Group #35214930 — subgroups

Subgroups — #35214930

Order 20864.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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	#23782143	C₂	1	1	yes	maximal, characteristic
2	#23781674	C₂	1	2	yes	maximal
3	C₂×#19104654	C₂	1	1	yes	maximal, characteristic
4	#23781425	C₂	1	2	yes	maximal
5	C₂×C₄⋊Q₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579683	C₂²	1	2	yes	—
7	Q₈×Dic₁₆₃	—	2	4	no	—
8	#19104694	—	2	2	no	—
9	#19104694	C₂²	1	2	yes	—
10	#19104659	—	2	4	no	—
11	#19104654	—	2	2	no	—
12	C₂²×Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₁₆₃	—	2	2	no	—
15	C₄⋊Q₈×C₁₆₃	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₄×C₁₆₃	—	2	2	no	—
17	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
18	#14579683	—	2	2	no	—
19	#14579683	—	2	2	no	—
20	C₂×C₁₆₃⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₆₃	—	2	4	no	—
22	C₄⋊Dic₁₆₃	D₄	1	4	yes	—
23	C₂²⋊Dic₁₆₃	—	2	2	no	—
24	C₄×Dic₁₆₃	—	2	4	no	—
25	C₄×Dic₁₆₃	—	2	2	no	—
26	C₂×Q₈×C₁₆₃	—	2	2	no	—
27	C₂×Q₈×C₁₆₃	—	2	1	no	—
28	C₂×Q₈×C₁₆₃	—	2	2	no	—
29	C₂×Q₈×C₁₆₃	—	2	2	no	—
30	C₂×Q₈×C₁₆₃	—	2	2	no	—
31	C₂×Q₈×C₁₆₃	C₂×C₄	1	2	yes	—
32	C₂×Q₈×C₁₆₃	C₂×C₄	1	2	yes	—
33	C₂×Q₈×C₁₆₃	D₄	1	2	yes	—
34	C₂²×C₄×C₁₆₃	—	2	1	no	—
35	C₂²×C₄×C₁₆₃	—	2	1	no	—
36	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
37	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₆₃	—	2	8	no	—
39	C₄²×C₁₆₃	—	2	1	no	—
40	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
41	C₂×Dic₁₆₃	—	2	4	no	—
42	C₁₆₃⋊C₈	—	4	2	no	—
43	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₁₆₃	—	2	4	no	—
45	Q₈×C₁₆₃	—	2	4	no	—
46	Q₈×C₁₆₃	—	2	4	no	—
47	Q₈×C₁₆₃	—	4	2	no	—
48	C₂×C₄×C₁₆₃	—	2	4	no	—
49	C₂×C₄×C₁₆₃	—	2	2	no	—
50	C₂×C₄×C₁₆₃	—	2	2	no	—
51	C₂×C₄×C₁₆₃	—	2	2	no	—
52	C₂×C₄×C₁₆₃	—	2	2	no	—
53	C₂×C₄×C₁₆₃	—	4	1	no	—
54	C₂×C₄×C₁₆₃	—	4	1	no	—
55	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₆₃	#41	1	2	yes	—
60	C₂²×C₁₆₃	—	2	2	no	—
61	C₂²×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₆₃	#120	1	1	yes	characteristic
63	C₂²×C₁₆₃	#123	1	1	yes	characteristic
64	C₄×C₁₆₃	—	2	4	no	—
65	C₄×C₁₆₃	—	2	4	no	—
66	C₄×C₁₆₃	—	4	2	no	—
67	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
68	C₄×C₁₆₃	#121	1	2	yes	—
69	Dic₁₆₃	—	4	4	no	—
70	C₂×C₁₆₃	—	2	1	no	—
71	C₂×C₁₆₃	—	2	1	no	—
72	C₂×C₁₆₃	#386	1	1	yes	characteristic
73	C₂×C₁₆₃	#456	1	2	yes	—
74	C₁₆₃	#1886	1	1	yes	characteristic, complemented, Sylow
75	#1886	—	163	1	no	maximal, Sylow
76	C₂×C₄⋊Q₈	D₁₆₃	1	1	yes	characteristic
77	#519	—	163	1	no	—
78	#419	—	163	2	no	—
79	C₂×#101	—	163	1	no	—
80	#339	—	163	2	no	—
81	C₂²×Q₈	C₂×D₁₆₃	1	1	yes	characteristic
82	C₂²×Q₈	C₂×D₁₆₃	1	1	yes	characteristic
83	C₂×M₁₆	—	163	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	C₄⋊Q₈	Dic₁₆₃	1	4	yes	complemented
86	C₄×Q₈	—	326	4	no	—
87	#117	—	326	2	no	—
88	#117	—	163	2	no	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
91	#103	—	326	4	no	—
92	#101	—	326	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂×Dic₁₆₃	1	2	yes	—
99	C₂×Q₈	C₂×Dic₁₆₃	1	2	yes	—
100	C₂×Q₈	C₂²⋊D₁₆₃	1	2	yes	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
104	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
105	M₁₆	—	326	2	no	—
106	M₁₆	—	326	2	no	—
107	C₂×C₈	—	326	2	no	—
108	C₄⋊C₄	—	2	8	no	—
109	C₄⋊C₄	—	326	4	no	—
110	C₄⋊C₄	—	163	4	no	—
111	C₂²⋊C₄	—	326	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	326	4	no	—
114	C₄²	—	326	2	no	—
115	C₄²	C₂×Dic₁₆₃	1	2	yes	—
116	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
117	Q₈	—	2	4	no	—
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	326	4	no	—
129	C₂×C₄	C₂²⋊Dic₁₆₃	1	4	yes	—
130	C₂×C₄	C₂²×Dic₁₆₃	1	1	yes	characteristic
131	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
133	C₂×C₄	#14579698	1	2	yes	—
134	C₈	—	652	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×C₂²⋊Dic₁₆₃	1	1	yes	characteristic
137	C₂²	#19104703	1	1	yes	characteristic
138	C₂²	#19104716	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	652	4	no	—
143	C₄	D₄×Dic₁₆₃	1	2	yes	—
144	C₄	#19104707	1	2	yes	—
145	C₂	—	2	1	no	—
146	C₂	—	2	1	no	—
147	C₂	#23781545	1	1	yes	characteristic
148	C₂	#23781847	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23782143

C₂

yes

maximal, characteristic

#23781674

C₂

yes

maximal

C₂×#19104654

C₂

yes

maximal, characteristic

#23781425

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

Q₈×Dic₁₆₃

—

#19104694

—

#19104694

C₂²

yes

—

#19104659

—

#19104654

—

C₂²×Q₈×C₁₆₃

C₂²

yes

characteristic

C₂²×Q₈×C₁₆₃

C₂²

yes

characteristic

C₄⋊Q₈×C₁₆₃

—

C₄⋊Q₈×C₁₆₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₆₃

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

#14579683

—

#14579683

—

C₂×C₁₆₃⋊C₈

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

D₄

yes

—

C₂²⋊Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

C₂×C₄

yes

—

C₂×Q₈×C₁₆₃

C₂×C₄

yes

—

C₂×Q₈×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

—

C₄²×C₁₆₃

C₂×C₄

yes

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂×C₂²⋊C₄

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄×D₄

yes

—

C₄×C₁₆₃

#121

yes

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#386

yes

characteristic

C₂×C₁₆₃

#456

yes

—

C₁₆₃

#1886

yes

characteristic, complemented, Sylow

#1886

—

163

maximal, Sylow

C₂×C₄⋊Q₈

D₁₆₃

yes

characteristic

#519

—

163

—

#419

—

163

—

C₂×#101

—

163

—

#339

—

163

—

C₂²×Q₈

C₂×D₁₆₃

yes

characteristic

C₂²×Q₈

C₂×D₁₆₃

yes

characteristic

C₂×M₁₆

—

163

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₆₃

yes

complemented

C₄×Q₈

—

326

—

#117

—

326

—

#117

—

163

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#103

—

326

—

#101

—

326

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₆₃

yes

—

C₂×Q₈

C₂×Dic₁₆₃

yes

—

100

C₂×Q₈

C₂²⋊D₁₆₃

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

104

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

105

M₁₆

—

326

—

106

M₁₆

—

326

—

107

C₂×C₈

—

326

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

326

—

110

C₄⋊C₄

—

163

—

111

C₂²⋊C₄

—

326

—

112

C₄²

—

113

C₄²

—

326

—

114

C₄²

—

326

—

115

C₄²

C₂×Dic₁₆₃

yes

—

116

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

326

—

129

C₂×C₄

C₂²⋊Dic₁₆₃

yes

—

130

C₂×C₄

C₂²×Dic₁₆₃

yes

characteristic

131

C₂×C₄

D₄×D₁₆₃

yes

—

132

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

133

C₂×C₄

#14579698

yes

—

134

C₈

—

652

—

135

C₂²

—

136

C₂²

C₂×C₂²⋊Dic₁₆₃

yes

characteristic

137

C₂²

#19104703

yes

characteristic

138

C₂²

#19104716

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

652

—

143

C₄

D₄×Dic₁₆₃

yes

—

144

C₄

#19104707

yes

—

145

C₂

—

146

C₂

—

147

C₂

#23781545

yes

characteristic

148

C₂

#23781847

yes

—

	subgroup	G/H
center	C₂²	#19104703
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	C₂×C₄⋊Q₈×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#120
prime radical	C₂³×C₁₆₃	C₂×D₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₆₃
hypocenter	C₁₆₃	#1886
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

#19104703

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

C₂×C₄⋊Q₈×C₁₆₃

C₂

socle

C₂²×C₁₆₃

#120

prime radical

C₂³×C₁₆₃

C₂×D₄

cosocle

C₂²×C₄×C₁₆₃

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₆₃

hypocenter

C₁₆₃

#1886

LCS

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
15. C₄⋊Q₈×C₁₆₃	C₄	semidirect
74. C₁₆₃	#1886	semidirect
85. C₄⋊Q₈	Dic₁₆₃	semidirect

H ≅ G/N

type

15. C₄⋊Q₈×C₁₆₃

C₄

semidirect

74. C₁₆₃

#1886

semidirect

85. C₄⋊Q₈

Dic₁₆₃

semidirect

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

subgroup	index
#23782143	2
#23781674	2
C₂×#19104654	2
#23781425	2
C₂×C₄⋊Q₈×C₁₆₃	2
#1886	163

subgroup

index

C₂×C₄⋊Q₈×C₁₆₃

163

Subgroups — #35214930

Conjugacy classes of subgroups

Complements

Maximal subgroups