Group #36324911 — subgroups

Subgroups — #36324911

Order 21376.

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

normal subgroups only

This group has 200 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×#19128767	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₁₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×#14581934	C₂	1	1	yes	maximal, characteristic
4	#24923033	C₂	1	2	yes	maximal
5	#423×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581934	C₂²	1	1	yes	characteristic
7	C₂×#14581934	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆₇⋊C₈	C₂²	1	2	yes	—
9	#19128767	—	2	4	no	—
10	C₄⋊(C₁₆₇⋊C₈)	C₄	1	2	yes	—
11	C₄⋊(C₁₆₇⋊C₈)	C₄	1	2	yes	—
12	#19128743	—	2	1	no	—
13	C₄×C₁₆₇⋊C₈	—	2	1	no	—
14	C₂×M₁₆×C₁₆₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₆₇	—	2	1	no	—
17	C₄⋊C₈×C₁₆₇	—	2	1	no	—
18	#14581934	—	2	1	no	—
19	#14581934	—	2	1	no	—
20	#14581934	D₄	1	2	yes	—
21	#14581934	Q₈	1	2	yes	—
22	C₂×C₁₆₇⋊C₈	—	2	2	no	—
23	C₂×C₁₆₇⋊C₈	—	2	1	no	—
24	C₂×C₁₆₇⋊C₈	—	2	1	no	—
25	C₂×C₁₆₇⋊C₈	—	2	2	no	—
26	C₂×C₁₆₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₆₇	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₆₇	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
30	M₁₆×C₁₆₇	—	2	2	no	—
31	M₁₆×C₁₆₇	—	2	2	no	—
32	C₂×C₈×C₁₆₇	—	2	2	no	—
33	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₆₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
36	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
37	C₁₆₇⋊C₈	—	2	4	no	—
38	C₁₆₇⋊C₈	—	2	2	no	—
39	C₁₆₇⋊C₈	—	2	2	no	—
40	C₂³×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₆₇	—	2	2	no	—
42	C₂×C₄×C₁₆₇	—	4	1	no	—
43	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
53	C₈×C₁₆₇	—	4	2	no	—
54	C₂²×C₁₆₇	—	2	1	no	—
55	C₂²×C₁₆₇	—	2	1	no	—
56	C₂²×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₆₇	#125	1	1	yes	characteristic
58	C₂²×C₁₆₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₄×C₁₆₇	—	2	1	no	—
60	C₄×C₁₆₇	—	2	1	no	—
61	C₄×C₁₆₇	—	2	1	no	—
62	C₄×C₁₆₇	#108	1	2	yes	complemented
63	C₄×C₁₆₇	C₄×D₄	1	1	yes	characteristic
64	C₄×C₁₆₇	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₁₆₇	#121	1	1	yes	characteristic
66	C₄×C₁₆₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₁₆₇	—	2	1	no	—
68	C₂×C₁₆₇	—	2	1	no	—
69	C₂×C₁₆₇	#389	1	1	yes	characteristic
70	C₂×C₁₆₇	C₂×#108	1	1	yes	characteristic
71	C₂×C₁₆₇	#430	1	1	yes	characteristic
72	C₁₆₇	#1950	1	1	yes	characteristic, complemented, Sylow
73	#1950	—	167	1	no	maximal, Sylow
74	C₂×#108	—	167	2	no	—
75	#423	D₁₆₇	1	1	yes	characteristic
76	C₂×C₄⋊C₈	—	167	1	no	—
77	C₄×M₁₆	—	167	1	no	—
78	#341	—	167	2	no	—
79	C₂×M₁₆	—	167	1	no	—
80	C₂×M₁₆	—	167	1	no	—
81	C₂×M₁₆	C₂×D₁₆₇	1	2	yes	—
82	C₂²×C₈	—	167	2	no	—
83	C₂×C₄²	C₂×D₁₆₇	1	1	yes	characteristic
84	#108	—	334	4	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	—	167	2	no	—
88	C₄⋊C₈	—	167	2	no	—
89	#97	—	334	1	no	—
90	C₄×C₈	—	334	1	no	—
91	C₂²×C₄	Dic₃₃₄	1	1	yes	characteristic
92	C₂²×C₄	Dic₃₃₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
94	M₁₆	—	2	2	no	—
95	M₁₆	—	2	2	no	—
96	M₁₆	—	334	1	no	—
97	M₁₆	—	334	1	no	—
98	M₁₆	—	167	2	no	—
99	M₁₆	—	167	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	334	2	no	—
102	C₂×C₈	—	334	1	no	—
103	C₂×C₈	—	334	1	no	—
104	C₂×C₈	—	334	2	no	—
105	C₂×C₈	—	167	4	no	—
106	C₄²	C₄×D₁₆₇	1	1	yes	characteristic
107	C₄²	C₄×D₁₆₇	1	1	yes	characteristic
108	C₄²	C₂²⋊D₁₆₇	1	1	yes	characteristic
109	C₄²	C₂²⋊D₁₆₇	1	1	yes	characteristic
110	C₂³	C₂×Dic₃₃₄	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	Dic₁₆₇⋊C₄	1	1	yes	characteristic
114	C₂×C₄	Dic₁₆₇⋊C₄	1	1	yes	characteristic
115	C₂×C₄	Dic₁₆₇⋊C₄	1	1	yes	characteristic
116	C₂×C₄	Dic₁₆₇⋊C₄	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
118	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₁₆₇	1	1	yes	characteristic
121	C₂×C₄	#14581949	1	1	yes	characteristic
122	C₂×C₄	#14581951	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	334	4	no	—
125	C₈	—	334	2	no	—
126	C₈	—	334	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂×Dic₁₆₇⋊C₄	1	1	yes	characteristic
130	C₂²	#19128823	1	1	yes	characteristic
131	C₂²	#19128831	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#19128767	1	2	yes	complemented
136	C₄	#19128794	1	1	yes	characteristic
137	C₄	#19128797	1	1	yes	characteristic
138	C₄	#19128803	1	1	yes	characteristic
139	C₄	#19128808	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#24923165	1	1	yes	characteristic
143	C₂	C₂×#19128767	1	1	yes	characteristic
144	C₂	#24923317	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128767

C₂

yes

maximal

C₂×C₄⋊(C₁₆₇⋊C₈)

C₂

yes

maximal, characteristic

C₄×#14581934

C₂

yes

maximal, characteristic

#24923033

C₂

yes

maximal

#423×C₁₆₇

C₂

yes

maximal, characteristic

C₂×#14581934

C₂²

yes

characteristic

C₂×#14581934

C₂²

yes

characteristic

C₂²×C₁₆₇⋊C₈

C₂²

yes

—

#19128767

—

C₄⋊(C₁₆₇⋊C₈)

C₄

yes

—

C₄⋊(C₁₆₇⋊C₈)

C₄

yes

—

#19128743

—

C₄×C₁₆₇⋊C₈

—

C₂×M₁₆×C₁₆₇

C₂²

yes

—

C₂×C₄²×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₇

—

C₄⋊C₈×C₁₆₇

—

#14581934

—

#14581934

—

#14581934

D₄

yes

—

#14581934

Q₈

yes

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₆₇

Q₈

yes

characteristic

C₂²×C₄×C₁₆₇

Q₈

yes

characteristic

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

M₁₆×C₁₆₇

—

M₁₆×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₄²×C₁₆₇

C₂×C₄

yes

characteristic

C₄²×C₁₆₇

C₂×C₄

yes

characteristic

C₄²×C₁₆₇

D₄

yes

characteristic

C₄²×C₁₆₇

D₄

yes

characteristic

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

characteristic

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×Q₈

yes

C₂×C₄×C₁₆₇

yes

C₂×C₄×C₁₆₇

yes

C₈×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₆₇

#125

yes

characteristic

C₂²×C₁₆₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

#108

yes

complemented

C₄×C₁₆₇

C₄×D₄

yes

characteristic

C₄×C₁₆₇

C₄×Q₈

yes

characteristic

C₄×C₁₆₇

#121

yes

characteristic

C₄×C₁₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#1950

—

167

maximal, Sylow

C₂×#108

—

167

—

#423

D₁₆₇

yes

characteristic

C₂×C₄⋊C₈

—

167

—

C₄×M₁₆

—

167

—

#341

—

167

—

C₂×M₁₆

—

167

—

C₂×M₁₆

—

167

—

C₂×M₁₆

C₂×D₁₆₇

yes

—

C₂²×C₈

—

167

—

C₂×C₄²

C₂×D₁₆₇

yes

characteristic

#108

—

334

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

167

—

C₄⋊C₈

—

167

—

#97

—

334

—

C₄×C₈

—

334

—

C₂²×C₄

Dic₃₃₄

yes

characteristic

C₂²×C₄

Dic₃₃₄

yes

characteristic

C₂²×C₄

C₂²×D₁₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

334

—

M₁₆

—

334

—

M₁₆

—

167

—

M₁₆

—

167

—

100

C₂×C₈

—

101

C₂×C₈

—

334

—

102

C₂×C₈

—

334

—

103

C₂×C₈

—

334

—

104

C₂×C₈

—

334

—

105

C₂×C₈

—

167

—

106

C₄²

C₄×D₁₆₇

yes

characteristic

107

C₄²

C₄×D₁₆₇

yes

characteristic

108

C₄²

C₂²⋊D₁₆₇

yes

characteristic

109

C₄²

C₂²⋊D₁₆₇

yes

characteristic

110

C₂³

C₂×Dic₃₃₄

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

Dic₁₆₇⋊C₄

yes

characteristic

114

C₂×C₄

Dic₁₆₇⋊C₄

yes

characteristic

115

C₂×C₄

Dic₁₆₇⋊C₄

yes

characteristic

116

C₂×C₄

Dic₁₆₇⋊C₄

yes

characteristic

117

C₂×C₄

C₂×C₄×D₁₆₇

yes

characteristic

118

C₂×C₄

D₄×D₁₆₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₁₆₇

yes

characteristic

120

C₂×C₄

Q₈×D₁₆₇

yes

characteristic

121

C₂×C₄

#14581949

yes

characteristic

122

C₂×C₄

#14581951

yes

characteristic

123

C₈

—

124

C₈

—

334

—

125

C₈

—

334

—

126

C₈

—

334

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×Dic₁₆₇⋊C₄

yes

characteristic

130

C₂²

#19128823

yes

characteristic

131

C₂²

#19128831

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#19128767

yes

complemented

136

C₄

#19128794

yes

characteristic

137

C₄

#19128797

yes

characteristic

138

C₄

#19128803

yes

characteristic

139

C₄

#19128808

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#24923165

yes

characteristic

143

C₂

C₂×#19128767

yes

characteristic

144

C₂

#24923317

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₆₇
commutator	C₂×C₄×C₁₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₇
Fitting	#423×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₆₇	C₂³
hypercenter	#423	D₁₆₇
hypocenter	C₁₆₇	#1950
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₆₇

commutator

C₂×C₄×C₁₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₇

Fitting

#423×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₆₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₆₇

C₂³

D₁₆₇

C₁₆₇

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
62. C₄×C₁₆₇	#108	semidirect
72. C₁₆₇	#1950	semidirect
135. C₄	#19128767	semidirect

H ≅ G/N

type

62. C₄×C₁₆₇

#108

semidirect

72. C₁₆₇

#1950

semidirect

135. C₄

#19128767

semidirect

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

subgroup	index
C₂×#19128767	2
C₂×C₄⋊(C₁₆₇⋊C₈)	2
C₄×#14581934	2
#24923033	2
#423×C₁₆₇	2
#1950	167

subgroup

index

C₂×C₄⋊(C₁₆₇⋊C₈)

#423×C₁₆₇

167

Subgroups — #36324911

Conjugacy classes of subgroups

Complements

Maximal subgroups