Group #35215059 — subgroups

Subgroups — #35215059

Order 20864.

312 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 93 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	#23782432	C₂	1	1	yes	maximal, characteristic
2	#23781674	C₂	1	2	yes	maximal, complemented
3	#23781639	C₂	1	1	yes	maximal, characteristic
4	#23781421	C₂	1	2	yes	maximal, complemented
5	#536×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	#19104742	—	2	4	no	—
7	C₂×#14579683	—	2	2	no	—
8	C₂×#14579683	C₂²	1	2	yes	—
9	#19104694	C₂²	1	2	yes	—
10	#19104659	—	2	4	no	—
11	#19104650	—	2	2	no	—
12	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₁₆₃	C₄	1	2	yes	complemented
15	#124×C₁₆₃	C₄	1	2	yes	complemented
16	#123×C₁₆₃	—	2	2	no	—
17	C₂²⋊Q₈×C₁₆₃	—	2	2	no	—
18	#117×C₁₆₃	C₂²	1	1	yes	characteristic
19	#14579683	—	2	4	no	—
20	#14579683	—	2	2	no	—
21	#14579683	—	2	2	no	—
22	C₂×C₁₆₃⋊C₈	—	2	4	no	—
23	C₂×C₁₆₃⋊C₈	D₄	1	4	yes	—
24	C₄⋊Dic₁₆₃	—	2	2	no	—
25	C₂²⋊Dic₁₆₃	—	2	2	no	—
26	C₄×Dic₁₆₃	—	2	2	no	—
27	#41×C₁₆₃	—	2	2	no	—
28	#41×C₁₆₃	—	2	2	no	—
29	C₂×Q₈×C₁₆₃	—	2	1	no	—
30	C₂×Q₈×C₁₆₃	—	2	2	no	—
31	C₂×Q₈×C₁₆₃	—	2	2	no	—
32	C₂×Q₈×C₁₆₃	C₂×C₄	1	2	yes	—
33	C₂×Q₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
34	C₂×D₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₁₆₃	D₄	1	2	yes	—
36	C₂²×C₄×C₁₆₃	—	2	1	no	—
37	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
38	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₆₃	—	2	4	no	—
40	C₄⋊C₄×C₁₆₃	—	2	1	no	—
41	C₂²⋊C₄×C₁₆₃	—	2	4	no	—
42	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
43	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
44	C₂×Dic₁₆₃	—	4	2	no	—
45	C₁₆₃⋊C₈	—	2	4	no	—
46	C₂³×C₁₆₃	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₁₆₃	—	2	2	no	—
49	Q₈×C₁₆₃	—	2	4	no	—
50	Q₈×C₁₆₃	—	4	1	no	—
51	Q₈×C₁₆₃	—	4	1	no	—
52	D₄×C₁₆₃	—	2	4	no	—
53	D₄×C₁₆₃	—	2	2	no	—
54	C₂×C₄×C₁₆₃	—	2	2	no	—
55	C₂×C₄×C₁₆₃	—	2	2	no	—
56	C₂×C₄×C₁₆₃	—	2	2	no	—
57	C₂×C₄×C₁₆₃	—	2	2	no	—
58	C₂×C₄×C₁₆₃	—	4	2	no	—
59	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₆₃	#41	1	2	yes	—
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	—	4	2	no	—
71	C₄×C₁₆₃	—	4	2	no	—
72	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
73	C₄×C₁₆₃	#120	1	1	yes	characteristic
74	C₄×C₁₆₃	#123	1	1	yes	characteristic
75	Dic₁₆₃	—	8	2	no	—
76	C₂×C₁₆₃	—	2	2	no	—
77	C₂×C₁₆₃	—	2	1	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	—	163	1	no	—
83	#536	D₁₆₃	1	1	yes	characteristic
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	C₂×D₁₆₃	1	1	yes	characteristic
89	#131	—	326	4	no	—
90	C₂×M₁₆	—	326	2	no	—
91	C₂×M₁₆	—	163	2	no	—
92	C₄⋊Q₈	Dic₁₆₃	1	2	yes	complemented
93	#124	Dic₁₆₃	1	2	yes	complemented
94	#123	—	2	2	no	—
95	C₂²⋊Q₈	—	2	2	no	—
96	#117	—	163	2	no	—
97	#117	C₂×D₁₆₃	1	1	yes	characteristic
98	#103	—	326	4	no	—
99	#99	—	326	2	no	—
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	C₂×Dic₁₆₃	1	2	yes	—
106	C₂×Q₈	C₂×Dic₁₆₃	1	1	yes	characteristic
107	C₂×D₄	C₂×Dic₁₆₃	1	1	yes	characteristic
108	C₂×D₄	C₂²⋊D₁₆₃	1	2	yes	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
112	M₁₆	—	326	4	no	—
113	M₁₆	—	326	2	no	—
114	M₁₆	—	326	2	no	—
115	C₂×C₈	—	326	4	no	—
116	C₂×C₈	—	163	4	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	326	2	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₂²⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	326	2	no	—
123	C₄²	—	326	2	no	—
124	C₄²	C₂×Dic₁₆₃	1	2	yes	—
125	C₂³	C₂²⋊Dic₁₆₃	1	2	yes	—
126	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	2	4	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	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	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	652	2	no	—
139	C₂×C₄	C₂²⋊Dic₁₆₃	1	2	yes	—
140	C₂×C₄	C₂²×Dic₁₆₃	1	1	yes	characteristic
141	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
143	C₂×C₄	#14579698	1	2	yes	—
144	C₈	—	326	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	C₂×C₂²⋊Dic₁₆₃	1	1	yes	characteristic
149	C₂²	#19104707	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	1304	2	no	—
154	C₄	D₄×Dic₁₆₃	1	2	yes	—
155	C₄	#19104703	1	1	yes	characteristic
156	C₄	#19104716	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#23781545	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23782432

C₂

yes

maximal, characteristic

#23781674

C₂

yes

maximal, complemented

#23781639

C₂

yes

maximal, characteristic

#23781421

C₂

yes

maximal, complemented

#536×C₁₆₃

C₂

yes

maximal, characteristic

#19104742

—

C₂×#14579683

—

C₂×#14579683

C₂²

yes

—

#19104694

C₂²

yes

—

#19104659

—

#19104650

—

C₂×#41×C₁₆₃

C₂²

yes

characteristic

C₂²×Q₈×C₁₆₃

C₂²

yes

characteristic

C₄⋊Q₈×C₁₆₃

C₄

yes

complemented

#124×C₁₆₃

C₄

yes

complemented

#123×C₁₆₃

—

C₂²⋊Q₈×C₁₆₃

—

#117×C₁₆₃

C₂²

yes

—

—

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

D₄

yes

—

C₄⋊Dic₁₆₃

—

C₂²⋊Dic₁₆₃

—

C₄×Dic₁₆₃

—

#41×C₁₆₃

—

#41×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

C₂×C₄

yes

—

C₂×Q₈×C₁₆₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂×C₄

yes

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂²⋊C₄

yes

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

D₄×C₁₆₃

—

D₄×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

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₆₃

#121

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄×D₄

yes

—

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#386

yes

characteristic

C₁₆₃

#1909

yes

characteristic, complemented, Sylow

#1909

—

163

maximal, Sylow

#568

—

163

—

#536

D₁₆₃

yes

characteristic

#419

—

163

—

#410

—

163

—

H(3,ℤ/4)

—

163

—

C₂×#41

C₂×D₁₆₃

yes

characteristic

C₂²×Q₈

C₂×D₁₆₃

yes

characteristic

#131

—

326

—

C₂×M₁₆

—

326

—

C₂×M₁₆

—

163

—

C₄⋊Q₈

Dic₁₆₃

yes

complemented

#124

Dic₁₆₃

yes

complemented

#123

—

C₂²⋊Q₈

—

#117

—

163

—

#117

C₂×D₁₆₃

yes

characteristic

#103

—

326

—

#99

—

326

—

100

#41

—

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₂×Dic₁₆₃

yes

—

106

C₂×Q₈

C₂×Dic₁₆₃

yes

characteristic

107

C₂×D₄

C₂×Dic₁₆₃

yes

characteristic

108

C₂×D₄

C₂²⋊D₁₆₃

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

111

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

112

M₁₆

—

326

—

113

M₁₆

—

326

—

114

M₁₆

—

326

—

115

C₂×C₈

—

326

—

116

C₂×C₈

—

163

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

326

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

326

—

123

C₄²

—

326

—

124

C₄²

C₂×Dic₁₆₃

yes

—

125

C₂³

C₂²⋊Dic₁₆₃

yes

—

126

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

652

—

139

C₂×C₄

C₂²⋊Dic₁₆₃

yes

—

140

C₂×C₄

C₂²×Dic₁₆₃

yes

characteristic

141

C₂×C₄

D₄×D₁₆₃

yes

—

142

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

143

C₂×C₄

#14579698

yes

—

144

C₈

—

326

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×C₂²⋊Dic₁₆₃

yes

characteristic

149

C₂²

#19104707

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

1304

—

154

C₄

D₄×Dic₁₆₃

yes

—

155

C₄

#19104703

yes

characteristic

156

C₄

#19104716

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#23781545

yes

characteristic

	subgroup	G/H
center	C₂	#23781545
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#536×C₁₆₃	C₂
socle	C₂×C₁₆₃	#386
prime radical	C₂×#41×C₁₆₃	C₂²
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	#536	D₁₆₃
hypocenter	C₁₆₃	#1909
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#23781545

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#536×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 ◁ #536

N	H ≅ G/N	type
2. #23781674	C₂	semidirect
4. #23781421	C₂	semidirect
14. C₄⋊Q₈×C₁₆₃	C₄	semidirect
15. #124×C₁₆₃	C₄	semidirect
80. C₁₆₃	#1909	semidirect
92. C₄⋊Q₈	Dic₁₆₃	semidirect
93. #124	Dic₁₆₃	semidirect

H ≅ G/N

type

2. #23781674

C₂

semidirect

4. #23781421

C₂

semidirect

14. C₄⋊Q₈×C₁₆₃

C₄

semidirect

15. #124×C₁₆₃

C₄

semidirect

80. C₁₆₃

#1909

semidirect

92. C₄⋊Q₈

Dic₁₆₃

semidirect

93. #124

Dic₁₆₃

semidirect

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

subgroup	index
#23782432	2
#23781674	2
#23781639	2
#23781421	2
#536×C₁₆₃	2
#1909	163

subgroup

index

#536×C₁₆₃

163

Subgroups — #35215059

Conjugacy classes of subgroups

Complements

Maximal subgroups