Group #36323596 — subgroups

Subgroups — #36323596

Order 21376.

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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	#24923620	C₂	1	1	yes	maximal, characteristic
2	#24923564	C₂	1	1	yes	maximal, characteristic
3	#24923554	C₂	1	1	yes	maximal, characteristic
4	#24923022	C₂	1	1	yes	maximal, characteristic
5	#24923017	C₂	1	1	yes	maximal, characteristic
6	#24923016	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
8	#19128832	C₂²	1	1	yes	characteristic
9	#19128832	C₂²	1	1	yes	characteristic
10	#19128765	—	2	1	no	—
11	#19128763	—	2	1	no	—
12	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	#19128757	—	2	1	no	—
15	#19128757	—	2	1	no	—
16	#19128756	—	2	1	no	—
17	#19128756	—	2	1	no	—
18	C₄×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₆₇	—	2	1	no	—
20	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₆₇	—	2	1	no	—
22	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₃₄	—	2	1	no	—
24	C₂×Dic₃₃₄	—	2	1	no	—
25	C₂×C₁₆₇⋊C₈	—	2	1	no	—
26	C₂×C₁₆₇⋊C₈	—	2	1	no	—
27	C₄⋊Dic₁₆₇	—	2	1	no	—
28	C₄⋊Dic₁₆₇	—	2	1	no	—
29	C₄⋊Dic₁₆₇	—	2	1	no	—
30	C₄⋊Dic₁₆₇	—	2	1	no	—
31	C₂×Q₈×C₁₆₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₆₇	—	2	1	no	—
33	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₁₆₇	—	2	1	no	—
36	C₄⋊C₄×C₁₆₇	D₄	1	1	yes	characteristic
37	C₄²×C₁₆₇	—	2	1	no	—
38	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
39	Dic₃₃₄	—	4	1	no	—
40	Dic₃₃₄	—	4	1	no	—
41	Dic₃₃₄	—	8	1	no	—
42	C₂×Dic₁₆₇	—	4	1	no	—
43	C₂×Dic₁₆₇	—	4	1	no	—
44	C₁₆₇⋊C₈	—	4	1	no	—
45	C₁₆₇⋊C₈	—	4	1	no	—
46	Q₈×C₁₆₇	—	2	1	no	—
47	Q₈×C₁₆₇	Q₁₆	1	2	yes	—
48	C₂×C₄×C₁₆₇	—	2	1	no	—
49	C₂×C₄×C₁₆₇	—	2	1	no	—
50	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
53	C₈×C₁₆₇	—	2	1	no	—
54	C₈×C₁₆₇	—	2	1	no	—
55	C₈×C₁₆₇	—	4	1	no	—
56	C₂²×C₁₆₇	#121	1	1	yes	characteristic
57	C₄×C₁₆₇	—	2	2	no	—
58	C₄×C₁₆₇	—	2	1	no	—
59	C₄×C₁₆₇	—	4	1	no	—
60	C₄×C₁₆₇	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₁₆₇	#135	1	1	yes	characteristic
62	C₄×C₁₆₇	#135	1	1	yes	characteristic
63	C₄×C₁₆₇	#137	1	1	yes	characteristic
64	Dic₁₆₇	—	8	1	no	—
65	Dic₁₆₇	—	8	1	no	—
66	C₂×C₁₆₇	#464	1	1	yes	characteristic
67	C₂×C₁₆₇	#467	1	1	yes	characteristic
68	C₂×C₁₆₇	#471	1	1	yes	characteristic
69	C₁₆₇	#1684	1	1	yes	characteristic, complemented, Sylow
70	#1684	—	167	1	no	maximal, Sylow
71	#487	—	167	1	no	—
72	#477	—	167	1	no	—
73	#475	—	167	1	no	—
74	C₈×Q₈	D₁₆₇	1	1	yes	characteristic
75	#335	—	167	1	no	—
76	#333	—	167	1	no	—
77	#332	—	167	1	no	—
78	C₄⋊Q₈	—	167	1	no	—
79	C₄⋊Q₈	—	167	1	no	—
80	C₄×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
81	#107	—	334	1	no	—
82	#106	—	334	1	no	—
83	C₄⋊C₈	—	2	1	no	—
84	C₄⋊C₈	—	167	1	no	—
85	C₄⋊C₈	—	167	1	no	—
86	C₄⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
87	#103	—	334	1	no	—
88	#103	—	334	1	no	—
89	#103	—	334	1	no	—
90	#103	—	334	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
93	C₂×Q₈	—	334	1	no	—
94	C₂×Q₈	—	334	1	no	—
95	C₂×Q₈	D₆₆₈	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	334	1	no	—
98	C₂×C₈	—	334	1	no	—
99	C₂×C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
100	C₂×C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	334	1	no	—
103	C₄⋊C₄	—	334	1	no	—
104	C₄⋊C₄	—	334	1	no	—
105	C₄⋊C₄	—	334	1	no	—
106	C₄⋊C₄	D₆₆₈	1	1	yes	characteristic
107	C₄²	—	2	1	no	—
108	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
109	Q₈	—	2	1	no	—
110	Q₈	—	1336	1	no	—
111	Q₈	—	668	1	no	—
112	Q₈	—	668	1	no	—
113	Q₈	Dic₆₆₈	1	2	yes	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	668	1	no	—
117	C₂×C₄	—	668	1	no	—
118	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
119	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
120	C₂×C₄	#14581950	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	668	1	no	—
125	C₈	—	668	1	no	—
126	C₂²	#19128806	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	1336	1	no	—
131	C₄	—	1336	1	no	—
132	C₄	C₂×Dic₆₆₈	1	1	yes	characteristic
133	C₄	#19128852	1	1	yes	characteristic
134	C₄	#19128854	1	1	yes	characteristic
135	C₄	#19128863	1	1	yes	characteristic
136	C₂	#24923491	1	1	yes	characteristic
137	C₂	#24923510	1	1	yes	characteristic
138	C₂	#24923532	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24923620

C₂

yes

maximal, characteristic

#24923564

C₂

yes

maximal, characteristic

#24923554

C₂

yes

maximal, characteristic

#24923022

C₂

yes

maximal, characteristic

#24923017

C₂

yes

maximal, characteristic

#24923016

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₆₇

C₂

yes

maximal, characteristic

#19128832

C₂²

yes

characteristic

#19128832

C₂²

yes

characteristic

#19128765

—

#19128763

—

C₄⋊(C₁₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₆₇⋊C₈)

C₂²

yes

—

—

—

—

C₄×Q₈×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₇

—

C₄⋊C₈×C₁₆₇

C₂²

yes

characteristic

C₄×C₈×C₁₆₇

—

C₄×C₈×C₁₆₇

C₂²

yes

characteristic

C₂×Dic₃₃₄

—

C₂×Dic₃₃₄

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₂×Q₈×C₁₆₇

D₄

yes

characteristic

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

D₄

yes

characteristic

C₂×C₈×C₁₆₇

D₄

yes

characteristic

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

D₄

yes

characteristic

C₄²×C₁₆₇

—

C₄²×C₁₆₇

C₂³

yes

characteristic

Dic₃₃₄

—

Dic₃₃₄

—

Dic₃₃₄

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

—

Q₈×C₁₆₇

—

Q₈×C₁₆₇

Q₁₆

yes

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

#41

yes

characteristic

C₈×C₁₆₇

—

C₈×C₁₆₇

—

C₈×C₁₆₇

—

C₂²×C₁₆₇

#121

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×Q₁₆

yes

C₄×C₁₆₇

yes

C₄×C₁₆₇

yes

C₄×C₁₆₇

yes

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#1684

—

167

maximal, Sylow

#487

—

167

—

#477

—

167

—

#475

—

167

—

C₈×Q₈

D₁₆₇

yes

characteristic

#335

—

167

—

#333

—

167

—

#332

—

167

—

C₄⋊Q₈

—

167

—

C₄⋊Q₈

—

167

—

C₄×Q₈

C₂×D₁₆₇

yes

characteristic

#107

—

334

—

#106

—

334

—

C₄⋊C₈

—

C₄⋊C₈

—

167

—

C₄⋊C₈

—

167

—

C₄⋊C₈

C₂×D₁₆₇

yes

characteristic

#103

—

334

—

#103

—

334

—

#103

—

334

—

#103

—

334

—

C₄×C₈

—

C₄×C₈

C₂×D₁₆₇

yes

characteristic

C₂×Q₈

—

334

—

C₂×Q₈

—

334

—

C₂×Q₈

D₆₆₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

334

—

C₂×C₈

—

334

—

C₂×C₈

C₂²⋊D₁₆₇

yes

characteristic

100

C₂×C₈

C₂²⋊D₁₆₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

334

—

103

C₄⋊C₄

—

334

—

104

C₄⋊C₄

—

334

—

105

C₄⋊C₄

—

334

—

106

C₄⋊C₄

D₆₆₈

yes

characteristic

107

C₄²

—

108

C₄²

C₂²×D₁₆₇

yes

characteristic

109

Q₈

—

110

Q₈

—

1336

—

111

Q₈

—

668

—

112

Q₈

—

668

—

113

Q₈

Dic₆₆₈

yes

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

668

—

117

C₂×C₄

—

668

—

118

C₂×C₄

C₂×C₂²⋊D₁₆₇

yes

characteristic

119

C₂×C₄

C₂×D₆₆₈

yes

characteristic

120

C₂×C₄

#14581950

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

668

—

125

C₈

—

668

—

126

C₂²

#19128806

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

1336

—

131

C₄

—

1336

—

132

C₄

C₂×Dic₆₆₈

yes

characteristic

133

C₄

#19128852

yes

characteristic

134

C₄

#19128854

yes

characteristic

135

C₄

#19128863

yes

characteristic

136

C₂

#24923491

yes

characteristic

137

C₂

#24923510

yes

characteristic

138

C₂

#24923532

yes

characteristic

	subgroup	G/H
center	C₂²	#19128806
commutator, cosocle	C₄²×C₁₆₇	C₂³
Frattini	C₄²	C₂²×D₁₆₇
Fitting	C₈×Q₈×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	#121
hypercenter	C₈×Q₈	D₁₆₇
hypocenter	C₁₆₇	#1684
LCS	G ▷ C₄²×C₁₆₇ ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#19128806

commutator, cosocle

C₄²×C₁₆₇

C₂³

Frattini

C₄²

C₂²×D₁₆₇

Fitting

C₈×Q₈×C₁₆₇

C₂

socle, prime radical

C₂²×C₁₆₇

#121

hypercenter

C₈×Q₈

D₁₆₇

hypocenter

C₁₆₇

#1684

LCS

G ▷ C₄²×C₁₆₇ ▷ C₂²×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
69. C₁₆₇	#1684	semidirect

H ≅ G/N

type

69. C₁₆₇

#1684

semidirect

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

subgroup	index
#24923620	2
#24923564	2
#24923554	2
#24923022	2
#24923017	2
#24923016	2
C₈×Q₈×C₁₆₇	2
#1684	167

subgroup

index

C₈×Q₈×C₁₆₇

167

Subgroups — #36323596

Conjugacy classes of subgroups

Complements

Maximal subgroups