Group #36320373 — subgroups

Subgroups — #2013×C₁₆₇ #36320373

Order 21376.

406 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 116 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×SD₁₆×C₁₆₇	C₂	1	2	yes	maximal
2	C₂×C₄⋊Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#103×C₁₆₇	C₂	1	2	yes	maximal, complemented
4	#390×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₁₆₇	C₂²	1	2	yes	—
7	C₂²×D₄×C₁₆₇	C₂²	1	1	yes	characteristic
8	C₂×SD₁₆×C₁₆₇	—	2	4	no	—
9	C₂×SD₁₆×C₁₆₇	—	2	4	no	—
10	C₂×SD₁₆×C₁₆₇	—	2	4	no	—
11	C₂²×C₈×C₁₆₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₆₇	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₆₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₁₆₇	—	2	2	no	—
16	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
17	#103×C₁₆₇	—	2	4	no	—
18	C₂⁴×C₁₆₇	—	2	1	no	—
19	C₂×Q₈×C₁₆₇	—	2	2	no	—
20	C₂×Q₈×C₁₆₇	—	2	2	no	—
21	C₂×Q₈×C₁₆₇	—	2	2	no	—
22	C₂×Q₈×C₁₆₇	—	2	2	no	—
23	C₂×Q₈×C₁₆₇	—	2	2	no	—
24	C₂×Q₈×C₁₆₇	D₄	1	4	yes	—
25	C₂×D₄×C₁₆₇	—	2	2	no	—
26	C₂×D₄×C₁₆₇	—	2	2	no	—
27	C₂×D₄×C₁₆₇	—	2	1	no	—
28	C₂×D₄×C₁₆₇	D₄	1	2	yes	—
29	C₂²×C₄×C₁₆₇	—	2	2	no	—
30	C₂²×C₄×C₁₆₇	—	2	1	no	—
31	C₂²×C₄×C₁₆₇	D₄	1	2	yes	—
32	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
33	SD₁₆×C₁₆₇	—	2	16	no	—
34	C₂×C₈×C₁₆₇	—	2	2	no	—
35	C₂×C₈×C₁₆₇	—	2	2	no	—
36	C₂×C₈×C₁₆₇	D₄	1	4	yes	—
37	C₄⋊C₄×C₁₆₇	—	2	8	no	—
38	C₄⋊C₄×C₁₆₇	—	2	2	no	—
39	C₂²⋊C₄×C₁₆₇	—	4	4	no	—
40	C₄²×C₁₆₇	—	2	2	no	—
41	C₂³×C₁₆₇	—	2	2	no	—
42	C₂³×C₁₆₇	—	4	1	no	—
43	C₂³×C₁₆₇	—	4	1	no	—
44	C₂³×C₁₆₇	—	4	2	no	—
45	C₂³×C₁₆₇	—	4	2	no	—
46	C₂³×C₁₆₇	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₁₆₇	—	2	4	no	—
48	Q₈×C₁₆₇	—	2	4	no	—
49	Q₈×C₁₆₇	—	4	2	no	—
50	Q₈×C₁₆₇	—	4	2	no	—
51	D₄×C₁₆₇	—	2	4	no	—
52	D₄×C₁₆₇	—	4	2	no	—
53	C₂×C₄×C₁₆₇	—	2	2	no	—
54	C₂×C₄×C₁₆₇	—	2	2	no	—
55	C₂×C₄×C₁₆₇	—	2	4	no	—
56	C₂×C₄×C₁₆₇	—	4	2	no	—
57	C₂×C₄×C₁₆₇	—	4	2	no	—
58	C₂×C₄×C₁₆₇	—	4	1	no	—
59	C₂×C₄×C₁₆₇	—	4	2	no	—
60	C₂×C₄×C₁₆₇	SD₁₆	1	4	yes	complemented
61	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
65	C₈×C₁₆₇	—	2	4	no	—
66	C₂²×C₁₆₇	—	4	2	no	—
67	C₂²×C₁₆₇	—	4	4	no	—
68	C₂²×C₁₆₇	—	8	2	no	—
69	C₂²×C₁₆₇	—	8	1	no	—
70	C₂²×C₁₆₇	—	8	1	no	—
71	C₂²×C₁₆₇	#120	1	1	yes	characteristic
72	C₂²×C₁₆₇	#121	1	1	yes	characteristic
73	C₂²×C₁₆₇	#127	1	1	yes	characteristic
74	C₂²×C₁₆₇	C₂×SD₁₆	1	2	yes	—
75	C₂²×C₁₆₇	#137	1	2	yes	—
76	C₄×C₁₆₇	—	2	4	no	—
77	C₄×C₁₆₇	—	4	4	no	—
78	C₄×C₁₆₇	—	8	1	no	—
79	C₄×C₁₆₇	#120	1	2	yes	—
80	C₄×C₁₆₇	#121	1	2	yes	—
81	C₂×C₁₆₇	—	8	2	no	—
82	C₂×C₁₆₇	#392	1	1	yes	characteristic
83	C₂×C₁₆₇	#448	1	2	yes	—
84	C₂×C₁₆₇	#461	1	2	yes	—
85	C₂×C₁₆₇	#492	1	1	yes	characteristic
86	C₂×C₁₆₇	#497	1	1	yes	characteristic
87	C₁₆₇	#2013	1	1	yes	characteristic, direct product factor, Sylow
88	#2013	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×SD₁₆	C₂×C₁₆₇	1	2	yes	—
90	C₂×C₄⋊Q₈	C₂×C₁₆₇	1	1	yes	characteristic, complemented
91	C₂×#103	C₂×C₁₆₇	1	2	yes	complemented
92	#390	C₂×C₁₆₇	1	1	yes	characteristic
93	#336	C₂×C₁₆₇	1	1	yes	characteristic, complemented
94	C₂²×Q₈	C₂²×C₁₆₇	1	2	yes	—
95	C₂²×D₄	C₂²×C₁₆₇	1	1	yes	characteristic
96	C₂×SD₁₆	—	2	4	no	—
97	C₂×SD₁₆	—	2	4	no	—
98	C₂×SD₁₆	—	2	4	no	—
99	C₂²×C₈	C₂²×C₁₆₇	1	2	yes	—
100	C₄⋊Q₈	—	2	4	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	—	2	2	no	—
104	C₂×C₄²	C₂²×C₁₆₇	1	1	yes	characteristic
105	#103	—	2	4	no	—
106	C₂⁴	—	2	1	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	D₄×C₁₆₇	1	4	yes	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	D₄×C₁₆₇	1	2	yes	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	D₄×C₁₆₇	1	2	yes	—
120	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
121	SD₁₆	—	2	16	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	D₄×C₁₆₇	1	4	yes	—
125	C₄⋊C₄	—	2	8	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	4	4	no	—
128	C₄²	—	2	2	no	—
129	C₂³	—	2	2	no	—
130	C₂³	—	4	1	no	—
131	C₂³	—	4	1	no	—
132	C₂³	—	4	2	no	—
133	C₂³	—	4	2	no	—
134	C₂³	C₂×D₄×C₁₆₇	1	1	yes	characteristic
135	Q₈	—	2	4	no	—
136	Q₈	—	2	4	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	4	2	no	—
139	D₄	—	2	4	no	—
140	D₄	—	4	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	SD₁₆×C₁₆₇	1	4	yes	complemented
149	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
150	C₂×C₄	C₂×D₄×C₁₆₇	1	2	yes	—
151	C₂×C₄	C₂×D₄×C₁₆₇	1	2	yes	—
152	C₂×C₄	#41×C₁₆₇	1	1	yes	characteristic
153	C₈	—	2	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	8	1	no	—
157	C₂²	—	8	1	no	—
158	C₂²	—	8	2	no	—
159	C₂²	#120×C₁₆₇	1	1	yes	characteristic
160	C₂²	#121×C₁₆₇	1	1	yes	characteristic
161	C₂²	#127×C₁₆₇	1	1	yes	characteristic
162	C₂²	C₂×SD₁₆×C₁₆₇	1	2	yes	—
163	C₂²	#137×C₁₆₇	1	2	yes	—
164	C₄	—	2	4	no	—
165	C₄	—	4	4	no	—
166	C₄	—	8	1	no	—
167	C₄	#120×C₁₆₇	1	2	yes	—
168	C₄	#121×C₁₆₇	1	2	yes	—
169	C₂	—	8	2	no	—
170	C₂	#392×C₁₆₇	1	1	yes	characteristic
171	C₂	#448×C₁₆₇	1	2	yes	—
172	C₂	#461×C₁₆₇	1	2	yes	—
173	C₂	#492×C₁₆₇	1	1	yes	characteristic
174	C₂	#497×C₁₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×SD₁₆×C₁₆₇

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#103×C₁₆₇

C₂

yes

maximal, complemented

#390×C₁₆₇

C₂

yes

maximal, characteristic

#336×C₁₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₆₇

C₂²

yes

—

C₂²×D₄×C₁₆₇

C₂²

yes

characteristic

C₂×SD₁₆×C₁₆₇

—

C₂×SD₁₆×C₁₆₇

—

C₂×SD₁₆×C₁₆₇

—

C₂²×C₈×C₁₆₇

C₂²

yes

—

C₄⋊Q₈×C₁₆₇

—

C₂×C₄⋊C₄×C₁₆₇

—

C₂×C₄⋊C₄×C₁₆₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₆₇

—

C₂×C₄²×C₁₆₇

C₂²

yes

characteristic

#103×C₁₆₇

—

C₂⁴×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

D₄

yes

—

C₂×D₄×C₁₆₇

—

C₂×D₄×C₁₆₇

—

C₂×D₄×C₁₆₇

—

C₂×D₄×C₁₆₇

D₄

yes

—

C₂²×C₄×C₁₆₇

—

C₂²×C₄×C₁₆₇

—

C₂²×C₄×C₁₆₇

D₄

yes

—

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

SD₁₆×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

D₄

yes

—

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₂²⋊C₄×C₁₆₇

—

C₄²×C₁₆₇

—

C₂³×C₁₆₇

—

C₂³×C₁₆₇

—

C₂³×C₁₆₇

—

C₂³×C₁₆₇

—

C₂³×C₁₆₇

—

C₂³×C₁₆₇

C₂×D₄

yes

characteristic

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

D₄×C₁₆₇

—

D₄×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

complemented

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

—

C₂×C₄×C₁₆₇

#41

yes

characteristic

C₈×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

yes

C₂²×C₁₆₇

yes

C₂²×C₁₆₇

yes

C₂²×C₁₆₇

C₂×SD₁₆

yes

—

C₂²×C₁₆₇

#137

yes

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

#120

yes

—

C₄×C₁₆₇

#121

yes

—

C₂×C₁₆₇

—

C₂×C₁₆₇

#392

yes

characteristic

C₂×C₁₆₇

#448

yes

—

C₂×C₁₆₇

#461

yes

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, direct product factor, Sylow

#2013

C₁₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×SD₁₆

C₂×C₁₆₇

yes

—

C₂×C₄⋊Q₈

C₂×C₁₆₇

yes

characteristic, complemented

C₂×#103

C₂×C₁₆₇

yes

complemented

#390

C₂×C₁₆₇

yes

characteristic

#336

C₂×C₁₆₇

yes

characteristic, complemented

C₂²×Q₈

C₂²×C₁₆₇

yes

—

C₂²×D₄

C₂²×C₁₆₇

yes

characteristic

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂²×C₈

C₂²×C₁₆₇

yes

—

100

C₄⋊Q₈

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂²×C₁₆₇

yes

characteristic

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

C₂²×C₁₆₇

yes

characteristic

105

#103

—

106

C₂⁴

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

D₄×C₁₆₇

yes

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

D₄×C₁₆₇

yes

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

D₄×C₁₆₇

yes

—

120

C₂²×C₄

C₂³×C₁₆₇

yes

characteristic

121

SD₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

D₄×C₁₆₇

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂×D₄×C₁₆₇

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

SD₁₆×C₁₆₇

yes

complemented

149

C₂×C₄

C₂×D₄×C₁₆₇

yes

characteristic

150

C₂×C₄

C₂×D₄×C₁₆₇

yes

—

151

C₂×C₄

C₂×D₄×C₁₆₇

yes

—

152

C₂×C₄

#41×C₁₆₇

yes

characteristic

153

C₈

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#120×C₁₆₇

yes

characteristic

160

C₂²

#121×C₁₆₇

yes

characteristic

161

C₂²

#127×C₁₆₇

yes

characteristic

162

C₂²

C₂×SD₁₆×C₁₆₇

yes

—

163

C₂²

#137×C₁₆₇

yes

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#120×C₁₆₇

yes

—

168

C₄

#121×C₁₆₇

yes

—

169

C₂

—

170

C₂

#392×C₁₆₇

yes

characteristic

171

C₂

#448×C₁₆₇

yes

—

172

C₂

#461×C₁₆₇

yes

—

173

C₂

#492×C₁₆₇

yes

characteristic

174

C₂

#497×C₁₆₇

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₆₇
prime radical	C₂²×D₄×C₁₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₆₇ ◁ C₂×C₄²×C₁₆₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₆₇

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₆₇

prime radical

C₂²×D₄×C₁₆₇

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₆₇ ◁ C₂×C₄²×C₁₆₇ ◁ G

N	H ≅ G/N	type
2. C₂×C₄⋊Q₈×C₁₆₇	C₂	semidirect
3. C₂×#103×C₁₆₇	C₂	semidirect
5. #336×C₁₆₇	C₂	semidirect
60. C₂×C₄×C₁₆₇	SD₁₆	semidirect
87. C₁₆₇	#2013	direct (N × H)
88. #2013	C₁₆₇	direct (N × H)
90. C₂×C₄⋊Q₈	C₂×C₁₆₇	semidirect
91. C₂×#103	C₂×C₁₆₇	semidirect
93. #336	C₂×C₁₆₇	semidirect
148. C₂×C₄	SD₁₆×C₁₆₇	semidirect

H ≅ G/N

type

2. C₂×C₄⋊Q₈×C₁₆₇

C₂

semidirect

3. C₂×#103×C₁₆₇

C₂

semidirect

5. #336×C₁₆₇

C₂

semidirect

60. C₂×C₄×C₁₆₇

SD₁₆

semidirect

87. C₁₆₇

#2013

direct (N × H)

88. #2013

C₁₆₇

direct (N × H)

90. C₂×C₄⋊Q₈

C₂×C₁₆₇

semidirect

91. C₂×#103

C₂×C₁₆₇

semidirect

93. #336

C₂×C₁₆₇

semidirect

148. C₂×C₄

SD₁₆×C₁₆₇

semidirect

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

subgroup	index
C₂²×SD₁₆×C₁₆₇	2
C₂×C₄⋊Q₈×C₁₆₇	2
C₂×#103×C₁₆₇	2
#390×C₁₆₇	2
#336×C₁₆₇	2
#2013	167

subgroup

index

C₂²×SD₁₆×C₁₆₇

C₂×C₄⋊Q₈×C₁₆₇

C₂×#103×C₁₆₇

#390×C₁₆₇

#336×C₁₆₇

#2013

167

Subgroups — #2013×C₁₆₇ #36320373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2013×C167 #36320373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2013×C₁₆₇ #36320373