Group #46860723 — subgroups

Subgroups — C₁₃₇×#11337 #46860723

Order 35072.

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

normal subgroups only

This group has 686 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	#2911×C₁₃₇	C₂	1	3	yes	maximal
2	C₂×#338×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1729×C₁₃₇	C₂	1	3	yes	maximal
4	C₂²×M₁₆×C₁₃₇	C₂²	1	3	yes	—
5	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₂×#117×C₁₃₇	C₂²	1	3	yes	—
7	#428×C₁₃₇	—	2	12	no	—
8	#427×C₁₃₇	—	2	6	no	—
9	#338×C₁₃₇	C₄	1	4	yes	complemented
10	#336×C₁₃₇	—	2	6	no	—
11	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
12	C₂×M₁₆×C₁₃₇	—	2	6	no	—
13	C₂×M₁₆×C₁₃₇	—	2	6	no	—
14	C₂×M₁₆×C₁₃₇	C₂×C₄	1	6	yes	—
15	C₂×M₁₆×C₁₃₇	Q₈	1	6	yes	—
16	C₂²×C₈×C₁₃₇	—	2	3	no	—
17	C₂²×C₈×C₁₃₇	—	2	3	no	—
18	C₂²×C₈×C₁₃₇	D₄	1	6	yes	—
19	C₂²×C₈×C₁₃₇	Q₈	1	2	yes	—
20	#117×C₁₃₇	—	2	6	no	—
21	#117×C₁₃₇	—	2	6	no	—
22	C₂×C₄⋊C₄×C₁₃₇	D₄	1	6	yes	—
23	C₂×C₂²⋊C₄×C₁₃₇	—	2	3	no	—
24	C₂×C₄²×C₁₃₇	—	2	3	no	—
25	#107×C₁₃₇	—	2	12	no	—
26	#106×C₁₃₇	—	2	12	no	—
27	C₂⁴×C₁₃₇	#41	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₇	—	2	6	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	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₇	#41	1	3	yes	—
35	M₁₆×C₁₃₇	—	2	12	no	—
36	M₁₆×C₁₃₇	—	2	12	no	—
37	C₂×C₈×C₁₃₇	—	2	6	no	—
38	C₂×C₈×C₁₃₇	—	2	6	no	—
39	C₂×C₈×C₁₃₇	—	2	6	no	—
40	C₂×C₈×C₁₃₇	—	4	3	no	—
41	C₂×C₈×C₁₃₇	—	4	1	no	—
42	C₂×C₈×C₁₃₇	C₄⋊C₄	1	12	yes	—
43	C₄⋊C₄×C₁₃₇	—	2	12	no	—
44	C₂²⋊C₄×C₁₃₇	—	4	6	no	—
45	C₄²×C₁₃₇	—	4	3	no	—
46	C₄²×C₁₃₇	—	4	3	no	—
47	C₂³×C₁₃₇	—	4	2	no	—
48	C₂³×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
49	C₂³×C₁₃₇	#121	1	3	yes	—
50	C₂³×C₁₃₇	#125	1	3	yes	—
51	C₂×C₄×C₁₃₇	—	4	12	no	—
52	C₂×C₄×C₁₃₇	—	4	6	no	—
53	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	3	yes	—
54	C₂×C₄×C₁₃₇	C₄×D₄	1	3	yes	—
55	C₂×C₄×C₁₃₇	#120	1	3	yes	—
56	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	6	yes	—
57	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	3	yes	—
58	C₂×C₄×C₁₃₇	#123	1	3	yes	—
59	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
60	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	3	yes	—
61	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	3	yes	—
62	C₈×C₁₃₇	—	2	12	no	—
63	C₈×C₁₃₇	—	4	2	no	—
64	C₂²×C₁₃₇	—	2	6	no	—
65	C₂²×C₁₃₇	—	4	1	no	—
66	C₂²×C₁₃₇	—	4	3	no	—
67	C₂²×C₁₃₇	#389	1	3	yes	—
68	C₂²×C₁₃₇	#392	1	1	yes	characteristic
69	C₂²×C₁₃₇	#398	1	3	yes	—
70	C₄×C₁₃₇	—	8	6	no	—
71	C₄×C₁₃₇	#384	1	1	yes	characteristic
72	C₄×C₁₃₇	#385	1	3	yes	—
73	C₄×C₁₃₇	#395	1	3	yes	—
74	C₄×C₁₃₇	#399	1	1	yes	characteristic
75	C₂×C₁₃₇	—	2	3	no	—
76	C₂×C₁₃₇	—	2	3	no	—
77	C₂×C₁₃₇	#1444	1	1	yes	characteristic
78	C₂×C₁₃₇	#2017	1	2	yes	—
79	#11337	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₁₃₇	#11337	1	1	yes	characteristic, direct product factor, Sylow
81	#2911	C₂×C₁₃₇	1	3	yes	—
82	C₂×#338	C₂×C₁₃₇	1	1	yes	characteristic
83	#1729	C₂×C₁₃₇	1	3	yes	—
84	C₂²×M₁₆	C₂²×C₁₃₇	1	3	yes	—
85	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
86	C₂×#117	C₂²×C₁₃₇	1	3	yes	—
87	#428	—	2	12	no	—
88	#427	—	2	6	no	—
89	#338	C₄×C₁₃₇	1	4	yes	complemented
90	#336	—	2	6	no	—
91	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
92	C₂×M₁₆	—	2	6	no	—
93	C₂×M₁₆	—	2	6	no	—
94	C₂×M₁₆	C₂×C₄×C₁₃₇	1	6	yes	—
95	C₂×M₁₆	Q₈×C₁₃₇	1	6	yes	—
96	C₂²×C₈	—	2	3	no	—
97	C₂²×C₈	—	2	3	no	—
98	C₂²×C₈	D₄×C₁₃₇	1	6	yes	—
99	C₂²×C₈	Q₈×C₁₃₇	1	2	yes	—
100	#117	—	2	6	no	—
101	#117	—	2	6	no	—
102	C₂×C₄⋊C₄	D₄×C₁₃₇	1	6	yes	—
103	C₂×C₂²⋊C₄	—	2	3	no	—
104	C₂×C₄²	—	2	3	no	—
105	#107	—	2	12	no	—
106	#106	—	2	12	no	—
107	C₂⁴	#41×C₁₃₇	1	1	yes	characteristic
108	C₂²×C₄	—	2	6	no	—
109	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
110	C₂²×C₄	C₂×D₄×C₁₃₇	1	3	yes	—
111	C₂²×C₄	C₂×D₄×C₁₃₇	1	3	yes	—
112	C₂²×C₄	C₂×Q₈×C₁₃₇	1	3	yes	—
113	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
114	C₂²×C₄	#41×C₁₃₇	1	3	yes	—
115	M₁₆	—	2	12	no	—
116	M₁₆	—	2	12	no	—
117	C₂×C₈	—	2	6	no	—
118	C₂×C₈	—	2	6	no	—
119	C₂×C₈	—	2	6	no	—
120	C₂×C₈	—	4	3	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	C₄⋊C₄×C₁₃₇	1	12	yes	—
123	C₄⋊C₄	—	2	12	no	—
124	C₂²⋊C₄	—	4	6	no	—
125	C₄²	—	4	3	no	—
126	C₄²	—	4	3	no	—
127	C₂³	—	4	2	no	—
128	C₂³	C₄×Q₈×C₁₃₇	1	1	yes	characteristic
129	C₂³	#121×C₁₃₇	1	3	yes	—
130	C₂³	#125×C₁₃₇	1	3	yes	—
131	C₂×C₄	—	4	12	no	—
132	C₂×C₄	—	4	6	no	—
133	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	3	yes	—
134	C₂×C₄	C₄×D₄×C₁₃₇	1	3	yes	—
135	C₂×C₄	#120×C₁₃₇	1	3	yes	—
136	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	6	yes	—
137	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	3	yes	—
138	C₂×C₄	#123×C₁₃₇	1	3	yes	—
139	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
140	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	3	yes	—
141	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	3	yes	—
142	C₈	—	2	12	no	—
143	C₈	—	4	2	no	—
144	C₂²	—	2	6	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	4	3	no	—
147	C₂²	#389×C₁₃₇	1	3	yes	—
148	C₂²	#392×C₁₃₇	1	1	yes	characteristic
149	C₂²	#398×C₁₃₇	1	3	yes	—
150	C₄	—	8	6	no	—
151	C₄	#384×C₁₃₇	1	1	yes	characteristic
152	C₄	#385×C₁₃₇	1	3	yes	—
153	C₄	#395×C₁₃₇	1	3	yes	—
154	C₄	#399×C₁₃₇	1	1	yes	characteristic
155	C₂	—	2	3	no	—
156	C₂	—	2	3	no	—
157	C₂	#1444×C₁₃₇	1	1	yes	characteristic
158	C₂	#2017×C₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#2911×C₁₃₇

C₂

yes

maximal

C₂×#338×C₁₃₇

C₂

yes

maximal, characteristic

#1729×C₁₃₇

C₂

yes

maximal

C₂²×M₁₆×C₁₃₇

C₂²

yes

—

C₂³×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×#117×C₁₃₇

C₂²

yes

—

#428×C₁₃₇

—

#427×C₁₃₇

—

#338×C₁₃₇

C₄

yes

complemented

#336×C₁₃₇

—

C₂³×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×M₁₆×C₁₃₇

—

C₂×M₁₆×C₁₃₇

—

C₂×M₁₆×C₁₃₇

C₂×C₄

yes

—

C₂×M₁₆×C₁₃₇

Q₈

yes

—

C₂²×C₈×C₁₃₇

—

C₂²×C₈×C₁₃₇

—

C₂²×C₈×C₁₃₇

D₄

yes

—

C₂²×C₈×C₁₃₇

Q₈

yes

—

#117×C₁₃₇

—

#117×C₁₃₇

—

C₂×C₄⋊C₄×C₁₃₇

D₄

yes

—

C₂×C₂²⋊C₄×C₁₃₇

—

C₂×C₄²×C₁₃₇

—

#107×C₁₃₇

—

#106×C₁₃₇

—

C₂⁴×C₁₃₇

#41

yes

characteristic

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

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃₇

#41

yes

—

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

—

C₄⋊C₄×C₁₃₇

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

—

C₂³×C₁₃₇

—

C₂³×C₁₃₇

C₄×Q₈

yes

characteristic

C₂³×C₁₃₇

#121

yes

—

C₂³×C₁₃₇

#125

yes

—

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

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃₇

#123

yes

—

C₂×C₄×C₁₃₇

#127

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#389

yes

—

C₂²×C₁₃₇

#392

yes

characteristic

C₂²×C₁₃₇

#398

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#384

yes

characteristic

C₄×C₁₃₇

#385

yes

—

C₄×C₁₃₇

#395

yes

—

C₄×C₁₃₇

#399

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#1444

yes

characteristic

C₂×C₁₃₇

#2017

yes

—

#11337

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#11337

yes

characteristic, direct product factor, Sylow

#2911

C₂×C₁₃₇

yes

—

C₂×#338

C₂×C₁₃₇

yes

characteristic

#1729

C₂×C₁₃₇

yes

—

C₂²×M₁₆

C₂²×C₁₃₇

yes

—

C₂³×C₈

C₂²×C₁₃₇

yes

characteristic

C₂×#117

C₂²×C₁₃₇

yes

—

#428

—

#427

—

#338

C₄×C₁₃₇

yes

complemented

#336

—

C₂³×C₄

C₂³×C₁₃₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄×C₁₃₇

yes

—

C₂×M₁₆

Q₈×C₁₃₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄×C₁₃₇

yes

—

C₂²×C₈

Q₈×C₁₃₇

yes

—

100

#117

—

101

#117

—

102

C₂×C₄⋊C₄

D₄×C₁₃₇

yes

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

—

105

#107

—

106

#106

—

107

C₂⁴

#41×C₁₃₇

yes

characteristic

108

C₂²×C₄

—

109

C₂²×C₄

C₂²×C₄×C₁₃₇

yes

characteristic

110

C₂²×C₄

C₂×D₄×C₁₃₇

yes

—

111

C₂²×C₄

C₂×D₄×C₁₃₇

yes

—

112

C₂²×C₄

C₂×Q₈×C₁₃₇

yes

—

113

C₂²×C₄

C₂×Q₈×C₁₃₇

yes

characteristic

114

C₂²×C₄

#41×C₁₃₇

yes

—

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₄⋊C₄×C₁₃₇

yes

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₂³

—

128

C₂³

C₄×Q₈×C₁₃₇

yes

characteristic

129

C₂³

#121×C₁₃₇

yes

—

130

C₂³

#125×C₁₃₇

yes

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×C₄⋊C₄×C₁₃₇

yes

—

134

C₂×C₄

C₄×D₄×C₁₃₇

yes

—

135

C₂×C₄

#120×C₁₃₇

yes

—

136

C₂×C₄

C₂²⋊Q₈×C₁₃₇

yes

—

137

C₂×C₄

C₂²⋊Q₈×C₁₃₇

yes

—

138

C₂×C₄

#123×C₁₃₇

yes

—

139

C₂×C₄

#127×C₁₃₇

yes

characteristic

140

C₂×C₄

C₄⋊Q₈×C₁₃₇

yes

—

141

C₂×C₄

C₄⋊Q₈×C₁₃₇

yes

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#389×C₁₃₇

yes

—

148

C₂²

#392×C₁₃₇

yes

characteristic

149

C₂²

#398×C₁₃₇

yes

—

150

C₄

—

151

C₄

#384×C₁₃₇

yes

characteristic

152

C₄

#385×C₁₃₇

yes

—

153

C₄

#395×C₁₃₇

yes

—

154

C₄

#399×C₁₃₇

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#1444×C₁₃₇

yes

characteristic

158

C₂

#2017×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	#392
prime radical	C₂⁴×C₁₃₇	#41
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

C₂²×C₁₃₇

C₂⁴×C₁₃₇

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

N	H ≅ G/N	type
9. #338×C₁₃₇	C₄	semidirect
79. #11337	C₁₃₇	direct (N × H)
80. C₁₃₇	#11337	direct (N × H)
89. #338	C₄×C₁₃₇	semidirect

H ≅ G/N

type

9. #338×C₁₃₇

C₄

semidirect

79. #11337

C₁₃₇

direct (N × H)

80. C₁₃₇

#11337

direct (N × H)

89. #338

C₄×C₁₃₇

semidirect

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

subgroup	index
#2911×C₁₃₇	2
C₂×#338×C₁₃₇	2
#1729×C₁₃₇	2
#11337	137

subgroup

index

#2911×C₁₃₇

C₂×#338×C₁₃₇

#1729×C₁₃₇

#11337

137

Subgroups — C₁₃₇×#11337 #46860723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#11337 #46860723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#11337 #46860723