Group #36325279 — subgroups

Subgroups — #36325279

Order 21376.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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×#19128832	C₂	1	1	yes	maximal, characteristic
2	C₂×#19128756	C₂	1	2	yes	maximal
3	#24923031	C₂	1	2	yes	maximal
4	#24923024	C₂	1	1	yes	maximal, characteristic
5	#391×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₃₄	—	2	1	no	—
7	C₂²×C₁₆₇⋊C₈	C₂²	1	2	yes	—
8	#19128832	—	2	4	no	—
9	C₂×C₄⋊Dic₁₆₇	—	2	1	no	—
10	C₂×C₄⋊Dic₁₆₇	C₂²	1	2	yes	—
11	#19128756	—	2	4	no	—
12	C₂²×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
15	#95×C₁₆₇	—	2	1	no	—
16	#95×C₁₆₇	—	2	1	no	—
17	C₂×Dic₃₃₄	—	2	4	no	—
18	C₂×Dic₃₃₄	—	4	2	no	—
19	C₂×Dic₃₃₄	—	4	2	no	—
20	C₂²×Dic₁₆₇	—	2	2	no	—
21	C₂×C₁₆₇⋊C₈	—	2	2	no	—
22	C₂×C₁₆₇⋊C₈	—	2	2	no	—
23	C₂×C₁₆₇⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₆₇	—	2	4	no	—
25	C₄⋊Dic₁₆₇	—	2	2	no	—
26	C₄⋊Dic₁₆₇	D₄	1	4	yes	—
27	C₂×Q₈×C₁₆₇	—	2	1	no	—
28	C₂×Q₈×C₁₆₇	—	2	2	no	—
29	C₂×Q₈×C₁₆₇	—	2	2	no	—
30	C₂×Q₈×C₁₆₇	Q₈	1	2	yes	complemented
31	C₂²×C₄×C₁₆₇	—	2	1	no	—
32	C₂²×C₄×C₁₆₇	—	2	1	no	—
33	C₂²×C₄×C₁₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₆₇	—	2	2	no	—
37	C₄²×C₁₆₇	—	2	2	no	—
38	Dic₃₃₄	—	4	8	no	—
39	C₂×Dic₁₆₇	—	2	4	no	—
40	C₂×Dic₁₆₇	—	4	2	no	—
41	C₂×Dic₁₆₇	—	4	2	no	—
42	C₁₆₇⋊C₈	—	4	2	no	—
43	C₂³×C₁₆₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₁₆₇	—	2	4	no	—
45	Q₈×C₁₆₇	—	4	2	no	—
46	C₂×C₄×C₁₆₇	—	2	1	no	—
47	C₂×C₄×C₁₆₇	—	2	1	no	—
48	C₂×C₄×C₁₆₇	—	2	1	no	—
49	C₂×C₄×C₁₆₇	—	2	1	no	—
50	C₂×C₄×C₁₆₇	—	2	2	no	—
51	C₂×C₄×C₁₆₇	—	4	1	no	—
52	C₂×C₄×C₁₆₇	—	4	1	no	—
53	C₂×C₄×C₁₆₇	—	4	1	no	—
54	C₂×C₄×C₁₆₇	—	4	1	no	—
55	C₂×C₄×C₁₆₇	—	4	1	no	—
56	C₂×C₄×C₁₆₇	SD₁₆	1	2	yes	—
57	C₂×C₄×C₁₆₇	Q₁₆	1	2	yes	—
58	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₁₆₇	#41	1	2	yes	—
61	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
62	C₂²×C₁₆₇	#120	1	1	yes	characteristic
63	C₂²×C₁₆₇	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₁₆₇	#124	1	1	yes	characteristic
65	C₂²×C₁₆₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₁₆₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₁₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₁₆₇	#137	1	1	yes	characteristic
69	C₄×C₁₆₇	—	2	2	no	—
70	C₄×C₁₆₇	—	2	2	no	—
71	C₄×C₁₆₇	—	4	2	no	—
72	C₄×C₁₆₇	—	8	1	no	—
73	C₄×C₁₆₇	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₁₆₇	#124	1	2	yes	—
75	Dic₁₆₇	—	4	4	no	—
76	C₂×C₁₆₇	#393	1	1	yes	characteristic
77	C₂×C₁₆₇	#450	1	1	yes	characteristic
78	C₂×C₁₆₇	#451	1	1	yes	characteristic
79	C₂×C₁₆₇	#475	1	1	yes	characteristic
80	C₂×C₁₆₇	#477	1	1	yes	characteristic
81	C₂×C₁₆₇	#487	1	1	yes	characteristic
82	C₂×C₁₆₇	#489	1	1	yes	characteristic
83	C₁₆₇	#2024	1	1	yes	characteristic, complemented, Sylow
84	#2024	—	167	1	no	maximal, Sylow
85	C₂×C₄⋊Q₈	—	167	1	no	—
86	C₂×#103	—	167	2	no	—
87	#391	D₁₆₇	1	1	yes	characteristic
88	#340	—	167	2	no	—
89	#336	—	167	1	no	—
90	C₂²×Q₈	—	334	1	no	—
91	C₂²×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
92	C₂²×C₈	—	167	2	no	—
93	C₄⋊Q₈	—	334	4	no	—
94	C₂×C₄⋊C₄	—	334	1	no	—
95	C₂×C₄⋊C₄	—	167	2	no	—
96	C₂×C₄⋊C₄	C₂×D₁₆₇	1	1	yes	characteristic
97	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
98	#103	—	334	4	no	—
99	#95	—	2	1	no	—
100	#95	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	334	4	no	—
105	C₂×Q₈	—	668	2	no	—
106	C₂×Q₈	—	668	2	no	—
107	C₂×Q₈	Dic₃₃₄	1	2	yes	complemented
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	334	2	no	—
111	C₂²×C₄	C₂²⋊D₁₆₇	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₁₆₇	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
114	C₂×C₈	—	334	2	no	—
115	C₂×C₈	—	334	2	no	—
116	C₂×C₈	—	334	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	334	4	no	—
119	C₄⋊C₄	—	334	2	no	—
120	C₄⋊C₄	—	167	4	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	668	8	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	334	4	no	—
137	C₂×C₄	—	668	2	no	—
138	C₂×C₄	—	668	2	no	—
139	C₂×C₄	#14581937	1	2	yes	—
140	C₂×C₄	#14581940	1	2	yes	—
141	C₂×C₄	D₄×D₁₆₇	1	2	yes	—
142	C₂×C₄	C₂×Dic₃₃₄	1	1	yes	characteristic
143	C₂×C₄	#14581949	1	2	yes	—
144	C₂×C₄	#14581950	1	1	yes	characteristic
145	C₈	—	668	2	no	—
146	C₂²	#19128800	1	1	yes	characteristic
147	C₂²	#19128811	1	1	yes	characteristic
148	C₂²	#19128819	1	1	yes	characteristic
149	C₂²	C₂×#14581937	1	1	yes	characteristic
150	C₂²	C₂×#14581940	1	1	yes	characteristic
151	C₂²	#19128857	1	1	yes	characteristic
152	C₂²	#19128862	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	—	8	1	no	—
157	C₄	—	668	4	no	—
158	C₄	#19128809	1	2	yes	—
159	C₄	#19128818	1	2	yes	—
160	C₂	#24923181	1	1	yes	characteristic
161	C₂	#24923419	1	1	yes	characteristic
162	C₂	#24923424	1	1	yes	characteristic
163	C₂	#24923554	1	1	yes	characteristic
164	C₂	#24923564	1	1	yes	characteristic
165	C₂	#24923619	1	1	yes	characteristic
166	C₂	#24923628	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128832

C₂

yes

maximal, characteristic

C₂×#19128756

C₂

yes

maximal

#24923031

C₂

yes

maximal

#24923024

C₂

yes

maximal, characteristic

#391×C₁₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₃₄

—

C₂²×C₁₆₇⋊C₈

C₂²

yes

—

#19128832

—

C₂×C₄⋊Dic₁₆₇

—

C₂×C₄⋊Dic₁₆₇

C₂²

yes

—

#19128756

—

C₂²×Q₈×C₁₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₆₇

C₂²

yes

characteristic

C₂×C₄²×C₁₆₇

C₂²

yes

characteristic

#95×C₁₆₇

—

#95×C₁₆₇

—

C₂×Dic₃₃₄

—

C₂×Dic₃₃₄

—

C₂×Dic₃₃₄

—

C₂²×Dic₁₆₇

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

D₄

yes

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

Q₈

yes

complemented

C₂²×C₄×C₁₆₇

—

C₂²×C₄×C₁₆₇

—

C₂²×C₄×C₁₆₇

D₄

yes

characteristic

C₂²×C₄×C₁₆₇

D₄

yes

characteristic

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₇

—

C₄²×C₁₆₇

—

Dic₃₃₄

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

C₂×D₄

yes

characteristic

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

SD₁₆

yes

—

C₂×C₄×C₁₆₇

Q₁₆

yes

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

—

C₂×C₄×C₁₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₇

#41

yes

—

C₂×C₄×C₁₆₇

yes

C₂²×C₁₆₇

yes

C₂²×C₁₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₆₇

#124

yes

characteristic

C₂²×C₁₆₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₆₇

C₂×Q₁₆

yes

C₂²×C₁₆₇

yes

C₂²×C₁₆₇

yes

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂²⋊Q₈

yes

—

C₄×C₁₆₇

#124

yes

—

Dic₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#2024

—

167

maximal, Sylow

C₂×C₄⋊Q₈

—

167

—

C₂×#103

—

167

—

#391

D₁₆₇

yes

characteristic

#340

—

167

—

#336

—

167

—

C₂²×Q₈

—

334

—

C₂²×Q₈

C₂×D₁₆₇

yes

characteristic

C₂²×C₈

—

167

—

C₄⋊Q₈

—

334

—

C₂×C₄⋊C₄

—

334

—

C₂×C₄⋊C₄

—

167

—

C₂×C₄⋊C₄

C₂×D₁₆₇

yes

characteristic

C₂×C₄²

C₂×D₁₆₇

yes

characteristic

#103

—

334

—

#95

—

100

#95

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

334

—

105

C₂×Q₈

—

668

—

106

C₂×Q₈

—

668

—

107

C₂×Q₈

Dic₃₃₄

yes

complemented

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

334

—

111

C₂²×C₄

C₂²⋊D₁₆₇

yes

characteristic

112

C₂²×C₄

C₂²⋊D₁₆₇

yes

characteristic

113

C₂²×C₄

C₂²×D₁₆₇

yes

characteristic

114

C₂×C₈

—

334

—

115

C₂×C₈

—

334

—

116

C₂×C₈

—

334

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

334

—

119

C₄⋊C₄

—

334

—

120

C₄⋊C₄

—

167

—

121

C₄²

—

122

C₂³

C₂×C₂²⋊D₁₆₇

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

668

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

334

—

137

C₂×C₄

—

668

—

138

C₂×C₄

—

668

—

139

C₂×C₄

#14581937

yes

—

140

C₂×C₄

#14581940

yes

—

141

C₂×C₄

D₄×D₁₆₇

yes

—

142

C₂×C₄

C₂×Dic₃₃₄

yes

characteristic

143

C₂×C₄

#14581949

yes

—

144

C₂×C₄

#14581950

yes

characteristic

145

C₈

—

668

—

146

C₂²

#19128800

yes

characteristic

147

C₂²

#19128811

yes

characteristic

148

C₂²

#19128819

yes

characteristic

149

C₂²

C₂×#14581937

yes

characteristic

150

C₂²

C₂×#14581940

yes

characteristic

151

C₂²

#19128857

yes

characteristic

152

C₂²

#19128862

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

668

—

158

C₄

#19128809

yes

—

159

C₄

#19128818

yes

—

160

C₂

#24923181

yes

characteristic

161

C₂

#24923419

yes

characteristic

162

C₂

#24923424

yes

characteristic

163

C₂

#24923554

yes

characteristic

164

C₂

#24923564

yes

characteristic

165

C₂

#24923619

yes

characteristic

166

C₂

#24923628

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₆₇
commutator, cosocle	C₂²×C₄×C₁₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₇
Fitting	#391×C₁₆₇	C₂
socle, prime radical	C₂³×C₁₆₇	C₂×D₄
hypercenter	#391	D₁₆₇
hypocenter	C₁₆₇	#2024
LCS	G ▷ C₂²×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₆₇

commutator, cosocle

C₂²×C₄×C₁₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₇

Fitting

#391×C₁₆₇

C₂

socle, prime radical

C₂³×C₁₆₇

C₂×D₄

D₁₆₇

C₁₆₇

G ▷ C₂²×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
30. C₂×Q₈×C₁₆₇	Q₈	semidirect
83. C₁₆₇	#2024	semidirect
107. C₂×Q₈	Dic₃₃₄	semidirect

H ≅ G/N

type

30. C₂×Q₈×C₁₆₇

Q₈

semidirect

83. C₁₆₇

#2024

semidirect

107. C₂×Q₈

Dic₃₃₄

semidirect

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

subgroup	index
C₂×#19128832	2
C₂×#19128756	2
#24923031	2
#24923024	2
#391×C₁₆₇	2
#2024	167

subgroup

index

#391×C₁₆₇

167

Subgroups — #36325279

Conjugacy classes of subgroups

Complements

Maximal subgroups