Group #36334651 — subgroups

Subgroups — #36334651

Order 21376.

356 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 141 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	#24923825	C₂	1	2	yes	maximal
2	#24923673	C₂	1	2	yes	maximal, complemented
3	#24923671	C₂	1	1	yes	maximal, characteristic, complemented
4	#24923668	C₂	1	2	yes	maximal, complemented
5	#24923666	C₂	1	2	yes	maximal, complemented
6	#24923306	C₂	1	2	yes	maximal
7	#24923304	C₂	1	2	yes	maximal
8	C₄×C₈×D₁₆₇	C₂	1	1	yes	maximal, characteristic
9	#499×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×D₁₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₆₇	C₂²	1	1	yes	characteristic
12	#19128831	C₂²	1	4	yes	—
13	#19128824	C₂²	1	2	yes	—
14	#19128810	—	2	4	no	—
15	#19128797	—	2	4	no	—
16	#19128790	C₂²	1	4	yes	—
17	C₄²×D₁₆₇	C₂²	1	1	yes	characteristic
18	#19128766	C₂²	1	2	yes	—
19	#19128765	C₂²	1	4	yes	—
20	#19128764	C₂²	1	2	yes	—
21	#19128763	C₂²	1	4	yes	—
22	C₄×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₈×Dic₁₆₇	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₆₇	C₂²	1	1	yes	characteristic
25	#125×C₁₆₇	C₂²	1	2	yes	—
26	#107×C₁₆₇	C₂²	1	2	yes	—
27	#106×C₁₆₇	C₂²	1	2	yes	—
28	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
29	C₂×Dic₃₃₄	—	2	4	no	—
30	C₂×C₄×D₁₆₇	D₄	1	2	yes	—
31	C₂×C₄×D₁₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
33	C₂×C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
34	C₈×D₁₆₇	Q₈	1	4	yes	complemented
35	C₈×D₁₆₇	Q₈	1	4	yes	complemented
36	C₄⋊Dic₁₆₇	C₂³	1	4	yes	—
37	Dic₁₆₇⋊C₄	—	2	4	no	—
38	Dic₁₆₇⋊C₄	—	2	4	no	—
39	(C₂×D₁₆₇)⋊C₄	—	2	4	no	—
40	C₄×Dic₁₆₇	—	2	4	no	—
41	C₄×Dic₁₆₇	D₄	1	2	yes	—
42	C₄×Dic₁₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₆₇	—	2	4	no	—
46	C₄⋊C₄×C₁₆₇	C₂³	1	4	yes	—
47	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
48	C₂²×D₁₆₇	C₂×D₄	1	1	yes	characteristic
49	Dic₃₃₄	—	4	8	no	—
50	C₂×Dic₁₆₇	—	2	4	no	—
51	C₂×Dic₁₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₁₆₇	C₂×D₄	1	2	yes	—
53	C₄×D₁₆₇	—	2	4	no	—
54	C₄×D₁₆₇	C₂×Q₈	1	2	yes	—
55	C₄×D₁₆₇	C₂×Q₈	1	2	yes	—
56	C₁₆₇⋊C₈	C₂×Q₈	1	2	yes	—
57	C₁₆₇⋊C₈	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₆₇	—	2	4	no	—
59	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₆₇	C₂×Q₈	1	2	yes	—
62	C₈×C₁₆₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₁₆₇	—	2	2	no	—
65	C₂×D₁₆₇	C₄⋊Q₈	1	2	yes	—
66	C₄×C₁₆₇	—	2	2	no	—
67	C₄×C₁₆₇	—	4	4	no	—
68	C₄×C₁₆₇	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₁₆₇	C₂²×Q₈	1	1	yes	characteristic
70	Dic₁₆₇	—	4	4	no	—
71	Dic₁₆₇	C₄⋊Q₈	1	2	yes	—
72	Dic₁₆₇	#135	1	4	yes	—
73	C₂×C₁₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₁₆₇	C₂×#135	1	2	yes	—
75	D₁₆₇	—	2	2	no	—
76	C₁₆₇	#3160	1	1	yes	characteristic, complemented, Sylow
77	#3160	—	167	1	no	maximal, Sylow
78	#533	—	167	2	no	—
79	#500	—	167	2	no	—
80	#499	—	167	2	no	—
81	#499	—	167	1	no	—
82	#499	D₁₆₇	1	1	yes	characteristic, complemented
83	#498	—	167	2	no	—
84	#427	—	167	2	no	—
85	#427	—	167	2	no	—
86	C₂×C₄×C₈	—	167	1	no	—
87	C₂²×C₈	—	167	1	no	—
88	C₂²×C₈	—	167	1	no	—
89	C₄⋊Q₈	—	167	4	no	—
90	#125	—	167	2	no	—
91	#125	C₂×D₁₆₇	1	2	yes	—
92	C₂²⋊Q₈	—	334	4	no	—
93	C₄×Q₈	—	334	4	no	—
94	#117	—	167	4	no	—
95	C₂×C₄²	—	167	1	no	—
96	#107	—	167	2	no	—
97	#107	—	167	4	no	—
98	#107	C₂×D₁₆₇	1	2	yes	—
99	#106	—	167	2	no	—
100	#106	—	167	4	no	—
101	#106	C₂×D₁₆₇	1	2	yes	—
102	C₄×C₈	—	167	1	no	—
103	C₄×C₈	—	167	1	no	—
104	C₄×C₈	—	167	1	no	—
105	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
106	C₂×Q₈	—	334	4	no	—
107	C₂²×C₄	—	167	2	no	—
108	C₂²×C₄	—	167	1	no	—
109	C₂×C₈	—	167	1	no	—
110	C₂×C₈	—	167	1	no	—
111	C₂×C₈	—	167	4	no	—
112	C₂×C₈	—	167	4	no	—
113	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	334	4	no	—
117	C₄⋊C₄	—	334	4	no	—
118	C₄⋊C₄	—	167	4	no	—
119	C₄⋊C₄	C₂²×D₁₆₇	1	4	yes	—
120	C₂²⋊C₄	—	334	4	no	—
121	C₄²	—	334	4	no	—
122	C₄²	—	167	2	no	—
123	C₄²	—	167	1	no	—
124	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
125	C₂³	—	167	1	no	—
126	Q₈	—	668	8	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	334	4	no	—
129	C₂×C₄	—	334	4	no	—
130	C₂×C₄	—	167	1	no	—
131	C₂×C₄	—	167	2	no	—
132	C₂×C₄	—	167	2	no	—
133	C₂×C₄	—	167	2	no	—
134	C₂×C₄	D₄×D₁₆₇	1	2	yes	—
135	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
136	C₈	—	167	2	no	—
137	C₈	—	167	2	no	—
138	C₈	Q₈×D₁₆₇	1	2	yes	—
139	C₈	Q₈×D₁₆₇	1	2	yes	—
140	C₂²	—	334	2	no	—
141	C₂²	—	167	2	no	—
142	C₂²	C₂×D₄×D₁₆₇	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	—	668	4	no	—
146	C₄	—	167	4	no	—
147	C₄	—	167	2	no	—
148	C₄	C₂×Q₈×D₁₆₇	1	1	yes	characteristic
149	C₄	C₂×Q₈×D₁₆₇	1	1	yes	characteristic
150	C₂	—	334	2	no	—
151	C₂	C₄⋊Q₈×D₁₆₇	1	1	yes	characteristic
152	C₂	#135×D₁₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24923825

C₂

yes

maximal

#24923673

C₂

yes

maximal, complemented

#24923671

C₂

yes

maximal, characteristic, complemented

#24923668

C₂

yes

maximal, complemented

#24923666

C₂

yes

maximal, complemented

#24923306

C₂

yes

maximal

#24923304

C₂

yes

maximal

C₄×C₈×D₁₆₇

C₂

yes

maximal, characteristic

#499×C₁₆₇

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₆₇

C₂²

yes

characteristic

C₂×C₈×D₁₆₇

C₂²

yes

characteristic

#19128831

C₂²

yes

—

#19128824

C₂²

yes

—

#19128810

—

#19128797

—

#19128790

C₂²

yes

—

C₄²×D₁₆₇

C₂²

yes

characteristic

#19128766

C₂²

yes

—

#19128765

C₂²

yes

—

#19128764

C₂²

yes

—

#19128763

C₂²

yes

—

C₄×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₆₇

C₂²

yes

characteristic

C₈×Dic₁₆₇

C₂²

yes

characteristic

#125×C₁₆₇

C₂²

yes

—

#107×C₁₆₇

C₂²

yes

—

#106×C₁₆₇

C₂²

yes

—

C₄×C₈×C₁₆₇

C₂²

yes

characteristic

C₂×Dic₃₃₄

—

C₂×C₄×D₁₆₇

D₄

yes

—

C₂×C₄×D₁₆₇

C₂³

yes

characteristic

C₂×C₁₆₇⋊C₈

C₂³

yes

characteristic

C₂×C₁₆₇⋊C₈

C₂³

yes

characteristic

C₈×D₁₆₇

Q₈

yes

complemented

C₈×D₁₆₇

Q₈

yes

complemented

C₄⋊Dic₁₆₇

C₂³

yes

—

Dic₁₆₇⋊C₄

—

Dic₁₆₇⋊C₄

—

(C₂×D₁₆₇)⋊C₄

—

C₄×Dic₁₆₇

—

C₄×Dic₁₆₇

D₄

yes

—

C₄×Dic₁₆₇

C₂³

yes

characteristic

C₂×C₈×C₁₆₇

C₂³

yes

characteristic

C₂×C₈×C₁₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

C₂³

yes

—

C₄²×C₁₆₇

C₂³

yes

characteristic

C₂²×D₁₆₇

C₂×D₄

yes

characteristic

Dic₃₃₄

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

C₂×D₄

yes

characteristic

C₂×Dic₁₆₇

C₂×D₄

yes

—

C₄×D₁₆₇

—

C₄×D₁₆₇

C₂×Q₈

yes

—

C₄×D₁₆₇

C₂×Q₈

yes

—

C₁₆₇⋊C₈

C₂×Q₈

yes

—

C₁₆₇⋊C₈

C₂×Q₈

yes

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

—

C₂×C₄×C₁₆₇

C₂⁴

yes

characteristic

C₈×C₁₆₇

C₂×Q₈

yes

—

C₈×C₁₆₇

C₂×Q₈

yes

—

C₂²×C₁₆₇

C₂²×D₄

yes

characteristic

C₂×D₁₆₇

—

C₂×D₁₆₇

C₄⋊Q₈

yes

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂²×Q₈

yes

characteristic

C₄×C₁₆₇

C₂²×Q₈

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

C₄⋊Q₈

yes

—

Dic₁₆₇

#135

yes

—

C₂×C₁₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₆₇

C₂×#135

yes

—

D₁₆₇

—

C₁₆₇

#3160

yes

characteristic, complemented, Sylow

#3160

—

167

maximal, Sylow

#533

—

167

—

#500

—

167

—

#499

—

167

—

#499

—

167

—

#499

D₁₆₇

yes

characteristic, complemented

#498

—

167

—

#427

—

167

—

#427

—

167

—

C₂×C₄×C₈

—

167

—

C₂²×C₈

—

167

—

C₂²×C₈

—

167

—

C₄⋊Q₈

—

167

—

#125

—

167

—

#125

C₂×D₁₆₇

yes

—

C₂²⋊Q₈

—

334

—

C₄×Q₈

—

334

—

#117

—

167

—

C₂×C₄²

—

167

—

#107

—

167

—

#107

—

167

—

#107

C₂×D₁₆₇

yes

—

#106

—

167

—

100

#106

—

167

—

101

#106

C₂×D₁₆₇

yes

—

102

C₄×C₈

—

167

—

103

C₄×C₈

—

167

—

104

C₄×C₈

—

167

—

105

C₄×C₈

C₂×D₁₆₇

yes

characteristic

106

C₂×Q₈

—

334

—

107

C₂²×C₄

—

167

—

108

C₂²×C₄

—

167

—

109

C₂×C₈

—

167

—

110

C₂×C₈

—

167

—

111

C₂×C₈

—

167

—

112

C₂×C₈

—

167

—

113

C₂×C₈

C₂²×D₁₆₇

yes

characteristic

114

C₂×C₈

C₂²×D₁₆₇

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

334

—

117

C₄⋊C₄

—

334

—

118

C₄⋊C₄

—

167

—

119

C₄⋊C₄

C₂²×D₁₆₇

yes

—

120

C₂²⋊C₄

—

334

—

121

C₄²

—

334

—

122

C₄²

—

167

—

123

C₄²

—

167

—

124

C₄²

C₂²×D₁₆₇

yes

characteristic

125

C₂³

—

167

—

126

Q₈

—

668

—

127

C₂×C₄

—

128

C₂×C₄

—

334

—

129

C₂×C₄

—

334

—

130

C₂×C₄

—

167

—

131

C₂×C₄

—

167

—

132

C₂×C₄

—

167

—

133

C₂×C₄

—

167

—

134

C₂×C₄

D₄×D₁₆₇

yes

—

135

C₂×C₄

C₂³×D₁₆₇

yes

characteristic

136

C₈

—

167

—

137

C₈

—

167

—

138

C₈

Q₈×D₁₆₇

yes

—

139

C₈

Q₈×D₁₆₇

yes

—

140

C₂²

—

334

—

141

C₂²

—

167

—

142

C₂²

C₂×D₄×D₁₆₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

668

—

146

C₄

—

167

—

147

C₄

—

167

—

148

C₄

C₂×Q₈×D₁₆₇

yes

characteristic

149

C₄

C₂×Q₈×D₁₆₇

yes

characteristic

150

C₂

—

334

—

151

C₂

C₄⋊Q₈×D₁₆₇

yes

characteristic

152

C₂

#135×D₁₆₇

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₆₇
commutator, cosocle	C₂×C₄×C₁₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₇
Fitting	#499×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂²×D₄
prime radical	C₂²×D₁₆₇	C₂×D₄
hypercenter	#499	D₁₆₇
hypocenter	C₁₆₇	#3160
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₁₆₇

commutator, cosocle

C₂×C₄×C₁₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₆₇

Fitting

#499×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₂²×D₄

prime radical

C₂²×D₁₆₇

C₂×D₄

D₁₆₇

C₁₆₇

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
2. #24923673	C₂	semidirect
3. #24923671	C₂	semidirect
4. #24923668	C₂	semidirect
5. #24923666	C₂	semidirect
9. #499×C₁₆₇	C₂	semidirect
34. C₈×D₁₆₇	Q₈	semidirect
35. C₈×D₁₆₇	Q₈	semidirect
76. C₁₆₇	#3160	semidirect
82. #499	D₁₆₇	semidirect

H ≅ G/N

type

2. #24923673

C₂

semidirect

3. #24923671

C₂

semidirect

4. #24923668

C₂

semidirect

5. #24923666

C₂

semidirect

9. #499×C₁₆₇

C₂

semidirect

34. C₈×D₁₆₇

Q₈

semidirect

35. C₈×D₁₆₇

Q₈

semidirect

76. C₁₆₇

#3160

semidirect

82. #499

D₁₆₇

semidirect

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

subgroup	index
#24923825	2
#24923673	2
#24923671	2
#24923668	2
#24923666	2
#24923306	2
#24923304	2
C₄×C₈×D₁₆₇	2
#499×C₁₆₇	2
#3160	167

subgroup

index

C₄×C₈×D₁₆₇

#499×C₁₆₇

#3160

167

Subgroups — #36334651

Conjugacy classes of subgroups

Complements

Maximal subgroups