Group #31790565 — subgroups

Subgroups — C₂×#22577709 #31790565

Order 17536.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984079	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984053	C₂	1	1	yes	maximal, characteristic
3	#22577709	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17984019	C₂	1	1	yes	maximal, characteristic
5	C₂×#17984017	C₂	1	1	yes	maximal, characteristic
6	C₂×#17984013	C₂	1	1	yes	maximal, characteristic
7	C₂×#17984011	C₂	1	1	yes	maximal, characteristic
8	C₂×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₇₄	C₂²	1	1	yes	characteristic
10	C₂²×#11764040	C₂²	1	1	yes	characteristic
11	#17984079	—	2	2	no	—
12	#17984079	C₂²	1	4	yes	—
13	#17984053	—	2	2	no	—
14	#17984053	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₁₃₇	—	2	1	no	—
16	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#14555489	—	2	1	no	—
18	C₂×#14555489	—	2	1	no	—
19	C₂×C₄×Dic₁₃₇	—	2	1	no	—
20	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
21	#17984019	C₂²	1	4	yes	—
22	#17984017	C₂²	1	4	yes	—
23	#17984013	C₂²	1	4	yes	—
24	#17984011	C₂²	1	4	yes	—
25	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
28	#107×C₁₃₇	C₂²	1	4	yes	—
29	C₂×Dic₂₇₄	—	2	1	no	—
30	C₂×Dic₂₇₄	—	2	2	no	—
31	C₂×Dic₂₇₄	Q₈	1	4	yes	—
32	C₂×Dic₂₇₄	C₂³	1	2	yes	—
33	C₂²×Dic₁₃₇	—	2	1	no	—
34	C₂²×Dic₁₃₇	—	2	1	no	—
35	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
36	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
37	C₂×#11764040	—	2	2	no	—
38	C₂×#11764040	C₂³	1	2	yes	—
39	C₄⋊Dic₁₃₇	—	2	4	no	—
40	C₄⋊Dic₁₃₇	—	2	1	no	—
41	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
42	#14555489	—	2	4	no	—
43	#14555489	—	2	4	no	—
44	C₄×Dic₁₃₇	—	2	4	no	—
45	C₄×Dic₁₃₇	D₄	1	2	yes	—
46	C₄×Dic₁₃₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃₇	—	2	1	no	—
48	C₂²×C₄×C₁₃₇	—	2	1	no	—
49	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₃₇	—	2	2	no	—
51	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
52	C₄⋊C₄×C₁₃₇	—	2	1	no	—
53	C₄⋊C₄×C₁₃₇	—	2	1	no	—
54	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
55	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
56	Dic₂₇₄	—	2	4	no	—
57	Dic₂₇₄	—	2	2	no	—
58	Dic₂₇₄	C₂×Q₈	1	4	yes	—
59	C₂×Dic₁₃₇	—	2	4	no	—
60	C₂×Dic₁₃₇	—	2	2	no	—
61	C₂×Dic₁₃₇	—	2	2	no	—
62	C₂×Dic₁₃₇	—	2	2	no	—
63	C₂×Dic₁₃₇	—	2	2	no	—
64	C₂×Dic₁₃₇	—	4	2	no	—
65	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
66	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
67	#11764040	—	2	2	no	—
68	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃₇	—	2	2	no	—
70	C₂×C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₄×C₁₃₇	—	4	2	no	—
72	C₂×C₄×C₁₃₇	—	4	2	no	—
73	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₁₃₇	#41	1	2	yes	—
76	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
77	C₈×C₁₃₇	—	2	2	no	—
78	C₂²×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
79	C₂²×C₁₃₇	#135	1	2	yes	—
80	C₂²×C₁₃₇	#137	1	2	yes	—
81	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₁₃₇	—	4	2	no	—
83	C₄×C₁₃₇	—	4	2	no	—
84	C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
85	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
87	Dic₁₃₇	—	2	4	no	—
88	Dic₁₃₇	—	2	2	no	—
89	Dic₁₃₇	—	2	2	no	—
90	Dic₁₃₇	—	4	2	no	—
91	C₂×C₁₃₇	#479	1	4	yes	complemented
92	C₂×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₁₃₇	C₂×#135	1	1	yes	characteristic
94	C₂×C₁₃₇	C₂×#137	1	1	yes	characteristic
95	C₁₃₇	C₂×#479	1	1	yes	characteristic, complemented, Sylow
96	C₂×#479	—	137	1	no	maximal, Sylow
97	C₂×#125	—	137	1	no	—
98	C₂×C₄×Q₈	—	137	1	no	—
99	#479	—	137	8	no	—
100	C₂×#107	D₁₃₇	1	1	yes	characteristic
101	C₂×#106	—	137	1	no	—
102	C₂×C₄⋊C₈	—	137	1	no	—
103	C₂×#103	—	137	1	no	—
104	C₂×#103	—	137	1	no	—
105	C₂²×Q₈	—	137	1	no	—
106	C₂²×C₈	—	137	1	no	—
107	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
108	#125	—	274	2	no	—
109	#125	—	137	4	no	—
110	C₄×Q₈	—	274	2	no	—
111	C₄×Q₈	—	137	4	no	—
112	C₂×C₄⋊C₄	—	274	1	no	—
113	C₂×C₄⋊C₄	—	274	1	no	—
114	C₂×C₄⋊C₄	—	274	1	no	—
115	C₂×C₄⋊C₄	—	137	1	no	—
116	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
118	C₂×C₄²	—	274	1	no	—
119	C₂×C₄²	—	137	1	no	—
120	#107	C₂×D₁₃₇	1	4	yes	—
121	#106	—	137	4	no	—
122	C₄⋊C₈	—	137	4	no	—
123	#103	—	137	4	no	—
124	#103	—	137	4	no	—
125	C₂×Q₈	—	274	1	no	—
126	C₂×Q₈	—	274	2	no	—
127	C₂×Q₈	—	137	4	no	—
128	C₂×Q₈	—	137	2	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	274	1	no	—
132	C₂²×C₄	—	274	1	no	—
133	C₂²×C₄	—	137	1	no	—
134	C₂²×C₄	—	137	1	no	—
135	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	274	2	no	—
138	C₂×C₈	—	137	2	no	—
139	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	274	4	no	—
143	C₄⋊C₄	—	274	4	no	—
144	C₄⋊C₄	—	274	4	no	—
145	C₄⋊C₄	—	274	1	no	—
146	C₄⋊C₄	—	137	2	no	—
147	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
148	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
149	C₄²	—	274	4	no	—
150	C₄²	—	137	2	no	—
151	C₄²	—	137	2	no	—
152	C₂³	D₄×D₁₃₇	1	1	yes	characteristic
153	Q₈	—	274	4	no	—
154	Q₈	—	274	2	no	—
155	Q₈	—	137	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	274	4	no	—
161	C₂×C₄	—	274	2	no	—
162	C₂×C₄	—	274	2	no	—
163	C₂×C₄	—	274	2	no	—
164	C₂×C₄	—	274	2	no	—
165	C₂×C₄	—	548	2	no	—
166	C₂×C₄	—	137	2	no	—
167	C₂×C₄	—	137	2	no	—
168	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
169	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
170	C₂×C₄	#14555510	1	2	yes	—
171	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	274	2	no	—
174	C₂²	#17984066	1	2	yes	—
175	C₂²	#17984107	1	2	yes	—
176	C₂²	#17984120	1	2	yes	—
177	C₂²	C₂×D₄×D₁₃₇	1	1	yes	characteristic
178	C₄	—	4	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	274	2	no	—
181	C₄	—	274	2	no	—
182	C₄	—	274	4	no	—
183	C₄	—	548	2	no	—
184	C₄	#17984066	1	2	yes	—
185	C₄	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
186	C₄	C₂×#14555510	1	1	yes	characteristic
187	C₂	#22577709	1	4	yes	direct product factor
188	C₂	C₂×#17984066	1	1	yes	characteristic
189	C₂	C₂×#17984107	1	1	yes	characteristic
190	C₂	C₂×#17984120	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984079

C₂

yes

maximal, characteristic

C₂×#17984053

C₂

yes

maximal, characteristic

#22577709

C₂

yes

maximal, direct product factor

C₂×#17984019

C₂

yes

maximal, characteristic

C₂×#17984017

C₂

yes

maximal, characteristic

C₂×#17984013

C₂

yes

maximal, characteristic

C₂×#17984011

C₂

yes

maximal, characteristic

C₂×#107×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₄

C₂²

yes

characteristic

C₂²×#11764040

C₂²

yes

characteristic

#17984079

—

#17984079

C₂²

yes

—

#17984053

—

#17984053

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

characteristic

C₂×#14555489

—

C₂×#14555489

—

C₂×C₄×Dic₁₃₇

—

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17984019

C₂²

yes

—

#17984017

C₂²

yes

—

#17984013

C₂²

yes

—

#17984011

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₇

C₂²

yes

characteristic

#107×C₁₃₇

C₂²

yes

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

Q₈

yes

—

C₂×Dic₂₇₄

C₂³

yes

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

D₄

yes

characteristic

C₂²×Dic₁₃₇

D₄

yes

characteristic

C₂×#11764040

—

C₂×#11764040

C₂³

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂³

yes

—

#14555489

—

#14555489

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

D₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

C₂³

yes

—

C₄⋊C₄×C₁₃₇

C₂³

yes

—

Dic₂₇₄

—

Dic₂₇₄

—

Dic₂₇₄

C₂×Q₈

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂×D₄

yes

—

C₂×Dic₁₃₇

C₂×D₄

yes

—

#11764040

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂²⋊Q₈

yes

—

C₂²×C₁₃₇

#135

yes

—

C₂²×C₁₃₇

#137

yes

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂²⋊Q₈

yes

—

C₄×C₁₃₇

C₂²×Q₈

yes

characteristic

C₄×C₁₃₇

C₂×#41

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

#479

yes

complemented

C₂×C₁₃₇

C₂×C₂²⋊Q₈

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

C₂×#479

—

137

maximal, Sylow

C₂×#125

—

137

—

C₂×C₄×Q₈

—

137

—

#479

—

137

—

100

C₂×#107

D₁₃₇

yes

characteristic

101

C₂×#106

—

137

—

102

C₂×C₄⋊C₈

—

137

—

103

C₂×#103

—

137

—

104

C₂×#103

—

137

—

105

C₂²×Q₈

—

137

—

106

C₂²×C₈

—

137

—

107

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

108

#125

—

274

—

109

#125

—

137

—

110

C₄×Q₈

—

274

—

111

C₄×Q₈

—

137

—

112

C₂×C₄⋊C₄

—

274

—

113

C₂×C₄⋊C₄

—

274

—

114

C₂×C₄⋊C₄

—

274

—

115

C₂×C₄⋊C₄

—

137

—

116

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

characteristic

117

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

characteristic

118

C₂×C₄²

—

274

—

119

C₂×C₄²

—

137

—

120

#107

C₂×D₁₃₇

yes

—

121

#106

—

137

—

122

C₄⋊C₈

—

137

—

123

#103

—

137

—

124

#103

—

137

—

125

C₂×Q₈

—

274

—

126

C₂×Q₈

—

274

—

127

C₂×Q₈

—

137

—

128

C₂×Q₈

—

137

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

274

—

132

C₂²×C₄

—

274

—

133

C₂²×C₄

—

137

—

134

C₂²×C₄

—

137

—

135

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

136

C₂×C₈

—

137

C₂×C₈

—

274

—

138

C₂×C₈

—

137

—

139

C₂×C₈

C₂²×D₁₃₇

yes

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

274

—

143

C₄⋊C₄

—

274

—

144

C₄⋊C₄

—

274

—

145

C₄⋊C₄

—

274

—

146

C₄⋊C₄

—

137

—

147

C₄⋊C₄

C₂²×D₁₃₇

yes

—

148

C₄⋊C₄

C₂²×D₁₃₇

yes

—

149

C₄²

—

274

—

150

C₄²

—

137

—

151

C₄²

—

137

—

152

C₂³

D₄×D₁₃₇

yes

characteristic

153

Q₈

—

274

—

154

Q₈

—

274

—

155

Q₈

—

137

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

274

—

161

C₂×C₄

—

274

—

162

C₂×C₄

—

274

—

163

C₂×C₄

—

274

—

164

C₂×C₄

—

274

—

165

C₂×C₄

—

548

—

166

C₂×C₄

—

137

—

167

C₂×C₄

—

137

—

168

C₂×C₄

D₄×D₁₃₇

yes

characteristic

169

C₂×C₄

Q₈×D₁₃₇

yes

—

170

C₂×C₄

#14555510

yes

—

171

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

172

C₈

—

173

C₈

—

274

—

174

C₂²

#17984066

yes

—

175

C₂²

#17984107

yes

—

176

C₂²

#17984120

yes

—

177

C₂²

C₂×D₄×D₁₃₇

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

274

—

181

C₄

—

274

—

182

C₄

—

274

—

183

C₄

—

548

—

184

C₄

#17984066

yes

—

185

C₄

C₂×Q₈×D₁₃₇

yes

characteristic

186

C₄

C₂×#14555510

yes

characteristic

187

C₂

#22577709

yes

direct product factor

188

C₂

C₂×#17984066

yes

characteristic

189

C₂

C₂×#17984107

yes

characteristic

190

C₂

C₂×#17984120

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₃₇
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×#107×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
hypercenter	C₂×#107	D₁₃₇
hypocenter	C₁₃₇	C₂×#479
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₃₇

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

C₂×#107×C₁₃₇

C₂

socle, prime radical

C₂³×C₁₃₇

C₂×D₄

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #22577709	C₂	direct (N × H)
91. C₂×C₁₃₇	#479	semidirect
95. C₁₃₇	C₂×#479	semidirect
187. C₂	#22577709	direct (N × H)

H ≅ G/N

type

3. #22577709

C₂

direct (N × H)

91. C₂×C₁₃₇

#479

semidirect

95. C₁₃₇

C₂×#479

semidirect

187. C₂

#22577709

direct (N × H)

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

subgroup	index
C₂×#17984079	2
C₂×#17984053	2
#22577709	2
C₂×#17984019	2
C₂×#17984017	2
C₂×#17984013	2
C₂×#17984011	2
C₂×#107×C₁₃₇	2
C₂×#479	137

subgroup

index

C₂×#107×C₁₃₇

137

Subgroups — C₂×#22577709 #31790565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22577709 #31790565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577709 #31790565