Group #46862827 — subgroups

Subgroups — C₁₃₇×#13441 #46862827

Order 35072.

378 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₂²×Q₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2151×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#362×C₁₃₇	C₂	1	2	yes	maximal
4	#1979×C₁₃₇	C₂	1	2	yes	maximal
5	#1381×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
7	C₂×Q₃₂×C₁₃₇	—	2	4	no	—
8	C₂×Q₃₂×C₁₃₇	—	2	2	no	—
9	C₂×M₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
12	#419×C₁₃₇	—	2	2	no	—
13	C₂×#101×C₁₃₇	—	2	2	no	—
14	#405×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₁₃₇	—	2	2	no	—
16	#362×C₁₃₇	—	2	4	no	—
17	C₂²×Q₈×C₁₃₇	—	2	2	no	—
18	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
19	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
20	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
21	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
22	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₃₇	D₄	1	4	yes	—
24	C₂×M₁₆×C₁₃₇	—	2	2	no	—
25	C₂×M₁₆×C₁₃₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
27	#117×C₁₃₇	D₄	1	1	yes	characteristic
28	Q₃₂×C₁₃₇	—	2	8	no	—
29	M₃₂×C₁₃₇	—	2	2	no	—
30	C₂×C₁₆×C₁₃₇	—	2	2	no	—
31	C₂×C₁₆×C₁₃₇	—	2	1	no	—
32	C₂×C₁₆×C₁₃₇	D₄	1	2	yes	—
33	#108×C₁₃₇	—	2	2	no	—
34	#108×C₁₃₇	—	2	2	no	—
35	#103×C₁₃₇	—	4	4	no	—
36	#101×C₁₃₇	—	4	4	no	—
37	#97×C₁₃₇	—	2	1	no	—
38	C₄×C₈×C₁₃₇	—	2	1	no	—
39	C₂×Q₈×C₁₃₇	—	2	4	no	—
40	C₂×Q₈×C₁₃₇	—	4	2	no	—
41	C₂×Q₈×C₁₃₇	—	4	2	no	—
42	C₂×Q₈×C₁₃₇	—	4	2	no	—
43	C₂×Q₈×C₁₃₇	—	4	2	no	—
44	C₂×Q₈×C₁₃₇	—	4	2	no	—
45	C₂²×C₄×C₁₃₇	—	4	2	no	—
46	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₁₃₇	—	2	4	no	—
48	Q₁₆×C₁₃₇	—	2	4	no	—
49	Q₁₆×C₁₃₇	—	4	2	no	—
50	Q₁₆×C₁₃₇	—	4	2	no	—
51	M₁₆×C₁₃₇	—	2	2	no	—
52	M₁₆×C₁₃₇	—	4	2	no	—
53	M₁₆×C₁₃₇	—	4	2	no	—
54	C₂×C₈×C₁₃₇	—	2	1	no	—
55	C₂×C₈×C₁₃₇	—	4	2	no	—
56	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
59	C₂×C₈×C₁₃₇	#41	1	2	yes	—
60	C₄⋊C₄×C₁₃₇	D₈	1	2	yes	—
61	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
62	C₄²×C₁₃₇	—	2	1	no	—
63	C₁₆×C₁₃₇	—	2	2	no	—
64	C₁₆×C₁₃₇	—	4	1	no	—
65	C₂³×C₁₃₇	#135	1	1	yes	characteristic
66	Q₈×C₁₃₇	—	4	4	no	—
67	Q₈×C₁₃₇	—	4	4	no	—
68	Q₈×C₁₃₇	—	8	2	no	—
69	Q₈×C₁₃₇	—	8	2	no	—
70	C₂×C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₄×C₁₃₇	—	4	4	no	—
72	C₂×C₄×C₁₃₇	—	8	2	no	—
73	C₂×C₄×C₁₃₇	—	8	2	no	—
74	C₂×C₄×C₁₃₇	#121	1	2	yes	—
75	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
78	C₈×C₁₃₇	—	4	1	no	—
79	C₈×C₁₃₇	—	8	2	no	—
80	C₈×C₁₃₇	#120	1	1	yes	characteristic
81	C₈×C₁₃₇	#123	1	2	yes	—
82	C₈×C₁₃₇	#124	1	1	yes	characteristic
83	C₂²×C₁₃₇	—	2	2	no	—
84	C₂²×C₁₃₇	#463	1	2	yes	—
85	C₂²×C₁₃₇	#466	1	1	yes	characteristic
86	C₄×C₁₃₇	—	4	2	no	—
87	C₄×C₁₃₇	—	8	4	no	—
88	C₄×C₁₃₇	#394	1	1	yes	characteristic
89	C₄×C₁₃₇	#459	1	2	yes	—
90	C₄×C₁₃₇	#469	1	1	yes	characteristic
91	C₂×C₁₃₇	—	2	2	no	—
92	C₂×C₁₃₇	#2054	1	1	yes	characteristic
93	C₂×C₁₃₇	#2222	1	2	yes	—
94	#13441	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₁₃₇	#13441	1	1	yes	characteristic, direct product factor, Sylow
96	C₂²×Q₃₂	C₂×C₁₃₇	1	1	yes	characteristic
97	#2151	C₂×C₁₃₇	1	1	yes	characteristic
98	C₂×#362	C₂×C₁₃₇	1	2	yes	—
99	#1979	C₂×C₁₃₇	1	2	yes	—
100	#1381	C₂×C₁₃₇	1	1	yes	characteristic
101	C₂²×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
102	C₂×Q₃₂	—	2	4	no	—
103	C₂×Q₃₂	—	2	2	no	—
104	C₂×M₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
105	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
106	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
107	#419	—	2	2	no	—
108	C₂×#101	—	2	2	no	—
109	#405	C₂²×C₁₃₇	1	1	yes	characteristic
110	C₄⋊C₁₆	—	2	2	no	—
111	#362	—	2	4	no	—
112	C₂²×Q₈	—	2	2	no	—
113	C₂×Q₁₆	—	2	2	no	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	D₄×C₁₃₇	1	4	yes	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	D₄×C₁₃₇	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
122	#117	D₄×C₁₃₇	1	1	yes	characteristic
123	Q₃₂	—	2	8	no	—
124	M₃₂	—	2	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	D₄×C₁₃₇	1	2	yes	—
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#103	—	4	4	no	—
131	#101	—	4	4	no	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
142	Q₁₆	—	2	4	no	—
143	Q₁₆	—	2	4	no	—
144	Q₁₆	—	4	2	no	—
145	Q₁₆	—	4	2	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	4	2	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
152	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
153	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
154	C₂×C₈	#41×C₁₃₇	1	2	yes	—
155	C₄⋊C₄	D₈×C₁₃₇	1	2	yes	—
156	C₂²⋊C₄	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	4	1	no	—
160	C₂³	#135×C₁₃₇	1	1	yes	characteristic
161	Q₈	—	4	4	no	—
162	Q₈	—	4	4	no	—
163	Q₈	—	8	2	no	—
164	Q₈	—	8	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	#121×C₁₃₇	1	2	yes	—
170	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
171	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
172	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
173	C₈	—	4	1	no	—
174	C₈	—	8	2	no	—
175	C₈	#120×C₁₃₇	1	1	yes	characteristic
176	C₈	#123×C₁₃₇	1	2	yes	—
177	C₈	#124×C₁₃₇	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#463×C₁₃₇	1	2	yes	—
180	C₂²	#466×C₁₃₇	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	—	8	4	no	—
183	C₄	#394×C₁₃₇	1	1	yes	characteristic
184	C₄	#459×C₁₃₇	1	2	yes	—
185	C₄	#469×C₁₃₇	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	#2054×C₁₃₇	1	1	yes	characteristic
188	C₂	#2222×C₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂×C₁₃₇

C₂

yes

maximal, characteristic

#2151×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#362×C₁₃₇

C₂

yes

maximal

#1979×C₁₃₇

C₂

yes

maximal

#1381×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₃₇

C₂²

yes

—

C₂×Q₃₂×C₁₃₇

—

C₂×Q₃₂×C₁₃₇

—

C₂×M₃₂×C₁₃₇

C₂²

yes

characteristic

C₂²×C₁₆×C₁₃₇

C₂²

yes

characteristic

C₂×#108×C₁₃₇

C₂²

yes

—

#419×C₁₃₇

—

C₂×#101×C₁₃₇

—

#405×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₃₇

—

#362×C₁₃₇

—

C₂²×Q₈×C₁₃₇

—

C₂×Q₁₆×C₁₃₇

—

C₂×Q₁₆×C₁₃₇

—

C₂×Q₁₆×C₁₃₇

—

C₂×Q₁₆×C₁₃₇

—

C₂×Q₁₆×C₁₃₇

—

C₂×Q₁₆×C₁₃₇

D₄

yes

—

C₂×M₁₆×C₁₃₇

—

C₂×M₁₆×C₁₃₇

D₄

yes

characteristic

C₂²×C₈×C₁₃₇

C₂³

yes

characteristic

#117×C₁₃₇

D₄

yes

characteristic

Q₃₂×C₁₃₇

—

M₃₂×C₁₃₇

—

C₂×C₁₆×C₁₃₇

—

C₂×C₁₆×C₁₃₇

—

C₂×C₁₆×C₁₃₇

D₄

yes

—

#108×C₁₃₇

—

#108×C₁₃₇

—

#103×C₁₃₇

—

#101×C₁₃₇

—

#97×C₁₃₇

—

C₄×C₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂×D₄

yes

characteristic

Q₁₆×C₁₃₇

—

Q₁₆×C₁₃₇

—

Q₁₆×C₁₃₇

—

Q₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×D₄

yes

—

C₂×C₈×C₁₃₇

C₂×D₄

yes

C₂×C₈×C₁₃₇

yes

C₂×C₈×C₁₃₇

yes

—

C₄⋊C₄×C₁₃₇

D₈

yes

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₁₆×C₁₃₇

—

C₁₆×C₁₃₇

—

C₂³×C₁₃₇

#135

yes

characteristic

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

#121

yes

—

C₂×C₄×C₁₃₇

#121

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃₇

AGL(1,ℤ/8)

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

#120

yes

characteristic

C₈×C₁₃₇

#123

yes

—

C₈×C₁₃₇

#124

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#463

yes

—

C₂²×C₁₃₇

#466

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#394

yes

characteristic

C₄×C₁₃₇

#459

yes

—

C₄×C₁₃₇

#469

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

#2054

yes

characteristic

C₂×C₁₃₇

#2222

yes

—

#13441

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#13441

yes

characteristic, direct product factor, Sylow

C₂²×Q₃₂

C₂×C₁₃₇

yes

characteristic

#2151

C₂×C₁₃₇

yes

characteristic

C₂×#362

C₂×C₁₃₇

yes

—

#1979

C₂×C₁₃₇

yes

—

100

#1381

C₂×C₁₃₇

yes

characteristic

101

C₂²×Q₁₆

C₂²×C₁₃₇

yes

—

102

C₂×Q₃₂

—

103

C₂×Q₃₂

—

104

C₂×M₃₂

C₂²×C₁₃₇

yes

characteristic

105

C₂²×C₁₆

C₂²×C₁₃₇

yes

characteristic

106

C₂×#108

C₂²×C₁₃₇

yes

—

107

#419

—

108

C₂×#101

—

109

#405

C₂²×C₁₃₇

yes

characteristic

110

C₄⋊C₁₆

—

111

#362

—

112

C₂²×Q₈

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

D₄×C₁₃₇

yes

—

119

C₂×M₁₆

—

120

C₂×M₁₆

D₄×C₁₃₇

yes

characteristic

121

C₂²×C₈

C₂³×C₁₃₇

yes

characteristic

122

#117

D₄×C₁₃₇

yes

characteristic

123

Q₃₂

—

124

M₃₂

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

D₄×C₁₃₇

yes

—

128

#108

—

129

#108

—

130

#103

—

131

#101

—

132

#97

—

133

C₄×C₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂×D₄×C₁₃₇

yes

characteristic

142

Q₁₆

—

143

Q₁₆

—

144

Q₁₆

—

145

Q₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

C₂×D₄×C₁₃₇

yes

—

152

C₂×C₈

C₂×D₄×C₁₃₇

yes

characteristic

153

C₂×C₈

#41×C₁₃₇

yes

characteristic

154

C₂×C₈

#41×C₁₃₇

yes

—

155

C₄⋊C₄

D₈×C₁₃₇

yes

—

156

C₂²⋊C₄

—

157

C₄²

—

158

C₁₆

—

159

C₁₆

—

160

C₂³

#135×C₁₃₇

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

#121×C₁₃₇

yes

—

170

C₂×C₄

#121×C₁₃₇

yes

characteristic

171

C₂×C₄

C₂×D₈×C₁₃₇

yes

characteristic

172

C₂×C₄

AGL(1,ℤ/8)×C₁₃₇

yes

—

173

C₈

—

174

C₈

—

175

C₈

#120×C₁₃₇

yes

characteristic

176

C₈

#123×C₁₃₇

yes

—

177

C₈

#124×C₁₃₇

yes

characteristic

178

C₂²

—

179

C₂²

#463×C₁₃₇

yes

—

180

C₂²

#466×C₁₃₇

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

#394×C₁₃₇

yes

characteristic

184

C₄

#459×C₁₃₇

yes

—

185

C₄

#469×C₁₃₇

yes

characteristic

186

C₂

—

187

C₂

#2054×C₁₃₇

yes

characteristic

188

C₂

#2222×C₁₃₇

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
prime radical	C₂³×C₁₃₇	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ #405×C₁₃₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃₇

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₃₇

prime radical

C₂³×C₁₃₇

#135

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ #405×C₁₃₇ ◁ G

N	H ≅ G/N	type
94. #13441	C₁₃₇	direct (N × H)
95. C₁₃₇	#13441	direct (N × H)

H ≅ G/N

type

94. #13441

C₁₃₇

direct (N × H)

95. C₁₃₇

#13441

direct (N × H)

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

subgroup	index
C₂²×Q₃₂×C₁₃₇	2
#2151×C₁₃₇	2
C₂×#362×C₁₃₇	2
#1979×C₁₃₇	2
#1381×C₁₃₇	2
#13441	137

subgroup

index

C₂²×Q₃₂×C₁₃₇

#2151×C₁₃₇

C₂×#362×C₁₃₇

#1979×C₁₃₇

#1381×C₁₃₇

#13441

137

Subgroups — C₁₃₇×#13441 #46862827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#13441 #46862827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13441 #46862827