Group #46856532 — subgroups

Subgroups — C₁₃₇×#7146 #46856532

Order 35072.

210 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 102 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	#2163×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#322×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
4	C₂×C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₄⋊C₁₆×C₁₃₇	—	2	2	no	—
7	C₂²⋊C₁₆×C₁₃₇	—	2	2	no	—
8	#346×C₁₃₇	C₄	1	4	yes	—
9	#322×C₁₃₇	—	2	2	no	—
10	C₂²×C₈×C₁₃₇	—	2	1	no	—
11	C₂²×C₈×C₁₃₇	C₈	1	2	yes	—
12	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆×C₁₃₇	—	2	4	no	—
16	C₄×C₈×C₁₃₇	—	2	1	no	—
17	C₄×C₈×C₁₃₇	—	2	1	no	—
18	C₄×C₈×C₁₃₇	—	2	1	no	—
19	C₄×C₈×C₁₃₇	C₈	1	2	yes	—
20	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
22	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
25	C₂×C₈×C₁₃₇	—	2	4	no	—
26	C₂×C₈×C₁₃₇	—	2	2	no	—
27	C₂×C₈×C₁₃₇	—	2	2	no	—
28	C₂×C₈×C₁₃₇	—	4	1	no	—
29	C₂×C₈×C₁₃₇	—	4	1	no	—
30	C₂×C₈×C₁₃₇	—	4	2	no	—
31	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
32	C₂×C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
33	C₄²×C₁₃₇	C₄²	1	1	yes	characteristic
34	C₄²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₄²×C₁₃₇	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₁₃₇	M₁₆	1	1	yes	characteristic
37	C₁₆×C₁₃₇	—	4	4	no	—
38	C₂³×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄×C₁₃₇	—	2	1	no	—
41	C₂×C₄×C₁₃₇	—	2	1	no	—
42	C₂×C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄×C₁₃₇	#95	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₃₇	#97	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
49	C₈×C₁₃₇	—	2	4	no	—
50	C₈×C₁₃₇	—	2	2	no	—
51	C₈×C₁₃₇	—	2	2	no	—
52	C₈×C₁₃₇	—	4	2	no	—
53	C₂²×C₁₃₇	—	2	1	no	—
54	C₂²×C₁₃₇	—	2	1	no	—
55	C₂²×C₁₃₇	#336	1	1	yes	characteristic
56	C₂²×C₁₃₇	#364	1	2	yes	—
57	C₄×C₁₃₇	—	2	1	no	—
58	C₄×C₁₃₇	—	2	1	no	—
59	C₄×C₁₃₇	—	2	1	no	—
60	C₄×C₁₃₇	—	2	1	no	—
61	C₄×C₁₃₇	H(3,ℤ/4)	1	1	yes	characteristic
62	C₄×C₁₃₇	#338	1	1	yes	characteristic
63	C₄×C₁₃₇	#349	1	2	yes	—
64	C₂×C₁₃₇	—	2	2	no	—
65	C₂×C₁₃₇	#1275	1	1	yes	characteristic
66	C₂×C₁₃₇	#1376	1	1	yes	characteristic
67	C₂×C₁₃₇	#1377	1	1	yes	characteristic
68	#7146	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₁₃₇	#7146	1	1	yes	characteristic, direct product factor, Sylow
70	#2163	C₂×C₁₃₇	1	2	yes	—
71	C₂×#322	C₂×C₁₃₇	1	1	yes	characteristic
72	C₂×C₄×C₈	C₄×C₁₃₇	1	1	yes	characteristic
73	C₂×C₄×C₈	C₄×C₁₃₇	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
75	C₄⋊C₁₆	—	2	2	no	—
76	C₂²⋊C₁₆	—	2	2	no	—
77	#346	C₄×C₁₃₇	1	4	yes	—
78	#322	—	2	2	no	—
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	C₈×C₁₃₇	1	2	yes	—
81	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
82	C₂²×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
83	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
84	C₂×C₁₆	—	2	4	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	C₈×C₁₃₇	1	2	yes	—
89	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
90	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
91	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
92	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
93	C₂²×C₄	M₁₆×C₁₃₇	1	1	yes	characteristic
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
101	C₂×C₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
102	C₄²	C₄²×C₁₃₇	1	1	yes	characteristic
103	C₄²	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
104	C₄²	C₂×C₈×C₁₃₇	1	1	yes	characteristic
105	C₄²	M₁₆×C₁₃₇	1	1	yes	characteristic
106	C₁₆	—	4	4	no	—
107	C₂³	C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	#95×C₁₃₇	1	1	yes	characteristic
113	C₂×C₄	C₄×C₈×C₁₃₇	1	1	yes	characteristic
114	C₂×C₄	#97×C₁₃₇	1	1	yes	characteristic
115	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
116	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
117	C₂×C₄	C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
118	C₈	—	2	4	no	—
119	C₈	—	2	2	no	—
120	C₈	—	2	2	no	—
121	C₈	—	4	2	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#336×C₁₃₇	1	1	yes	characteristic
125	C₂²	#364×C₁₃₇	1	2	yes	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	H(3,ℤ/4)×C₁₃₇	1	1	yes	characteristic
131	C₄	#338×C₁₃₇	1	1	yes	characteristic
132	C₄	#349×C₁₃₇	1	2	yes	—
133	C₂	—	2	2	no	—
134	C₂	#1275×C₁₃₇	1	1	yes	characteristic
135	C₂	#1376×C₁₃₇	1	1	yes	characteristic
136	C₂	#1377×C₁₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2163×C₁₃₇

C₂

yes

maximal

C₂×#322×C₁₃₇

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃₇

C₄

yes

characteristic

C₂×C₄×C₈×C₁₃₇

C₄

yes

characteristic

C₂×C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₃₇

—

C₂²⋊C₁₆×C₁₃₇

—

#346×C₁₃₇

C₄

yes

—

#322×C₁₃₇

—

C₂²×C₈×C₁₃₇

—

C₂²×C₈×C₁₃₇

C₈

yes

—

C₂²×C₈×C₁₃₇

D₄

yes

characteristic

C₂²×C₈×C₁₃₇

Q₈

yes

characteristic

C₂×C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₃₇

—

C₄×C₈×C₁₃₇

—

C₄×C₈×C₁₃₇

—

C₄×C₈×C₁₃₇

—

C₄×C₈×C₁₃₇

C₈

yes

—

C₄×C₈×C₁₃₇

C₂×C₄

yes

—

C₄×C₈×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃₇

M₁₆

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃₇

C₄⋊C₄

yes

—

C₄²×C₁₃₇

C₄²

yes

characteristic

C₄²×C₁₃₇

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₈

yes

characteristic

C₄²×C₁₃₇

M₁₆

yes

characteristic

C₁₆×C₁₃₇

—

C₂³×C₁₃₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

#95

yes

characteristic

C₂×C₄×C₁₃₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₃₇

#97

yes

characteristic

C₂×C₄×C₁₃₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊C₈

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#336

yes

characteristic

C₂²×C₁₃₇

#364

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#7146

yes

characteristic, direct product factor, Sylow

#2163

C₂×C₁₃₇

yes

—

C₂×#322

C₂×C₁₃₇

yes

characteristic

C₂×C₄×C₈

C₄×C₁₃₇

yes

characteristic

C₂×C₄×C₈

C₄×C₁₃₇

yes

characteristic

C₂×C₄×C₈

C₂²×C₁₃₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

C₄×C₁₃₇

yes

—

#322

—

C₂²×C₈

—

C₂²×C₈

C₈×C₁₃₇

yes

—

C₂²×C₈

D₄×C₁₃₇

yes

characteristic

C₂²×C₈

Q₈×C₁₃₇

yes

characteristic

C₂×C₄²

C₂×C₄×C₁₃₇

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₈×C₁₃₇

yes

—

C₄×C₈

C₂×C₄×C₁₃₇

yes

—

C₄×C₈

D₄×C₁₃₇

yes

—

C₂²×C₄

C₄⋊C₄×C₁₃₇

yes

characteristic

C₂²×C₄

C₂×C₈×C₁₃₇

yes

characteristic

C₂²×C₄

M₁₆×C₁₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂²⋊C₄×C₁₃₇

yes

—

101

C₂×C₈

C₄⋊C₄×C₁₃₇

yes

—

102

C₄²

C₄²×C₁₃₇

yes

characteristic

103

C₄²

C₂²⋊C₄×C₁₃₇

yes

characteristic

104

C₄²

C₂×C₈×C₁₃₇

yes

characteristic

105

C₄²

M₁₆×C₁₃₇

yes

characteristic

106

C₁₆

—

107

C₂³

C₄⋊C₈×C₁₃₇

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#95×C₁₃₇

yes

characteristic

113

C₂×C₄

C₄×C₈×C₁₃₇

yes

characteristic

114

C₂×C₄

#97×C₁₃₇

yes

characteristic

115

C₂×C₄

C₂²⋊C₈×C₁₃₇

yes

characteristic

116

C₂×C₄

C₂²⋊C₈×C₁₃₇

yes

characteristic

117

C₂×C₄

C₄⋊C₈×C₁₃₇

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

#336×C₁₃₇

yes

characteristic

125

C₂²

#364×C₁₃₇

yes

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

H(3,ℤ/4)×C₁₃₇

yes

characteristic

131

C₄

#338×C₁₃₇

yes

characteristic

132

C₄

#349×C₁₃₇

yes

—

133

C₂

—

134

C₂

#1275×C₁₃₇

yes

characteristic

135

C₂

#1376×C₁₃₇

yes

characteristic

136

C₂

#1377×C₁₃₇

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#95
commutator	C₂×C₄	C₄×C₈×C₁₃₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₃₇
socle	C₂²×C₁₃₇	#336
prime radical	C₂³×C₁₃₇	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₃₇

#95

commutator

C₂×C₄

C₄×C₈×C₁₃₇

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₁₃₇

socle

C₂²×C₁₃₇

#336

prime radical

C₂³×C₁₃₇

C₄⋊C₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

N	H ≅ G/N	type
68. #7146	C₁₃₇	direct (N × H)
69. C₁₃₇	#7146	direct (N × H)

H ≅ G/N

type

68. #7146

C₁₃₇

direct (N × H)

69. C₁₃₇

#7146

direct (N × H)

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

subgroup	index
#2163×C₁₃₇	2
C₂×#322×C₁₃₇	2
#7146	137

subgroup

index

#2163×C₁₃₇

C₂×#322×C₁₃₇

#7146

137

Subgroups — C₁₃₇×#7146 #46856532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#7146 #46856532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7146 #46856532