Group #46860724 — subgroups

Subgroups — C₁₃₇×#11338 #46860724

Order 35072.

702 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 254 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×#430×C₁₃₇	C₂	1	3	yes	maximal
2	C₂×#338×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1728×C₁₃₇	C₂	1	3	yes	maximal
4	C₂²×M₁₆×C₁₃₇	C₂²	1	3	yes	—
5	C₂²×M₁₆×C₁₃₇	C₂²	1	3	yes	—
6	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
7	#430×C₁₃₇	—	2	12	no	—
8	C₂×#108×C₁₃₇	—	2	6	no	—
9	#338×C₁₃₇	C₄	1	4	yes	complemented
10	#336×C₁₃₇	—	2	6	no	—
11	C₂³×C₄×C₁₃₇	Q₈	1	2	yes	—
12	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
13	C₂×M₁₆×C₁₃₇	—	2	6	no	—
14	C₂×M₁₆×C₁₃₇	—	2	6	no	—
15	C₂×M₁₆×C₁₃₇	—	2	3	no	—
16	C₂×M₁₆×C₁₃₇	—	2	6	no	—
17	C₂×M₁₆×C₁₃₇	—	2	6	no	—
18	C₂×M₁₆×C₁₃₇	C₂×C₄	1	6	yes	—
19	C₂×M₁₆×C₁₃₇	D₄	1	6	yes	—
20	C₂×M₁₆×C₁₃₇	Q₈	1	6	yes	—
21	C₂²×C₈×C₁₃₇	—	2	3	no	—
22	C₂²×C₈×C₁₃₇	D₄	1	6	yes	—
23	C₂×C₄²×C₁₃₇	—	2	3	no	—
24	C₂×C₄²×C₁₃₇	—	2	3	no	—
25	#108×C₁₃₇	—	2	24	no	—
26	C₂⁴×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₃₇	—	2	6	no	—
28	C₂²×C₄×C₁₃₇	—	4	4	no	—
29	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
31	C₂²×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
32	C₂²×C₄×C₁₃₇	C₂×Q₈	1	3	yes	—
33	C₂²×C₄×C₁₃₇	#41	1	3	yes	—
34	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
35	M₁₆×C₁₃₇	—	2	12	no	—
36	M₁₆×C₁₃₇	—	2	12	no	—
37	M₁₆×C₁₃₇	—	2	12	no	—
38	M₁₆×C₁₃₇	—	4	6	no	—
39	C₂×C₈×C₁₃₇	—	2	6	no	—
40	C₂×C₈×C₁₃₇	—	2	6	no	—
41	C₂×C₈×C₁₃₇	—	2	6	no	—
42	C₂×C₈×C₁₃₇	—	4	3	no	—
43	C₂×C₈×C₁₃₇	—	4	3	no	—
44	C₂×C₈×C₁₃₇	C₄⋊C₄	1	12	yes	—
45	C₄²×C₁₃₇	—	4	3	no	—
46	C₄²×C₁₃₇	—	4	1	no	—
47	C₂³×C₁₃₇	—	4	2	no	—
48	C₂³×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
49	C₂³×C₁₃₇	C₂²⋊Q₈	1	3	yes	—
50	C₂³×C₁₃₇	C₄⋊Q₈	1	3	yes	—
51	C₂×C₄×C₁₃₇	—	2	12	no	—
52	C₂×C₄×C₁₃₇	—	4	6	no	—
53	C₂×C₄×C₁₃₇	—	4	2	no	—
54	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	3	yes	—
55	C₂×C₄×C₁₃₇	C₄×D₄	1	3	yes	—
56	C₂×C₄×C₁₃₇	#120	1	3	yes	—
57	C₂×C₄×C₁₃₇	#121	1	3	yes	—
58	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	6	yes	—
59	C₂×C₄×C₁₃₇	#123	1	3	yes	—
60	C₂×C₄×C₁₃₇	#125	1	3	yes	—
61	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
62	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	3	yes	—
63	C₈×C₁₃₇	—	2	12	no	—
64	C₈×C₁₃₇	—	4	6	no	—
65	C₂²×C₁₃₇	—	2	6	no	—
66	C₂²×C₁₃₇	—	4	1	no	—
67	C₂²×C₁₃₇	—	4	3	no	—
68	C₂²×C₁₃₇	#389	1	3	yes	—
69	C₂²×C₁₃₇	#395	1	3	yes	—
70	C₂²×C₁₃₇	#399	1	1	yes	characteristic
71	C₄×C₁₃₇	—	4	4	no	—
72	C₄×C₁₃₇	#384	1	1	yes	characteristic
73	C₄×C₁₃₇	#385	1	3	yes	—
74	C₄×C₁₃₇	#392	1	1	yes	characteristic
75	C₄×C₁₃₇	#398	1	3	yes	—
76	C₂×C₁₃₇	—	2	3	no	—
77	C₂×C₁₃₇	—	2	3	no	—
78	C₂×C₁₃₇	#1444	1	1	yes	characteristic
79	C₂×C₁₃₇	#2091	1	2	yes	—
80	#11338	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₁₃₇	#11338	1	1	yes	characteristic, direct product factor, Sylow
82	C₂×#430	C₂×C₁₃₇	1	3	yes	—
83	C₂×#338	C₂×C₁₃₇	1	1	yes	characteristic
84	#1728	C₂×C₁₃₇	1	3	yes	—
85	C₂²×M₁₆	C₂²×C₁₃₇	1	3	yes	—
86	C₂²×M₁₆	C₂²×C₁₃₇	1	3	yes	—
87	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
88	#430	—	2	12	no	—
89	C₂×#108	—	2	6	no	—
90	#338	C₄×C₁₃₇	1	4	yes	complemented
91	#336	—	2	6	no	—
92	C₂³×C₄	Q₈×C₁₃₇	1	2	yes	—
93	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
94	C₂×M₁₆	—	2	6	no	—
95	C₂×M₁₆	—	2	6	no	—
96	C₂×M₁₆	—	2	3	no	—
97	C₂×M₁₆	—	2	6	no	—
98	C₂×M₁₆	—	2	6	no	—
99	C₂×M₁₆	C₂×C₄×C₁₃₇	1	6	yes	—
100	C₂×M₁₆	D₄×C₁₃₇	1	6	yes	—
101	C₂×M₁₆	Q₈×C₁₃₇	1	6	yes	—
102	C₂²×C₈	—	2	3	no	—
103	C₂²×C₈	D₄×C₁₃₇	1	6	yes	—
104	C₂×C₄²	—	2	3	no	—
105	C₂×C₄²	—	2	3	no	—
106	#108	—	2	24	no	—
107	C₂⁴	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
108	C₂²×C₄	—	2	6	no	—
109	C₂²×C₄	—	4	4	no	—
110	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₄×C₁₃₇	1	3	yes	—
112	C₂²×C₄	C₂×D₄×C₁₃₇	1	3	yes	—
113	C₂²×C₄	C₂×Q₈×C₁₃₇	1	3	yes	—
114	C₂²×C₄	#41×C₁₃₇	1	3	yes	—
115	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
116	M₁₆	—	2	12	no	—
117	M₁₆	—	2	12	no	—
118	M₁₆	—	2	12	no	—
119	M₁₆	—	4	6	no	—
120	C₂×C₈	—	2	6	no	—
121	C₂×C₈	—	2	6	no	—
122	C₂×C₈	—	2	6	no	—
123	C₂×C₈	—	4	3	no	—
124	C₂×C₈	—	4	3	no	—
125	C₂×C₈	C₄⋊C₄×C₁₃₇	1	12	yes	—
126	C₄²	—	4	3	no	—
127	C₄²	—	4	1	no	—
128	C₂³	—	4	2	no	—
129	C₂³	C₄×Q₈×C₁₃₇	1	1	yes	characteristic
130	C₂³	C₂²⋊Q₈×C₁₃₇	1	3	yes	—
131	C₂³	C₄⋊Q₈×C₁₃₇	1	3	yes	—
132	C₂×C₄	—	2	12	no	—
133	C₂×C₄	—	4	6	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	3	yes	—
136	C₂×C₄	C₄×D₄×C₁₃₇	1	3	yes	—
137	C₂×C₄	#120×C₁₃₇	1	3	yes	—
138	C₂×C₄	#121×C₁₃₇	1	3	yes	—
139	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	6	yes	—
140	C₂×C₄	#123×C₁₃₇	1	3	yes	—
141	C₂×C₄	#125×C₁₃₇	1	3	yes	—
142	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
143	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	3	yes	—
144	C₈	—	2	12	no	—
145	C₈	—	4	6	no	—
146	C₂²	—	2	6	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	3	no	—
149	C₂²	#389×C₁₃₇	1	3	yes	—
150	C₂²	#395×C₁₃₇	1	3	yes	—
151	C₂²	#399×C₁₃₇	1	1	yes	characteristic
152	C₄	—	4	4	no	—
153	C₄	#384×C₁₃₇	1	1	yes	characteristic
154	C₄	#385×C₁₃₇	1	3	yes	—
155	C₄	#392×C₁₃₇	1	1	yes	characteristic
156	C₄	#398×C₁₃₇	1	3	yes	—
157	C₂	—	2	3	no	—
158	C₂	—	2	3	no	—
159	C₂	#1444×C₁₃₇	1	1	yes	characteristic
160	C₂	#2091×C₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#430×C₁₃₇

C₂

yes

maximal

C₂×#338×C₁₃₇

C₂

yes

maximal, characteristic

#1728×C₁₃₇

C₂

yes

maximal

C₂²×M₁₆×C₁₃₇

C₂²

yes

—

C₂²×M₁₆×C₁₃₇

C₂²

yes

—

C₂²×C₄²×C₁₃₇

C₂²

yes

characteristic

#430×C₁₃₇

—

C₂×#108×C₁₃₇

—

#338×C₁₃₇

C₄

yes

complemented

#336×C₁₃₇

—

C₂³×C₄×C₁₃₇

Q₈

yes

—

C₂³×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×M₁₆×C₁₃₇

—

C₂×M₁₆×C₁₃₇

—

C₂×M₁₆×C₁₃₇

—

C₂×M₁₆×C₁₃₇

—

C₂×M₁₆×C₁₃₇

—

C₂×M₁₆×C₁₃₇

C₂×C₄

yes

—

C₂×M₁₆×C₁₃₇

D₄

yes

—

C₂×M₁₆×C₁₃₇

Q₈

yes

—

C₂²×C₈×C₁₃₇

—

C₂²×C₈×C₁₃₇

D₄

yes

—

C₂×C₄²×C₁₃₇

—

C₂×C₄²×C₁₃₇

—

#108×C₁₃₇

—

C₂⁴×C₁₃₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×D₄

yes

—

C₂²×C₄×C₁₃₇

C₂×D₄

yes

—

C₂²×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂²×C₄×C₁₃₇

#41

yes

—

C₂²×C₄×C₁₃₇

#41

yes

characteristic

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₄⋊C₄

yes

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

—

C₂³×C₁₃₇

—

C₂³×C₁₃₇

C₄×Q₈

yes

characteristic

C₂³×C₁₃₇

C₂²⋊Q₈

yes

—

C₂³×C₁₃₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₄×D₄

yes

—

C₂×C₄×C₁₃₇

#120

yes

—

C₂×C₄×C₁₃₇

#121

yes

—

C₂×C₄×C₁₃₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃₇

#123

yes

—

C₂×C₄×C₁₃₇

#125

yes

—

C₂×C₄×C₁₃₇

#127

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#389

yes

—

C₂²×C₁₃₇

#395

yes

—

C₂²×C₁₃₇

#399

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

#384

yes

characteristic

C₄×C₁₃₇

#385

yes

—

C₄×C₁₃₇

#392

yes

characteristic

C₄×C₁₃₇

#398

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#1444

yes

characteristic

C₂×C₁₃₇

#2091

yes

—

#11338

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#11338

yes

characteristic, direct product factor, Sylow

C₂×#430

C₂×C₁₃₇

yes

—

C₂×#338

C₂×C₁₃₇

yes

characteristic

#1728

C₂×C₁₃₇

yes

—

C₂²×M₁₆

C₂²×C₁₃₇

yes

—

C₂²×M₁₆

C₂²×C₁₃₇

yes

—

C₂²×C₄²

C₂²×C₁₃₇

yes

—

—

C₄×C₁₃₇

yes

complemented

#336

—

C₂³×C₄

Q₈×C₁₃₇

yes

—

C₂³×C₄

C₂³×C₁₃₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄×C₁₃₇

yes

—

100

C₂×M₁₆

D₄×C₁₃₇

yes

—

101

C₂×M₁₆

Q₈×C₁₃₇

yes

—

102

C₂²×C₈

—

103

C₂²×C₈

D₄×C₁₃₇

yes

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

#108

—

107

C₂⁴

C₂×Q₈×C₁₃₇

yes

characteristic

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²×C₄×C₁₃₇

yes

characteristic

111

C₂²×C₄

C₂×D₄×C₁₃₇

yes

—

112

C₂²×C₄

C₂×D₄×C₁₃₇

yes

—

113

C₂²×C₄

C₂×Q₈×C₁₃₇

yes

—

114

C₂²×C₄

#41×C₁₃₇

yes

—

115

C₂²×C₄

#41×C₁₃₇

yes

characteristic

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

C₄⋊C₄×C₁₃₇

yes

—

126

C₄²

—

127

C₄²

—

128

C₂³

—

129

C₂³

C₄×Q₈×C₁₃₇

yes

characteristic

130

C₂³

C₂²⋊Q₈×C₁₃₇

yes

—

131

C₂³

C₄⋊Q₈×C₁₃₇

yes

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄⋊C₄×C₁₃₇

yes

—

136

C₂×C₄

C₄×D₄×C₁₃₇

yes

—

137

C₂×C₄

#120×C₁₃₇

yes

—

138

C₂×C₄

#121×C₁₃₇

yes

—

139

C₂×C₄

C₂²⋊Q₈×C₁₃₇

yes

—

140

C₂×C₄

#123×C₁₃₇

yes

—

141

C₂×C₄

#125×C₁₃₇

yes

—

142

C₂×C₄

#127×C₁₃₇

yes

characteristic

143

C₂×C₄

C₄⋊Q₈×C₁₃₇

yes

—

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#389×C₁₃₇

yes

—

150

C₂²

#395×C₁₃₇

yes

—

151

C₂²

#399×C₁₃₇

yes

characteristic

152

C₄

—

153

C₄

#384×C₁₃₇

yes

characteristic

154

C₄

#385×C₁₃₇

yes

—

155

C₄

#392×C₁₃₇

yes

characteristic

156

C₄

#398×C₁₃₇

yes

—

157

C₂

—

158

C₂

—

159

C₂

#1444×C₁₃₇

yes

characteristic

160

C₂

#2091×C₁₃₇

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#127
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#399
prime radical	C₂⁴×C₁₃₇	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₃₇

#127

commutator

C₂²×C₄

C₂²×C₄×C₁₃₇

Frattini, cosocle

C₂³×C₄

C₂³×C₁₃₇

socle

C₂²×C₁₃₇

#399

prime radical

C₂⁴×C₁₃₇

C₂×Q₈

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

N	H ≅ G/N	type
9. #338×C₁₃₇	C₄	semidirect
80. #11338	C₁₃₇	direct (N × H)
81. C₁₃₇	#11338	direct (N × H)
90. #338	C₄×C₁₃₇	semidirect

H ≅ G/N

type

9. #338×C₁₃₇

C₄

semidirect

80. #11338

C₁₃₇

direct (N × H)

81. C₁₃₇

#11338

direct (N × H)

90. #338

C₄×C₁₃₇

semidirect

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

subgroup	index
C₂×#430×C₁₃₇	2
C₂×#338×C₁₃₇	2
#1728×C₁₃₇	2
#11338	137

subgroup

index

C₂×#430×C₁₃₇

C₂×#338×C₁₃₇

#1728×C₁₃₇

#11338

137

Subgroups — C₁₃₇×#11338 #46860724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#11338 #46860724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#11338 #46860724