Group #36472659 — subgroups

Subgroups — #36472659

Order 22784.

590 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 159 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×#19151975	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151398	C₂	1	4	yes	maximal, complemented
3	C₂×#19151374	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₇₁₂	—	2	1	no	—
6	C₂²×D₇₁₂	C₂²	1	2	yes	—
7	C₂×#15668009	—	2	1	no	—
8	C₂²×#13083549	C₂²	1	2	yes	—
9	#19151975	—	2	2	no	—
10	#19151975	C₄	1	4	yes	complemented
11	#19151398	—	2	8	no	—
12	#19151374	—	2	2	no	—
13	C₂×#107×C₈₉	C₂²	1	2	yes	—
14	C₂×#106×C₈₉	—	2	1	no	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#382×C₈₉	—	2	1	no	—
17	C₂²×D₃₅₆	—	2	2	no	—
18	C₂²×D₃₅₆	—	4	2	no	—
19	C₂×D₇₁₂	—	2	4	no	—
20	C₂×D₇₁₂	—	2	4	no	—
21	C₂×D₇₁₂	—	2	2	no	—
22	C₂×D₇₁₂	—	2	4	no	—
23	C₂×D₇₁₂	—	4	2	no	—
24	C₂×D₇₁₂	—	4	2	no	—
25	C₂×D₇₁₂	C₂×C₄	1	4	yes	—
26	#15668009	—	2	4	no	—
27	#15668009	—	4	2	no	—
28	C₂×#13083549	—	2	2	no	—
29	C₂×#13083549	—	2	2	no	—
30	C₂×#13083549	D₄	1	4	yes	complemented
31	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
35	#107×C₈₉	—	2	4	no	—
36	#106×C₈₉	—	4	2	no	—
37	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₈₉	D₄	1	2	yes	—
39	#95×C₈₉	—	4	2	no	—
40	C₂³×D₈₉	—	4	2	no	—
41	C₂×D₃₅₆	—	2	4	no	—
42	C₂×D₃₅₆	—	4	2	no	—
43	C₂×D₃₅₆	—	4	2	no	—
44	C₂×D₃₅₆	—	4	2	no	—
45	C₂×D₃₅₆	—	4	2	no	—
46	C₂×D₃₅₆	—	4	8	no	—
47	C₂×D₃₅₆	—	4	2	no	—
48	C₂×D₃₅₆	—	8	8	no	—
49	D₇₁₂	—	2	8	no	—
50	D₇₁₂	—	4	8	no	—
51	D₇₁₂	—	4	4	no	—
52	#13083549	—	2	4	no	—
53	C₂²×C₄×C₈₉	—	4	2	no	—
54	C₂²×C₄×C₈₉	D₈	1	2	yes	—
55	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₈₉	D₈	1	2	yes	—
58	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
59	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	#41	1	2	yes	—
63	C₄⋊C₄×C₈₉	—	4	2	no	—
64	C₄⋊C₄×C₈₉	—	4	2	no	—
65	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₈₉	SD₁₆	1	2	yes	—
67	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
68	C₂²×D₈₉	—	4	4	no	—
69	C₂²×D₈₉	—	8	2	no	—
70	C₂²×D₈₉	—	8	2	no	—
71	C₂²×D₈₉	—	8	8	no	—
72	C₂×C₄×C₈₉	—	2	4	no	—
73	C₂×C₄×C₈₉	—	8	2	no	—
74	C₂×C₄×C₈₉	—	8	4	no	—
75	C₂×C₄×C₈₉	#102	1	4	yes	—
76	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
78	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
82	D₃₅₆	—	4	4	no	—
83	D₃₅₆	—	4	4	no	—
84	D₃₅₆	—	8	16	no	—
85	D₃₅₆	—	8	2	no	—
86	D₃₅₆	—	8	2	no	—
87	C₈×C₈₉	#102	1	4	yes	—
88	C₈×C₈₉	C₄×D₄	1	2	yes	—
89	C₈×C₈₉	#121	1	2	yes	—
90	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
91	C₂²×C₈₉	#486	1	1	yes	characteristic
92	C₂²×C₈₉	#493	1	1	yes	characteristic
93	C₂²×C₈₉	#509	1	4	yes	—
94	C₂×D₈₉	—	8	8	no	—
95	C₂×D₈₉	—	8	4	no	—
96	C₂×D₈₉	—	16	4	no	—
97	C₂×D₈₉	—	16	4	no	—
98	C₄×C₈₉	—	2	4	no	—
99	C₄×C₈₉	—	16	2	no	—
100	C₄×C₈₉	#390	1	1	yes	characteristic
101	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
102	C₄×C₈₉	#486	1	1	yes	characteristic
103	C₄×C₈₉	#492	1	1	yes	characteristic
104	#13206	—	89	1	no	maximal, Sylow
105	C₂×C₈₉	#1957	1	1	yes	characteristic
106	C₂×C₈₉	#2140	1	2	yes	—
107	C₂×C₈₉	#2244	1	2	yes	—
108	C₂×C₈₉	#2250	1	2	yes	—
109	D₈₉	—	16	4	no	—
110	C₂×#493	—	89	1	no	—
111	C₂×#357	—	89	4	no	—
112	C₂×#346	—	89	1	no	—
113	#1846	D₈₉	1	1	yes	characteristic, complemented
114	C₈₉	#13206	1	1	yes	characteristic, complemented, Sylow
115	C₂²×D₈	—	89	2	no	—
116	C₂²×D₈	—	178	1	no	—
117	C₂×#127	—	178	1	no	—
118	C₂²×C₁₆	—	89	2	no	—
119	#493	—	89	4	no	—
120	#493	—	178	2	no	—
121	C₂×#107	C₂×D₈₉	1	2	yes	—
122	C₂×#106	—	2	1	no	—
123	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
124	#382	—	2	1	no	—
125	#357	—	178	8	no	—
126	#346	—	178	2	no	—
127	C₂²×D₄	—	356	2	no	—
128	C₂²×D₄	—	178	2	no	—
129	C₂×D₈	—	89	4	no	—
130	C₂×D₈	—	356	2	no	—
131	C₂×D₈	—	356	2	no	—
132	C₂×D₈	—	178	4	no	—
133	C₂×D₈	—	178	4	no	—
134	C₂×D₈	—	178	2	no	—
135	C₂×D₈	—	178	4	no	—
136	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
138	#127	—	356	2	no	—
139	#127	—	178	4	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
142	C₂×C₁₆	—	89	4	no	—
143	C₂×C₁₆	—	178	2	no	—
144	C₂×C₁₆	—	178	2	no	—
145	#107	—	2	4	no	—
146	#106	—	4	2	no	—
147	C₄×C₈	C₄×D₈₉	1	2	yes	—
148	C₄×C₈	D₃₅₆	1	2	yes	—
149	#95	—	4	2	no	—
150	C₂⁴	—	356	2	no	—
151	C₂×D₄	—	356	2	no	—
152	C₂×D₄	—	356	2	no	—
153	C₂×D₄	—	356	2	no	—
154	C₂×D₄	—	356	2	no	—
155	C₂×D₄	—	356	2	no	—
156	C₂×D₄	—	356	8	no	—
157	C₂×D₄	—	178	4	no	—
158	C₂×D₄	—	712	8	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	#13083559	1	2	yes	—
161	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
162	D₈	—	356	4	no	—
163	D₈	—	356	8	no	—
164	D₈	—	178	8	no	—
165	C₂×C₈	#13083551	1	2	yes	—
166	C₂×C₈	#13083559	1	2	yes	—
167	C₂×C₈	#13083561	1	2	yes	—
168	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
169	C₂×C₈	D₄×D₈₉	1	2	yes	—
170	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
171	C₂×C₈	#13083575	1	2	yes	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	#13083551	1	2	yes	—
175	C₄²	#13083561	1	2	yes	—
176	C₁₆	—	178	4	no	—
177	C₂³	—	356	4	no	—
178	C₂³	—	712	2	no	—
179	C₂³	—	712	2	no	—
180	C₂³	—	712	8	no	—
181	C₂³	C₂×#13083559	1	1	yes	characteristic
182	D₄	—	356	4	no	—
183	D₄	—	356	4	no	—
184	D₄	—	712	16	no	—
185	D₄	—	712	2	no	—
186	D₄	—	712	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	—	8	4	no	—
190	C₂×C₄	#15667932	1	4	yes	—
191	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
192	C₂×C₄	#15668001	1	1	yes	characteristic
193	C₂×C₄	#15668008	1	1	yes	characteristic
194	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
195	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
196	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
197	C₈	#15667932	1	4	yes	—
198	C₈	#15667974	1	2	yes	—
199	C₈	#15667987	1	2	yes	—
200	C₂²	—	1424	4	no	—
201	C₂²	—	1424	4	no	—
202	C₂²	—	712	8	no	—
203	C₂²	—	712	4	no	—
204	C₂²	C₂×#15667932	1	1	yes	characteristic
205	C₂²	#19151946	1	1	yes	characteristic
206	C₂²	#19151974	1	1	yes	characteristic
207	C₂²	#19152033	1	4	yes	—
208	C₄	—	2	4	no	—
209	C₄	—	16	2	no	—
210	C₄	#19151499	1	1	yes	characteristic
211	C₄	C₂×#15667932	1	1	yes	characteristic
212	C₄	#19151946	1	1	yes	characteristic
213	C₄	#19151972	1	1	yes	characteristic
214	C₂	—	1424	4	no	—
215	C₂	#26021057	1	1	yes	characteristic
216	C₂	#26021967	1	2	yes	—
217	C₂	#26022495	1	2	yes	—
218	C₂	#26022519	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151975

C₂

yes

maximal, characteristic

C₂×#19151398

C₂

yes

maximal, complemented

C₂×#19151374

C₂

yes

maximal, characteristic, complemented

C₈₉×#1846

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₁₂

—

C₂²×D₇₁₂

C₂²

yes

—

C₂×#15668009

—

C₂²×#13083549

C₂²

yes

—

#19151975

—

#19151975

C₄

yes

complemented

#19151398

—

#19151374

—

C₂×#107×C₈₉

C₂²

yes

—

C₂×#106×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#382×C₈₉

—

C₂²×D₃₅₆

—

C₂²×D₃₅₆

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

C₂×C₄

yes

—

—

—

—

—

D₄

yes

complemented

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

#107×C₈₉

—

#106×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#95×C₈₉

—

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₈

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

#102

yes

—

C₈×C₈₉

C₄×D₄

yes

—

C₈×C₈₉

#121

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

100

C₄×C₈₉

#390

yes

characteristic

101

C₄×C₈₉

C₂×#102

yes

characteristic

102

C₄×C₈₉

#486

yes

characteristic

103

C₄×C₈₉

yes

104

—

105

C₂×C₈₉

#1957

yes

characteristic

106

C₂×C₈₉

#2140

yes

—

107

C₂×C₈₉

#2244

yes

—

108

C₂×C₈₉

#2250

yes

—

109

D₈₉

—

110

C₂×#493

—

111

C₂×#357

—

112

C₂×#346

—

113

#1846

D₈₉

yes

characteristic, complemented

114

C₈₉

#13206

yes

characteristic, complemented, Sylow

115

C₂²×D₈

—

116

C₂²×D₈

—

178

—

117

C₂×#127

—

178

—

118

C₂²×C₁₆

—

119

#493

—

120

#493

—

178

—

121

C₂×#107

C₂×D₈₉

yes

—

122

C₂×#106

—

123

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

124

#382

—

125

#357

—

178

—

126

#346

—

178

—

127

C₂²×D₄

—

356

—

128

C₂²×D₄

—

178

—

129

C₂×D₈

—

130

C₂×D₈

—

356

—

131

C₂×D₈

—

356

—

132

C₂×D₈

—

178

—

133

C₂×D₈

—

178

—

134

C₂×D₈

—

178

—

135

C₂×D₈

—

178

—

136

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

137

C₂²×C₈

C₂²×D₈₉

yes

characteristic

138

#127

—

356

—

139

#127

—

178

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

142

C₂×C₁₆

—

143

C₂×C₁₆

—

178

—

144

C₂×C₁₆

—

178

—

145

#107

—

146

#106

—

147

C₄×C₈

C₄×D₈₉

yes

—

148

C₄×C₈

D₃₅₆

yes

—

149

#95

—

150

C₂⁴

—

356

—

151

C₂×D₄

—

356

—

152

C₂×D₄

—

356

—

153

C₂×D₄

—

356

—

154

C₂×D₄

—

356

—

155

C₂×D₄

—

356

—

156

C₂×D₄

—

356

—

157

C₂×D₄

—

178

—

158

C₂×D₄

—

712

—

159

C₂²×C₄

—

160

C₂²×C₄

#13083559

yes

—

161

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

162

D₈

—

356

—

163

D₈

—

356

—

164

D₈

—

178

—

165

C₂×C₈

#13083551

yes

—

166

C₂×C₈

#13083559

yes

—

167

C₂×C₈

#13083561

yes

—

168

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

169

C₂×C₈

D₄×D₈₉

yes

—

170

C₂×C₈

C₂×D₃₅₆

yes

characteristic

171

C₂×C₈

#13083575

yes

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄²

#13083551

yes

—

175

C₄²

#13083561

yes

—

176

C₁₆

—

178

—

177

C₂³

—

356

—

178

C₂³

—

712

—

179

C₂³

—

712

—

180

C₂³

—

712

—

181

C₂³

C₂×#13083559

yes

characteristic

182

D₄

—

356

—

183

D₄

—

356

—

184

D₄

—

712

—

185

D₄

—

712

—

186

D₄

—

712

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

#15667932

yes

—

191

C₂×C₄

C₂×#13083551

yes

characteristic

192

C₂×C₄

#15668001

yes

characteristic

193

C₂×C₄

#15668008

yes

characteristic

194

C₂×C₄

C₂×#13083559

yes

characteristic

195

C₂×C₄

C₂×#13083561

yes

characteristic

196

C₂×C₄

C₂×#13083561

yes

characteristic

197

C₈

#15667932

yes

—

198

C₈

#15667974

yes

—

199

C₈

#15667987

yes

—

200

C₂²

—

1424

—

201

C₂²

—

1424

—

202

C₂²

—

712

—

203

C₂²

—

712

—

204

C₂²

C₂×#15667932

yes

characteristic

205

C₂²

#19151946

yes

characteristic

206

C₂²

#19151974

yes

characteristic

207

C₂²

#19152033

yes

—

208

C₄

—

209

C₄

—

210

C₄

#19151499

yes

characteristic

211

C₄

C₂×#15667932

yes

characteristic

212

C₄

#19151946

yes

characteristic

213

C₄

#19151972

yes

characteristic

214

C₂

—

1424

—

215

C₂

#26021057

yes

characteristic

216

C₂

#26021967

yes

—

217

C₂

#26022495

yes

—

218

C₂

#26022519

yes

—

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1846	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂×#19151975	C₂
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1846	D₈₉
hypocenter	C₈₉	#13206
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#13083559

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1846

C₂

socle

C₂³×C₈₉

C₂×D₈

prime radical

C₂×#19151975

C₂

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
2. C₂×#19151398	C₂	semidirect
3. C₂×#19151374	C₂	semidirect
4. C₈₉×#1846	C₂	semidirect
10. #19151975	C₄	semidirect
30. C₂×#13083549	D₄	semidirect
113. #1846	D₈₉	semidirect
114. C₈₉	#13206	semidirect

H ≅ G/N

type

2. C₂×#19151398

C₂

semidirect

3. C₂×#19151374

C₂

semidirect

4. C₈₉×#1846

C₂

semidirect

10. #19151975

C₄

semidirect

30. C₂×#13083549

D₄

semidirect

113. #1846

D₈₉

semidirect

114. C₈₉

#13206

semidirect

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

subgroup	index
C₂×#19151975	2
C₂×#19151398	2
C₂×#19151374	2
C₈₉×#1846	2
#13206	89

subgroup

index

C₈₉×#1846

Subgroups — #36472659

Conjugacy classes of subgroups

Complements

Maximal subgroups