Group #46864049 — subgroups

Subgroups — C₁₃₇×#14663 #46864049

Order 35072.

1462 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 1462 conjugacy classes of subgroups — too many to draw. Only its 462 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×C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#383×C₁₃₇	C₂	1	2	yes	maximal
3	#1446×C₁₃₇	C₂	1	4	yes	maximal
4	#1445×C₁₃₇	C₂	1	8	yes	maximal
5	C₂³×Q₈×C₁₃₇	C₂²	1	2	yes	—
6	C₂×C₄⋊Q₈×C₁₃₇	—	2	8	no	—
7	C₂×C₄⋊Q₈×C₁₃₇	C₄	1	8	yes	complemented
8	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
9	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
10	#383×C₁₃₇	—	2	8	no	—
11	C₂×#95×C₁₃₇	C₂²	1	8	yes	—
12	C₂×#95×C₁₃₇	C₂²	1	4	yes	—
13	C₂×#95×C₁₃₇	C₂²	1	16	yes	—
14	C₂²×Q₈×C₁₃₇	—	2	8	no	—
15	C₂²×Q₈×C₁₃₇	—	4	8	no	—
16	C₂²×Q₈×C₁₃₇	C₂×C₄	1	8	yes	—
17	C₂³×C₄×C₁₃₇	C₂³	1	8	yes	—
18	C₂³×C₄×C₁₃₇	C₂³	1	4	yes	—
19	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
20	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈×C₁₃₇	—	2	32	no	—
22	C₂×C₄⋊C₄×C₁₃₇	—	2	16	no	—
23	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	16	yes	—
24	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₃₇	D₄	1	8	yes	—
26	#95×C₁₃₇	—	2	16	no	—
27	#95×C₁₃₇	—	2	8	no	—
28	#95×C₁₃₇	—	4	16	no	—
29	C₂⁴×C₁₃₇	C₂⁴	1	1	yes	characteristic
30	C₂×Q₈×C₁₃₇	—	2	16	no	—
31	C₂×Q₈×C₁₃₇	—	4	16	no	—
32	C₂×Q₈×C₁₃₇	—	4	32	no	—
33	C₂²×C₄×C₁₃₇	—	2	4	no	—
34	C₂²×C₄×C₁₃₇	—	2	8	no	—
35	C₂²×C₄×C₁₃₇	—	2	8	no	—
36	C₂²×C₄×C₁₃₇	—	2	16	no	—
37	C₂²×C₄×C₁₃₇	—	2	8	no	—
38	C₂²×C₄×C₁₃₇	—	2	16	no	—
39	C₂²×C₄×C₁₃₇	—	2	32	no	—
40	C₂²×C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
41	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₁₃₇	C₂×D₄	1	8	yes	—
44	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
45	C₄⋊C₄×C₁₃₇	—	2	32	no	—
46	C₄²×C₁₃₇	C₂²⋊C₄	1	16	yes	—
47	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
48	C₂³×C₁₃₇	C₂²×D₄	1	2	yes	—
49	C₂³×C₁₃₇	H(5,𝔽₂)	1	4	yes	—
50	C₂³×C₁₃₇	2₋¹⁺⁴	1	8	yes	—
51	Q₈×C₁₃₇	—	4	32	no	—
52	C₂×C₄×C₁₃₇	—	2	4	no	—
53	C₂×C₄×C₁₃₇	—	2	8	no	—
54	C₂×C₄×C₁₃₇	—	2	8	no	—
55	C₂×C₄×C₁₃₇	—	2	16	no	—
56	C₂×C₄×C₁₃₇	—	2	8	no	—
57	C₂×C₄×C₁₃₇	—	2	16	no	—
58	C₂×C₄×C₁₃₇	—	4	16	no	—
59	C₂×C₄×C₁₃₇	—	4	8	no	—
60	C₂×C₄×C₁₃₇	—	4	16	no	—
61	C₂×C₄×C₁₃₇	—	4	32	no	—
62	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	8	yes	—
63	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
64	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₁₃₇	#518	1	2	yes	—
66	C₂²×C₁₃₇	#519	1	4	yes	—
67	C₂²×C₁₃₇	#535	1	4	yes	—
68	C₂²×C₁₃₇	#536	1	8	yes	—
69	C₂²×C₁₃₇	#563	1	16	yes	—
70	C₄×C₁₃₇	—	2	4	no	—
71	C₄×C₁₃₇	—	2	8	no	—
72	C₄×C₁₃₇	—	4	8	no	—
73	C₄×C₁₃₇	—	8	8	no	—
74	C₂×C₁₃₇	#2309	1	1	yes	characteristic
75	C₂×C₁₃₇	#2310	1	2	yes	—
76	C₂×C₁₃₇	#2381	1	4	yes	—
77	C₂×C₁₃₇	#2821	1	8	yes	—
78	#14663	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₁₃₇	#14663	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²×C₄⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic
81	C₂×#383	C₂×C₁₃₇	1	2	yes	—
82	#1446	C₂×C₁₃₇	1	4	yes	—
83	#1445	C₂×C₁₃₇	1	8	yes	—
84	C₂³×Q₈	C₂²×C₁₃₇	1	2	yes	—
85	C₂×C₄⋊Q₈	—	2	8	no	—
86	C₂×C₄⋊Q₈	C₄×C₁₃₇	1	8	yes	complemented
87	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
88	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
89	#383	—	2	8	no	—
90	C₂×#95	C₂²×C₁₃₇	1	8	yes	—
91	C₂×#95	C₂²×C₁₃₇	1	4	yes	—
92	C₂×#95	C₂²×C₁₃₇	1	16	yes	—
93	C₂²×Q₈	—	2	8	no	—
94	C₂²×Q₈	—	4	8	no	—
95	C₂²×Q₈	C₂×C₄×C₁₃₇	1	8	yes	—
96	C₂³×C₄	C₂³×C₁₃₇	1	8	yes	—
97	C₂³×C₄	C₂³×C₁₃₇	1	4	yes	—
98	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
99	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
100	C₄⋊Q₈	—	2	32	no	—
101	C₂×C₄⋊C₄	—	2	16	no	—
102	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	16	yes	—
103	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
104	C₂×C₄²	D₄×C₁₃₇	1	8	yes	—
105	#95	—	2	16	no	—
106	#95	—	2	8	no	—
107	#95	—	4	16	no	—
108	C₂⁴	C₂⁴×C₁₃₇	1	1	yes	characteristic
109	C₂×Q₈	—	2	16	no	—
110	C₂×Q₈	—	4	16	no	—
111	C₂×Q₈	—	4	32	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	2	16	no	—
116	C₂²×C₄	—	2	8	no	—
117	C₂²×C₄	—	2	16	no	—
118	C₂²×C₄	—	2	32	no	—
119	C₂²×C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
120	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
121	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
122	C₂²×C₄	C₂×D₄×C₁₃₇	1	8	yes	—
123	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
124	C₄⋊C₄	—	2	32	no	—
125	C₄²	C₂²⋊C₄×C₁₃₇	1	16	yes	—
126	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
127	C₂³	C₂²×D₄×C₁₃₇	1	2	yes	—
128	C₂³	H(5,𝔽₂)×C₁₃₇	1	4	yes	—
129	C₂³	2₋¹⁺⁴×C₁₃₇	1	8	yes	—
130	Q₈	—	4	32	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	16	no	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	2	16	no	—
137	C₂×C₄	—	4	16	no	—
138	C₂×C₄	—	4	8	no	—
139	C₂×C₄	—	4	16	no	—
140	C₂×C₄	—	4	32	no	—
141	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	8	yes	—
142	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
143	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
144	C₂²	#518×C₁₃₇	1	2	yes	—
145	C₂²	#519×C₁₃₇	1	4	yes	—
146	C₂²	#535×C₁₃₇	1	4	yes	—
147	C₂²	#536×C₁₃₇	1	8	yes	—
148	C₂²	#563×C₁₃₇	1	16	yes	—
149	C₄	—	2	4	no	—
150	C₄	—	2	8	no	—
151	C₄	—	4	8	no	—
152	C₄	—	8	8	no	—
153	C₂	#2309×C₁₃₇	1	1	yes	characteristic
154	C₂	#2310×C₁₃₇	1	2	yes	—
155	C₂	#2381×C₁₃₇	1	4	yes	—
156	C₂	#2821×C₁₃₇	1	8	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Q₈×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#383×C₁₃₇

C₂

yes

maximal

#1446×C₁₃₇

C₂

yes

maximal

#1445×C₁₃₇

C₂

yes

maximal

C₂³×Q₈×C₁₃₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₁₃₇

—

C₂×C₄⋊Q₈×C₁₃₇

C₄

yes

complemented

C₂²×C₄⋊C₄×C₁₃₇

C₂²

yes

—

C₂²×C₄²×C₁₃₇

C₂²

yes

characteristic

#383×C₁₃₇

—

C₂×#95×C₁₃₇

C₂²

yes

—

C₂×#95×C₁₃₇

C₂²

yes

—

C₂×#95×C₁₃₇

C₂²

yes

—

C₂²×Q₈×C₁₃₇

—

C₂²×Q₈×C₁₃₇

—

C₂²×Q₈×C₁₃₇

C₂×C₄

yes

—

C₂³×C₄×C₁₃₇

C₂³

yes

—

C₂³×C₄×C₁₃₇

C₂³

yes

—

C₂³×C₄×C₁₃₇

C₂³

yes

—

C₂³×C₄×C₁₃₇

C₂³

yes

characteristic

C₄⋊Q₈×C₁₃₇

—

C₂×C₄⋊C₄×C₁₃₇

—

C₂×C₄⋊C₄×C₁₃₇

C₂×C₄

yes

—

C₂×C₄²×C₁₃₇

C₂×C₄

yes

—

C₂×C₄²×C₁₃₇

D₄

yes

—

#95×C₁₃₇

—

#95×C₁₃₇

—

#95×C₁₃₇

—

C₂⁴×C₁₃₇

C₂⁴

yes

characteristic

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂×D₄

yes

—

C₂²×C₄×C₁₃₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂²⋊C₄

yes

—

C₂³×C₁₃₇

C₂³×C₄

yes

characteristic

C₂³×C₁₃₇

C₂²×D₄

yes

—

C₂³×C₁₃₇

H(5,𝔽₂)

yes

—

C₂³×C₁₃₇

2₋¹⁺⁴

yes

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₃₇

C₂²×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₇

#518

yes

—

C₂²×C₁₃₇

#519

yes

—

C₂²×C₁₃₇

#535

yes

—

C₂²×C₁₃₇

#536

yes

—

C₂²×C₁₃₇

#563

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₂×C₁₃₇

#2309

yes

characteristic

C₂×C₁₃₇

#2310

yes

—

C₂×C₁₃₇

#2381

yes

—

C₂×C₁₃₇

#2821

yes

—

#14663

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#14663

yes

characteristic, direct product factor, Sylow

C₂²×C₄⋊Q₈

C₂×C₁₃₇

yes

characteristic

C₂×#383

C₂×C₁₃₇

yes

—

#1446

C₂×C₁₃₇

yes

—

#1445

C₂×C₁₃₇

yes

—

C₂³×Q₈

C₂²×C₁₃₇

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₄×C₁₃₇

yes

complemented

C₂²×C₄⋊C₄

C₂²×C₁₃₇

yes

—

C₂²×C₄²

C₂²×C₁₃₇

yes

characteristic

#383

—

C₂×#95

C₂²×C₁₃₇

yes

—

C₂×#95

C₂²×C₁₃₇

yes

—

C₂×#95

C₂²×C₁₃₇

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×C₄×C₁₃₇

yes

—

C₂³×C₄

C₂³×C₁₃₇

yes

—

C₂³×C₄

C₂³×C₁₃₇

yes

—

C₂³×C₄

C₂³×C₁₃₇

yes

—

C₂³×C₄

C₂³×C₁₃₇

yes

characteristic

100

C₄⋊Q₈

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×C₄×C₁₃₇

yes

—

103

C₂×C₄²

C₂×C₄×C₁₃₇

yes

—

104

C₂×C₄²

D₄×C₁₃₇

yes

—

105

#95

—

106

#95

—

107

#95

—

108

C₂⁴

C₂⁴×C₁₃₇

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²×C₄×C₁₃₇

yes

—

120

C₂²×C₄

C₂²×C₄×C₁₃₇

yes

—

121

C₂²×C₄

C₂²×C₄×C₁₃₇

yes

—

122

C₂²×C₄

C₂×D₄×C₁₃₇

yes

—

123

C₂²×C₄

C₂×D₄×C₁₃₇

yes

—

124

C₄⋊C₄

—

125

C₄²

C₂²⋊C₄×C₁₃₇

yes

—

126

C₂³

C₂³×C₄×C₁₃₇

yes

characteristic

127

C₂³

C₂²×D₄×C₁₃₇

yes

—

128

C₂³

H(5,𝔽₂)×C₁₃₇

yes

—

129

C₂³

2₋¹⁺⁴×C₁₃₇

yes

—

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₂×C₂²⋊C₄×C₁₃₇

yes

—

142

C₂×C₄

C₂×C₂²⋊C₄×C₁₃₇

yes

—

143

C₂²

C₂²×C₂²⋊C₄×C₁₃₇

yes

characteristic

144

C₂²

#518×C₁₃₇

yes

—

145

C₂²

#519×C₁₃₇

yes

—

146

C₂²

#535×C₁₃₇

yes

—

147

C₂²

#536×C₁₃₇

yes

—

148

C₂²

#563×C₁₃₇

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₂

#2309×C₁₃₇

yes

characteristic

154

C₂

#2310×C₁₃₇

yes

—

155

C₂

#2381×C₁₃₇

yes

—

156

C₂

#2821×C₁₃₇

yes

—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂⁴	C₂⁴×C₁₃₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂⁴×C₁₃₇

C₂⁴

commutator

C₂³

C₂³×C₄×C₁₃₇

Frattini, cosocle

C₂⁴

C₂⁴×C₁₃₇

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₁₃₇ ◁ G

N	H ≅ G/N	type
7. C₂×C₄⋊Q₈×C₁₃₇	C₄	semidirect
78. #14663	C₁₃₇	direct (N × H)
79. C₁₃₇	#14663	direct (N × H)
86. C₂×C₄⋊Q₈	C₄×C₁₃₇	semidirect

H ≅ G/N

type

7. C₂×C₄⋊Q₈×C₁₃₇

C₄

semidirect

78. #14663

C₁₃₇

direct (N × H)

79. C₁₃₇

#14663

direct (N × H)

86. C₂×C₄⋊Q₈

C₄×C₁₃₇

semidirect

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

subgroup	index
C₂²×C₄⋊Q₈×C₁₃₇	2
C₂×#383×C₁₃₇	2
#1446×C₁₃₇	2
#1445×C₁₃₇	2
#14663	137

subgroup

index

C₂²×C₄⋊Q₈×C₁₃₇

C₂×#383×C₁₃₇

#1446×C₁₃₇

#1445×C₁₃₇

#14663

137

Subgroups — C₁₃₇×#14663 #46864049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#14663 #46864049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#14663 #46864049