Group #36337143 — subgroups

Subgroups — #36337143

Order 21376.

348 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 125 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	#24923997	C₂	1	2	yes	maximal
2	#24923649	C₂	1	1	yes	maximal, characteristic
3	#24923647	C₂	1	2	yes	maximal
4	#24923479	C₂	1	2	yes	maximal
5	#24923478	C₂	1	4	yes	maximal
6	C₈×Dic₃₃₄	C₂	1	1	yes	maximal, characteristic
7	#24923363	C₂	1	2	yes	maximal
8	C₄×Q₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₆₆₈	C₂²	1	2	yes	—
10	C₂×Dic₆₆₈	C₂²	1	2	yes	—
11	C₂×#14581940	C₂²	1	4	yes	—
12	#19128832	C₂²	1	2	yes	—
13	#19128830	C₂²	1	4	yes	—
14	#19128824	—	2	2	no	—
15	#19128822	—	2	4	no	—
16	C₄×Dic₃₃₄	—	2	2	no	—
17	C₄×Dic₃₃₄	C₂²	1	1	yes	characteristic
18	#19128797	C₂²	1	4	yes	—
19	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₁₆₇⋊C₈	C₂²	1	2	yes	—
21	#19128755	C₂²	1	4	yes	—
22	C₈×Dic₁₆₇	C₂²	1	2	yes	—
23	C₂×Q₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₆₇	C₂²	1	2	yes	—
25	#107×C₁₆₇	C₂²	1	1	yes	characteristic
26	#103×C₁₆₇	C₂²	1	2	yes	—
27	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
28	C₂×Dic₃₃₄	D₄	1	1	yes	characteristic
29	C₂×Dic₃₃₄	C₂³	1	4	yes	—
30	Dic₆₆₈	—	2	4	no	—
31	Dic₆₆₈	—	4	2	no	—
32	#14581940	—	4	4	no	—
33	C₂×C₁₆₇⋊C₈	C₂³	1	2	yes	—
34	C₄⋊Dic₁₆₇	—	2	2	no	—
35	C₄⋊Dic₁₆₇	—	2	2	no	—
36	C₄⋊Dic₁₆₇	D₄	1	1	yes	characteristic
37	Dic₁₆₇⋊C₄	—	2	4	no	—
38	Dic₁₆₇⋊C₄	—	2	4	no	—
39	Dic₁₆₇⋊C₄	—	2	4	no	—
40	Dic₁₆₇⋊C₄	D₄	1	2	yes	—
41	C₄×Dic₁₆₇	—	2	4	no	—
42	C₄×Dic₁₆₇	C₂³	1	2	yes	—
43	C₂×Q₈×C₁₆₇	C₂³	1	2	yes	—
44	Q₁₆×C₁₆₇	—	2	2	no	—
45	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₆₇	—	2	2	no	—
48	C₄⋊C₄×C₁₆₇	C₂³	1	2	yes	—
49	C₄²×C₁₆₇	—	2	2	no	—
50	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
51	Dic₃₃₄	—	2	4	no	—
52	Dic₃₃₄	—	2	4	no	—
53	Dic₃₃₄	Q₁₆	1	4	yes	complemented
54	C₂×Dic₁₆₇	—	2	4	no	—
55	C₂×Dic₁₆₇	C₂×D₄	1	2	yes	—
56	C₂×Dic₁₆₇	C₂×D₄	1	2	yes	—
57	C₁₆₇⋊C₈	—	2	2	no	—
58	Q₈×C₁₆₇	—	2	2	no	—
59	Q₈×C₁₆₇	—	2	2	no	—
60	C₂×C₄×C₁₆₇	—	2	2	no	—
61	C₂×C₄×C₁₆₇	—	2	2	no	—
62	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
65	C₈×C₁₆₇	—	2	1	no	—
66	C₈×C₁₆₇	#41	1	2	yes	—
67	C₂²×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₁₆₇	—	2	1	no	—
69	C₄×C₁₆₇	—	4	2	no	—
70	C₄×C₁₆₇	—	4	2	no	—
71	C₄×C₁₆₇	C₂×Q₁₆	1	2	yes	—
72	C₄×C₁₆₇	C₂×#41	1	1	yes	characteristic
73	C₄×C₁₆₇	H(5,𝔽₂)	1	1	yes	characteristic
74	Dic₁₆₇	—	2	2	no	—
75	Dic₁₆₇	—	4	4	no	—
76	Dic₁₆₇	C₂×Q₁₆	1	4	yes	—
77	C₂×C₁₆₇	#550	1	1	yes	characteristic
78	C₂×C₁₆₇	C₂²×Q₁₆	1	1	yes	characteristic
79	C₂×C₁₆₇	#578	1	1	yes	characteristic
80	C₁₆₇	#3395	1	1	yes	characteristic, complemented, Sylow
81	#3395	—	167	1	no	maximal, Sylow
82	#557	—	167	2	no	—
83	#494	—	167	2	no	—
84	#494	—	167	1	no	—
85	#462	—	167	2	no	—
86	#462	—	167	4	no	—
87	C₈×Q₈	—	167	1	no	—
88	C₄×Q₁₆	—	167	2	no	—
89	C₄×Q₁₆	D₁₆₇	1	1	yes	characteristic
90	C₂×Q₁₆	—	167	4	no	—
91	C₂×Q₁₆	—	167	2	no	—
92	C₂×Q₁₆	—	167	2	no	—
93	C₂×Q₁₆	C₂×D₁₆₇	1	1	yes	characteristic
94	C₄⋊Q₈	—	167	4	no	—
95	C₄⋊Q₈	—	167	2	no	—
96	#125	—	334	2	no	—
97	#125	—	334	4	no	—
98	C₄×Q₈	—	334	2	no	—
99	C₄×Q₈	—	167	4	no	—
100	C₄×Q₈	—	167	1	no	—
101	C₄×Q₈	C₂×D₁₆₇	1	2	yes	—
102	#107	C₂×D₁₆₇	1	1	yes	characteristic
103	C₄⋊C₈	—	167	2	no	—
104	C₄⋊C₈	—	167	1	no	—
105	#103	—	167	4	no	—
106	#103	C₂×D₁₆₇	1	2	yes	—
107	C₄×C₈	—	167	2	no	—
108	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
109	C₂×Q₈	—	167	4	no	—
110	C₂×Q₈	—	167	1	no	—
111	C₂×Q₈	C₂²×D₁₆₇	1	2	yes	—
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	334	4	no	—
114	Q₁₆	—	668	2	no	—
115	Q₁₆	—	668	4	no	—
116	C₂×C₈	—	167	2	no	—
117	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	334	4	no	—
121	C₄⋊C₄	—	334	4	no	—
122	C₄⋊C₄	—	334	4	no	—
123	C₄⋊C₄	—	334	2	no	—
124	C₄⋊C₄	—	334	2	no	—
125	C₄⋊C₄	—	167	2	no	—
126	C₄⋊C₄	—	167	1	no	—
127	C₄⋊C₄	C₂²×D₁₆₇	1	2	yes	—
128	C₄²	—	2	2	no	—
129	C₄²	—	334	4	no	—
130	C₄²	—	167	2	no	—
131	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
132	Q₈	—	2	2	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	334	4	no	—
135	Q₈	—	334	4	no	—
136	Q₈	—	167	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	334	4	no	—
140	C₂×C₄	—	167	2	no	—
141	C₂×C₄	—	167	2	no	—
142	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
143	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
144	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	334	2	no	—
147	C₈	#14581950	1	2	yes	—
148	C₂²	C₂×D₄×D₁₆₇	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	334	2	no	—
153	C₄	—	668	4	no	—
154	C₄	—	167	4	no	—
155	C₄	Q₁₆×D₁₆₇	1	2	yes	—
156	C₄	C₂×#14581950	1	1	yes	characteristic
157	C₄	#19128878	1	1	yes	characteristic
158	C₂	#24923940	1	1	yes	characteristic
159	C₂	C₂×Q₁₆×D₁₆₇	1	1	yes	characteristic
160	C₂	#24924083	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24923997

C₂

yes

maximal

#24923649

C₂

yes

maximal, characteristic

#24923647

C₂

yes

maximal

#24923479

C₂

yes

maximal

#24923478

C₂

yes

maximal

C₈×Dic₃₃₄

C₂

yes

maximal, characteristic

#24923363

C₂

yes

maximal

C₄×Q₁₆×C₁₆₇

C₂

yes

maximal, characteristic

C₂×Dic₆₆₈

C₂²

yes

—

C₂×Dic₆₆₈

C₂²

yes

—

C₂×#14581940

C₂²

yes

—

#19128832

C₂²

yes

—

#19128830

C₂²

yes

—

#19128824

—

#19128822

—

C₄×Dic₃₃₄

—

C₄×Dic₃₃₄

C₂²

yes

characteristic

#19128797

C₂²

yes

—

C₄⋊(C₁₆₇⋊C₈)

C₂²

yes

characteristic

Dic₁₆₇⋊C₈

C₂²

yes

—

#19128755

C₂²

yes

—

C₈×Dic₁₆₇

C₂²

yes

—

C₂×Q₁₆×C₁₆₇

C₂²

yes

characteristic

C₄×Q₈×C₁₆₇

C₂²

yes

—

#107×C₁₆₇

C₂²

yes

characteristic

#103×C₁₆₇

C₂²

yes

—

C₄×C₈×C₁₆₇

C₂²

yes

characteristic

C₂×Dic₃₃₄

D₄

yes

characteristic

C₂×Dic₃₃₄

C₂³

yes

—

Dic₆₆₈

—

Dic₆₆₈

—

#14581940

—

C₂×C₁₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

D₄

yes

characteristic

Dic₁₆₇⋊C₄

—

Dic₁₆₇⋊C₄

—

Dic₁₆₇⋊C₄

—

Dic₁₆₇⋊C₄

D₄

yes

—

C₄×Dic₁₆₇

—

C₄×Dic₁₆₇

C₂³

yes

—

C₂×Q₈×C₁₆₇

C₂³

yes

—

Q₁₆×C₁₆₇

—

C₂×C₈×C₁₆₇

C₂³

yes

characteristic

C₂×C₈×C₁₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

C₂³

yes

—

C₄²×C₁₆₇

—

C₄²×C₁₆₇

C₂³

yes

characteristic

Dic₃₃₄

—

Dic₃₃₄

—

Dic₃₃₄

Q₁₆

yes

complemented

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

C₂×D₄

yes

—

C₂×Dic₁₆₇

C₂×D₄

yes

—

C₁₆₇⋊C₈

—

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂⁴

yes

characteristic

C₈×C₁₆₇

—

C₈×C₁₆₇

#41

yes

—

C₂²×C₁₆₇

C₂²×D₄

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×Q₁₆

yes

—

C₄×C₁₆₇

C₂×#41

yes

characteristic

C₄×C₁₆₇

H(5,𝔽₂)

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

Dic₁₆₇

C₂×Q₁₆

yes

—

C₂×C₁₆₇

#550

yes

characteristic

C₂×C₁₆₇

C₂²×Q₁₆

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#3395

—

167

maximal, Sylow

#557

—

167

—

#494

—

167

—

#494

—

167

—

#462

—

167

—

#462

—

167

—

C₈×Q₈

—

167

—

C₄×Q₁₆

—

167

—

C₄×Q₁₆

D₁₆₇

yes

characteristic

C₂×Q₁₆

—

167

—

C₂×Q₁₆

—

167

—

C₂×Q₁₆

—

167

—

C₂×Q₁₆

C₂×D₁₆₇

yes

characteristic

C₄⋊Q₈

—

167

—

C₄⋊Q₈

—

167

—

#125

—

334

—

#125

—

334

—

C₄×Q₈

—

334

—

C₄×Q₈

—

167

—

100

C₄×Q₈

—

167

—

101

C₄×Q₈

C₂×D₁₆₇

yes

—

102

#107

C₂×D₁₆₇

yes

characteristic

103

C₄⋊C₈

—

167

—

104

C₄⋊C₈

—

167

—

105

#103

—

167

—

106

#103

C₂×D₁₆₇

yes

—

107

C₄×C₈

—

167

—

108

C₄×C₈

C₂×D₁₆₇

yes

characteristic

109

C₂×Q₈

—

167

—

110

C₂×Q₈

—

167

—

111

C₂×Q₈

C₂²×D₁₆₇

yes

—

112

Q₁₆

—

113

Q₁₆

—

334

—

114

Q₁₆

—

668

—

115

Q₁₆

—

668

—

116

C₂×C₈

—

167

—

117

C₂×C₈

C₂²×D₁₆₇

yes

characteristic

118

C₂×C₈

C₂²×D₁₆₇

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

334

—

121

C₄⋊C₄

—

334

—

122

C₄⋊C₄

—

334

—

123

C₄⋊C₄

—

334

—

124

C₄⋊C₄

—

334

—

125

C₄⋊C₄

—

167

—

126

C₄⋊C₄

—

167

—

127

C₄⋊C₄

C₂²×D₁₆₇

yes

—

128

C₄²

—

129

C₄²

—

334

—

130

C₄²

—

167

—

131

C₄²

C₂²×D₁₆₇

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

334

—

135

Q₈

—

334

—

136

Q₈

—

167

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

334

—

140

C₂×C₄

—

167

—

141

C₂×C₄

—

167

—

142

C₂×C₄

D₄×D₁₆₇

yes

characteristic

143

C₂×C₄

D₄×D₁₆₇

yes

characteristic

144

C₂×C₄

C₂³×D₁₆₇

yes

characteristic

145

C₈

—

146

C₈

—

334

—

147

C₈

#14581950

yes

—

148

C₂²

C₂×D₄×D₁₆₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

334

—

153

C₄

—

668

—

154

C₄

—

167

—

155

C₄

Q₁₆×D₁₆₇

yes

—

156

C₄

C₂×#14581950

yes

characteristic

157

C₄

#19128878

yes

characteristic

158

C₂

#24923940

yes

characteristic

159

C₂

C₂×Q₁₆×D₁₆₇

yes

characteristic

160

C₂

#24924083

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₆₇
commutator, cosocle	C₂×C₄×C₁₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₇
Fitting	C₄×Q₁₆×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₂²×D₄
hypercenter	C₄×Q₁₆	D₁₆₇
hypocenter	C₁₆₇	#3395
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₁₆₇

commutator, cosocle

C₂×C₄×C₁₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₆₇

Fitting

C₄×Q₁₆×C₁₆₇

C₂

socle, prime radical

C₂²×C₁₆₇

C₂²×D₄

hypercenter

C₄×Q₁₆

D₁₆₇

hypocenter

C₁₆₇

#3395

LCS

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
53. Dic₃₃₄	Q₁₆	semidirect
80. C₁₆₇	#3395	semidirect

H ≅ G/N

type

53. Dic₃₃₄

Q₁₆

semidirect

80. C₁₆₇

#3395

semidirect

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

subgroup	index
#24923997	2
#24923649	2
#24923647	2
#24923479	2
#24923478	2
C₈×Dic₃₃₄	2
#24923363	2
C₄×Q₁₆×C₁₆₇	2
#3395	167

subgroup

index

C₈×Dic₃₃₄

C₄×Q₁₆×C₁₆₇

167

Subgroups — #36337143

Conjugacy classes of subgroups

Complements

Maximal subgroups