Group #36320331 — subgroups

Subgroups — #1971×C₁₆₇ #36320331

Order 21376.

304 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 128 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×Q₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#106×C₁₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#103×C₁₆₇	C₂	1	2	yes	maximal
4	C₂×#97×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	#391×C₁₆₇	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₁₆₇	C₂²	1	2	yes	—
7	C₂×Q₁₆×C₁₆₇	—	2	2	no	—
8	C₂×Q₁₆×C₁₆₇	—	2	2	no	—
9	C₂×Q₁₆×C₁₆₇	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₆₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
14	#106×C₁₆₇	—	2	2	no	—
15	#103×C₁₆₇	—	2	4	no	—
16	#97×C₁₆₇	—	2	2	no	—
17	#95×C₁₆₇	—	2	4	no	—
18	C₂×Q₈×C₁₆₇	—	2	2	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	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₆₇	—	2	2	no	—
25	C₂²×C₄×C₁₆₇	—	2	2	no	—
26	C₂²×C₄×C₁₆₇	D₄	1	2	yes	—
27	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
28	Q₁₆×C₁₆₇	—	2	8	no	—
29	C₂×C₈×C₁₆₇	—	2	1	no	—
30	C₂×C₈×C₁₆₇	—	2	1	no	—
31	C₂×C₈×C₁₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
33	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
34	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
35	C₄⋊C₄×C₁₆₇	—	2	2	no	—
36	C₄⋊C₄×C₁₆₇	—	2	2	no	—
37	C₄²×C₁₆₇	—	2	1	no	—
38	C₄²×C₁₆₇	—	2	1	no	—
39	C₂³×C₁₆₇	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₆₇	—	2	4	no	—
41	Q₈×C₁₆₇	—	2	4	no	—
42	Q₈×C₁₆₇	—	4	2	no	—
43	Q₈×C₁₆₇	—	4	2	no	—
44	C₂×C₄×C₁₆₇	—	2	4	no	—
45	C₂×C₄×C₁₆₇	—	2	2	no	—
46	C₂×C₄×C₁₆₇	—	2	4	no	—
47	C₂×C₄×C₁₆₇	—	4	2	no	—
48	C₂×C₄×C₁₆₇	—	4	2	no	—
49	C₂×C₄×C₁₆₇	—	4	2	no	—
50	C₂×C₄×C₁₆₇	—	4	4	no	—
51	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
57	C₈×C₁₆₇	—	2	2	no	—
58	C₈×C₁₆₇	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₁₆₇	C₄×D₄	1	1	yes	characteristic
60	C₂²×C₁₆₇	#121	1	1	yes	characteristic
61	C₂²×C₁₆₇	#127	1	1	yes	characteristic
62	C₂²×C₁₆₇	#137	1	4	yes	—
63	C₄×C₁₆₇	—	4	4	no	—
64	C₄×C₁₆₇	—	4	2	no	—
65	C₄×C₁₆₇	—	8	2	no	—
66	C₄×C₁₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₁₆₇	C₄×D₄	1	1	yes	characteristic
68	C₄×C₁₆₇	#121	1	1	yes	characteristic
69	C₄×C₁₆₇	#124	1	1	yes	characteristic
70	C₂×C₁₆₇	#390	1	1	yes	characteristic
71	C₂×C₁₆₇	#441	1	2	yes	—
72	C₂×C₁₆₇	#470	1	2	yes	—
73	C₂×C₁₆₇	#497	1	2	yes	—
74	C₁₆₇	#1971	1	1	yes	characteristic, direct product factor, Sylow
75	#1971	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×Q₁₆	C₂×C₁₆₇	1	1	yes	characteristic
77	C₂×#106	C₂×C₁₆₇	1	1	yes	characteristic
78	C₂×#103	C₂×C₁₆₇	1	2	yes	—
79	C₂×#97	C₂×C₁₆₇	1	1	yes	characteristic
80	#391	C₂×C₁₆₇	1	2	yes	—
81	C₂²×Q₈	C₂²×C₁₆₇	1	2	yes	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×Q₁₆	—	2	2	no	—
84	C₂×Q₁₆	C₄×C₁₆₇	1	4	yes	complemented
85	C₂²×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
86	C₂²×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂²×C₁₆₇	1	2	yes	—
88	C₂×C₄²	C₂²×C₁₆₇	1	1	yes	characteristic
89	#106	—	2	2	no	—
90	#103	—	2	4	no	—
91	#97	—	2	2	no	—
92	#95	—	2	4	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂×C₄×C₁₆₇	1	4	yes	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	D₄×C₁₆₇	1	2	yes	—
102	C₂²×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
103	Q₁₆	—	2	8	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	C₂×C₄×C₁₆₇	1	2	yes	—
107	C₂×C₈	D₄×C₁₆₇	1	2	yes	—
108	C₂×C₈	D₄×C₁₆₇	1	2	yes	—
109	C₂×C₈	D₄×C₁₆₇	1	2	yes	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₄×C₁₆₇	1	1	yes	characteristic
115	Q₈	—	2	4	no	—
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	C₂²×C₄×C₁₆₇	1	1	yes	characteristic
127	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
128	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
129	C₂×C₄	C₂×D₄×C₁₆₇	1	1	yes	characteristic
130	C₂×C₄	#41×C₁₆₇	1	1	yes	characteristic
131	C₂×C₄	#41×C₁₆₇	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	C₂²⋊C₄×C₁₆₇	1	4	yes	—
134	C₂²	C₄×D₄×C₁₆₇	1	1	yes	characteristic
135	C₂²	#121×C₁₆₇	1	1	yes	characteristic
136	C₂²	#127×C₁₆₇	1	1	yes	characteristic
137	C₂²	#137×C₁₆₇	1	4	yes	—
138	C₄	—	4	4	no	—
139	C₄	—	4	2	no	—
140	C₄	—	8	2	no	—
141	C₄	C₂×C₂²⋊C₄×C₁₆₇	1	1	yes	characteristic
142	C₄	C₄×D₄×C₁₆₇	1	1	yes	characteristic
143	C₄	#121×C₁₆₇	1	1	yes	characteristic
144	C₄	#124×C₁₆₇	1	1	yes	characteristic
145	C₂	#390×C₁₆₇	1	1	yes	characteristic
146	C₂	#441×C₁₆₇	1	2	yes	—
147	C₂	#470×C₁₆₇	1	2	yes	—
148	C₂	#497×C₁₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₁₆×C₁₆₇

C₂

yes

maximal, characteristic

C₂×#106×C₁₆₇

C₂

yes

maximal, characteristic

C₂×#103×C₁₆₇

C₂

yes

maximal

C₂×#97×C₁₆₇

C₂

yes

maximal, characteristic

#391×C₁₆₇

C₂

yes

maximal

C₂²×Q₈×C₁₆₇

C₂²

yes

—

C₂×Q₁₆×C₁₆₇

—

C₂×Q₁₆×C₁₆₇

—

C₂×Q₁₆×C₁₆₇

C₄

yes

complemented

C₂²×C₈×C₁₆₇

C₂²

yes

characteristic

C₂²×C₈×C₁₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₆₇

C₂²

yes

—

C₂×C₄²×C₁₆₇

C₂²

yes

characteristic

#106×C₁₆₇

—

#103×C₁₆₇

—

#97×C₁₆₇

—

#95×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

C₂×C₄

yes

—

C₂²×C₄×C₁₆₇

—

C₂²×C₄×C₁₆₇

—

C₂²×C₄×C₁₆₇

D₄

yes

—

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

Q₁₆×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

C₂×C₄

yes

—

C₂×C₈×C₁₆₇

D₄

yes

—

C₂×C₈×C₁₆₇

D₄

yes

—

C₂×C₈×C₁₆₇

D₄

yes

—

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₄²×C₁₆₇

—

C₄²×C₁₆₇

—

C₂³×C₁₆₇

C₂×D₄

yes

characteristic

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

C₂×C₄×C₁₆₇

yes

C₂×C₄×C₁₆₇

yes

C₈×C₁₆₇

—

C₈×C₁₆₇

C₂²⋊C₄

yes

—

C₂²×C₁₆₇

C₄×D₄

yes

C₂²×C₁₆₇

yes

C₂²×C₁₆₇

yes

C₂²×C₁₆₇

yes

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₆₇

C₄×D₄

yes

C₄×C₁₆₇

yes

C₄×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

—

C₂×C₁₆₇

#470

yes

—

C₂×C₁₆₇

#497

yes

—

C₁₆₇

#1971

yes

characteristic, direct product factor, Sylow

#1971

C₁₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂×C₁₆₇

yes

characteristic

C₂×#106

C₂×C₁₆₇

yes

characteristic

C₂×#103

C₂×C₁₆₇

yes

—

C₂×#97

C₂×C₁₆₇

yes

characteristic

#391

C₂×C₁₆₇

yes

—

C₂²×Q₈

C₂²×C₁₆₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₄×C₁₆₇

yes

complemented

C₂²×C₈

C₂²×C₁₆₇

yes

characteristic

C₂²×C₈

C₂²×C₁₆₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₆₇

yes

—

C₂×C₄²

C₂²×C₁₆₇

yes

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₁₆₇

yes

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

D₄×C₁₆₇

yes

—

102

C₂²×C₄

C₂³×C₁₆₇

yes

characteristic

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×C₄×C₁₆₇

yes

—

107

C₂×C₈

D₄×C₁₆₇

yes

—

108

C₂×C₈

D₄×C₁₆₇

yes

—

109

C₂×C₈

D₄×C₁₆₇

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₄×C₁₆₇

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂²×C₄×C₁₆₇

yes

characteristic

127

C₂×C₄

C₂×D₄×C₁₆₇

yes

characteristic

128

C₂×C₄

C₂×D₄×C₁₆₇

yes

characteristic

129

C₂×C₄

C₂×D₄×C₁₆₇

yes

characteristic

130

C₂×C₄

#41×C₁₆₇

yes

characteristic

131

C₂×C₄

#41×C₁₆₇

yes

characteristic

132

C₈

—

133

C₈

C₂²⋊C₄×C₁₆₇

yes

—

134

C₂²

C₄×D₄×C₁₆₇

yes

characteristic

135

C₂²

#121×C₁₆₇

yes

characteristic

136

C₂²

#127×C₁₆₇

yes

characteristic

137

C₂²

#137×C₁₆₇

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

C₂×C₂²⋊C₄×C₁₆₇

yes

characteristic

142

C₄

C₄×D₄×C₁₆₇

yes

characteristic

143

C₄

#121×C₁₆₇

yes

characteristic

144

C₄

#124×C₁₆₇

yes

characteristic

145

C₂

#390×C₁₆₇

yes

characteristic

146

C₂

#441×C₁₆₇

yes

—

147

C₂

#470×C₁₆₇

yes

—

148

C₂

#497×C₁₆₇

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³×C₁₆₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₆₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₆₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₆₇ ◁ C₂×C₄²×C₁₆₇ ◁ G

N	H ≅ G/N	type
9. C₂×Q₁₆×C₁₆₇	C₄	semidirect
74. C₁₆₇	#1971	direct (N × H)
75. #1971	C₁₆₇	direct (N × H)
84. C₂×Q₁₆	C₄×C₁₆₇	semidirect

H ≅ G/N

type

9. C₂×Q₁₆×C₁₆₇

C₄

semidirect

74. C₁₆₇

#1971

direct (N × H)

75. #1971

C₁₆₇

direct (N × H)

84. C₂×Q₁₆

C₄×C₁₆₇

semidirect

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

subgroup	index
C₂²×Q₁₆×C₁₆₇	2
C₂×#106×C₁₆₇	2
C₂×#103×C₁₆₇	2
C₂×#97×C₁₆₇	2
#391×C₁₆₇	2
#1971	167

subgroup

index

C₂²×Q₁₆×C₁₆₇

C₂×#106×C₁₆₇

C₂×#103×C₁₆₇

C₂×#97×C₁₆₇

#391×C₁₆₇

#1971

167

Subgroups — #1971×C₁₆₇ #36320331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1971×C167 #36320331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1971×C₁₆₇ #36320331