Group #36709747 — subgroups

Subgroups — C₂×#26022499 #36709747

Order 22784.

582 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 203 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×#19151978	C₂	1	1	yes	maximal, characteristic
2	#26022499	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151401	C₂	1	4	yes	maximal
4	C₂×#19151374	C₂	1	1	yes	maximal, characteristic
5	C₂×#499×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	—	2	1	no	—
7	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
8	C₂×#15668012	—	2	1	no	—
9	C₂²×#13083549	C₂²	1	2	yes	—
10	#19151978	—	2	2	no	—
11	#19151978	C₂²	1	4	yes	—
12	#19151401	C₂²	1	16	yes	—
13	#19151374	C₂²	1	4	yes	—
14	C₂×#125×C₈₉	—	2	1	no	—
15	#499×C₈₉	—	2	2	no	—
16	#499×C₈₉	C₂²	1	4	yes	—
17	C₂×#107×C₈₉	C₂²	1	2	yes	—
18	C₂×#106×C₈₉	—	2	1	no	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₃₅₆	—	2	2	no	—
22	C₂×Dic₃₅₆	—	2	4	no	—
23	C₂×Dic₃₅₆	—	2	8	no	—
24	C₂×Dic₃₅₆	—	4	4	no	—
25	C₂×Dic₃₅₆	C₂³	1	4	yes	—
26	#15668012	—	2	4	no	—
27	#15668012	—	4	2	no	—
28	C₂×C₄⋊Dic₈₉	—	4	2	no	—
29	C₂×#13083549	—	2	4	no	—
30	C₂×#13083549	C₂³	1	4	yes	—
31	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	#125×C₈₉	—	2	4	no	—
34	#125×C₈₉	—	4	2	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	—	4	2	no	—
37	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
38	#107×C₈₉	—	2	2	no	—
39	#107×C₈₉	C₂³	1	4	yes	—
40	#106×C₈₉	—	2	4	no	—
41	C₄×C₈×C₈₉	D₄	1	2	yes	—
42	C₄×C₈×C₈₉	C₂³	1	2	yes	—
43	C₂×Dic₁₇₈	—	2	4	no	—
44	C₂×Dic₁₇₈	—	4	4	no	—
45	C₂×Dic₁₇₈	—	4	4	no	—
46	C₂×Dic₁₇₈	—	4	2	no	—
47	C₂²×Dic₈₉	—	4	2	no	—
48	Dic₃₅₆	—	2	8	no	—
49	Dic₃₅₆	—	4	4	no	—
50	Dic₃₅₆	—	4	8	no	—
51	C₄⋊Dic₈₉	—	4	8	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	SD₁₆	1	4	yes	—
57	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₈₉	#41	1	4	yes	—
60	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₈₉	—	2	4	no	—
62	C₄⋊C₄×C₈₉	—	4	2	no	—
63	C₄⋊C₄×C₈₉	—	4	8	no	—
64	C₄²×C₈₉	D₈	1	2	yes	—
65	C₄²×C₈₉	C₂×D₄	1	2	yes	—
66	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	—	2	4	no	—
68	C₂×C₄×C₈₉	—	4	4	no	—
69	C₂×C₄×C₈₉	—	8	4	no	—
70	C₂×C₄×C₈₉	#124	1	2	yes	—
71	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
72	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
74	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₈₉	—	4	4	no	—
76	C₂×Dic₈₉	—	8	4	no	—
77	Dic₁₇₈	—	4	4	no	—
78	Dic₁₇₈	—	4	4	no	—
79	Dic₁₇₈	—	8	4	no	—
80	C₈×C₈₉	#124	1	2	yes	—
81	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
82	C₈×C₈₉	C₂×#41	1	2	yes	—
83	C₂²×C₈₉	#486	1	2	yes	—
84	C₂²×C₈₉	#510	1	4	yes	—
85	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
86	C₄×C₈₉	—	2	4	no	—
87	C₄×C₈₉	—	8	4	no	—
88	C₄×C₈₉	#486	1	2	yes	—
89	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
90	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
91	Dic₈₉	—	8	4	no	—
92	C₂×#2245	—	89	1	no	maximal, Sylow
93	C₂×C₈₉	#2245	1	4	yes	complemented
94	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
95	C₂×C₈₉	C₂×#510	1	2	yes	—
96	C₂×#499	D₈₉	1	1	yes	characteristic
97	C₂×#494	—	89	1	no	—
98	#2245	—	89	8	no	—
99	C₂×#358	—	89	4	no	—
100	C₂×#346	—	89	1	no	—
101	C₈₉	C₂×#2245	1	1	yes	characteristic, complemented, Sylow
102	C₂²×Q₁₆	—	89	2	no	—
103	C₂²×Q₁₆	—	178	1	no	—
104	C₂×C₄⋊Q₈	—	178	1	no	—
105	C₂×#125	—	2	1	no	—
106	C₂²×C₁₆	—	89	2	no	—
107	#499	—	2	2	no	—
108	#499	C₂×D₈₉	1	4	yes	—
109	#494	—	89	4	no	—
110	#494	—	178	2	no	—
111	C₂×#107	C₂×D₈₉	1	2	yes	—
112	C₂×#106	—	2	1	no	—
113	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
114	#358	—	89	16	no	—
115	#346	—	89	4	no	—
116	C₂²×Q₈	—	178	2	no	—
117	C₂×Q₁₆	—	89	4	no	—
118	C₂×Q₁₆	—	356	4	no	—
119	C₂×Q₁₆	—	178	2	no	—
120	C₂×Q₁₆	—	178	4	no	—
121	C₂×Q₁₆	—	178	8	no	—
122	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
124	C₄⋊Q₈	—	356	2	no	—
125	C₄⋊Q₈	—	178	4	no	—
126	#125	—	2	4	no	—
127	#125	—	4	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	4	2	no	—
130	C₂×C₄⋊C₄	—	356	2	no	—
131	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
132	C₂×C₁₆	—	89	4	no	—
133	C₂×C₁₆	—	178	4	no	—
134	#107	—	2	2	no	—
135	#107	C₂²×D₈₉	1	4	yes	—
136	#106	—	2	4	no	—
137	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
138	C₄×C₈	C₂²×D₈₉	1	2	yes	—
139	C₂×Q₈	—	356	2	no	—
140	C₂×Q₈	—	356	4	no	—
141	C₂×Q₈	—	356	4	no	—
142	C₂×Q₈	—	178	4	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	356	2	no	—
145	C₂²×C₄	#13083559	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
147	Q₁₆	—	356	4	no	—
148	Q₁₆	—	356	8	no	—
149	Q₁₆	—	178	8	no	—
150	C₂×C₈	#13083561	1	4	yes	—
151	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
153	C₂×C₈	#13083575	1	4	yes	—
154	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	8	no	—
158	C₄⋊C₄	—	356	8	no	—
159	C₄²	#13083559	1	2	yes	—
160	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
161	C₁₆	—	178	4	no	—
162	C₂³	C₂×#13083559	1	1	yes	characteristic
163	Q₈	—	356	4	no	—
164	Q₈	—	356	4	no	—
165	Q₈	—	712	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	356	4	no	—
170	C₂×C₄	—	712	4	no	—
171	C₂×C₄	#15668001	1	2	yes	—
172	C₂×C₄	C₂×#13083559	1	4	yes	—
173	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
174	C₂×C₄	C₂×#13083561	1	2	yes	—
175	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
176	C₈	#15668001	1	2	yes	—
177	C₈	C₂×#13083561	1	4	yes	—
178	C₈	C₂×#13083575	1	2	yes	—
179	C₂²	#19151946	1	2	yes	—
180	C₂²	#19152038	1	4	yes	—
181	C₂²	C₂²×#13083559	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	8	4	no	—
184	C₄	—	712	4	no	—
185	C₄	#19151946	1	2	yes	—
186	C₄	C₂×#15668001	1	1	yes	characteristic
187	C₄	C₂²×#13083561	1	1	yes	characteristic
188	C₂	#26022499	1	4	yes	direct product factor
189	C₂	C₂×#19151946	1	1	yes	characteristic
190	C₂	C₂×#19152038	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151978

C₂

yes

maximal, characteristic

#26022499

C₂

yes

maximal, direct product factor

C₂×#19151401

C₂

yes

maximal

C₂×#19151374

C₂

yes

maximal, characteristic

C₂×#499×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₆

—

C₂²×Dic₃₅₆

C₂²

yes

—

C₂×#15668012

—

C₂²×#13083549

C₂²

yes

—

#19151978

—

#19151978

C₂²

yes

—

#19151401

C₂²

yes

—

#19151374

C₂²

yes

—

C₂×#125×C₈₉

—

#499×C₈₉

—

#499×C₈₉

C₂²

yes

—

C₂×#107×C₈₉

C₂²

yes

—

C₂×#106×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂²×Dic₁₇₈

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

C₂³

yes

—

#15668012

—

#15668012

—

C₂×C₄⋊Dic₈₉

—

C₂×#13083549

—

C₂×#13083549

C₂³

yes

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

#125×C₈₉

—

#125×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

#107×C₈₉

—

#107×C₈₉

C₂³

yes

—

#106×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₈

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₈

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

#124

yes

—

C₈×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

#486

yes

—

C₂²×C₈₉

#510

yes

—

C₂²×C₈₉

C₂²×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#486

yes

—

C₄×C₈₉

C₂×#124

yes

characteristic

C₄×C₈₉

C₂²×SD₁₆

yes

characteristic

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×#499

D₈₉

yes

—

—

—

100

C₂×#346

—

101

C₈₉

C₂×#2245

yes

characteristic, complemented, Sylow

102

C₂²×Q₁₆

—

103

C₂²×Q₁₆

—

178

—

104

C₂×C₄⋊Q₈

—

178

—

105

C₂×#125

—

106

C₂²×C₁₆

—

107

#499

—

108

#499

C₂×D₈₉

yes

—

109

#494

—

110

#494

—

178

—

111

C₂×#107

C₂×D₈₉

yes

—

112

C₂×#106

—

113

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

114

#358

—

115

#346

—

116

C₂²×Q₈

—

178

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

356

—

119

C₂×Q₁₆

—

178

—

120

C₂×Q₁₆

—

178

—

121

C₂×Q₁₆

—

178

—

122

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

123

C₂²×C₈

C₂²×D₈₉

yes

characteristic

124

C₄⋊Q₈

—

356

—

125

C₄⋊Q₈

—

178

—

126

#125

—

127

#125

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

356

—

131

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

132

C₂×C₁₆

—

133

C₂×C₁₆

—

178

—

134

#107

—

135

#107

C₂²×D₈₉

yes

—

136

#106

—

137

C₄×C₈

C₂²⋊D₈₉

yes

—

138

C₄×C₈

C₂²×D₈₉

yes

—

139

C₂×Q₈

—

356

—

140

C₂×Q₈

—

356

—

141

C₂×Q₈

—

356

—

142

C₂×Q₈

—

178

—

143

C₂²×C₄

—

144

C₂²×C₄

—

356

—

145

C₂²×C₄

#13083559

yes

—

146

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

147

Q₁₆

—

356

—

148

Q₁₆

—

356

—

149

Q₁₆

—

178

—

150

C₂×C₈

#13083561

yes

—

151

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

152

C₂×C₈

C₂×C₂²⋊D₈₉

yes

—

153

C₂×C₈

#13083575

yes

—

154

C₂×C₈

C₂³×D₈₉

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

356

—

159

C₄²

#13083559

yes

—

160

C₄²

C₂×C₂²⋊D₈₉

yes

—

161

C₁₆

—

178

—

162

C₂³

C₂×#13083559

yes

characteristic

163

Q₈

—

356

—

164

Q₈

—

356

—

165

Q₈

—

712

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

356

—

170

C₂×C₄

—

712

—

171

C₂×C₄

#15668001

yes

—

172

C₂×C₄

C₂×#13083559

yes

—

173

C₂×C₄

C₂×#13083559

yes

characteristic

174

C₂×C₄

C₂×#13083561

yes

—

175

C₂×C₄

C₂²×C₂²⋊D₈₉

yes

characteristic

176

C₈

#15668001

yes

—

177

C₈

C₂×#13083561

yes

—

178

C₈

C₂×#13083575

yes

—

179

C₂²

#19151946

yes

—

180

C₂²

#19152038

yes

—

181

C₂²

C₂²×#13083559

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

712

—

185

C₄

#19151946

yes

—

186

C₄

C₂×#15668001

yes

characteristic

187

C₄

C₂²×#13083561

yes

characteristic

188

C₂

#26022499

yes

direct product factor

189

C₂

C₂×#19151946

yes

characteristic

190

C₂

C₂×#19152038

yes

—

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#499×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	C₂×#499	D₈₉
hypocenter	C₈₉	C₂×#2245
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂³

C₂×#13083559

commutator, cosocle

C₂×C₈×C₈₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₈₉

Fitting

C₂×#499×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₈

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
2. #26022499	C₂	direct (N × H)
93. C₂×C₈₉	#2245	semidirect
101. C₈₉	C₂×#2245	semidirect
188. C₂	#26022499	direct (N × H)

H ≅ G/N

type

2. #26022499

C₂

direct (N × H)

93. C₂×C₈₉

#2245

semidirect

101. C₈₉

C₂×#2245

semidirect

188. C₂

#26022499

direct (N × H)

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

subgroup	index
C₂×#19151978	2
#26022499	2
C₂×#19151401	2
C₂×#19151374	2
C₂×#499×C₈₉	2
C₂×#2245	89

subgroup

index

C₂×#499×C₈₉

Subgroups — C₂×#26022499 #36709747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#26022499 #36709747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022499 #36709747