Group #35216005 — subgroups

Subgroups — #35216005

Order 20864.

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

normal subgroups only

This group has 184 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	#23781713	C₂	1	1	yes	maximal, characteristic
2	#23781673	C₂	1	1	yes	maximal, characteristic
3	#23781673	C₂	1	1	yes	maximal, characteristic
4	C₂×#19104653	C₂	1	1	yes	maximal, characteristic
5	#23781435	C₂	1	1	yes	maximal, characteristic
6	#23781435	C₂	1	1	yes	maximal, characteristic
7	H(3,ℤ/4)×C₁₆₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₂₆	C₂²	1	1	yes	characteristic
9	C₂×#14579683	C₂²	1	1	yes	characteristic
10	C₂×#14579683	C₂²	1	1	yes	characteristic
11	C₂×#14579683	C₂²	1	1	yes	characteristic
12	#19104670	—	2	2	no	—
13	#19104658	—	2	2	no	—
14	#19104658	—	2	2	no	—
15	#19104653	—	2	2	no	—
16	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
17	#117×C₁₆₃	C₂²	1	1	yes	characteristic
18	#117×C₁₆₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₂₆	—	2	1	no	—
20	C₂×Dic₃₂₆	—	2	1	no	—
21	C₂×Dic₃₂₆	—	2	1	no	—
22	C₂×Dic₃₂₆	—	2	1	no	—
23	C₂×Dic₃₂₆	—	2	1	no	—
24	C₂×Dic₃₂₆	D₄	1	2	yes	—
25	C₂²×Dic₁₆₃	—	2	1	no	—
26	#14579683	—	2	1	no	—
27	#14579683	—	2	1	no	—
28	#14579683	—	2	1	no	—
29	#14579683	—	2	1	no	—
30	#14579683	—	2	1	no	—
31	#14579683	—	2	1	no	—
32	C₂×C₁₆₃⋊C₈	—	2	1	no	—
33	C₂×C₁₆₃⋊C₈	—	2	1	no	—
34	C₂×C₁₆₃⋊C₈	D₄	1	2	yes	—
35	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₆₃	—	2	1	no	—
37	M₁₆×C₁₆₃	—	2	1	no	—
38	C₂×C₈×C₁₆₃	—	2	1	no	—
39	C₄⋊C₄×C₁₆₃	—	2	1	no	—
40	C₄⋊C₄×C₁₆₃	—	2	1	no	—
41	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
42	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
43	C₄²×C₁₆₃	—	2	1	no	—
44	C₄²×C₁₆₃	—	2	1	no	—
45	Dic₃₂₆	—	2	2	no	—
46	Dic₃₂₆	—	2	2	no	—
47	Dic₃₂₆	—	4	1	no	—
48	Dic₃₂₆	—	4	1	no	—
49	C₂×Dic₁₆₃	—	2	2	no	—
50	C₂×Dic₁₆₃	—	4	1	no	—
51	C₂×Dic₁₆₃	—	4	1	no	—
52	C₁₆₃⋊C₈	—	2	2	no	—
53	C₁₆₃⋊C₈	—	4	1	no	—
54	C₁₆₃⋊C₈	—	4	1	no	—
55	C₂³×C₁₆₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₆₃	—	4	1	no	—
57	C₂×C₄×C₁₆₃	—	4	1	no	—
58	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
61	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
62	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
63	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
64	C₈×C₁₆₃	—	4	1	no	—
65	C₂²×C₁₆₃	—	4	1	no	—
66	C₂²×C₁₆₃	#123	1	1	yes	characteristic
67	C₂²×C₁₆₃	#124	1	1	yes	characteristic
68	C₂²×C₁₆₃	#126	1	1	yes	characteristic
69	C₄×C₁₆₃	—	8	1	no	—
70	C₄×C₁₆₃	—	8	1	no	—
71	C₄×C₁₆₃	#121	1	1	yes	characteristic
72	C₄×C₁₆₃	#123	1	1	yes	characteristic
73	C₄×C₁₆₃	#123	1	1	yes	characteristic
74	C₄×C₁₆₃	#126	1	1	yes	characteristic
75	Dic₁₆₃	—	4	2	no	—
76	C₂×C₁₆₃	—	2	1	no	—
77	C₂×C₁₆₃	—	2	1	no	—
78	C₂×C₁₆₃	—	2	1	no	—
79	C₂×C₁₆₃	#397	1	1	yes	characteristic
80	C₁₆₃	#2082	1	1	yes	characteristic, complemented, Sylow
81	#2082	—	163	1	no	maximal, Sylow
82	#430	—	163	1	no	—
83	#419	—	163	1	no	—
84	#419	—	163	1	no	—
85	C₂×#101	—	163	1	no	—
86	#344	—	163	1	no	—
87	#344	—	163	1	no	—
88	H(3,ℤ/4)	D₁₆₃	1	1	yes	characteristic
89	C₂²×Q₈	—	163	1	no	—
90	C₂×M₁₆	—	163	1	no	—
91	C₂×M₁₆	—	163	1	no	—
92	C₂×M₁₆	—	163	1	no	—
93	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
94	#117	C₂×D₁₆₃	1	1	yes	characteristic
95	#117	C₂×D₁₆₃	1	1	yes	characteristic
96	#108	—	326	2	no	—
97	#103	—	326	2	no	—
98	#103	—	326	2	no	—
99	#101	—	326	2	no	—
100	C₂×Q₈	—	326	1	no	—
101	C₂×Q₈	—	326	1	no	—
102	C₂×Q₈	—	326	1	no	—
103	C₂×Q₈	—	326	1	no	—
104	C₂×Q₈	—	326	1	no	—
105	C₂×Q₈	—	163	2	no	—
106	C₂²×C₄	—	326	1	no	—
107	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	—	2	1	no	—
110	M₁₆	—	326	1	no	—
111	M₁₆	—	326	1	no	—
112	M₁₆	—	326	1	no	—
113	M₁₆	—	326	1	no	—
114	M₁₆	—	326	1	no	—
115	M₁₆	—	326	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	326	1	no	—
118	C₂×C₈	—	326	1	no	—
119	C₂×C₈	—	163	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₂³	#14579698	1	1	yes	characteristic
127	Q₈	—	326	2	no	—
128	Q₈	—	326	2	no	—
129	Q₈	—	652	1	no	—
130	Q₈	—	652	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	326	2	no	—
134	C₂×C₄	—	652	1	no	—
135	C₂×C₄	—	652	1	no	—
136	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
137	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
138	C₂×C₄	#14579699	1	1	yes	characteristic
139	C₂×C₄	#14579699	1	1	yes	characteristic
140	C₂×C₄	#14579699	1	1	yes	characteristic
141	C₂×C₄	#14579700	1	1	yes	characteristic
142	C₈	—	4	1	no	—
143	C₈	—	326	2	no	—
144	C₈	—	652	1	no	—
145	C₈	—	652	1	no	—
146	C₂²	—	4	1	no	—
147	C₂²	#19104717	1	1	yes	characteristic
148	C₂²	#19104721	1	1	yes	characteristic
149	C₂²	#19104730	1	1	yes	characteristic
150	C₄	—	8	1	no	—
151	C₄	—	8	1	no	—
152	C₄	—	652	2	no	—
153	C₄	#19104708	1	1	yes	characteristic
154	C₄	#19104720	1	1	yes	characteristic
155	C₄	#19104720	1	1	yes	characteristic
156	C₄	#19104729	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#23781594	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23781713

C₂

yes

maximal, characteristic

#23781673

C₂

yes

maximal, characteristic

#23781673

C₂

yes

maximal, characteristic

C₂×#19104653

C₂

yes

maximal, characteristic

#23781435

C₂

yes

maximal, characteristic

#23781435

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₂₆

C₂²

yes

characteristic

C₂×#14579683

C₂²

yes

characteristic

C₂×#14579683

C₂²

yes

characteristic

C₂×#14579683

C₂²

yes

—

—

—

—

C₂×M₁₆×C₁₆₃

C₂²

yes

characteristic

#117×C₁₆₃

C₂²

yes

characteristic

#117×C₁₆₃

C₂²

yes

characteristic

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

D₄

yes

—

C₂²×Dic₁₆₃

—

—

—

—

—

—

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

M₁₆×C₁₆₃

—

M₁₆×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

—

C₄²×C₁₆₃

—

Dic₃₂₆

—

Dic₃₂₆

—

Dic₃₂₆

—

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

#41

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

C₂×C₄×C₁₆₃

yes

C₂×C₄×C₁₆₃

yes

C₂×C₄×C₁₆₃

yes

C₂×C₄×C₁₆₃

yes

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#397

yes

characteristic

C₁₆₃

#2082

yes

characteristic, complemented, Sylow

#2082

—

163

maximal, Sylow

#430

—

163

—

#419

—

163

—

#419

—

163

—

C₂×#101

—

163

—

#344

—

163

—

#344

—

163

—

H(3,ℤ/4)

D₁₆₃

yes

characteristic

C₂²×Q₈

—

163

—

C₂×M₁₆

—

163

—

C₂×M₁₆

—

163

—

C₂×M₁₆

—

163

—

C₂×M₁₆

C₂×D₁₆₃

yes

characteristic

#117

C₂×D₁₆₃

yes

characteristic

#117

C₂×D₁₆₃

yes

characteristic

#108

—

326

—

#103

—

326

—

#103

—

326

—

#101

—

326

—

100

C₂×Q₈

—

326

—

101

C₂×Q₈

—

326

—

102

C₂×Q₈

—

326

—

103

C₂×Q₈

—

326

—

104

C₂×Q₈

—

326

—

105

C₂×Q₈

—

163

—

106

C₂²×C₄

—

326

—

107

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

326

—

111

M₁₆

—

326

—

112

M₁₆

—

326

—

113

M₁₆

—

326

—

114

M₁₆

—

326

—

115

M₁₆

—

326

—

116

C₂×C₈

—

117

C₂×C₈

—

326

—

118

C₂×C₈

—

326

—

119

C₂×C₈

—

163

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₂³

#14579698

yes

characteristic

127

Q₈

—

326

—

128

Q₈

—

326

—

129

Q₈

—

652

—

130

Q₈

—

652

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

326

—

134

C₂×C₄

—

652

—

135

C₂×C₄

—

652

—

136

C₂×C₄

D₄×D₁₆₃

yes

characteristic

137

C₂×C₄

D₄×D₁₆₃

yes

characteristic

138

C₂×C₄

#14579699

yes

characteristic

139

C₂×C₄

#14579699

yes

characteristic

140

C₂×C₄

#14579699

yes

characteristic

141

C₂×C₄

#14579700

yes

characteristic

142

C₈

—

143

C₈

—

326

—

144

C₈

—

652

—

145

C₈

—

652

—

146

C₂²

—

147

C₂²

#19104717

yes

characteristic

148

C₂²

#19104721

yes

characteristic

149

C₂²

#19104730

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

652

—

153

C₄

#19104708

yes

characteristic

154

C₄

#19104720

yes

characteristic

155

C₄

#19104720

yes

characteristic

156

C₄

#19104729

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#23781594

yes

characteristic

	subgroup	G/H
center	C₂	#23781594
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	H(3,ℤ/4)×C₁₆₃	C₂
socle	C₂×C₁₆₃	#397
prime radical	C₂³×C₁₆₃	#41
hypercenter	H(3,ℤ/4)	D₁₆₃
hypocenter	C₁₆₃	#2082
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₂

#23781594

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

H(3,ℤ/4)×C₁₆₃

C₂

C₂×C₁₆₃

C₂³×C₁₆₃

D₁₆₃

C₁₆₃

G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
80. C₁₆₃	#2082	semidirect

H ≅ G/N

type

80. C₁₆₃

#2082

semidirect

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

subgroup	index
#23781713	2
#23781673	2
C₂×#19104653	2
#23781435	2
H(3,ℤ/4)×C₁₆₃	2
#2082	163

subgroup

index

H(3,ℤ/4)×C₁₆₃

163

Subgroups — #35216005

Conjugacy classes of subgroups

Complements

Maximal subgroups