Group #46863004 — subgroups

Subgroups — C₁₃₇×#13618 #46863004

Order 35072.

290 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₂×M₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2158×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1416×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₁₆×C₁₃₇	—	2	1	no	—
6	C₂×Q₃₂×C₁₃₇	—	2	1	no	—
7	C₂×Q₃₂×C₁₃₇	—	2	2	no	—
8	C₂×Q₃₂×C₁₃₇	C₄	1	4	yes	complemented
9	C₂×Q₃₂×C₁₃₇	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
11	#427×C₁₃₇	—	2	1	no	—
12	M₆₄×C₁₃₇	—	2	2	no	—
13	C₂×C₃₂×C₁₃₇	C₂²	1	2	yes	—
14	#367×C₁₃₇	—	2	1	no	—
15	#366×C₁₃₇	C₂²	1	2	yes	—
16	C₂²×Q₈×C₁₃₇	—	4	1	no	—
17	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
18	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
19	C₂×Q₁₆×C₁₃₇	—	4	1	no	—
20	C₂×Q₁₆×C₁₃₇	—	4	2	no	—
21	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
22	#117×C₁₃₇	—	4	1	no	—
23	Q₃₂×C₁₃₇	—	2	4	no	—
24	Q₃₂×C₁₃₇	—	2	2	no	—
25	Q₃₂×C₁₃₇	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₃₇	D₄	1	2	yes	—
29	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
30	#107×C₁₃₇	—	2	2	no	—
31	#106×C₁₃₇	—	4	1	no	—
32	C₃₂×C₁₃₇	—	2	2	no	—
33	C₂×Q₈×C₁₃₇	—	4	4	no	—
34	C₂×Q₈×C₁₃₇	—	4	2	no	—
35	C₂×Q₈×C₁₃₇	—	8	1	no	—
36	C₂×Q₈×C₁₃₇	—	8	2	no	—
37	C₂²×C₄×C₁₃₇	—	8	1	no	—
38	C₂²×C₄×C₁₃₇	D₈	1	1	yes	characteristic
39	Q₁₆×C₁₃₇	—	2	4	no	—
40	Q₁₆×C₁₃₇	—	4	4	no	—
41	Q₁₆×C₁₃₇	—	4	2	no	—
42	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₃₇	D₈	1	1	yes	characteristic
44	C₂×C₈×C₁₃₇	SD₁₆	1	2	yes	—
45	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₁₃₇	—	4	2	no	—
47	C₂²⋊C₄×C₁₃₇	—	8	1	no	—
48	C₄²×C₁₃₇	—	8	1	no	—
49	C₁₆×C₁₃₇	C₂²⋊C₄	1	2	yes	—
50	C₁₆×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₁₆×C₁₃₇	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₁₃₇	D₁₆	1	1	yes	characteristic
53	Q₈×C₁₃₇	—	4	4	no	—
54	Q₈×C₁₃₇	—	8	4	no	—
55	Q₈×C₁₃₇	—	8	2	no	—
56	C₂×C₄×C₁₃₇	—	8	4	no	—
57	C₂×C₄×C₁₃₇	—	8	2	no	—
58	C₂×C₄×C₁₃₇	—	8	2	no	—
59	C₂×C₄×C₁₃₇	#102	1	2	yes	—
60	C₂×C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
61	C₂×C₄×C₁₃₇	SD₃₂	1	2	yes	—
62	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
63	C₈×C₁₃₇	#102	1	2	yes	—
64	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₁₃₇	—	2	2	no	—
67	C₂²×C₁₃₇	#357	1	2	yes	—
68	C₂²×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₁₃₇	—	8	4	no	—
70	C₄×C₁₃₇	—	16	2	no	—
71	C₄×C₁₃₇	#357	1	2	yes	—
72	C₄×C₁₃₇	C₂×#102	1	1	yes	characteristic
73	C₄×C₁₃₇	C₂×SD₃₂	1	1	yes	characteristic
74	C₂×C₁₃₇	—	2	2	no	—
75	C₂×C₁₃₇	C₂×#357	1	1	yes	characteristic
76	C₂×C₁₃₇	#2264	1	2	yes	—
77	#13618	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₁₃₇	#13618	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²×Q₃₂	C₂×C₁₃₇	1	1	yes	characteristic
80	C₂×M₆₄	C₂×C₁₃₇	1	1	yes	characteristic
81	#2158	C₂×C₁₃₇	1	1	yes	characteristic
82	#1416	C₂×C₁₃₇	1	4	yes	complemented
83	C₂²×Q₁₆	—	2	1	no	—
84	C₂×Q₃₂	—	2	1	no	—
85	C₂×Q₃₂	—	2	2	no	—
86	C₂×Q₃₂	C₄×C₁₃₇	1	4	yes	complemented
87	C₂×Q₃₂	C₂²×C₁₃₇	1	2	yes	—
88	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
89	#427	—	2	1	no	—
90	M₆₄	—	2	2	no	—
91	C₂×C₃₂	C₂²×C₁₃₇	1	2	yes	—
92	#367	—	2	1	no	—
93	#366	C₂²×C₁₃₇	1	2	yes	—
94	C₂²×Q₈	—	4	1	no	—
95	C₂×Q₁₆	—	2	4	no	—
96	C₂×Q₁₆	—	2	2	no	—
97	C₂×Q₁₆	—	4	1	no	—
98	C₂×Q₁₆	—	4	2	no	—
99	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
100	#117	—	4	1	no	—
101	Q₃₂	—	2	4	no	—
102	Q₃₂	—	2	2	no	—
103	Q₃₂	C₂×C₄×C₁₃₇	1	4	yes	—
104	C₂×C₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
105	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
106	C₂×C₁₆	D₄×C₁₃₇	1	2	yes	—
107	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
108	#107	—	2	2	no	—
109	#106	—	4	1	no	—
110	C₃₂	—	2	2	no	—
111	C₂×Q₈	—	4	4	no	—
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	8	1	no	—
114	C₂×Q₈	—	8	2	no	—
115	C₂²×C₄	—	8	1	no	—
116	C₂²×C₄	D₈×C₁₃₇	1	1	yes	characteristic
117	Q₁₆	—	2	4	no	—
118	Q₁₆	—	4	4	no	—
119	Q₁₆	—	4	2	no	—
120	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
121	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
122	C₂×C₈	SD₁₆×C₁₃₇	1	2	yes	—
123	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
124	C₄⋊C₄	—	4	2	no	—
125	C₂²⋊C₄	—	8	1	no	—
126	C₄²	—	8	1	no	—
127	C₁₆	C₂²⋊C₄×C₁₃₇	1	2	yes	—
128	C₁₆	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
129	C₁₆	C₂×D₄×C₁₃₇	1	1	yes	characteristic
130	C₂³	D₁₆×C₁₃₇	1	1	yes	characteristic
131	Q₈	—	4	4	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	8	4	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	8	4	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	#102×C₁₃₇	1	2	yes	—
138	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
139	C₂×C₄	SD₃₂×C₁₃₇	1	2	yes	—
140	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
141	C₈	#102×C₁₃₇	1	2	yes	—
142	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
143	C₈	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
144	C₂²	—	2	2	no	—
145	C₂²	#357×C₁₃₇	1	2	yes	—
146	C₂²	C₂×D₁₆×C₁₃₇	1	1	yes	characteristic
147	C₄	—	8	4	no	—
148	C₄	—	16	2	no	—
149	C₄	#357×C₁₃₇	1	2	yes	—
150	C₄	C₂×#102×C₁₃₇	1	1	yes	characteristic
151	C₄	C₂×SD₃₂×C₁₃₇	1	1	yes	characteristic
152	C₂	—	2	2	no	—
153	C₂	C₂×#357×C₁₃₇	1	1	yes	characteristic
154	C₂	#2264×C₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₃₂×C₁₃₇

C₂

yes

maximal, characteristic

C₂×M₆₄×C₁₃₇

C₂

yes

maximal, characteristic

#2158×C₁₃₇

C₂

yes

maximal, characteristic

#1416×C₁₃₇

C₂

yes

maximal, complemented

C₂²×Q₁₆×C₁₃₇

—

C₂×Q₃₂×C₁₃₇

—

C₂×Q₃₂×C₁₃₇

—

C₂×Q₃₂×C₁₃₇

C₄

yes

complemented

C₂×Q₃₂×C₁₃₇

C₂²

yes

—

C₂²×C₁₆×C₁₃₇

C₂²

yes

characteristic

#427×C₁₃₇

—

M₆₄×C₁₃₇

—

C₂×C₃₂×C₁₃₇

C₂²

yes

—

#367×C₁₃₇

—

#366×C₁₃₇

C₂²

yes

—

C₂²×Q₈×C₁₃₇

—

C₂×Q₁₆×C₁₃₇

—

C₂×Q₁₆×C₁₃₇

—

C₂×Q₁₆×C₁₃₇

—

C₂×Q₁₆×C₁₃₇

—

C₂²×C₈×C₁₃₇

D₄

yes

characteristic

#117×C₁₃₇

—

Q₃₂×C₁₃₇

—

Q₃₂×C₁₃₇

—

Q₃₂×C₁₃₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₃₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₃₇

D₄

yes

characteristic

C₂×C₁₆×C₁₃₇

D₄

yes

—

C₂×C₁₆×C₁₃₇

C₂³

yes

characteristic

#107×C₁₃₇

—

#106×C₁₃₇

—

C₃₂×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₈

yes

characteristic

Q₁₆×C₁₃₇

—

Q₁₆×C₁₃₇

—

Q₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃₇

D₈

yes

characteristic

C₂×C₈×C₁₃₇

SD₁₆

yes

—

C₂×C₈×C₁₃₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₁₆×C₁₃₇

C₂²⋊C₄

yes

—

C₁₆×C₁₃₇

C₂²×C₄

yes

characteristic

C₁₆×C₁₃₇

C₂×D₄

yes

characteristic

C₂³×C₁₃₇

D₁₆

yes

characteristic

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

#102

yes

—

C₂×C₄×C₁₃₇

D₁₆

yes

characteristic

C₂×C₄×C₁₃₇

SD₃₂

yes

—

C₂×C₄×C₁₃₇

C₂×D₈

yes

characteristic

C₈×C₁₃₇

#102

yes

—

C₈×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₃₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#357

yes

—

C₂²×C₁₃₇

C₂×D₁₆

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#357

yes

—

C₄×C₁₃₇

C₂×#102

yes

characteristic

C₄×C₁₃₇

C₂×SD₃₂

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×#357

yes

characteristic

C₂×C₁₃₇

#2264

yes

—

#13618

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#13618

yes

characteristic, direct product factor, Sylow

C₂²×Q₃₂

C₂×C₁₃₇

yes

characteristic

C₂×M₆₄

C₂×C₁₃₇

yes

characteristic

#2158

C₂×C₁₃₇

yes

characteristic

#1416

C₂×C₁₃₇

yes

complemented

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₄×C₁₃₇

yes

complemented

C₂×Q₃₂

C₂²×C₁₃₇

yes

—

C₂²×C₁₆

C₂²×C₁₃₇

yes

characteristic

#427

—

M₆₄

—

C₂×C₃₂

C₂²×C₁₃₇

yes

—

#367

—

#366

C₂²×C₁₃₇

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₄×C₁₃₇

yes

characteristic

100

#117

—

101

Q₃₂

—

102

Q₃₂

—

103

Q₃₂

C₂×C₄×C₁₃₇

yes

—

104

C₂×C₁₆

C₂×C₄×C₁₃₇

yes

—

105

C₂×C₁₆

D₄×C₁₃₇

yes

characteristic

106

C₂×C₁₆

D₄×C₁₃₇

yes

—

107

C₂×C₁₆

C₂³×C₁₃₇

yes

characteristic

108

#107

—

109

#106

—

110

C₃₂

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂²×C₄

—

116

C₂²×C₄

D₈×C₁₃₇

yes

characteristic

117

Q₁₆

—

118

Q₁₆

—

119

Q₁₆

—

120

C₂×C₈

C₂²⋊C₄×C₁₃₇

yes

—

121

C₂×C₈

D₈×C₁₃₇

yes

characteristic

122

C₂×C₈

SD₁₆×C₁₃₇

yes

—

123

C₂×C₈

C₂×D₄×C₁₃₇

yes

characteristic

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₁₆

C₂²⋊C₄×C₁₃₇

yes

—

128

C₁₆

C₂²×C₄×C₁₃₇

yes

characteristic

129

C₁₆

C₂×D₄×C₁₃₇

yes

characteristic

130

C₂³

D₁₆×C₁₃₇

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#102×C₁₃₇

yes

—

138

C₂×C₄

D₁₆×C₁₃₇

yes

characteristic

139

C₂×C₄

SD₃₂×C₁₃₇

yes

—

140

C₂×C₄

C₂×D₈×C₁₃₇

yes

characteristic

141

C₈

#102×C₁₃₇

yes

—

142

C₈

C₂×C₂²⋊C₄×C₁₃₇

yes

characteristic

143

C₈

C₂×SD₁₆×C₁₃₇

yes

characteristic

144

C₂²

—

145

C₂²

#357×C₁₃₇

yes

—

146

C₂²

C₂×D₁₆×C₁₃₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

#357×C₁₃₇

yes

—

150

C₄

C₂×#102×C₁₃₇

yes

characteristic

151

C₄

C₂×SD₃₂×C₁₃₇

yes

characteristic

152

C₂

—

153

C₂

C₂×#357×C₁₃₇

yes

characteristic

154

C₂

#2264×C₁₃₇

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃₇
prime radical	C₂³×C₁₃₇	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂²×C₈×C₁₃₇ ◁ C₂²×C₁₆×C₁₃₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃₇

C₂×D₁₆

commutator

C₁₆

C₂²×C₄×C₁₃₇

Frattini, cosocle

C₂×C₁₆

C₂³×C₁₃₇

prime radical

C₂³×C₁₃₇

D₁₆

LCS

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂²×C₈×C₁₃₇ ◁ C₂²×C₁₆×C₁₃₇ ◁ G

N	H ≅ G/N	type
4. #1416×C₁₃₇	C₂	semidirect
8. C₂×Q₃₂×C₁₃₇	C₄	semidirect
77. #13618	C₁₃₇	direct (N × H)
78. C₁₃₇	#13618	direct (N × H)
82. #1416	C₂×C₁₃₇	semidirect
86. C₂×Q₃₂	C₄×C₁₃₇	semidirect

H ≅ G/N

type

4. #1416×C₁₃₇

C₂

semidirect

8. C₂×Q₃₂×C₁₃₇

C₄

semidirect

77. #13618

C₁₃₇

direct (N × H)

78. C₁₃₇

#13618

direct (N × H)

82. #1416

C₂×C₁₃₇

semidirect

86. C₂×Q₃₂

C₄×C₁₃₇

semidirect

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

subgroup	index
C₂²×Q₃₂×C₁₃₇	2
C₂×M₆₄×C₁₃₇	2
#2158×C₁₃₇	2
#1416×C₁₃₇	2
#13618	137

subgroup

index

C₂²×Q₃₂×C₁₃₇

C₂×M₆₄×C₁₃₇

#2158×C₁₃₇

#1416×C₁₃₇

#13618

137

Subgroups — C₁₃₇×#13618 #46863004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#13618 #46863004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13618 #46863004