Group #36709908 — subgroups

Subgroups — C₂×#26022529 #36709908

Order 22784.

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

normal subgroups only

This group has 494 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×#19151999	C₂	1	2	yes	maximal
2	#26022529	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151425	C₂	1	2	yes	maximal
4	C₂×#19151424	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#366×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#15668003	—	2	2	no	—
8	C₂²×#13083549	C₂²	1	1	yes	characteristic
9	#19151999	—	2	4	no	—
10	#19151999	C₂²	1	8	yes	—
11	C₂×#15667946	C₂²	1	2	yes	—
12	C₂×#15667943	—	2	2	no	—
13	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
14	#19151425	C₂²	1	8	yes	—
15	#19151424	C₂²	1	4	yes	—
16	C₁₆×Dic₈₉	C₂²	1	4	yes	—
17	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#107×C₈₉	C₂²	1	2	yes	—
19	#366×C₈₉	C₂²	1	4	yes	—
20	C₂²×#10623494	D₄	1	1	yes	characteristic
21	#15668003	—	2	8	no	—
22	#15668003	—	4	4	no	—
23	C₂×C₄⋊Dic₈₉	—	2	2	no	—
24	C₂×C₄⋊Dic₈₉	—	4	2	no	—
25	C₂×#13083553	—	4	2	no	—
26	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
27	C₂×#13083549	Q₈	1	4	yes	—
28	C₂×#13083549	C₂³	1	2	yes	—
29	#15667946	—	2	2	no	—
30	#15667946	C₂³	1	4	yes	—
31	#15667943	—	2	8	no	—
32	C₈×Dic₈₉	D₄	1	2	yes	—
33	C₈×Dic₈₉	C₂³	1	2	yes	—
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	Q₈	1	4	yes	—
37	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
38	#107×C₈₉	—	2	2	no	—
39	#107×C₈₉	C₂³	1	4	yes	—
40	C₂²×Dic₈₉	—	4	2	no	—
41	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
42	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
43	C₂×#10623494	Q₁₆	1	4	yes	—
44	C₂×#10623494	C₂×D₄	1	2	yes	—
45	C₄⋊Dic₈₉	—	2	4	no	—
46	C₄⋊Dic₈₉	—	4	2	no	—
47	C₄⋊Dic₈₉	—	4	8	no	—
48	#13083553	—	4	8	no	—
49	C₄×Dic₈₉	D₈	1	2	yes	—
50	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
51	#13083549	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₈₉	—	4	2	no	—
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₈₉	—	2	4	no	—
59	C₄⋊C₄×C₈₉	—	4	2	no	—
60	C₁₆×C₈₉	C₂×Q₈	1	4	yes	—
61	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₈₉	—	4	4	no	—
63	C₂×C₄×C₈₉	—	8	4	no	—
64	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
67	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
68	C₂×Dic₈₉	—	2	2	no	—
69	C₂×Dic₈₉	—	2	2	no	—
70	C₂×Dic₈₉	—	4	4	no	—
71	C₂×Dic₈₉	—	8	4	no	—
72	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
73	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
74	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
75	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
76	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
77	#10623494	C₂×Q₁₆	1	4	yes	—
78	C₂²×C₈₉	#500	1	2	yes	—
79	C₂²×C₈₉	#508	1	2	yes	—
80	C₂²×C₈₉	#508	1	2	yes	—
81	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
82	C₄×C₈₉	—	8	4	no	—
83	C₄×C₈₉	#500	1	2	yes	—
84	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
85	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
86	Dic₈₉	—	2	2	no	—
87	Dic₈₉	—	2	2	no	—
88	Dic₈₉	—	8	4	no	—
89	C₂×#2253	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	#2253	1	4	yes	complemented
91	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
92	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
93	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
94	C₂×#499	—	89	2	no	—
95	#2253	—	89	8	no	—
96	C₂×#367	—	89	2	no	—
97	C₂×#366	—	89	1	no	—
98	C₂×#366	D₈₉	1	1	yes	characteristic
99	C₂×C₄×C₁₆	—	89	1	no	—
100	C₈₉	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
101	C₂×#125	—	178	2	no	—
102	C₂²×C₁₆	—	89	1	no	—
103	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
104	#499	—	89	8	no	—
105	#499	—	178	4	no	—
106	C₂×#107	—	89	2	no	—
107	C₂×#107	C₂×D₈₉	1	2	yes	—
108	C₂×#106	—	178	2	no	—
109	C₂×C₄×C₈	—	89	1	no	—
110	#367	—	89	8	no	—
111	#366	—	89	4	no	—
112	#366	C₂×D₈₉	1	4	yes	—
113	C₄×C₁₆	—	89	4	no	—
114	C₂²×C₈	—	89	1	no	—
115	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
116	#125	—	356	4	no	—
117	#125	—	178	8	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	356	2	no	—
120	C₂×C₄⋊C₄	—	356	2	no	—
121	C₂×C₄⋊C₄	—	178	2	no	—
122	C₂×C₄²	—	89	1	no	—
123	C₂×C₁₆	—	89	4	no	—
124	C₂×C₁₆	—	89	2	no	—
125	C₂×C₁₆	Dic₁₇₈	1	4	yes	—
126	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	89	4	no	—
129	#107	—	178	2	no	—
130	#107	C₂²×D₈₉	1	4	yes	—
131	#106	—	178	8	no	—
132	C₄×C₈	—	89	2	no	—
133	C₄×C₈	—	89	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	89	1	no	—
136	C₂²×C₄	—	89	1	no	—
137	C₂²×C₄	—	356	2	no	—
138	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
139	C₂×C₈	—	89	4	no	—
140	C₂×C₈	—	89	2	no	—
141	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
142	C₂×C₈	Q₈×D₈₉	1	2	yes	—
143	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
144	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	356	8	no	—
148	C₄⋊C₄	—	356	8	no	—
149	C₄⋊C₄	—	356	2	no	—
150	C₄⋊C₄	—	178	4	no	—
151	C₄²	—	89	2	no	—
152	C₄²	—	89	2	no	—
153	C₁₆	—	89	4	no	—
154	C₁₆	C₂×Dic₁₇₈	1	4	yes	—
155	C₂³	D₈×D₈₉	1	1	yes	characteristic
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	8	4	no	—
158	C₂×C₄	—	89	2	no	—
159	C₂×C₄	—	89	2	no	—
160	C₂×C₄	—	356	4	no	—
161	C₂×C₄	—	178	2	no	—
162	C₂×C₄	—	178	2	no	—
163	C₂×C₄	—	712	4	no	—
164	C₂×C₄	#15668011	1	2	yes	—
165	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
167	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
168	C₈	—	89	4	no	—
169	C₈	#15668011	1	2	yes	—
170	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
171	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
172	C₂²	#19152002	1	2	yes	—
173	C₂²	#19152027	1	2	yes	—
174	C₂²	#19152031	1	2	yes	—
175	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
176	C₄	—	8	4	no	—
177	C₄	—	178	2	no	—
178	C₄	—	178	2	no	—
179	C₄	—	712	4	no	—
180	C₄	#19152002	1	2	yes	—
181	C₄	C₂×#15668011	1	1	yes	characteristic
182	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
183	C₂	#26022529	1	4	yes	direct product factor
184	C₂	C₂×#19152002	1	1	yes	characteristic
185	C₂	C₂×#19152027	1	1	yes	characteristic
186	C₂	C₂×#19152031	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151999

C₂

yes

maximal

#26022529

C₂

yes

maximal, direct product factor

C₂×#19151425

C₂

yes

maximal

C₂×#19151424

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₈₉

C₂

yes

maximal, characteristic

C₂×#366×C₈₉

C₂

yes

maximal, characteristic

C₂×#15668003

—

C₂²×#13083549

C₂²

yes

characteristic

#19151999

—

#19151999

C₂²

yes

—

C₂×#15667946

C₂²

yes

—

C₂×#15667943

—

C₂×C₈×Dic₈₉

C₂²

yes

characteristic

#19151425

C₂²

yes

—

#19151424

C₂²

yes

—

C₁₆×Dic₈₉

C₂²

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#107×C₈₉

C₂²

yes

—

#366×C₈₉

C₂²

yes

—

C₂²×#10623494

D₄

yes

characteristic

#15668003

—

#15668003

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×#13083553

—

C₂×C₄×Dic₈₉

D₄

yes

characteristic

C₂×#13083549

Q₈

yes

—

C₂×#13083549

C₂³

yes

—

#15667946

—

#15667946

C₂³

yes

—

#15667943

—

C₈×Dic₈₉

D₄

yes

—

C₈×Dic₈₉

C₂³

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₁₆×C₈₉

Q₈

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

—

#107×C₈₉

—

#107×C₈₉

C₂³

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₈

yes

characteristic

C₂²×Dic₈₉

D₈

yes

characteristic

C₂×#10623494

Q₁₆

yes

—

C₂×#10623494

C₂×D₄

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

D₈

yes

—

C₄×Dic₈₉

C₂×D₄

yes

—

#13083549

C₂×Q₈

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

C₂×Q₈

yes

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

—

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₈

yes

—

C₂×Dic₈₉

C₂×D₈

yes

—

C₈×C₈₉

C₄⋊Q₈

yes

—

C₈×C₈₉

C₂²×Q₈

yes

characteristic

C₈×C₈₉

C₂²×Q₈

yes

characteristic

#10623494

C₂×Q₁₆

yes

—

C₂²×C₈₉

#500

yes

—

C₂²×C₈₉

#508

yes

—

C₂²×C₈₉

#508

yes

—

C₂²×C₈₉

C₂²×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#500

yes

—

C₄×C₈₉

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₈₉

C₂²×Q₁₆

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

—

—

D₈₉

yes

characteristic

C₂×C₄×C₁₆

—

100

C₈₉

C₂×#2253

yes

characteristic, complemented, Sylow

101

C₂×#125

—

178

—

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

104

#499

—

105

#499

—

178

—

106

C₂×#107

—

107

C₂×#107

C₂×D₈₉

yes

—

108

C₂×#106

—

178

—

109

C₂×C₄×C₈

—

110

#367

—

111

#366

—

112

#366

C₂×D₈₉

yes

—

113

C₄×C₁₆

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²×D₈₉

yes

characteristic

116

#125

—

356

—

117

#125

—

178

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

356

—

120

C₂×C₄⋊C₄

—

356

—

121

C₂×C₄⋊C₄

—

178

—

122

C₂×C₄²

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₂×C₁₆

Dic₁₇₈

yes

—

126

C₂×C₁₆

C₂²×D₈₉

yes

—

127

#107

—

128

#107

—

129

#107

—

178

—

130

#107

C₂²×D₈₉

yes

—

131

#106

—

178

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

356

—

138

C₂²×C₄

D₄×D₈₉

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

D₄×D₈₉

yes

characteristic

142

C₂×C₈

Q₈×D₈₉

yes

—

143

C₂×C₈

C₂×Dic₁₇₈

yes

—

144

C₂×C₈

C₂³×D₈₉

yes

characteristic

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

356

—

148

C₄⋊C₄

—

356

—

149

C₄⋊C₄

—

356

—

150

C₄⋊C₄

—

178

—

151

C₄²

—

152

C₄²

—

153

C₁₆

—

154

C₁₆

C₂×Dic₁₇₈

yes

—

155

C₂³

D₈×D₈₉

yes

characteristic

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

356

—

161

C₂×C₄

—

178

—

162

C₂×C₄

—

178

—

163

C₂×C₄

—

712

—

164

C₂×C₄

#15668011

yes

—

165

C₂×C₄

D₈×D₈₉

yes

characteristic

166

C₂×C₄

Q₁₆×D₈₉

yes

—

167

C₂×C₄

C₂×D₄×D₈₉

yes

characteristic

168

C₈

—

169

C₈

#15668011

yes

—

170

C₈

C₂×Q₈×D₈₉

yes

characteristic

171

C₈

C₂²×Dic₁₇₈

yes

characteristic

172

C₂²

#19152002

yes

—

173

C₂²

#19152027

yes

—

174

C₂²

#19152031

yes

—

175

C₂²

C₂×D₈×D₈₉

yes

characteristic

176

C₄

—

177

C₄

—

178

—

178

C₄

—

178

—

179

C₄

—

712

—

180

C₄

#19152002

yes

—

181

C₄

C₂×#15668011

yes

characteristic

182

C₄

C₂×Q₁₆×D₈₉

yes

characteristic

183

C₂

#26022529

yes

direct product factor

184

C₂

C₂×#19152002

yes

characteristic

185

C₂

C₂×#19152027

yes

characteristic

186

C₂

C₂×#19152031

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#366×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	C₂×#366	D₈₉
hypocenter	C₈₉	C₂×#2253
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂³

D₈×D₈₉

commutator, cosocle

C₂×C₈×C₈₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₈₉

Fitting

C₂×#366×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₈

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
2. #26022529	C₂	direct (N × H)
90. C₂×C₈₉	#2253	semidirect
100. C₈₉	C₂×#2253	semidirect
183. C₂	#26022529	direct (N × H)

H ≅ G/N

type

2. #26022529

C₂

direct (N × H)

90. C₂×C₈₉

#2253

semidirect

100. C₈₉

C₂×#2253

semidirect

183. C₂

#26022529

direct (N × H)

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

subgroup	index
C₂×#19151999	2
#26022529	2
C₂×#19151425	2
C₂×#19151424	2
C₂×C₁₆×Dic₈₉	2
C₂×#366×C₈₉	2
C₂×#2253	89

subgroup

index

C₂×C₁₆×Dic₈₉

C₂×#366×C₈₉

C₂×#2253

Subgroups — C₂×#26022529 #36709908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#26022529 #36709908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022529 #36709908