Group #36323357 — subgroups

Subgroups — #36323357

Order 21376.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₃₃₄	C₂	1	1	yes	maximal, characteristic
2	#24923621	C₂	1	1	yes	maximal, characteristic
3	#24923551	C₂	1	1	yes	maximal, characteristic
4	#24923017	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₁₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#24922999	C₂	1	1	yes	maximal, characteristic
7	#462×C₁₆₇	C₂	1	1	yes	maximal, characteristic
8	#19128832	—	2	1	no	—
9	#19128832	C₂²	1	1	yes	characteristic
10	#19128831	—	2	2	no	—
11	#19128831	—	2	1	no	—
12	Q₈×Dic₁₆₇	—	2	1	no	—
13	C₄×Dic₃₃₄	—	2	1	no	—
14	C₄×Dic₃₃₄	C₂²	1	1	yes	characteristic
15	#19128763	—	2	1	no	—
16	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
17	#19128755	—	2	1	no	—
18	#19128755	—	2	2	no	—
19	C₄×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₁₆₇	—	2	1	no	—
21	C₄⋊Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
24	#103×C₁₆₇	—	2	1	no	—
25	C₂×Dic₃₃₄	—	2	2	no	—
26	C₂×Dic₃₃₄	D₄	1	1	yes	characteristic
27	C₂×Dic₃₃₄	D₄	1	2	yes	—
28	C₂×C₁₆₇⋊C₈	—	2	1	no	—
29	C₂×C₁₆₇⋊C₈	—	2	1	no	—
30	C₄⋊Dic₁₆₇	—	2	2	no	—
31	C₄⋊Dic₁₆₇	—	2	1	no	—
32	C₄⋊Dic₁₆₇	—	2	1	no	—
33	C₄⋊Dic₁₆₇	D₄	1	1	yes	characteristic
34	Dic₁₆₇⋊C₄	—	2	2	no	—
35	Dic₁₆₇⋊C₄	—	2	1	no	—
36	C₄×Dic₁₆₇	—	2	2	no	—
37	C₄×Dic₁₆₇	—	2	1	no	—
38	C₂×Q₈×C₁₆₇	—	2	1	no	—
39	C₂×Q₈×C₁₆₇	D₄	1	1	yes	characteristic
40	Q₁₆×C₁₆₇	—	4	2	no	—
41	C₂×C₈×C₁₆₇	—	2	1	no	—
42	C₄⋊C₄×C₁₆₇	—	2	1	no	—
43	C₄⋊C₄×C₁₆₇	—	2	1	no	—
44	C₄⋊C₄×C₁₆₇	—	2	1	no	—
45	C₄⋊C₄×C₁₆₇	D₄	1	1	yes	characteristic
46	C₄²×C₁₆₇	—	2	1	no	—
47	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
48	Dic₃₃₄	—	2	1	no	—
49	Dic₃₃₄	—	4	2	no	—
50	Dic₃₃₄	—	8	2	no	—
51	Dic₃₃₄	Q₁₆	1	2	yes	complemented
52	C₂×Dic₁₆₇	—	2	1	no	—
53	C₂×Dic₁₆₇	—	2	2	no	—
54	C₂×Dic₁₆₇	—	2	1	no	—
55	C₁₆₇⋊C₈	—	4	1	no	—
56	C₁₆₇⋊C₈	—	4	1	no	—
57	Q₈×C₁₆₇	—	2	1	no	—
58	Q₈×C₁₆₇	—	4	1	no	—
59	Q₈×C₁₆₇	—	4	1	no	—
60	Q₈×C₁₆₇	SD₁₆	1	2	yes	—
61	C₂×C₄×C₁₆₇	—	2	1	no	—
62	C₂×C₄×C₁₆₇	—	2	1	no	—
63	C₂×C₄×C₁₆₇	—	4	1	no	—
64	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
67	C₈×C₁₆₇	—	4	1	no	—
68	C₂²×C₁₆₇	#120	1	1	yes	characteristic
69	C₄×C₁₆₇	—	2	2	no	—
70	C₄×C₁₆₇	—	2	1	no	—
71	C₄×C₁₆₇	—	4	1	no	—
72	C₄×C₁₆₇	—	8	1	no	—
73	C₄×C₁₆₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₄×C₁₆₇	C₂×Q₁₆	1	1	yes	characteristic
75	C₄×C₁₆₇	#137	1	1	yes	characteristic
76	C₄×C₁₆₇	#137	1	1	yes	characteristic
77	Dic₁₆₇	—	2	2	no	—
78	Dic₁₆₇	—	4	2	no	—
79	Dic₁₆₇	—	4	1	no	—
80	C₂×C₁₆₇	#448	1	1	yes	characteristic
81	C₂×C₁₆₇	#451	1	1	yes	characteristic
82	C₂×C₁₆₇	#455	1	1	yes	characteristic
83	C₁₆₇	#1645	1	1	yes	characteristic, complemented, Sylow
84	#1645	—	167	1	no	maximal, Sylow
85	Q₈²	—	167	1	no	—
86	#487	—	167	1	no	—
87	#475	—	167	1	no	—
88	#462	D₁₆₇	1	1	yes	characteristic
89	#333	—	167	1	no	—
90	Q₈⋊C₈	—	167	1	no	—
91	Q₈⋊C₈	—	167	1	no	—
92	C₂×Q₁₆	—	2	1	no	—
93	C₄⋊Q₈	—	334	1	no	—
94	C₄⋊Q₈	—	334	2	no	—
95	C₄⋊Q₈	—	334	1	no	—
96	C₄⋊Q₈	—	167	1	no	—
97	C₄⋊Q₈	C₂×D₁₆₇	1	1	yes	characteristic
98	C₄×Q₈	—	334	1	no	—
99	C₄×Q₈	—	334	1	no	—
100	C₄×Q₈	—	167	1	no	—
101	C₄×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
102	#106	—	334	1	no	—
103	C₄⋊C₈	—	167	1	no	—
104	C₄⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
105	#103	—	2	1	no	—
106	#103	—	334	1	no	—
107	#103	—	334	2	no	—
108	C₄×C₈	—	167	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	334	2	no	—
111	C₂×Q₈	—	167	1	no	—
112	C₂×Q₈	—	167	2	no	—
113	C₂×Q₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
114	Q₁₆	—	4	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	334	1	no	—
117	C₂×C₈	—	334	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	334	2	no	—
122	C₄⋊C₄	—	334	2	no	—
123	C₄⋊C₄	—	334	1	no	—
124	C₄⋊C₄	—	334	1	no	—
125	C₄⋊C₄	—	334	1	no	—
126	C₄⋊C₄	—	167	1	no	—
127	C₄⋊C₄	C₂²⋊D₁₆₇	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	334	2	no	—
130	C₄²	—	334	1	no	—
131	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	1336	2	no	—
136	Q₈	—	334	1	no	—
137	Q₈	—	668	2	no	—
138	Q₈	—	167	2	no	—
139	Q₈	#14581938	1	2	yes	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	334	1	no	—
144	C₂×C₄	—	334	2	no	—
145	C₂×C₄	—	334	1	no	—
146	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
147	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	668	1	no	—
151	C₈	—	668	1	no	—
152	C₂²	#19128800	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	334	2	no	—
158	C₄	—	668	1	no	—
159	C₄	—	668	2	no	—
160	C₄	C₂×#14581938	1	1	yes	characteristic
161	C₄	Q₁₆×D₁₆₇	1	1	yes	characteristic
162	C₄	#19128862	1	1	yes	characteristic
163	C₄	#19128864	1	1	yes	characteristic
164	C₂	#24923407	1	1	yes	characteristic
165	C₂	#24923423	1	1	yes	characteristic
166	C₂	#24923445	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₃₃₄

C₂

yes

maximal, characteristic

#24923621

C₂

yes

maximal, characteristic

#24923551

C₂

yes

maximal, characteristic

#24923017

C₂

yes

maximal, characteristic

Q₈⋊(C₁₆₇⋊C₈)

C₂

yes

maximal, characteristic

#24922999

C₂

yes

maximal, characteristic

#462×C₁₆₇

C₂

yes

maximal, characteristic

#19128832

—

#19128832

C₂²

yes

characteristic

#19128831

—

#19128831

—

Q₈×Dic₁₆₇

—

C₄×Dic₃₃₄

—

C₄×Dic₃₃₄

C₂²

yes

characteristic

#19128763

—

C₄⋊(C₁₆₇⋊C₈)

C₂²

yes

characteristic

#19128755

—

#19128755

—

C₄×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₁₆₇

—

C₄⋊Q₈×C₁₆₇

C₂²

yes

characteristic

C₄×Q₈×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₇

C₂²

yes

characteristic

#103×C₁₆₇

—

C₂×Dic₃₃₄

—

C₂×Dic₃₃₄

D₄

yes

characteristic

C₂×Dic₃₃₄

D₄

yes

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

D₄

yes

characteristic

Dic₁₆₇⋊C₄

—

Dic₁₆₇⋊C₄

—

C₄×Dic₁₆₇

—

C₄×Dic₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

D₄

yes

characteristic

Q₁₆×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

D₄

yes

characteristic

C₄²×C₁₆₇

—

C₄²×C₁₆₇

C₂³

yes

characteristic

Dic₃₃₄

—

Dic₃₃₄

—

Dic₃₃₄

—

Dic₃₃₄

Q₁₆

yes

complemented

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

—

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

Q₈×C₁₆₇

SD₁₆

yes

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₈×C₁₆₇

—

C₂²×C₁₆₇

#120

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₆₇

C₂×Q₁₆

yes

C₄×C₁₆₇

yes

C₄×C₁₆₇

yes

Dic₁₆₇

—

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#1645

—

167

maximal, Sylow

Q₈²

—

167

—

#487

—

167

—

#475

—

167

—

#462

D₁₆₇

yes

characteristic

#333

—

167

—

Q₈⋊C₈

—

167

—

Q₈⋊C₈

—

167

—

C₂×Q₁₆

—

C₄⋊Q₈

—

334

—

C₄⋊Q₈

—

334

—

C₄⋊Q₈

—

334

—

C₄⋊Q₈

—

167

—

C₄⋊Q₈

C₂×D₁₆₇

yes

characteristic

C₄×Q₈

—

334

—

C₄×Q₈

—

334

—

100

C₄×Q₈

—

167

—

101

C₄×Q₈

C₂×D₁₆₇

yes

characteristic

102

#106

—

334

—

103

C₄⋊C₈

—

167

—

104

C₄⋊C₈

C₂×D₁₆₇

yes

characteristic

105

#103

—

106

#103

—

334

—

107

#103

—

334

—

108

C₄×C₈

—

167

—

109

C₂×Q₈

—

110

C₂×Q₈

—

334

—

111

C₂×Q₈

—

167

—

112

C₂×Q₈

—

167

—

113

C₂×Q₈

C₂²⋊D₁₆₇

yes

characteristic

114

Q₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

334

—

117

C₂×C₈

—

334

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

334

—

122

C₄⋊C₄

—

334

—

123

C₄⋊C₄

—

334

—

124

C₄⋊C₄

—

334

—

125

C₄⋊C₄

—

334

—

126

C₄⋊C₄

—

167

—

127

C₄⋊C₄

C₂²⋊D₁₆₇

yes

characteristic

128

C₄²

—

129

C₄²

—

334

—

130

C₄²

—

334

—

131

C₄²

C₂²×D₁₆₇

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

1336

—

136

Q₈

—

334

—

137

Q₈

—

668

—

138

Q₈

—

167

—

139

Q₈

#14581938

yes

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

334

—

144

C₂×C₄

—

334

—

145

C₂×C₄

—

334

—

146

C₂×C₄

D₄×D₁₆₇

yes

characteristic

147

C₂×C₄

D₄×D₁₆₇

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₁₆₇

yes

characteristic

149

C₈

—

150

C₈

—

668

—

151

C₈

—

668

—

152

C₂²

#19128800

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

334

—

158

C₄

—

668

—

159

C₄

—

668

—

160

C₄

C₂×#14581938

yes

characteristic

161

C₄

Q₁₆×D₁₆₇

yes

characteristic

162

C₄

#19128862

yes

characteristic

163

C₄

#19128864

yes

characteristic

164

C₂

#24923407

yes

characteristic

165

C₂

#24923423

yes

characteristic

166

C₂

#24923445

yes

characteristic

	subgroup	G/H
center	C₂²	#19128800
commutator, cosocle	C₄²×C₁₆₇	C₂³
Frattini	C₄²	C₂²×D₁₆₇
Fitting	#462×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	#120
hypercenter	#462	D₁₆₇
hypocenter	C₁₆₇	#1645
LCS	G ▷ C₄²×C₁₆₇ ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#19128800

commutator, cosocle

C₄²×C₁₆₇

C₂³

Frattini

C₄²

C₂²×D₁₆₇

Fitting

#462×C₁₆₇

C₂

C₂²×C₁₆₇

D₁₆₇

C₁₆₇

G ▷ C₄²×C₁₆₇ ▷ C₂²×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
51. Dic₃₃₄	Q₁₆	semidirect
83. C₁₆₇	#1645	semidirect

H ≅ G/N

type

51. Dic₃₃₄

Q₁₆

semidirect

83. C₁₆₇

#1645

semidirect

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

subgroup	index
Q₈×Dic₃₃₄	2
#24923621	2
#24923551	2
#24923017	2
Q₈⋊(C₁₆₇⋊C₈)	2
#24922999	2
#462×C₁₆₇	2
#1645	167

subgroup

index

Q₈×Dic₃₃₄

Q₈⋊(C₁₆₇⋊C₈)

#462×C₁₆₇

167

Subgroups — #36323357

Conjugacy classes of subgroups

Complements

Maximal subgroups