Group #35224936 — subgroups

Subgroups — #35224936

Order 20864.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×#19104727	C₂	1	1	yes	maximal, characteristic, complemented
2	#23782033	C₂	1	2	yes	maximal, complemented
3	#23782029	C₂	1	2	yes	maximal, complemented
4	#23782019	C₂	1	4	yes	maximal, complemented
5	#23781666	C₂	1	4	yes	maximal, complemented
6	C₄×#14579683	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₁₆₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14579683	C₂²	1	2	yes	—
9	#19104727	—	2	2	no	—
10	#19104727	C₂²	1	2	yes	—
11	#19104727	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
13	C₂×Dic₁₆₃⋊C₄	—	2	2	no	—
14	#19104659	C₂²	1	8	yes	—
15	#19104656	C₂²	1	8	yes	—
16	#19104646	C₂²	1	2	yes	—
17	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	—
18	C₂×#41×C₁₆₃	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
20	#127×C₁₆₃	C₂²	1	1	yes	characteristic
21	#124×C₁₆₃	C₂²	1	2	yes	—
22	#121×C₁₆₃	—	2	2	no	—
23	C₄×D₄×C₁₆₃	—	2	2	no	—
24	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₆₃	—	2	2	no	—
26	#14579683	—	2	4	no	—
27	C₂×C₁₆₃⋊C₈	C₂³	1	4	yes	—
28	C₄⋊Dic₁₆₃	—	2	2	no	—
29	C₄⋊Dic₁₆₃	C₂³	1	4	yes	—
30	Dic₁₆₃⋊C₄	—	2	4	no	—
31	Dic₁₆₃⋊C₄	—	2	4	no	—
32	#41×C₁₆₃	—	2	8	no	—
33	C₂×Q₈×C₁₆₃	C₂³	1	2	yes	—
34	C₂×D₄×C₁₆₃	—	2	2	no	—
35	C₂×D₄×C₁₆₃	—	2	2	no	—
36	C₂×D₄×C₁₆₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₆₃	—	2	2	no	—
38	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
39	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₆₃	—	2	2	no	—
41	C₂²⋊C₄×C₁₆₃	—	2	4	no	—
42	C₄²×C₁₆₃	D₄	1	2	yes	—
43	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
44	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₆₃	—	2	4	no	—
46	C₂×Dic₁₆₃	—	4	4	no	—
47	C₁₆₃⋊C₈	—	2	4	no	—
48	C₂³×C₁₆₃	—	2	2	no	—
49	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₁₆₃	—	2	4	no	—
51	D₄×C₁₆₃	—	2	4	no	—
52	D₄×C₁₆₃	—	4	4	no	—
53	D₄×C₁₆₃	—	4	4	no	—
54	C₂×C₄×C₁₆₃	—	2	2	no	—
55	C₂×C₄×C₁₆₃	—	4	2	no	—
56	C₂×C₄×C₁₆₃	—	4	4	no	—
57	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₆₃	#41	1	4	yes	—
61	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₆₃	—	2	2	no	—
63	C₂²×C₁₆₃	—	4	2	no	—
64	C₂²×C₁₆₃	—	4	4	no	—
65	C₂²×C₁₆₃	#124	1	2	yes	—
66	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
67	C₄×C₁₆₃	—	2	2	no	—
68	C₄×C₁₆₃	—	2	2	no	—
69	C₄×C₁₆₃	—	4	2	no	—
70	C₄×C₁₆₃	#124	1	2	yes	—
71	C₄×C₁₆₃	C₂×#41	1	2	yes	—
72	Dic₁₆₃	—	4	4	no	—
73	C₂×C₁₆₃	—	2	2	no	—
74	C₂×C₁₆₃	—	8	2	no	—
75	C₂×C₁₆₃	C₂×#124	1	1	yes	characteristic
76	C₂×C₁₆₃	#575	1	2	yes	—
77	C₁₆₃	#3142	1	1	yes	characteristic, complemented, Sylow
78	#3142	—	163	1	no	maximal, Sylow
79	#532	D₁₆₃	1	1	yes	characteristic
80	C₂×#125	—	163	1	no	—
81	#491	—	163	2	no	—
82	#490	—	163	2	no	—
83	#488	—	163	4	no	—
84	#417	—	163	4	no	—
85	C₄×M₁₆	—	163	1	no	—
86	C₂×#41	C₂×D₁₆₃	1	2	yes	—
87	C₂×M₁₆	—	163	2	no	—
88	C₄⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
89	#127	C₂×D₁₆₃	1	1	yes	characteristic
90	#125	—	326	2	no	—
91	#125	—	163	2	no	—
92	#125	—	163	2	no	—
93	#124	C₂×D₁₆₃	1	2	yes	—
94	#121	—	2	2	no	—
95	C₄×D₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	326	2	no	—
97	C₂×C₄⋊C₄	—	163	2	no	—
98	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
99	#103	—	163	8	no	—
100	#102	—	163	8	no	—
101	#97	—	163	2	no	—
102	C₄×C₈	—	163	2	no	—
103	#41	—	2	8	no	—
104	C₂×Q₈	C₂²×D₁₆₃	1	2	yes	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	C₂²×D₁₆₃	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	326	2	no	—
110	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
112	M₁₆	—	326	4	no	—
113	C₂×C₈	—	163	4	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	326	4	no	—
116	C₄⋊C₄	—	326	4	no	—
117	C₄⋊C₄	—	326	2	no	—
118	C₄⋊C₄	—	163	4	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₄²	C₂²⋊D₁₆₃	1	2	yes	—
121	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
122	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
123	C₂³	—	2	2	no	—
124	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
125	Q₈	—	2	4	no	—
126	D₄	—	2	4	no	—
127	D₄	—	4	4	no	—
128	D₄	—	4	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	326	4	no	—
133	C₂×C₄	—	652	4	no	—
134	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
137	C₂×C₄	#14579698	1	4	yes	—
138	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
139	C₈	—	326	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	#19104722	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₁₆₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	652	4	no	—
149	C₄	#19104722	1	2	yes	—
150	C₄	C₂×#14579698	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	8	2	no	—
153	C₂	C₂×#19104722	1	1	yes	characteristic
154	C₂	#23782463	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104727

C₂

yes

maximal, characteristic, complemented

#23782033

C₂

yes

maximal, complemented

#23782029

C₂

yes

maximal, complemented

#23782019

C₂

yes

maximal, complemented

#23781666

C₂

yes

maximal, complemented

C₄×#14579683

C₂

yes

maximal, characteristic, complemented

#532×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

#19104727

—

#19104727

C₂²

yes

—

#19104727

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

C₂×Dic₁₆₃⋊C₄

—

#19104659

C₂²

yes

—

#19104656

C₂²

yes

—

#19104646

C₂²

yes

—

C₄×C₁₆₃⋊C₈

C₂²

yes

—

C₂×#41×C₁₆₃

C₂²

yes

—

C₄⋊Q₈×C₁₆₃

C₂²

yes

characteristic

#127×C₁₆₃

C₂²

yes

characteristic

#124×C₁₆₃

C₂²

yes

—

#121×C₁₆₃

—

C₄×D₄×C₁₆₃

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₂²×Dic₁₆₃

—

#14579683

—

C₂×C₁₆₃⋊C₈

C₂³

yes

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

C₂³

yes

—

Dic₁₆₃⋊C₄

—

Dic₁₆₃⋊C₄

—

#41×C₁₆₃

—

C₂×Q₈×C₁₆₃

C₂³

yes

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

C₂³

yes

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

D₄

yes

—

C₄²×C₁₆₃

C₂³

yes

characteristic

C₄²×C₁₆₃

C₂³

yes

characteristic

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

Q₈×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

C₂⁴

yes

characteristic

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#124

yes

—

C₂²×C₁₆₃

C₂²×D₄

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#124

yes

—

C₄×C₁₆₃

C₂×#41

yes

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

C₂×#124

yes

characteristic

C₂×C₁₆₃

#575

yes

—

C₁₆₃

#3142

yes

characteristic, complemented, Sylow

#3142

—

163

maximal, Sylow

#532

D₁₆₃

yes

characteristic

C₂×#125

—

163

—

#491

—

163

—

#490

—

163

—

#488

—

163

—

#417

—

163

—

C₄×M₁₆

—

163

—

C₂×#41

C₂×D₁₆₃

yes

—

C₂×M₁₆

—

163

—

C₄⋊Q₈

C₂×D₁₆₃

yes

characteristic

#127

C₂×D₁₆₃

yes

characteristic

#125

—

326

—

#125

—

163

—

#125

—

163

—

#124

C₂×D₁₆₃

yes

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#103

—

163

—

100

#102

—

163

—

101

#97

—

163

—

102

C₄×C₈

—

163

—

103

#41

—

104

C₂×Q₈

C₂²×D₁₆₃

yes

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₁₆₃

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

326

—

110

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

111

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

112

M₁₆

—

326

—

113

C₂×C₈

—

163

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

326

—

116

C₄⋊C₄

—

326

—

117

C₄⋊C₄

—

326

—

118

C₄⋊C₄

—

163

—

119

C₂²⋊C₄

—

120

C₄²

C₂²⋊D₁₆₃

yes

—

121

C₄²

C₂²×D₁₆₃

yes

characteristic

122

C₄²

C₂²×D₁₆₃

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

326

—

133

C₂×C₄

—

652

—

134

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

135

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

136

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

137

C₂×C₄

#14579698

yes

—

138

C₂×C₄

C₂³×D₁₆₃

yes

characteristic

139

C₈

—

326

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#19104722

yes

—

144

C₂²

C₂²×C₂²⋊D₁₆₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

652

—

149

C₄

#19104722

yes

—

150

C₄

C₂×#14579698

yes

—

151

C₂

—

152

C₂

—

153

C₂

C₂×#19104722

yes

characteristic

154

C₂

#23782463

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₆₃
commutator, cosocle	C₂×C₄×C₁₆₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting, prime radical	#532×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²×D₄
hypercenter	#532	D₁₆₃
hypocenter	C₁₆₃	#3142
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₆₃

commutator, cosocle

C₂×C₄×C₁₆₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₆₃

Fitting, prime radical

#532×C₁₆₃

C₂

socle

C₂²×C₁₆₃

C₂²×D₄

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. C₂×#19104727	C₂	semidirect
2. #23782033	C₂	semidirect
3. #23782029	C₂	semidirect
4. #23782019	C₂	semidirect
5. #23781666	C₂	semidirect
6. C₄×#14579683	C₂	semidirect
77. C₁₆₃	#3142	semidirect

H ≅ G/N

type

1. C₂×#19104727

C₂

semidirect

2. #23782033

C₂

semidirect

3. #23782029

C₂

semidirect

4. #23782019

C₂

semidirect

5. #23781666

C₂

semidirect

6. C₄×#14579683

C₂

semidirect

77. C₁₆₃

#3142

semidirect

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

subgroup	index
C₂×#19104727	2
#23782033	2
#23782029	2
#23782019	2
#23781666	2
C₄×#14579683	2
#532×C₁₆₃	2
#3142	163

subgroup

index

#532×C₁₆₃

163

Subgroups — #35224936

Conjugacy classes of subgroups

Complements

Maximal subgroups