Group #46861918 — subgroups

Subgroups — C₁₃₇×#12532 #46861918

Order 35072.

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

normal subgroups only

This group has 272 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	#2254×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#2240×C₁₃₇	C₂	1	2	yes	maximal
3	#1713×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1706×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1312×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#500×C₁₃₇	—	2	1	no	—
7	#500×C₁₃₇	C₂²	1	1	yes	characteristic
8	#500×C₁₃₇	C₂²	1	2	yes	—
9	#494×C₁₃₇	—	2	1	no	—
10	#493×C₁₃₇	C₂²	1	1	yes	characteristic
11	#492×C₁₃₇	—	2	1	no	—
12	#486×C₁₃₇	—	2	1	no	—
13	#367×C₁₃₇	—	2	4	no	—
14	#357×C₁₃₇	—	2	4	no	—
15	C₄×C₁₆×C₁₃₇	C₂²	1	2	yes	—
16	C₈²×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₃₇	—	4	2	no	—
18	C₂×SD₁₆×C₁₃₇	—	4	2	no	—
19	C₂×D₈×C₁₃₇	—	2	2	no	—
20	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
21	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
22	#127×C₁₃₇	—	2	1	no	—
23	C₂×C₁₆×C₁₃₇	D₄	1	4	yes	—
24	#107×C₁₃₇	—	2	2	no	—
25	#107×C₁₃₇	—	2	2	no	—
26	#107×C₁₃₇	—	2	2	no	—
27	#107×C₁₃₇	—	4	2	no	—
28	#102×C₁₃₇	—	4	2	no	—
29	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₁₃₇	—	4	1	no	—
33	C₂×Q₈×C₁₃₇	—	4	2	no	—
34	C₂×D₄×C₁₃₇	—	4	1	no	—
35	C₂×D₄×C₁₃₇	—	4	1	no	—
36	C₂×D₄×C₁₃₇	—	4	1	no	—
37	Q₁₆×C₁₃₇	—	8	4	no	—
38	SD₁₆×C₁₃₇	—	8	4	no	—
39	D₈×C₁₃₇	—	4	4	no	—
40	C₂×C₈×C₁₃₇	D₈	1	2	yes	—
41	C₂×C₈×C₁₃₇	D₈	1	2	yes	—
42	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
43	C₄⋊C₄×C₁₃₇	—	4	1	no	—
44	C₄⋊C₄×C₁₃₇	—	4	2	no	—
45	C₄⋊C₄×C₁₃₇	—	4	1	no	—
46	C₄⋊C₄×C₁₃₇	—	4	2	no	—
47	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₁₃₇	—	2	4	no	—
49	C₂³×C₁₃₇	—	8	1	no	—
50	Q₈×C₁₃₇	—	8	2	no	—
51	Q₈×C₁₃₇	—	8	4	no	—
52	D₄×C₁₃₇	—	8	2	no	—
53	D₄×C₁₃₇	—	8	2	no	—
54	D₄×C₁₃₇	—	8	2	no	—
55	C₂×C₄×C₁₃₇	—	8	1	no	—
56	C₂×C₄×C₁₃₇	—	8	2	no	—
57	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
60	C₈×C₁₃₇	—	2	2	no	—
61	C₈×C₁₃₇	SD₃₂	1	4	yes	—
62	C₈×C₁₃₇	#137	1	4	yes	—
63	C₂²×C₁₃₇	—	16	1	no	—
64	C₂²×C₁₃₇	—	16	1	no	—
65	C₂²×C₁₃₇	—	16	1	no	—
66	C₂²×C₁₃₇	#493	1	1	yes	characteristic
67	C₄×C₁₃₇	—	16	1	no	—
68	C₄×C₁₃₇	—	16	2	no	—
69	C₄×C₁₃₇	#497	1	2	yes	—
70	C₄×C₁₃₇	C₂×SD₃₂	1	2	yes	—
71	C₄×C₁₃₇	#509	1	2	yes	—
72	C₂×C₁₃₇	—	32	1	no	—
73	C₂×C₁₃₇	#1722	1	1	yes	characteristic
74	C₂×C₁₃₇	#2248	1	1	yes	characteristic
75	C₂×C₁₃₇	#2250	1	1	yes	characteristic
76	#12532	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₁₃₇	#12532	1	1	yes	characteristic, direct product factor, Sylow
78	#2254	C₂×C₁₃₇	1	2	yes	complemented
79	#2240	C₂×C₁₃₇	1	2	yes	—
80	#1713	C₂×C₁₃₇	1	1	yes	characteristic, complemented
81	#1706	C₂×C₁₃₇	1	1	yes	characteristic
82	#1312	C₂×C₁₃₇	1	1	yes	characteristic, complemented
83	#500	—	2	1	no	—
84	#500	C₂²×C₁₃₇	1	1	yes	characteristic
85	#500	C₂²×C₁₃₇	1	2	yes	—
86	#494	—	2	1	no	—
87	#493	C₂²×C₁₃₇	1	1	yes	characteristic
88	#492	—	2	1	no	—
89	#486	—	2	1	no	—
90	#367	—	2	4	no	—
91	#357	—	2	4	no	—
92	C₄×C₁₆	C₂²×C₁₃₇	1	2	yes	—
93	C₈²	C₂²×C₁₃₇	1	1	yes	characteristic
94	C₂×Q₁₆	—	4	2	no	—
95	C₂×SD₁₆	—	4	2	no	—
96	C₂×D₈	—	2	2	no	—
97	C₄⋊Q₈	—	2	1	no	—
98	C₄⋊Q₈	—	2	2	no	—
99	#127	—	2	1	no	—
100	C₂×C₁₆	D₄×C₁₃₇	1	4	yes	—
101	#107	—	2	2	no	—
102	#107	—	2	2	no	—
103	#107	—	2	2	no	—
104	#107	—	4	2	no	—
105	#102	—	4	2	no	—
106	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
107	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
108	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
109	C₂×Q₈	—	4	1	no	—
110	C₂×Q₈	—	4	2	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	4	1	no	—
113	C₂×D₄	—	4	1	no	—
114	Q₁₆	—	8	4	no	—
115	SD₁₆	—	8	4	no	—
116	D₈	—	4	4	no	—
117	C₂×C₈	D₈×C₁₃₇	1	2	yes	—
118	C₂×C₈	D₈×C₁₃₇	1	2	yes	—
119	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
125	C₁₆	—	2	4	no	—
126	C₂³	—	8	1	no	—
127	Q₈	—	8	2	no	—
128	Q₈	—	8	4	no	—
129	D₄	—	8	2	no	—
130	D₄	—	8	2	no	—
131	D₄	—	8	2	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
135	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
136	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	SD₃₂×C₁₃₇	1	4	yes	—
139	C₈	#137×C₁₃₇	1	4	yes	—
140	C₂²	—	16	1	no	—
141	C₂²	—	16	1	no	—
142	C₂²	—	16	1	no	—
143	C₂²	#493×C₁₃₇	1	1	yes	characteristic
144	C₄	—	16	1	no	—
145	C₄	—	16	2	no	—
146	C₄	#497×C₁₃₇	1	2	yes	—
147	C₄	C₂×SD₃₂×C₁₃₇	1	2	yes	—
148	C₄	#509×C₁₃₇	1	2	yes	—
149	C₂	—	32	1	no	—
150	C₂	#1722×C₁₃₇	1	1	yes	characteristic
151	C₂	#2248×C₁₃₇	1	1	yes	characteristic
152	C₂	#2250×C₁₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2254×C₁₃₇

C₂

yes

maximal, complemented

#2240×C₁₃₇

C₂

yes

maximal

#1713×C₁₃₇

C₂

yes

maximal, characteristic, complemented

#1706×C₁₃₇

C₂

yes

maximal, characteristic

#1312×C₁₃₇

C₂

yes

maximal, characteristic, complemented

#500×C₁₃₇

—

#500×C₁₃₇

C₂²

yes

characteristic

#500×C₁₃₇

C₂²

yes

—

#494×C₁₃₇

—

#493×C₁₃₇

C₂²

yes

characteristic

#492×C₁₃₇

—

#486×C₁₃₇

—

#367×C₁₃₇

—

#357×C₁₃₇

—

C₄×C₁₆×C₁₃₇

C₂²

yes

—

C₈²×C₁₃₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₃₇

—

C₂×SD₁₆×C₁₃₇

—

C₂×D₈×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

—

#127×C₁₃₇

—

C₂×C₁₆×C₁₃₇

D₄

yes

—

#107×C₁₃₇

—

#107×C₁₃₇

—

#107×C₁₃₇

—

#107×C₁₃₇

—

#102×C₁₃₇

—

C₄×C₈×C₁₃₇

D₄

yes

characteristic

C₄×C₈×C₁₃₇

D₄

yes

characteristic

C₄×C₈×C₁₃₇

C₂³

yes

characteristic

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

Q₁₆×C₁₃₇

—

SD₁₆×C₁₃₇

—

D₈×C₁₃₇

—

C₂×C₈×C₁₃₇

D₈

yes

—

C₂×C₈×C₁₃₇

D₈

yes

—

C₂×C₈×C₁₃₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂×D₄

yes

characteristic

C₁₆×C₁₃₇

—

C₂³×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

#127

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₈

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

SD₃₂

yes

—

C₈×C₁₃₇

#137

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#493

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#497

yes

—

C₄×C₁₃₇

C₂×SD₃₂

yes

—

C₄×C₁₃₇

#509

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#12532

yes

characteristic, direct product factor, Sylow

#2254

C₂×C₁₃₇

yes

complemented

#2240

C₂×C₁₃₇

yes

—

#1713

C₂×C₁₃₇

yes

characteristic, complemented

#1706

C₂×C₁₃₇

yes

characteristic

#1312

C₂×C₁₃₇

yes

characteristic, complemented

#500

—

#500

C₂²×C₁₃₇

yes

characteristic

#500

C₂²×C₁₃₇

yes

—

#494

—

#493

C₂²×C₁₃₇

yes

—

—

—

—

C₄×C₁₆

C₂²×C₁₃₇

yes

—

C₈²

C₂²×C₁₃₇

yes

characteristic

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

100

C₂×C₁₆

D₄×C₁₃₇

yes

—

101

#107

—

102

#107

—

103

#107

—

104

#107

—

105

#102

—

106

C₄×C₈

D₄×C₁₃₇

yes

characteristic

107

C₄×C₈

D₄×C₁₃₇

yes

characteristic

108

C₄×C₈

C₂³×C₁₃₇

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

Q₁₆

—

115

SD₁₆

—

116

D₈

—

117

C₂×C₈

D₈×C₁₃₇

yes

—

118

C₂×C₈

D₈×C₁₃₇

yes

—

119

C₂×C₈

C₂×D₄×C₁₃₇

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

C₂×D₄×C₁₃₇

yes

characteristic

125

C₁₆

—

126

C₂³

—

127

Q₈

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#127×C₁₃₇

yes

characteristic

135

C₂×C₄

C₂×D₈×C₁₃₇

yes

characteristic

136

C₂×C₄

C₂×D₈×C₁₃₇

yes

characteristic

137

C₈

—

138

C₈

SD₃₂×C₁₃₇

yes

—

139

C₈

#137×C₁₃₇

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#493×C₁₃₇

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

#497×C₁₃₇

yes

—

147

C₄

C₂×SD₃₂×C₁₃₇

yes

—

148

C₄

#509×C₁₃₇

yes

—

149

C₂

—

150

C₂

#1722×C₁₃₇

yes

characteristic

151

C₂

#2248×C₁₃₇

yes

characteristic

152

C₂

#2250×C₁₃₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃₇
prime radical	#493×C₁₃₇	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ C₈²×C₁₃₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃₇

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₃₇

prime radical

#493×C₁₃₇

C₂²

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ C₈²×C₁₃₇ ◁ G

N	H ≅ G/N	type
1. #2254×C₁₃₇	C₂	semidirect
3. #1713×C₁₃₇	C₂	semidirect
5. #1312×C₁₃₇	C₂	semidirect
76. #12532	C₁₃₇	direct (N × H)
77. C₁₃₇	#12532	direct (N × H)
78. #2254	C₂×C₁₃₇	semidirect
80. #1713	C₂×C₁₃₇	semidirect
82. #1312	C₂×C₁₃₇	semidirect

H ≅ G/N

type

1. #2254×C₁₃₇

C₂

semidirect

3. #1713×C₁₃₇

C₂

semidirect

5. #1312×C₁₃₇

C₂

semidirect

76. #12532

C₁₃₇

direct (N × H)

77. C₁₃₇

#12532

direct (N × H)

78. #2254

C₂×C₁₃₇

semidirect

80. #1713

C₂×C₁₃₇

semidirect

82. #1312

C₂×C₁₃₇

semidirect

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

subgroup	index
#2254×C₁₃₇	2
#2240×C₁₃₇	2
#1713×C₁₃₇	2
#1706×C₁₃₇	2
#1312×C₁₃₇	2
#12532	137

subgroup

index

#2254×C₁₃₇

#2240×C₁₃₇

#1713×C₁₃₇

#1706×C₁₃₇

#1312×C₁₃₇

#12532

137

Subgroups — C₁₃₇×#12532 #46861918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#12532 #46861918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#12532 #46861918