Group #36324888 — subgroups

Subgroups — #36324888

Order 21376.

200 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 83 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	1	yes	maximal, characteristic
2	#24923305	C₂	1	1	yes	maximal, characteristic
3	#24923295	C₂	1	1	yes	maximal, characteristic
4	C₂×#19128742	C₂	1	1	yes	maximal, characteristic
5	#24923030	C₂	1	1	yes	maximal, characteristic
6	#24923028	C₂	1	1	yes	maximal, characteristic
7	#424×C₁₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14581934	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	#19128791	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₆₇	C₂²	1	1	yes	characteristic
12	#19128767	—	2	2	no	—
13	#19128765	C₄	1	2	yes	complemented
14	#19128763	C₄	1	2	yes	complemented
15	Dic₁₆₇⋊C₈	—	2	2	no	—
16	#19128742	—	2	2	no	—
17	C₂×M₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
19	#117×C₁₆₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₁₆₇	—	2	2	no	—
21	C₂²×Dic₁₆₇	—	2	1	no	—
22	#14581934	—	2	1	no	—
23	#14581934	—	2	1	no	—
24	C₂×C₁₆₇⋊C₈	—	2	1	no	—
25	C₂×C₁₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆₇⋊C₈	D₄	1	2	yes	—
28	C₂×C₁₆₇⋊C₈	Q₈	1	2	yes	—
29	C₄⋊Dic₁₆₇	C₂×C₄	1	2	yes	—
30	C₂²⋊Dic₁₆₇	—	2	1	no	—
31	C₄×Dic₁₆₇	—	2	1	no	—
32	C₄×Dic₁₆₇	—	2	1	no	—
33	C₄×Dic₁₆₇	Q₈	1	2	yes	—
34	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₆₇	—	2	1	no	—
36	M₁₆×C₁₆₇	—	2	1	no	—
37	C₂×C₈×C₁₆₇	—	2	1	no	—
38	C₂×C₈×C₁₆₇	—	2	1	no	—
39	C₂×C₈×C₁₆₇	—	2	1	no	—
40	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
41	C₄⋊C₄×C₁₆₇	C₂×C₄	1	2	yes	—
42	C₂²⋊C₄×C₁₆₇	—	2	1	no	—
43	C₄²×C₁₆₇	—	2	1	no	—
44	C₂×Dic₁₆₇	—	2	2	no	—
45	C₂×Dic₁₆₇	—	2	2	no	—
46	C₂×Dic₁₆₇	—	4	2	no	—
47	C₁₆₇⋊C₈	—	4	1	no	—
48	C₁₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₁₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
50	C₂³×C₁₆₇	#41	1	1	yes	characteristic
51	C₂×C₄×C₁₆₇	—	2	2	no	—
52	C₂×C₄×C₁₆₇	C₂²×C₄	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	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
58	C₈×C₁₆₇	—	2	2	no	—
59	C₈×C₁₆₇	—	4	1	no	—
60	C₂²×C₁₆₇	—	2	2	no	—
61	C₂²×C₁₆₇	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₁₆₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₁₆₇	#125	1	1	yes	characteristic
64	C₄×C₁₆₇	—	4	2	no	—
65	C₄×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₁₆₇	C₄×Q₈	1	1	yes	characteristic
67	C₄×C₁₆₇	#121	1	1	yes	characteristic
68	C₄×C₁₆₇	C₄⋊Q₈	1	1	yes	characteristic
69	Dic₁₆₇	—	4	2	no	—
70	Dic₁₆₇	—	4	2	no	—
71	C₂×C₁₆₇	—	2	1	no	—
72	C₂×C₁₆₇	—	2	1	no	—
73	C₂×C₁₆₇	#389	1	1	yes	characteristic
74	C₂×C₁₆₇	#444	1	2	yes	—
75	C₁₆₇	#1946	1	1	yes	characteristic, complemented, Sylow
76	#1946	—	167	1	no	maximal, Sylow
77	C₂×#108	—	167	1	no	—
78	#427	—	167	1	no	—
79	#424	—	167	1	no	—
80	#424	D₁₆₇	1	1	yes	characteristic
81	C₂×#97	—	167	1	no	—
82	#339	—	167	1	no	—
83	#339	—	167	1	no	—
84	C₂×M₁₆	—	167	1	no	—
85	C₂×M₁₆	C₂×D₁₆₇	1	1	yes	characteristic
86	C₂²×C₈	—	167	1	no	—
87	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
88	#117	—	167	1	no	—
89	#117	C₂×D₁₆₇	1	1	yes	characteristic
90	C₂×C₄²	—	167	1	no	—
91	#108	—	334	2	no	—
92	#107	—	167	2	no	—
93	#106	—	167	2	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	334	2	no	—
96	#97	—	334	2	no	—
97	C₂²×C₄	—	334	1	no	—
98	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	334	1	no	—
102	M₁₆	—	334	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	334	1	no	—
107	C₂×C₈	—	167	1	no	—
108	C₂×C₈	—	167	1	no	—
109	C₂×C₈	—	167	2	no	—
110	C₂×C₈	—	167	2	no	—
111	C₂×C₈	C₂²⋊D₁₆₇	1	2	yes	—
112	C₄⋊C₄	—	167	2	no	—
113	C₄⋊C₄	C₄×D₁₆₇	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	334	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	334	1	no	—
118	C₄²	—	334	1	no	—
119	C₄²	—	167	2	no	—
120	C₂³	#14581950	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	334	2	no	—
123	C₂×C₄	—	334	2	no	—
124	C₂×C₄	—	668	2	no	—
125	C₂×C₄	C₂×C₄×D₁₆₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₆₇	1	1	yes	characteristic
129	C₂×C₄	Q₈×D₁₆₇	1	1	yes	characteristic
130	C₂×C₄	#14581949	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	4	1	no	—
133	C₈	—	668	1	no	—
134	C₈	—	167	2	no	—
135	C₈	—	167	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	C₄×C₂²⋊D₁₆₇	1	1	yes	characteristic
138	C₂²	#19128810	1	1	yes	characteristic
139	C₂²	#19128822	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	668	2	no	—
142	C₄	—	668	2	no	—
143	C₄	C₄⋊C₄×D₁₆₇	1	1	yes	characteristic
144	C₄	#19128797	1	1	yes	characteristic
145	C₄	#19128804	1	1	yes	characteristic
146	C₄	#19128830	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#24923164	1	1	yes	characteristic
150	C₂	#24923389	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128767

C₂

yes

maximal, characteristic

#24923305

C₂

yes

maximal, characteristic

#24923295

C₂

yes

maximal, characteristic

C₂×#19128742

C₂

yes

maximal, characteristic

#24923030

C₂

yes

maximal, characteristic

#24923028

C₂

yes

maximal, characteristic

#424×C₁₆₇

C₂

yes

maximal, characteristic

C₂×#14581934

C₂²

yes

characteristic

C₂²×C₁₆₇⋊C₈

C₂²

yes

characteristic

#19128791

C₂²

yes

characteristic

C₂×C₄×Dic₁₆₇

C₂²

yes

characteristic

#19128767

—

#19128765

C₄

yes

complemented

#19128763

C₄

yes

complemented

Dic₁₆₇⋊C₈

—

#19128742

—

C₂×M₁₆×C₁₆₇

C₂²

yes

characteristic

C₂²×C₈×C₁₆₇

C₂²

yes

characteristic

#117×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₇

—

C₂²×Dic₁₆₇

—

#14581934

—

#14581934

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₆₇⋊C₈

D₄

yes

—

C₂×C₁₆₇⋊C₈

Q₈

yes

—

C₄⋊Dic₁₆₇

C₂×C₄

yes

—

C₂²⋊Dic₁₆₇

—

C₄×Dic₁₆₇

—

C₄×Dic₁₆₇

—

C₄×Dic₁₆₇

Q₈

yes

—

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

M₁₆×C₁₆₇

—

M₁₆×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

D₄

yes

—

C₄⋊C₄×C₁₆₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₆₇

—

C₄²×C₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

C₄⋊C₄

yes

—

C₁₆₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₁₆₇

#41

yes

characteristic

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×Q₈

yes

characteristic

C₂×C₄×C₁₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₇

#41

yes

characteristic

C₈×C₁₆₇

—

C₈×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₄×D₄

yes

characteristic

C₂²×C₁₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₆₇

#125

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₆₇

C₄×Q₈

yes

characteristic

C₄×C₁₆₇

#121

yes

characteristic

C₄×C₁₆₇

C₄⋊Q₈

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

#389

yes

characteristic

C₂×C₁₆₇

#444

yes

—

C₁₆₇

#1946

yes

characteristic, complemented, Sylow

#1946

—

167

maximal, Sylow

C₂×#108

—

167

—

#427

—

167

—

#424

—

167

—

#424

D₁₆₇

yes

characteristic

C₂×#97

—

167

—

#339

—

167

—

#339

—

167

—

C₂×M₁₆

—

167

—

C₂×M₁₆

C₂×D₁₆₇

yes

characteristic

C₂²×C₈

—

167

—

C₂²×C₈

C₂×D₁₆₇

yes

characteristic

#117

—

167

—

#117

C₂×D₁₆₇

yes

characteristic

C₂×C₄²

—

167

—

#108

—

334

—

#107

—

167

—

#106

—

167

—

C₄⋊C₈

—

C₄⋊C₈

—

334

—

#97

—

334

—

C₂²×C₄

—

334

—

C₂²×C₄

C₂²×D₁₆₇

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

334

—

102

M₁₆

—

334

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

334

—

107

C₂×C₈

—

167

—

108

C₂×C₈

—

167

—

109

C₂×C₈

—

167

—

110

C₂×C₈

—

167

—

111

C₂×C₈

C₂²⋊D₁₆₇

yes

—

112

C₄⋊C₄

—

167

—

113

C₄⋊C₄

C₄×D₁₆₇

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

334

—

116

C₄²

—

117

C₄²

—

334

—

118

C₄²

—

334

—

119

C₄²

—

167

—

120

C₂³

#14581950

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

334

—

123

C₂×C₄

—

334

—

124

C₂×C₄

—

668

—

125

C₂×C₄

C₂×C₄×D₁₆₇

yes

characteristic

126

C₂×C₄

D₄×D₁₆₇

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₁₆₇

yes

characteristic

128

C₂×C₄

Q₈×D₁₆₇

yes

characteristic

129

C₂×C₄

Q₈×D₁₆₇

yes

characteristic

130

C₂×C₄

#14581949

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

668

—

134

C₈

—

167

—

135

C₈

—

167

—

136

C₂²

—

137

C₂²

C₄×C₂²⋊D₁₆₇

yes

characteristic

138

C₂²

#19128810

yes

characteristic

139

C₂²

#19128822

yes

characteristic

140

C₄

—

141

C₄

—

668

—

142

C₄

—

668

—

143

C₄

C₄⋊C₄×D₁₆₇

yes

characteristic

144

C₄

#19128797

yes

characteristic

145

C₄

#19128804

yes

characteristic

146

C₄

#19128830

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#24923164

yes

characteristic

150

C₂

#24923389

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₆₇
commutator	C₂×C₄×C₁₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₇
Fitting	#424×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₄×D₄
prime radical	C₂³×C₁₆₇	#41
cosocle	C₂²×C₄×C₁₆₇	C₂³
hypercenter	#424	D₁₆₇
hypocenter	C₁₆₇	#1946
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₆₇

commutator

C₂×C₄×C₁₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₇

Fitting

#424×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₄×D₄

prime radical

C₂³×C₁₆₇

#41

cosocle

C₂²×C₄×C₁₆₇

C₂³

D₁₆₇

C₁₆₇

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

N	H ≅ G/N	type
13. #19128765	C₄	semidirect
14. #19128763	C₄	semidirect
75. C₁₆₇	#1946	semidirect

H ≅ G/N

type

13. #19128765

C₄

semidirect

14. #19128763

C₄

semidirect

75. C₁₆₇

#1946

semidirect

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

subgroup	index
C₂×#19128767	2
#24923305	2
#24923295	2
C₂×#19128742	2
#24923030	2
#24923028	2
#424×C₁₆₇	2
#1946	167

subgroup

index

#424×C₁₆₇

167

Subgroups — #36324888

Conjugacy classes of subgroups

Complements

Maximal subgroups