Group #36324854 — subgroups

Subgroups — #36324854

Order 21376.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×Dic₁₆₇⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₁₆₇	C₂	1	1	yes	maximal, characteristic
4	#24923023	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₆₇	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₆₇	C₂²	1	2	yes	—
11	Dic₁₆₇⋊C₈	—	2	4	no	—
12	C₄×C₁₆₇⋊C₈	—	2	2	no	—
13	C₁₆₇⋊#95	C₄	1	2	yes	—
14	C₂²×C₈×C₁₆₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₆₇	—	2	2	no	—
17	C₂²×Dic₁₆₇	C₂×C₄	1	2	yes	—
18	C₂²×Dic₁₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₁₆₇⋊C₈	—	2	1	no	—
20	C₂×C₁₆₇⋊C₈	—	2	1	no	—
21	C₂×C₁₆₇⋊C₈	—	2	1	no	—
22	C₂×C₁₆₇⋊C₈	—	2	1	no	—
23	C₂×C₁₆₇⋊C₈	D₄	1	2	yes	—
24	C₂×C₁₆₇⋊C₈	Q₈	1	2	yes	—
25	C₄⋊Dic₁₆₇	—	2	2	no	—
26	C₄⋊Dic₁₆₇	C₈	1	4	yes	complemented
27	C₄×Dic₁₆₇	—	2	2	no	—
28	C₄×Dic₁₆₇	—	2	2	no	—
29	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₆₇	—	2	2	no	—
33	C₂×C₈×C₁₆₇	—	2	2	no	—
34	C₂×C₈×C₁₆₇	—	2	2	no	—
35	C₄²×C₁₆₇	D₄	1	2	yes	—
36	C₄²×C₁₆₇	Q₈	1	2	yes	—
37	C₂×Dic₁₆₇	—	2	2	no	—
38	C₂×Dic₁₆₇	—	2	2	no	—
39	C₂×Dic₁₆₇	—	2	2	no	—
40	C₂×Dic₁₆₇	—	2	2	no	—
41	C₂×Dic₁₆₇	—	2	2	no	—
42	C₂×Dic₁₆₇	C₂×C₈	1	4	yes	—
43	C₁₆₇⋊C₈	—	2	2	no	—
44	C₁₆₇⋊C₈	—	2	2	no	—
45	C₂³×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₆₇	—	2	1	no	—
47	C₂×C₄×C₁₆₇	—	2	1	no	—
48	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₆₇	C₂×C₈	1	2	yes	—
51	C₂×C₄×C₁₆₇	M₁₆	1	2	yes	—
52	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
57	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
58	C₈×C₁₆₇	—	4	2	no	—
59	C₂²×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₆₇	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₁₆₇	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₁₆₇	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₁₆₇	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₁₆₇	#131	1	1	yes	characteristic
65	C₂²×C₁₆₇	#131	1	1	yes	characteristic
66	C₄×C₁₆₇	—	2	2	no	—
67	C₄×C₁₆₇	C₄⋊C₈	1	4	yes	complemented
68	C₄×C₁₆₇	#121	1	1	yes	characteristic
69	C₄×C₁₆₇	C₂²⋊Q₈	1	1	yes	characteristic
70	C₄×C₁₆₇	#125	1	1	yes	characteristic
71	C₄×C₁₆₇	C₄⋊Q₈	1	1	yes	characteristic
72	Dic₁₆₇	—	2	4	no	—
73	Dic₁₆₇	—	4	2	no	—
74	C₂×C₁₆₇	#389	1	1	yes	characteristic
75	C₂×C₁₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₁₆₇	#424	1	1	yes	characteristic
77	C₂×C₁₆₇	C₈×D₄	1	1	yes	characteristic
78	C₂×C₁₆₇	#436	1	1	yes	characteristic
79	C₂×C₁₆₇	C₈×Q₈	1	1	yes	characteristic
80	C₂×C₁₆₇	#446	1	1	yes	characteristic
81	C₁₆₇	#1939	1	1	yes	characteristic, complemented, Sylow
82	#1939	—	167	1	no	maximal, Sylow
83	C₂×C₄⋊C₈	—	167	2	no	—
84	C₂×C₄⋊C₈	D₁₆₇	1	1	yes	characteristic
85	C₂×C₄×C₈	—	167	1	no	—
86	C₄×C₄⋊C₄	—	167	1	no	—
87	#336	—	167	2	no	—
88	C₂²×C₈	—	167	1	no	—
89	C₂²×C₈	—	167	1	no	—
90	C₂²×C₈	C₂×D₁₆₇	1	2	yes	—
91	C₂×C₄⋊C₄	—	167	1	no	—
92	C₂×C₄⋊C₄	—	167	1	no	—
93	C₂×C₄²	—	167	2	no	—
94	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	334	4	no	—
97	C₄×C₈	—	334	2	no	—
98	#95	—	167	2	no	—
99	C₂²×C₄	—	167	2	no	—
100	C₂²×C₄	—	167	2	no	—
101	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
102	C₂²×C₄	C₄×D₁₆₇	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	334	1	no	—
108	C₂×C₈	—	334	1	no	—
109	C₂×C₈	—	334	1	no	—
110	C₂×C₈	—	334	1	no	—
111	C₂×C₈	—	167	2	no	—
112	C₂×C₈	—	167	2	no	—
113	C₄⋊C₄	—	334	2	no	—
114	C₄⋊C₄	—	167	4	no	—
115	C₄²	—	334	2	no	—
116	C₄²	—	334	2	no	—
117	C₄²	C₂²⋊D₁₆₇	1	2	yes	—
118	C₄²	Dic₃₃₄	1	2	yes	—
119	C₂³	C₂×C₄×D₁₆₇	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	334	2	no	—
123	C₂×C₄	—	334	2	no	—
124	C₂×C₄	—	334	2	no	—
125	C₂×C₄	—	334	2	no	—
126	C₂×C₄	—	334	2	no	—
127	C₂×C₄	—	167	4	no	—
128	C₂×C₄	Dic₁₆₇⋊C₄	1	2	yes	—
129	C₂×C₄	Dic₁₆₇⋊C₄	1	2	yes	—
130	C₂×C₄	C₈×D₁₆₇	1	2	yes	—
131	C₂×C₄	#14581933	1	2	yes	—
132	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
134	C₂×C₄	Q₈×D₁₆₇	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₃₃₄	1	1	yes	characteristic
136	C₂×C₄	#14581949	1	1	yes	characteristic
137	C₂×C₄	#14581951	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	334	2	no	—
140	C₈	—	334	2	no	—
141	C₂²	C₂×Dic₁₆₇⋊C₄	1	1	yes	characteristic
142	C₂²	#19128794	1	1	yes	characteristic
143	C₂²	#19128797	1	1	yes	characteristic
144	C₂²	C₂×C₈×D₁₆₇	1	1	yes	characteristic
145	C₂²	C₂×#14581933	1	1	yes	characteristic
146	C₂²	#19128840	1	1	yes	characteristic
147	C₂²	#19128840	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	334	4	no	—
150	C₄	—	668	2	no	—
151	C₄	Dic₁₆₇⋊C₈	1	4	yes	complemented
152	C₄	#19128803	1	1	yes	characteristic
153	C₄	#19128808	1	1	yes	characteristic
154	C₄	#19128823	1	1	yes	characteristic
155	C₄	#19128831	1	1	yes	characteristic
156	C₂	#24923165	1	1	yes	characteristic
157	C₂	C₂×Dic₁₆₇⋊C₈	1	1	yes	characteristic
158	C₂	#24923297	1	1	yes	characteristic
159	C₂	#24923334	1	1	yes	characteristic
160	C₂	#24923346	1	1	yes	characteristic
161	C₂	#24923393	1	1	yes	characteristic
162	C₂	#24923396	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₆₇⋊C₈

C₂

yes

maximal

C₂×C₄×C₁₆₇⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₆₇

C₂

yes

maximal, characteristic

#24923023

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₆₇

C₂

yes

maximal, characteristic

C₂²×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₂²×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₆₇

C₄

yes

characteristic

C₂×C₄⋊Dic₁₆₇

C₄

yes

characteristic

C₂×C₄×Dic₁₆₇

C₂²

yes

—

Dic₁₆₇⋊C₈

—

C₄×C₁₆₇⋊C₈

—

C₁₆₇⋊#95

C₄

yes

—

C₂²×C₈×C₁₆₇

C₂²

yes

—

C₂×C₄²×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₇

—

C₂²×Dic₁₆₇

C₂×C₄

yes

—

C₂²×Dic₁₆₇

C₂×C₄

yes

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

D₄

yes

—

C₂×C₁₆₇⋊C₈

Q₈

yes

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

C₈

yes

complemented

C₄×Dic₁₆₇

—

C₄×Dic₁₆₇

—

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₄²×C₁₆₇

D₄

yes

—

C₄²×C₁₆₇

Q₈

yes

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

C₂×C₈

yes

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₇

C₂×C₈

yes

—

C₂×C₄×C₁₆₇

M₁₆

yes

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₇

C₂×Q₈

yes

C₂×C₄×C₁₆₇

yes

C₂×C₄×C₁₆₇

yes

C₈×C₁₆₇

—

C₂²×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₆₇

C₄×D₄

yes

characteristic

C₂²×C₁₆₇

C₄×Q₈

yes

characteristic

C₂²×C₁₆₇

C₂²×C₈

yes

characteristic

C₂²×C₁₆₇

C₂×M₁₆

yes

C₂²×C₁₆₇

yes

C₂²×C₁₆₇

yes

C₄×C₁₆₇

—

C₄×C₁₆₇

C₄⋊C₈

yes

complemented

C₄×C₁₆₇

#121

yes

characteristic

C₄×C₁₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₆₇

#125

yes

characteristic

C₄×C₁₆₇

C₄⋊Q₈

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

#389

yes

characteristic

C₂×C₁₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₆₇

#424

yes

characteristic

C₂×C₁₆₇

C₈×D₄

yes

characteristic

C₂×C₁₆₇

#436

yes

characteristic

C₂×C₁₆₇

C₈×Q₈

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#1939

—

167

maximal, Sylow

C₂×C₄⋊C₈

—

167

—

C₂×C₄⋊C₈

D₁₆₇

yes

characteristic

C₂×C₄×C₈

—

167

—

C₄×C₄⋊C₄

—

167

—

#336

—

167

—

C₂²×C₈

—

167

—

C₂²×C₈

—

167

—

C₂²×C₈

C₂×D₁₆₇

yes

—

C₂×C₄⋊C₄

—

167

—

C₂×C₄⋊C₄

—

167

—

C₂×C₄²

—

167

—

C₂×C₄²

C₂×D₁₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

334

—

C₄×C₈

—

334

—

#95

—

167

—

C₂²×C₄

—

167

—

100

C₂²×C₄

—

167

—

101

C₂²×C₄

C₄×D₁₆₇

yes

characteristic

102

C₂²×C₄

C₄×D₁₆₇

yes

characteristic

103

C₂²×C₄

C₂²×D₁₆₇

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

334

—

108

C₂×C₈

—

334

—

109

C₂×C₈

—

334

—

110

C₂×C₈

—

334

—

111

C₂×C₈

—

167

—

112

C₂×C₈

—

167

—

113

C₄⋊C₄

—

334

—

114

C₄⋊C₄

—

167

—

115

C₄²

—

334

—

116

C₄²

—

334

—

117

C₄²

C₂²⋊D₁₆₇

yes

—

118

C₄²

Dic₃₃₄

yes

—

119

C₂³

C₂×C₄×D₁₆₇

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

334

—

123

C₂×C₄

—

334

—

124

C₂×C₄

—

334

—

125

C₂×C₄

—

334

—

126

C₂×C₄

—

334

—

127

C₂×C₄

—

167

—

128

C₂×C₄

Dic₁₆₇⋊C₄

yes

—

129

C₂×C₄

Dic₁₆₇⋊C₄

yes

—

130

C₂×C₄

C₈×D₁₆₇

yes

—

131

C₂×C₄

#14581933

yes

—

132

C₂×C₄

D₄×D₁₆₇

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₁₆₇

yes

characteristic

134

C₂×C₄

Q₈×D₁₆₇

yes

characteristic

135

C₂×C₄

C₂×Dic₃₃₄

yes

characteristic

136

C₂×C₄

#14581949

yes

characteristic

137

C₂×C₄

#14581951

yes

characteristic

138

C₈

—

139

C₈

—

334

—

140

C₈

—

334

—

141

C₂²

C₂×Dic₁₆₇⋊C₄

yes

characteristic

142

C₂²

#19128794

yes

characteristic

143

C₂²

#19128797

yes

characteristic

144

C₂²

C₂×C₈×D₁₆₇

yes

characteristic

145

C₂²

C₂×#14581933

yes

characteristic

146

C₂²

#19128840

yes

characteristic

147

C₂²

#19128840

yes

characteristic

148

C₄

—

149

C₄

—

334

—

150

C₄

—

668

—

151

C₄

Dic₁₆₇⋊C₈

yes

complemented

152

C₄

#19128803

yes

characteristic

153

C₄

#19128808

yes

characteristic

154

C₄

#19128823

yes

characteristic

155

C₄

#19128831

yes

characteristic

156

C₂

#24923165

yes

characteristic

157

C₂

C₂×Dic₁₆₇⋊C₈

yes

characteristic

158

C₂

#24923297

yes

characteristic

159

C₂

#24923334

yes

characteristic

160

C₂

#24923346

yes

characteristic

161

C₂

#24923393

yes

characteristic

162

C₂

#24923396

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₆₇
commutator	C₂²×C₁₆₇	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₁₆₇	C₂
socle, prime radical	C₂³×C₁₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₆₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₆₇
hypocenter	C₁₆₇	#1939
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₆₇

commutator

C₂²×C₁₆₇

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₁₆₇

C₂

socle, prime radical

C₂³×C₁₆₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₆₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₆₇

C₁₆₇

G ▷ C₂²×C₁₆₇ ▷ C₁₆₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
26. C₄⋊Dic₁₆₇	C₈	semidirect
67. C₄×C₁₆₇	C₄⋊C₈	semidirect
81. C₁₆₇	#1939	semidirect
151. C₄	Dic₁₆₇⋊C₈	semidirect

H ≅ G/N

type

26. C₄⋊Dic₁₆₇

C₈

semidirect

67. C₄×C₁₆₇

C₄⋊C₈

semidirect

81. C₁₆₇

#1939

semidirect

151. C₄

Dic₁₆₇⋊C₈

semidirect

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

subgroup	index
C₂×Dic₁₆₇⋊C₈	2
C₂×C₄×C₁₆₇⋊C₈	2
C₄×C₄⋊Dic₁₆₇	2
#24923023	2
C₂×C₄⋊C₈×C₁₆₇	2
#1939	167

subgroup

index

C₂×Dic₁₆₇⋊C₈

C₂×C₄×C₁₆₇⋊C₈

C₄×C₄⋊Dic₁₆₇

#24923023

C₂×C₄⋊C₈×C₁₆₇

#1939

167

Subgroups — #36324854

Conjugacy classes of subgroups

Complements

Maximal subgroups