Group #31790559 — subgroups

Subgroups — C₂×#22577697 #31790559

Order 17536.

416 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 157 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×#17984087	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984053	C₂	1	1	yes	maximal, characteristic
3	#22577697	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17984021	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	—	2	1	no	—
10	C₂²×Dic₂₇₄	C₂²	1	1	yes	characteristic
11	C₂²×#11764040	C₂²	1	1	yes	characteristic
12	#17984087	—	2	2	no	—
13	#17984087	C₂²	1	4	yes	—
14	#17984053	—	2	2	no	—
15	#17984053	C₂²	1	4	yes	—
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	#17984021	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	4	no	—
30	C₂×Dic₂₇₄	—	2	1	no	—
31	C₂×Dic₂₇₄	—	2	2	no	—
32	C₂×Dic₂₇₄	—	4	4	no	—
33	C₂×Dic₂₇₄	Q₈	1	4	yes	complemented
34	C₂×Dic₂₇₄	C₂³	1	2	yes	—
35	C₂²×Dic₁₃₇	—	2	1	no	—
36	C₂²×Dic₁₃₇	—	2	1	no	—
37	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
38	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
39	C₂×#11764040	—	2	2	no	—
40	C₂×#11764040	C₂³	1	2	yes	—
41	C₄⋊Dic₁₃₇	—	2	1	no	—
42	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
43	#14555489	—	2	4	no	—
44	#14555489	—	2	4	no	—
45	C₄×Dic₁₃₇	—	2	4	no	—
46	C₄×Dic₁₃₇	D₄	1	2	yes	—
47	C₄×Dic₁₃₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃₇	—	2	1	no	—
49	C₂²×C₄×C₁₃₇	—	2	1	no	—
50	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₃₇	—	2	2	no	—
52	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
53	C₄⋊C₄×C₁₃₇	—	2	1	no	—
54	C₄⋊C₄×C₁₃₇	—	2	1	no	—
55	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
56	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
57	Dic₂₇₄	—	2	4	no	—
58	Dic₂₇₄	—	2	2	no	—
59	Dic₂₇₄	—	4	8	no	—
60	Dic₂₇₄	C₂×Q₈	1	4	yes	complemented
61	C₂×Dic₁₃₇	—	2	4	no	—
62	C₂×Dic₁₃₇	—	2	2	no	—
63	C₂×Dic₁₃₇	—	2	2	no	—
64	C₂×Dic₁₃₇	—	2	2	no	—
65	C₂×Dic₁₃₇	—	4	2	no	—
66	C₂×Dic₁₃₇	Q₁₆	1	4	yes	—
67	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
68	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
69	#11764040	—	2	2	no	—
70	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃₇	—	2	2	no	—
72	C₂×C₄×C₁₃₇	—	2	2	no	—
73	C₂×C₄×C₁₃₇	—	4	2	no	—
74	C₂×C₄×C₁₃₇	—	4	2	no	—
75	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₁₃₇	#41	1	2	yes	—
78	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
79	C₈×C₁₃₇	—	2	2	no	—
80	C₂²×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
81	C₂²×C₁₃₇	C₂×Q₁₆	1	2	yes	—
82	C₂²×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
83	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₁₃₇	—	4	2	no	—
85	C₄×C₁₃₇	—	4	2	no	—
86	C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
87	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
88	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
89	Dic₁₃₇	—	2	4	no	—
90	Dic₁₃₇	—	2	2	no	—
91	Dic₁₃₇	—	4	2	no	—
92	Dic₁₃₇	C₂×Q₁₆	1	4	yes	—
93	C₂×C₁₃₇	#477	1	4	yes	complemented
94	C₂×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₁₃₇	C₂²×Q₁₆	1	1	yes	characteristic
96	C₂×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₁₃₇	C₂×#477	1	1	yes	characteristic, complemented, Sylow
98	C₂×#477	—	137	1	no	maximal, Sylow
99	C₂×C₄⋊Q₈	—	137	1	no	—
100	C₂×C₄×Q₈	—	137	1	no	—
101	#477	—	137	8	no	—
102	C₂×#107	—	137	1	no	—
103	C₂×#107	D₁₃₇	1	1	yes	characteristic
104	C₂×C₄⋊C₈	—	137	1	no	—
105	C₂×#103	—	137	1	no	—
106	C₂×#103	—	137	1	no	—
107	C₂²×Q₈	—	274	1	no	—
108	C₂²×Q₈	—	137	1	no	—
109	C₂²×C₈	—	137	1	no	—
110	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
111	C₄⋊Q₈	—	274	2	no	—
112	C₄⋊Q₈	—	137	4	no	—
113	C₄×Q₈	—	274	2	no	—
114	C₄×Q₈	—	137	4	no	—
115	C₂×C₄⋊C₄	—	274	1	no	—
116	C₂×C₄⋊C₄	—	274	1	no	—
117	C₂×C₄⋊C₄	—	137	1	no	—
118	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
120	C₂×C₄²	—	274	1	no	—
121	C₂×C₄²	—	137	1	no	—
122	#107	—	137	4	no	—
123	#107	C₂×D₁₃₇	1	4	yes	—
124	C₄⋊C₈	—	137	4	no	—
125	#103	—	137	4	no	—
126	#103	—	137	4	no	—
127	C₂×Q₈	—	274	4	no	—
128	C₂×Q₈	—	274	1	no	—
129	C₂×Q₈	—	274	2	no	—
130	C₂×Q₈	—	548	4	no	—
131	C₂×Q₈	—	137	4	no	—
132	C₂×Q₈	—	137	2	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	274	1	no	—
136	C₂²×C₄	—	274	1	no	—
137	C₂²×C₄	—	137	1	no	—
138	C₂²×C₄	—	137	1	no	—
139	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	274	2	no	—
142	C₂×C₈	—	137	2	no	—
143	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	274	4	no	—
147	C₄⋊C₄	—	274	4	no	—
148	C₄⋊C₄	—	274	1	no	—
149	C₄⋊C₄	—	137	2	no	—
150	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
151	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
152	C₄²	—	274	4	no	—
153	C₄²	—	137	2	no	—
154	C₄²	—	137	2	no	—
155	C₂³	D₄×D₁₃₇	1	1	yes	characteristic
156	Q₈	—	274	4	no	—
157	Q₈	—	274	2	no	—
158	Q₈	—	548	8	no	—
159	Q₈	—	137	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	274	4	no	—
165	C₂×C₄	—	274	2	no	—
166	C₂×C₄	—	274	2	no	—
167	C₂×C₄	—	274	2	no	—
168	C₂×C₄	—	548	2	no	—
169	C₂×C₄	—	137	2	no	—
170	C₂×C₄	—	137	4	no	—
171	C₂×C₄	—	137	2	no	—
172	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
173	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
174	C₂×C₄	#14555510	1	2	yes	—
175	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	274	2	no	—
178	C₂²	#17984066	1	2	yes	—
179	C₂²	Q₁₆×D₁₃₇	1	2	yes	—
180	C₂²	#17984112	1	2	yes	—
181	C₂²	C₂×D₄×D₁₃₇	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	274	2	no	—
185	C₄	—	274	4	no	—
186	C₄	—	548	2	no	—
187	C₄	—	137	4	no	—
188	C₄	#17984066	1	2	yes	—
189	C₄	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
190	C₄	C₂×#14555510	1	1	yes	characteristic
191	C₂	#22577697	1	4	yes	direct product factor
192	C₂	C₂×#17984066	1	1	yes	characteristic
193	C₂	C₂×Q₁₆×D₁₃₇	1	1	yes	characteristic
194	C₂	C₂×#17984112	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984087

C₂

yes

maximal, characteristic

C₂×#17984053

C₂

yes

maximal, characteristic

#22577697

C₂

yes

maximal, direct product factor

C₂×#17984021

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×Dic₂₇₄

C₂²

yes

characteristic

C₂²×#11764040

C₂²

yes

characteristic

#17984087

—

#17984087

C₂²

yes

—

#17984053

—

#17984053

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

characteristic

C₂×#14555489

—

C₂×#14555489

—

C₂×C₄×Dic₁₃₇

—

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17984021

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

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

Q₈

yes

complemented

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

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

—

Dic₂₇₄

C₂×Q₈

yes

complemented

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

Q₁₆

yes

—

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

C₂×Q₁₆

yes

—

C₂²×C₁₃₇

AGL(1,ℤ/8)

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×Q₁₆

yes

—

C₂×C₁₃₇

#477

yes

complemented

C₂×C₁₃₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₁₃₇

C₂²×Q₁₆

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

C₂×#477

—

137

maximal, Sylow

C₂×C₄⋊Q₈

—

137

—

100

C₂×C₄×Q₈

—

137

—

101

#477

—

137

—

102

C₂×#107

—

137

—

103

C₂×#107

D₁₃₇

yes

characteristic

104

C₂×C₄⋊C₈

—

137

—

105

C₂×#103

—

137

—

106

C₂×#103

—

137

—

107

C₂²×Q₈

—

274

—

108

C₂²×Q₈

—

137

—

109

C₂²×C₈

—

137

—

110

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

111

C₄⋊Q₈

—

274

—

112

C₄⋊Q₈

—

137

—

113

C₄×Q₈

—

274

—

114

C₄×Q₈

—

137

—

115

C₂×C₄⋊C₄

—

274

—

116

C₂×C₄⋊C₄

—

274

—

117

C₂×C₄⋊C₄

—

137

—

118

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

characteristic

120

C₂×C₄²

—

274

—

121

C₂×C₄²

—

137

—

122

#107

—

137

—

123

#107

C₂×D₁₃₇

yes

—

124

C₄⋊C₈

—

137

—

125

#103

—

137

—

126

#103

—

137

—

127

C₂×Q₈

—

274

—

128

C₂×Q₈

—

274

—

129

C₂×Q₈

—

274

—

130

C₂×Q₈

—

548

—

131

C₂×Q₈

—

137

—

132

C₂×Q₈

—

137

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

274

—

136

C₂²×C₄

—

274

—

137

C₂²×C₄

—

137

—

138

C₂²×C₄

—

137

—

139

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

274

—

142

C₂×C₈

—

137

—

143

C₂×C₈

C₂²×D₁₃₇

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

274

—

147

C₄⋊C₄

—

274

—

148

C₄⋊C₄

—

274

—

149

C₄⋊C₄

—

137

—

150

C₄⋊C₄

C₂²×D₁₃₇

yes

—

151

C₄⋊C₄

C₂²×D₁₃₇

yes

—

152

C₄²

—

274

—

153

C₄²

—

137

—

154

C₄²

—

137

—

155

C₂³

D₄×D₁₃₇

yes

characteristic

156

Q₈

—

274

—

157

Q₈

—

274

—

158

Q₈

—

548

—

159

Q₈

—

137

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

274

—

165

C₂×C₄

—

274

—

166

C₂×C₄

—

274

—

167

C₂×C₄

—

274

—

168

C₂×C₄

—

548

—

169

C₂×C₄

—

137

—

170

C₂×C₄

—

137

—

171

C₂×C₄

—

137

—

172

C₂×C₄

D₄×D₁₃₇

yes

characteristic

173

C₂×C₄

Q₈×D₁₃₇

yes

—

174

C₂×C₄

#14555510

yes

—

175

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

176

C₈

—

177

C₈

—

274

—

178

C₂²

#17984066

yes

—

179

C₂²

Q₁₆×D₁₃₇

yes

—

180

C₂²

#17984112

yes

—

181

C₂²

C₂×D₄×D₁₃₇

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

274

—

185

C₄

—

274

—

186

C₄

—

548

—

187

C₄

—

137

—

188

C₄

#17984066

yes

—

189

C₄

C₂×Q₈×D₁₃₇

yes

characteristic

190

C₄

C₂×#14555510

yes

characteristic

191

C₂

#22577697

yes

direct product factor

192

C₂

C₂×#17984066

yes

characteristic

193

C₂

C₂×Q₁₆×D₁₃₇

yes

characteristic

194

C₂

C₂×#17984112

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×#477
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. #22577697	C₂	direct (N × H)
33. C₂×Dic₂₇₄	Q₈	semidirect
60. Dic₂₇₄	C₂×Q₈	semidirect
93. C₂×C₁₃₇	#477	semidirect
97. C₁₃₇	C₂×#477	semidirect
191. C₂	#22577697	direct (N × H)

H ≅ G/N

type

3. #22577697

C₂

direct (N × H)

33. C₂×Dic₂₇₄

Q₈

semidirect

60. Dic₂₇₄

C₂×Q₈

semidirect

93. C₂×C₁₃₇

#477

semidirect

97. C₁₃₇

C₂×#477

semidirect

191. C₂

#22577697

direct (N × H)

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

subgroup	index
C₂×#17984087	2
C₂×#17984053	2
#22577697	2
C₂×#17984021	2
C₂×#17984017	2
C₂×#17984013	2
C₂×#17984011	2
C₂×#107×C₁₃₇	2
C₂×#477	137

subgroup

index

C₂×#107×C₁₃₇

137

Subgroups — C₂×#22577697 #31790559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22577697 #31790559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577697 #31790559