Group #46862231 — subgroups

Subgroups — C₁₃₇×#12845 #46862231

Order 35072.

454 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 182 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×#405×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₁₆×C₁₃₇	C₂	1	2	yes	maximal
3	#1374×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₂×#117×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₃₇	C₂²	1	4	yes	—
8	#405×C₁₃₇	—	2	4	no	—
9	C₂×#97×C₁₃₇	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂²⋊C₁₆×C₁₃₇	—	2	4	no	—
12	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
13	C₂×M₁₆×C₁₃₇	—	2	2	no	—
14	C₂×M₁₆×C₁₃₇	—	2	4	no	—
15	C₂²×C₈×C₁₃₇	—	2	4	no	—
16	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
19	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
20	#117×C₁₃₇	—	2	2	no	—
21	#117×C₁₃₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₃₇	C₈	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₃₇	C₈	1	2	yes	—
24	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₁₃₇	—	2	4	no	—
26	C₂×C₁₆×C₁₃₇	—	2	8	no	—
27	#97×C₁₃₇	—	2	4	no	—
28	C₄×C₈×C₁₃₇	—	2	4	no	—
29	C₂⁴×C₁₃₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₇	—	2	2	no	—
31	C₂²×C₄×C₁₃₇	—	2	2	no	—
32	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₁₃₇	C₂×C₈	1	4	yes	—
34	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₇	M₁₆	1	2	yes	—
36	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
37	M₁₆×C₁₃₇	—	2	8	no	—
38	C₂×C₈×C₁₃₇	—	2	2	no	—
39	C₂×C₈×C₁₃₇	—	2	2	no	—
40	C₂×C₈×C₁₃₇	—	2	2	no	—
41	C₂×C₈×C₁₃₇	—	2	8	no	—
42	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₃₇	M₁₆	1	4	yes	—
44	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₁₃₇	#41	1	4	yes	—
46	C₄⋊C₄×C₁₃₇	—	2	4	no	—
47	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
48	C₄²×C₁₃₇	—	2	2	no	—
49	C₄²×C₁₃₇	—	2	2	no	—
50	C₁₆×C₁₃₇	—	4	4	no	—
51	C₂³×C₁₃₇	—	2	2	no	—
52	C₂³×C₁₃₇	—	2	4	no	—
53	C₂³×C₁₃₇	C₂²⋊C₈	1	2	yes	—
54	C₂³×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	—	2	2	no	—
56	C₂×C₄×C₁₃₇	—	2	2	no	—
57	C₂×C₄×C₁₃₇	—	2	2	no	—
58	C₂×C₄×C₁₃₇	—	2	4	no	—
59	C₂×C₄×C₁₃₇	—	2	4	no	—
60	C₂×C₄×C₁₃₇	—	2	8	no	—
61	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃₇	#117	1	2	yes	—
64	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₁₃₇	—	2	2	no	—
67	C₈×C₁₃₇	—	2	4	no	—
68	C₈×C₁₃₇	#123	1	4	yes	—
69	C₂²×C₁₃₇	—	2	2	no	—
70	C₂²×C₁₃₇	—	2	4	no	—
71	C₂²×C₁₃₇	—	4	4	no	—
72	C₂²×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
73	C₂²×C₁₃₇	#431	1	2	yes	—
74	C₂²×C₁₃₇	#504	1	4	yes	—
75	C₄×C₁₃₇	—	2	2	no	—
76	C₄×C₁₃₇	—	4	4	no	—
77	C₄×C₁₃₇	#381	1	1	yes	characteristic
78	C₄×C₁₃₇	#407	1	1	yes	characteristic
79	C₄×C₁₃₇	#432	1	2	yes	—
80	C₂×C₁₃₇	—	4	2	no	—
81	C₂×C₁₃₇	#1817	1	1	yes	characteristic
82	C₂×C₁₃₇	#2113	1	2	yes	—
83	C₂×C₁₃₇	#2164	1	4	yes	—
84	#12845	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₁₃₇	#12845	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×#405	C₂×C₁₃₇	1	1	yes	characteristic
87	C₂×C₂²⋊C₁₆	C₂×C₁₃₇	1	2	yes	—
88	#1374	C₂×C₁₃₇	1	4	yes	complemented
89	C₂²×M₁₆	C₄×C₁₃₇	1	1	yes	characteristic
90	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
91	C₂×#117	C₄×C₁₃₇	1	1	yes	characteristic
92	C₂²×C₁₆	C₂²×C₁₃₇	1	4	yes	—
93	#405	—	2	4	no	—
94	C₂×#97	C₄×C₁₃₇	1	2	yes	—
95	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
96	C₂²⋊C₁₆	—	2	4	no	—
97	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	4	no	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
102	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
103	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
104	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	C₂×C₄⋊C₄	C₈×C₁₃₇	1	2	yes	—
108	C₂×C₂²⋊C₄	C₈×C₁₃₇	1	2	yes	—
109	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
110	C₂×C₁₆	—	2	4	no	—
111	C₂×C₁₆	—	2	8	no	—
112	#97	—	2	4	no	—
113	C₄×C₈	—	2	4	no	—
114	C₂⁴	C₂×C₈×C₁₃₇	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
118	C₂²×C₄	C₂×C₈×C₁₃₇	1	4	yes	—
119	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
120	C₂²×C₄	M₁₆×C₁₃₇	1	2	yes	—
121	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
122	M₁₆	—	2	8	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
128	C₂×C₈	M₁₆×C₁₃₇	1	4	yes	—
129	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
130	C₂×C₈	#41×C₁₃₇	1	4	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₁₆	—	4	4	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	4	no	—
138	C₂³	C₂²⋊C₈×C₁₃₇	1	2	yes	—
139	C₂³	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
148	C₂×C₄	#117×C₁₃₇	1	2	yes	—
149	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
150	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	4	no	—
153	C₈	#123×C₁₃₇	1	4	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
158	C₂²	#431×C₁₃₇	1	2	yes	—
159	C₂²	#504×C₁₃₇	1	4	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#381×C₁₃₇	1	1	yes	characteristic
163	C₄	#407×C₁₃₇	1	1	yes	characteristic
164	C₄	#432×C₁₃₇	1	2	yes	—
165	C₂	—	4	2	no	—
166	C₂	#1817×C₁₃₇	1	1	yes	characteristic
167	C₂	#2113×C₁₃₇	1	2	yes	—
168	C₂	#2164×C₁₃₇	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#405×C₁₃₇

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₁₆×C₁₃₇

C₂

yes

maximal

#1374×C₁₃₇

C₂

yes

maximal, complemented

C₂²×M₁₆×C₁₃₇

C₄

yes

characteristic

C₂³×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×#117×C₁₃₇

C₄

yes

characteristic

C₂²×C₁₆×C₁₃₇

C₂²

yes

—

#405×C₁₃₇

—

C₂×#97×C₁₃₇

C₄

yes

—

C₂×C₄×C₈×C₁₃₇

C₂²

yes

—

C₂²⋊C₁₆×C₁₃₇

—

C₂³×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₃₇

—

C₂×M₁₆×C₁₃₇

—

C₂²×C₈×C₁₃₇

—

C₂²×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃₇

C₂×C₄

yes

—

C₂²×C₈×C₁₃₇

D₄

yes

—

C₂²×C₈×C₁₃₇

C₂³

yes

characteristic

#117×C₁₃₇

—

#117×C₁₃₇

—

C₂×C₄⋊C₄×C₁₃₇

C₈

yes

—

C₂×C₂²⋊C₄×C₁₃₇

C₈

yes

—

C₂×C₄²×C₁₃₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₃₇

—

C₂×C₁₆×C₁₃₇

—

#97×C₁₃₇

—

C₄×C₈×C₁₃₇

—

C₂⁴×C₁₃₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₈

yes

—

C₂²×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃₇

M₁₆

yes

—

C₂²×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃₇

M₁₆

yes

—

C₂×C₈×C₁₃₇

C₂×D₄

yes

—

C₂×C₈×C₁₃₇

#41

yes

—

C₄⋊C₄×C₁₃₇

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

—

C₁₆×C₁₃₇

—

C₂³×C₁₃₇

—

C₂³×C₁₃₇

—

C₂³×C₁₃₇

C₂²⋊C₈

yes

—

C₂³×C₁₃₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

#117

yes

—

C₂×C₄×C₁₃₇

C₂×M₁₆

yes

—

C₂×C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

#123

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₁₃₇

#431

yes

—

C₂²×C₁₃₇

#504

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#1817

yes

characteristic

C₂×C₁₃₇

#2113

yes

—

C₂×C₁₃₇

#2164

yes

—

#12845

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#12845

yes

characteristic, direct product factor, Sylow

C₂×#405

C₂×C₁₃₇

yes

characteristic

C₂×C₂²⋊C₁₆

C₂×C₁₃₇

yes

—

#1374

C₂×C₁₃₇

yes

complemented

C₂²×M₁₆

C₄×C₁₃₇

yes

characteristic

C₂³×C₈

C₂²×C₁₃₇

yes

characteristic

C₂×#117

C₄×C₁₃₇

yes

characteristic

C₂²×C₁₆

C₂²×C₁₃₇

yes

—

#405

—

C₂×#97

C₄×C₁₃₇

yes

—

C₂×C₄×C₈

C₂²×C₁₃₇

yes

—

C₂²⋊C₁₆

—

C₂³×C₄

C₂×C₄×C₁₃₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×C₄×C₁₃₇

yes

characteristic

102

C₂²×C₈

C₂×C₄×C₁₃₇

yes

—

103

C₂²×C₈

D₄×C₁₃₇

yes

—

104

C₂²×C₈

C₂³×C₁₃₇

yes

characteristic

105

#117

—

106

#117

—

107

C₂×C₄⋊C₄

C₈×C₁₃₇

yes

—

108

C₂×C₂²⋊C₄

C₈×C₁₃₇

yes

—

109

C₂×C₄²

C₂×C₄×C₁₃₇

yes

—

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

#97

—

113

C₄×C₈

—

114

C₂⁴

C₂×C₈×C₁₃₇

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²⋊C₄×C₁₃₇

yes

—

118

C₂²×C₄

C₂×C₈×C₁₃₇

yes

—

119

C₂²×C₄

C₂×C₈×C₁₃₇

yes

characteristic

120

C₂²×C₄

M₁₆×C₁₃₇

yes

—

121

C₂²×C₄

C₂²×C₄×C₁₃₇

yes

characteristic

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂²⋊C₄×C₁₃₇

yes

—

128

C₂×C₈

M₁₆×C₁₃₇

yes

—

129

C₂×C₈

C₂×D₄×C₁₃₇

yes

—

130

C₂×C₈

#41×C₁₃₇

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₁₆

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂²⋊C₈×C₁₃₇

yes

—

139

C₂³

C₂²×C₈×C₁₃₇

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊C₈×C₁₃₇

yes

—

147

C₂×C₄

C₂×C₂²⋊C₄×C₁₃₇

yes

characteristic

148

C₂×C₄

#117×C₁₃₇

yes

—

149

C₂×C₄

C₂×M₁₆×C₁₃₇

yes

—

150

C₂×C₄

C₂×M₁₆×C₁₃₇

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

#123×C₁₃₇

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₂²⋊C₈×C₁₃₇

yes

characteristic

158

C₂²

#431×C₁₃₇

yes

—

159

C₂²

#504×C₁₃₇

yes

—

160

C₄

—

161

C₄

—

162

C₄

#381×C₁₃₇

yes

characteristic

163

C₄

#407×C₁₃₇

yes

characteristic

164

C₄

#432×C₁₃₇

yes

—

165

C₂

—

166

C₂

#1817×C₁₃₇

yes

characteristic

167

C₂

#2113×C₁₃₇

yes

—

168

C₂

#2164×C₁₃₇

yes

—

	subgroup	G/H
center	C₂²×C₈×C₁₃₇	C₂³
commutator	C₂³	C₂²×C₈×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
socle	C₂³×C₁₃₇	C₂²×C₈
prime radical	C₂⁴×C₁₃₇	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₃₇ ◁ G

subgroup

G/H

center

C₂²×C₈×C₁₃₇

C₂³

commutator

C₂³

C₂²×C₈×C₁₃₇

Frattini, cosocle

C₂²×C₈

C₂³×C₁₃₇

socle

C₂³×C₁₃₇

C₂²×C₈

prime radical

C₂⁴×C₁₃₇

C₂×C₈

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₈×C₁₃₇ ◁ G

N	H ≅ G/N	type
3. #1374×C₁₃₇	C₂	semidirect
84. #12845	C₁₃₇	direct (N × H)
85. C₁₃₇	#12845	direct (N × H)
88. #1374	C₂×C₁₃₇	semidirect

H ≅ G/N

type

3. #1374×C₁₃₇

C₂

semidirect

84. #12845

C₁₃₇

direct (N × H)

85. C₁₃₇

#12845

direct (N × H)

88. #1374

C₂×C₁₃₇

semidirect

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

subgroup	index
C₂×#405×C₁₃₇	2
C₂×C₂²⋊C₁₆×C₁₃₇	2
#1374×C₁₃₇	2
#12845	137

subgroup

index

C₂×#405×C₁₃₇

C₂×C₂²⋊C₁₆×C₁₃₇

#1374×C₁₃₇

#12845

137

Subgroups — C₁₃₇×#12845 #46862231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#12845 #46862231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#12845 #46862231