no	—
134	C₂²×C₄	—	89	1	no	—
135	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	89	8	no	—
143	C₂×C₈	—	89	4	no	—
144	C₂×C₈	—	178	4	no	—
145	C₂×C₈	—	178	2	no	—
146	C₂×C₈	—	178	2	no	—
147	C₂×C₈	—	178	4	no	—
148	C₂×C₈	#13083557	1	4	yes	—
149	C₄²	—	89	4	no	—
150	C₄²	—	178	4	no	—
151	C₄²	—	178	2	no	—
152	C₄²	—	178	4	no	—
153	C₄²	D₄×D₈₉	1	1	yes	characteristic
154	C₄²	Q₈×D₈₉	1	1	yes	characteristic
155	C₄²	#13083573	1	2	yes	—
156	C₂³	C₄²×D₈₉	1	1	yes	characteristic
157	C₂×C₄	—	89	4	no	—
158	C₂×C₄	—	178	1	no	—
159	C₂×C₄	—	178	1	no	—
160	C₂×C₄	—	178	2	no	—
161	C₂×C₄	—	178	2	no	—
162	C₂×C₄	—	178	2	no	—
163	C₂×C₄	—	178	2	no	—
164	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
165	C₂×C₄	#15667968	1	1	yes	characteristic
166	C₂×C₄	#15667972	1	2	yes	—
167	C₂×C₄	#15667977	1	2	yes	—
168	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
169	C₂×C₄	M₁₆×D₈₉	1	4	yes	—
170	C₂×C₄	C₂×#13083557	1	2	yes	—
171	C₂×C₄	#15668018	1	4	yes	—
172	C₈	—	2	4	no	—
173	C₈	—	4	4	no	—
174	C₈	—	178	4	no	—
175	C₈	—	178	8	no	—
176	C₂²	#19151449	1	1	yes	characteristic
177	C₂²	#97×D₈₉	1	1	yes	characteristic
178	C₂²	C₄×#13083557	1	1	yes	characteristic
179	C₂²	#19151550	1	2	yes	—
180	C₂²	#19151552	1	2	yes	—
181	C₄	—	356	1	no	—
182	C₄	—	356	1	no	—
183	C₄	—	178	4	no	—
184	C₄	#19151620	1	2	yes	—
185	C₄	#19151652	1	2	yes	—
186	C₄	#19151672	1	4	yes	—
187	C₄	#19151727	1	4	yes	—
188	C₂	#26018540	1	2	yes	—
189	C₂	#26018546	1	1	yes	characteristic
190	C₂	#26018556	1	1	yes	characteristic
191	C₂	#26020104	1	1	yes	characteristic
192	C₂	#26020125	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#15667941

C₂

yes

maximal

C₄×#15667922

C₂

yes

maximal, characteristic

C₂×#19151320

C₂

yes

maximal

#26018079

C₂

yes

maximal, characteristic

C₈₉×#1273

C₂

yes

maximal, characteristic

C₂×#15667941

C₄

yes

—

C₂×#15667922

C₄

yes

—

C₂×#15667922

C₄

yes

—

C₂×C₄×#10623494

C₂²

yes

—

C₂×C₄×#10623494

C₂²

yes

characteristic

C₄²×Dic₈₉

C₂²

yes

characteristic

#19151352

C₄

yes

—

#19151320

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

—

C₂²×#10623494

C₂×C₄

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#10623494

Q₈

yes

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₂²×Dic₈₉

C₄²

yes

characteristic

C₂²×Dic₈₉

C₄²

yes

characteristic

C₂²×Dic₈₉

C₄²

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#10623494

M₁₆

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

M₁₆

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#117

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₄×Q₈

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#97

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

C₂×#97

yes

characteristic

C₂²×C₈₉

C₄×M₁₆

yes

—

C₂²×C₈₉

C₄×M₁₆

yes

characteristic

C₂²×C₈₉

#405

yes

—

C₄×C₈₉

#423

yes

—

C₄×C₈₉

#432

yes

—

C₄×C₈₉

#435

yes

—

C₄×C₈₉

#446

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

#8263

—

maximal, Sylow

C₂×C₈₉

#1451

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₄×C₄⋊C₈

—

C₄×#97

—

C₂×#322

—

100

#1273

—

101

#1273

D₈₉

yes

characteristic

102

C₈₉

#8263

yes

characteristic, complemented, Sylow

103

C₂×C₄⋊C₈

—

104

C₂×#97

—

105

C₂×#97

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

C₂×D₈₉

yes

—

109

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

110

C₄³

—

111

#336

—

112

#322

—

178

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₄×D₈₉

yes

—

116

C₂²×C₈

C₄×D₈₉

yes

—

117

C₂×C₄²

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

C₂×C₄²

C₂²×D₈₉

yes

characteristic

122

C₄⋊C₈

—

178

—

123

#97

—

178

—

124

#97

—

178

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

178

—

131

C₄×C₈

—

178

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

136

C₂²×C₄

C₂×C₄×D₈₉

yes

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

178

—

145

C₂×C₈

—

178

—

146

C₂×C₈

—

178

—

147

C₂×C₈

—

178

—

148

C₂×C₈

#13083557

yes

—

149

C₄²

—

150

C₄²

—

178

—

151

C₄²

—

178

—

152

C₄²

—

178

—

153

C₄²

D₄×D₈₉

yes

characteristic

154

C₄²

Q₈×D₈₉

yes

characteristic

155

C₄²

#13083573

yes

—

156

C₂³

C₄²×D₈₉

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

178

—

159

C₂×C₄

—

178

—

160

C₂×C₄

—

178

—

161

C₂×C₄

—

178

—

162

C₂×C₄

—

178

—

163

C₂×C₄

—

178

—

164

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

165

C₂×C₄

#15667968

yes

characteristic

166

C₂×C₄

#15667972

yes

—

167

C₂×C₄

#15667977

yes

—

168

C₂×C₄

M₁₆×D₈₉

yes

—

169

C₂×C₄

M₁₆×D₈₉

yes

—

170

C₂×C₄

C₂×#13083557

yes

—

171

C₂×C₄

#15668018

yes

—

172

C₈

—

173

C₈

—

174

C₈

—

178

—

175

C₈

—

178

—

176

C₂²

#19151449

yes

characteristic

177

C₂²

#97×D₈₉

yes

characteristic

178

C₂²

C₄×#13083557

yes

characteristic

179

C₂²

#19151550

yes

—

180

C₂²

#19151552

yes

—

181

C₄

—

356

—

182

C₄

—

356

—

183

C₄

—

178

—

184

C₄

#19151620

yes

—

185

C₄

#19151652

yes

—

186

C₄

#19151672

yes

—

187

C₄

#19151727

yes

—

188

C₂

#26018540

yes

—

189

C₂

#26018546

yes

characteristic

190

C₂

#26018556

yes

characteristic

191

C₂

#26020104

yes

characteristic

192

C₂

#26020125

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂×C₄²
Fitting	C₈₉×#1273	C₂
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1273	D₈₉
hypocenter	C₈₉	#8263
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₈₉

commutator, socle, prime radical

C₂³×C₈₉

C₂×C₄²

Fitting

C₈₉×#1273

C₂

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
102. C₈₉	#8263	semidirect

H ≅ G/N

type

102. C₈₉

#8263

semidirect

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

subgroup	index
C₄×#15667941	2
C₄×#15667922	2
C₂×#19151320	2
#26018079	2
C₈₉×#1273	2
#8263	89

subgroup

index

C₈₉×#1273

Subgroups — #36445568

Conjugacy classes of subgroups

Complements

Maximal subgroups