Group #35215058 — subgroups

Subgroups — #35215058

Order 20864.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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	#23782236	C₂	1	1	yes	maximal, characteristic
2	#23781675	C₂	1	2	yes	maximal, complemented
3	#23781639	C₂	1	1	yes	maximal, characteristic
4	#23781422	C₂	1	2	yes	maximal, complemented
5	#568×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₂₆	C₂²	1	1	yes	characteristic
7	#19104733	C₄	1	2	yes	complemented
8	#19104722	C₄	1	2	yes	complemented
9	#19104716	—	2	2	no	—
10	#19104714	—	2	2	no	—
11	#19104694	C₂²	1	2	yes	—
12	#19104694	C₂²	1	1	yes	characteristic
13	#19104660	—	2	4	no	—
14	#19104650	—	2	2	no	—
15	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
16	#131×C₁₆₃	—	2	4	no	—
17	C₂×M₁₆×C₁₆₃	—	2	2	no	—
18	C₂×M₁₆×C₁₆₃	C₂²	1	2	yes	—
19	C₂×Dic₃₂₆	—	2	1	no	—
20	C₂×Dic₃₂₆	—	2	2	no	—
21	C₂×Dic₃₂₆	—	2	2	no	—
22	C₂×Dic₃₂₆	C₂×C₄	1	2	yes	—
23	C₂²×Dic₁₆₃	—	2	1	no	—
24	C₄⋊Dic₁₆₃	—	2	1	no	—
25	C₄⋊Dic₁₆₃	—	2	2	no	—
26	Dic₁₆₃⋊C₄	—	2	4	no	—
27	C₂²⋊Dic₁₆₃	—	2	4	no	—
28	C₂²⋊Dic₁₆₃	—	2	1	no	—
29	C₂²⋊Dic₁₆₃	—	2	2	no	—
30	C₄×Dic₁₆₃	—	2	2	no	—
31	C₄×Dic₁₆₃	C₂×C₄	1	2	yes	—
32	#41×C₁₆₃	—	2	2	no	—
33	#41×C₁₆₃	—	2	2	no	—
34	C₂×Q₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
36	C₂×D₄×C₁₆₃	D₄	1	2	yes	—
37	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
38	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
39	M₁₆×C₁₆₃	—	2	4	no	—
40	M₁₆×C₁₆₃	—	2	2	no	—
41	M₁₆×C₁₆₃	—	2	2	no	—
42	C₂×C₈×C₁₆₃	—	2	4	no	—
43	C₂×C₈×C₁₆₃	D₄	1	4	yes	—
44	Dic₃₂₆	—	2	4	no	—
45	Dic₃₂₆	—	4	1	no	—
46	Dic₃₂₆	—	4	1	no	—
47	C₂×Dic₁₆₃	—	2	2	no	—
48	C₂×Dic₁₆₃	—	2	2	no	—
49	C₂×Dic₁₆₃	—	4	2	no	—
50	C₂×Dic₁₆₃	—	4	2	no	—
51	C₂³×C₁₆₃	C₂²⋊C₄	1	2	yes	—
52	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
53	Q₈×C₁₆₃	—	2	2	no	—
54	D₄×C₁₆₃	—	2	4	no	—
55	D₄×C₁₆₃	—	2	2	no	—
56	C₂×C₄×C₁₆₃	—	2	2	no	—
57	C₂×C₄×C₁₆₃	—	2	2	no	—
58	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₆₃	#41	1	2	yes	—
63	C₈×C₁₆₃	—	2	4	no	—
64	C₂²×C₁₆₃	—	2	2	no	—
65	C₂²×C₁₆₃	—	4	1	no	—
66	C₂²×C₁₆₃	—	4	2	no	—
67	C₂²×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₁₆₃	#121	1	2	yes	—
69	C₄×C₁₆₃	—	2	2	no	—
70	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
71	C₄×C₁₆₃	#120	1	1	yes	characteristic
72	C₄×C₁₆₃	#123	1	1	yes	characteristic
73	Dic₁₆₃	—	4	2	no	—
74	Dic₁₆₃	—	4	2	no	—
75	Dic₁₆₃	—	8	2	no	—
76	C₂×C₁₆₃	—	2	1	no	—
77	C₂×C₁₆₃	—	2	2	no	—
78	C₂×C₁₆₃	—	4	2	no	—
79	C₂×C₁₆₃	#386	1	1	yes	characteristic
80	C₁₆₃	#1909	1	1	yes	characteristic, complemented, Sylow
81	#1909	—	163	1	no	maximal, Sylow
82	#568	D₁₆₃	1	1	yes	characteristic
83	#536	—	163	1	no	—
84	#419	—	163	2	no	—
85	#410	—	163	1	no	—
86	H(3,ℤ/4)	—	163	2	no	—
87	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
88	C₂²×Q₈	—	163	1	no	—
89	#131	—	2	4	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	C₂×D₁₆₃	1	2	yes	—
92	C₄⋊Q₈	—	163	2	no	—
93	#124	—	163	2	no	—
94	#123	—	326	2	no	—
95	C₂²⋊Q₈	—	326	2	no	—
96	#117	—	163	2	no	—
97	#117	—	163	1	no	—
98	#103	—	326	4	no	—
99	#99	—	326	2	no	—
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	—	326	1	no	—
103	C₂×Q₈	—	326	2	no	—
104	C₂×Q₈	—	326	2	no	—
105	C₂×Q₈	—	163	2	no	—
106	C₂×Q₈	C₄×D₁₆₃	1	1	yes	characteristic
107	C₂×D₄	C₄×D₁₆₃	1	1	yes	characteristic
108	C₂×D₄	D₆₅₂	1	2	yes	—
109	C₂²×C₄	—	326	1	no	—
110	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	C₂²⋊D₁₆₃	1	4	yes	—
117	C₄⋊C₄	—	326	4	no	—
118	C₄⋊C₄	—	326	1	no	—
119	C₄⋊C₄	—	326	2	no	—
120	C₂²⋊C₄	—	326	4	no	—
121	C₂²⋊C₄	—	326	1	no	—
122	C₂²⋊C₄	—	326	2	no	—
123	C₄²	—	326	2	no	—
124	C₄²	—	163	2	no	—
125	C₂³	(C₂×D₁₆₃)⋊C₄	1	2	yes	—
126	C₂³	C₂×D₆₅₂	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	326	4	no	—
129	Q₈	—	652	1	no	—
130	Q₈	—	652	1	no	—
131	D₄	—	2	4	no	—
132	D₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	326	2	no	—
136	C₂×C₄	—	326	2	no	—
137	C₂×C₄	—	652	2	no	—
138	C₂×C₄	—	652	2	no	—
139	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
143	C₂×C₄	#14579699	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	C₂×(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
149	C₂²	#19104709	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	1304	2	no	—
152	C₄	—	652	2	no	—
153	C₄	—	652	2	no	—
154	C₄	C₄×C₂²⋊D₁₆₃	1	2	yes	—
155	C₄	#19104704	1	1	yes	characteristic
156	C₄	#19104718	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	2	1	no	—
159	C₂	—	4	2	no	—
160	C₂	#23781544	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23782236

C₂

yes

maximal, characteristic

#23781675

C₂

yes

maximal, complemented

#23781639

C₂

yes

maximal, characteristic

#23781422

C₂

yes

maximal, complemented

#568×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₂₆

C₂²

yes

characteristic

#19104733

C₄

yes

complemented

#19104722

C₄

yes

—

—

C₂²

yes

—

#19104694

C₂²

yes

characteristic

#19104660

—

#19104650

—

C₂×#41×C₁₆₃

C₂²

yes

characteristic

#131×C₁₆₃

—

C₂×M₁₆×C₁₆₃

—

C₂×M₁₆×C₁₆₃

C₂²

yes

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

C₂×C₄

yes

—

C₂²×Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

Dic₁₆₃⋊C₄

—

C₂²⋊Dic₁₆₃

—

C₂²⋊Dic₁₆₃

—

C₂²⋊Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

C₂×C₄

yes

—

#41×C₁₆₃

—

#41×C₁₆₃

—

C₂×Q₈×C₁₆₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

M₁₆×C₁₆₃

—

M₁₆×C₁₆₃

—

M₁₆×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

D₄

yes

—

Dic₃₂₆

—

Dic₃₂₆

—

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂³×C₁₆₃

C₂²⋊C₄

yes

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

Q₈×C₁₆₃

—

D₄×C₁₆₃

—

D₄×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

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₆₃

#121

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄×D₄

yes

—

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

Dic₁₆₃

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#386

yes

characteristic

C₁₆₃

#1909

yes

characteristic, complemented, Sylow

#1909

—

163

maximal, Sylow

#568

D₁₆₃

yes

characteristic

#536

—

163

—

#419

—

163

—

#410

—

163

—

H(3,ℤ/4)

—

163

—

C₂×#41

C₂×D₁₆₃

yes

characteristic

C₂²×Q₈

—

163

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₆₃

yes

—

C₄⋊Q₈

—

163

—

#124

—

163

—

#123

—

326

—

C₂²⋊Q₈

—

326

—

#117

—

163

—

#117

—

163

—

#103

—

326

—

#99

—

326

—

100

#41

—

101

#41

—

102

C₂×Q₈

—

326

—

103

C₂×Q₈

—

326

—

104

C₂×Q₈

—

326

—

105

C₂×Q₈

—

163

—

106

C₂×Q₈

C₄×D₁₆₃

yes

characteristic

107

C₂×D₄

C₄×D₁₆₃

yes

characteristic

108

C₂×D₄

D₆₅₂

yes

—

109

C₂²×C₄

—

326

—

110

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

111

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

C₂²⋊D₁₆₃

yes

—

117

C₄⋊C₄

—

326

—

118

C₄⋊C₄

—

326

—

119

C₄⋊C₄

—

326

—

120

C₂²⋊C₄

—

326

—

121

C₂²⋊C₄

—

326

—

122

C₂²⋊C₄

—

326

—

123

C₄²

—

326

—

124

C₄²

—

163

—

125

C₂³

(C₂×D₁₆₃)⋊C₄

yes

—

126

C₂³

C₂×D₆₅₂

yes

characteristic

127

Q₈

—

128

Q₈

—

326

—

129

Q₈

—

652

—

130

Q₈

—

652

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

326

—

136

C₂×C₄

—

326

—

137

C₂×C₄

—

652

—

138

C₂×C₄

—

652

—

139

C₂×C₄

(C₂×D₁₆₃)⋊C₄

yes

—

140

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

142

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

143

C₂×C₄

#14579699

yes

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×(C₂×D₁₆₃)⋊C₄

yes

characteristic

149

C₂²

#19104709

yes

—

150

C₄

—

151

C₄

—

1304

—

152

C₄

—

652

—

153

C₄

—

652

—

154

C₄

C₄×C₂²⋊D₁₆₃

yes

—

155

C₄

#19104704

yes

characteristic

156

C₄

#19104718

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#23781544

yes

characteristic

	subgroup	G/H
center	C₂	#23781544
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#568×C₁₆₃	C₂
socle	C₂×C₁₆₃	#386
prime radical	C₂×#41×C₁₆₃	C₂²
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	#568	D₁₆₃
hypocenter	C₁₆₃	#1909
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#23781544

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#568×C₁₆₃

C₂

socle

C₂×C₁₆₃

#386

prime radical

C₂×#41×C₁₆₃

C₂²

cosocle

C₂²×C₄×C₁₆₃

C₂³

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
2. #23781675	C₂	semidirect
4. #23781422	C₂	semidirect
7. #19104733	C₄	semidirect
8. #19104722	C₄	semidirect
80. C₁₆₃	#1909	semidirect

H ≅ G/N

type

2. #23781675

C₂

semidirect

4. #23781422

C₂

semidirect

7. #19104733

C₄

semidirect

8. #19104722

C₄

semidirect

80. C₁₆₃

#1909

semidirect

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

subgroup	index
#23782236	2
#23781675	2
#23781639	2
#23781422	2
#568×C₁₆₃	2
#1909	163

subgroup

index

#568×C₁₆₃

163

Subgroups — #35215058

Conjugacy classes of subgroups

Complements

Maximal subgroups