Group #35216007 — subgroups

Subgroups — #35216007

Order 20864.

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

normal subgroups only

This group has 230 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	C₂×#19104668	C₂	1	2	yes	maximal
2	#23781535	C₂	1	1	yes	maximal, characteristic
3	#23781427	C₂	1	2	yes	maximal
4	#23781420	C₂	1	1	yes	maximal, characteristic
5	#389×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₆₃	—	2	1	no	—
8	C₂×C₄⋊Dic₁₆₃	—	2	1	no	—
9	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
10	#19104668	—	2	4	no	—
11	C₂×C₄⋊C₄×C₁₆₃	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
15	#95×C₁₆₃	—	2	1	no	—
16	C₂²×Dic₁₆₃	—	2	2	no	—
17	C₂×C₁₆₃⋊C₈	—	2	2	no	—
18	C₂×C₁₆₃⋊C₈	—	2	2	no	—
19	C₂×C₁₆₃⋊C₈	Q₈	1	4	yes	—
20	C₄⋊Dic₁₆₃	—	2	2	no	—
21	C₄⋊Dic₁₆₃	—	2	2	no	—
22	C₄⋊Dic₁₆₃	—	4	2	no	—
23	C₄⋊Dic₁₆₃	—	4	2	no	—
24	C₂²×C₄×C₁₆₃	—	2	1	no	—
25	C₂²×C₄×C₁₆₃	—	2	1	no	—
26	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₆₃	—	2	1	no	—
30	C₄⋊C₄×C₁₆₃	—	2	2	no	—
31	C₄⋊C₄×C₁₆₃	—	4	2	no	—
32	C₄⋊C₄×C₁₆₃	D₄	1	2	yes	—
33	C₄²×C₁₆₃	—	2	2	no	—
34	C₂×Dic₁₆₃	—	4	2	no	—
35	C₂×Dic₁₆₃	—	4	2	no	—
36	C₂×Dic₁₆₃	—	4	2	no	—
37	C₁₆₃⋊C₈	—	2	4	no	—
38	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₆₃	—	2	1	no	—
40	C₂×C₄×C₁₆₃	—	2	1	no	—
41	C₂×C₄×C₁₆₃	—	2	1	no	—
42	C₂×C₄×C₁₆₃	—	2	1	no	—
43	C₂×C₄×C₁₆₃	—	2	2	no	—
44	C₂×C₄×C₁₆₃	—	4	1	no	—
45	C₂×C₄×C₁₆₃	—	4	1	no	—
46	C₂×C₄×C₁₆₃	—	4	1	no	—
47	C₂×C₄×C₁₆₃	—	4	1	no	—
48	C₂×C₄×C₁₆₃	—	4	1	no	—
49	C₂×C₄×C₁₆₃	D₈	1	2	yes	—
50	C₂×C₄×C₁₆₃	Q₁₆	1	2	yes	—
51	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₆₃	#41	1	2	yes	—
54	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
55	C₂²×C₁₆₃	#121	1	1	yes	characteristic
56	C₂²×C₁₆₃	#123	1	1	yes	characteristic
57	C₂²×C₁₆₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₁₆₃	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₁₆₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₁₆₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₁₆₃	#137	1	1	yes	characteristic
62	C₄×C₁₆₃	—	2	2	no	—
63	C₄×C₁₆₃	—	2	2	no	—
64	C₄×C₁₆₃	—	4	2	no	—
65	C₄×C₁₆₃	—	8	1	no	—
66	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₁₆₃	#125	1	2	yes	—
68	Dic₁₆₃	—	8	2	no	—
69	C₂×C₁₆₃	#398	1	1	yes	characteristic
70	C₂×C₁₆₃	#459	1	1	yes	characteristic
71	C₂×C₁₆₃	#462	1	1	yes	characteristic
72	C₂×C₁₆₃	#480	1	1	yes	characteristic
73	C₂×C₁₆₃	#484	1	1	yes	characteristic
74	C₂×C₁₆₃	#500	1	1	yes	characteristic
75	C₂×C₁₆₃	#501	1	1	yes	characteristic
76	C₁₆₃	#2083	1	1	yes	characteristic, complemented, Sylow
77	#2083	—	163	1	no	maximal, Sylow
78	C₂×#107	—	163	2	no	—
79	#389	D₁₆₃	1	1	yes	characteristic
80	#384	—	163	1	no	—
81	#340	—	163	2	no	—
82	#336	—	163	1	no	—
83	C₂²×C₈	—	163	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	326	1	no	—
86	C₂×C₄⋊C₄	—	326	1	no	—
87	C₂×C₄⋊C₄	—	163	2	no	—
88	C₂×C₄⋊C₄	C₂×D₁₆₃	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×D₁₆₃	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
91	#107	—	326	4	no	—
92	#95	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	326	2	no	—
96	C₂²×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
99	C₂×C₈	—	326	2	no	—
100	C₂×C₈	—	326	2	no	—
101	C₂×C₈	—	163	4	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	326	2	no	—
106	C₄⋊C₄	—	326	2	no	—
107	C₄⋊C₄	—	652	2	no	—
108	C₄⋊C₄	—	652	2	no	—
109	C₄⋊C₄	D₆₅₂	1	2	yes	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	652	2	no	—
123	C₂×C₄	—	652	2	no	—
124	C₂×C₄	—	652	2	no	—
125	C₂×C₄	#14579684	1	2	yes	—
126	C₂×C₄	#14579689	1	2	yes	—
127	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₆₃	1	2	yes	—
129	C₂×C₄	#14579698	1	2	yes	—
130	C₂×C₄	#14579699	1	1	yes	characteristic
131	C₈	—	326	4	no	—
132	C₂²	#19104709	1	1	yes	characteristic
133	C₂²	#19104716	1	1	yes	characteristic
134	C₂²	#19104733	1	1	yes	characteristic
135	C₂²	C₂×#14579684	1	1	yes	characteristic
136	C₂²	C₂×#14579689	1	1	yes	characteristic
137	C₂²	#19104760	1	1	yes	characteristic
138	C₂²	#19104765	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	8	1	no	—
143	C₄	—	1304	2	no	—
144	C₄	#19104715	1	2	yes	—
145	C₄	#19104725	1	2	yes	—
146	C₂	#23781595	1	1	yes	characteristic
147	C₂	#23781860	1	1	yes	characteristic
148	C₂	#23781875	1	1	yes	characteristic
149	C₂	#23781976	1	1	yes	characteristic
150	C₂	#23781998	1	1	yes	characteristic
151	C₂	#23782068	1	1	yes	characteristic
152	C₂	#23782075	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104668

C₂

yes

maximal

#23781535

C₂

yes

maximal, characteristic

#23781427

C₂

yes

maximal

#23781420

C₂

yes

maximal, characteristic

#389×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

—

C₂×C₄⋊Dic₁₆₃

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

#19104668

—

C₂×C₄⋊C₄×C₁₆₃

—

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

characteristic

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

#95×C₁₆₃

—

C₂²×Dic₁₆₃

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

Q₈

yes

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

characteristic

C₂²×C₄×C₁₆₃

D₄

yes

characteristic

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

D₄

yes

—

C₄²×C₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×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

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

D₈

yes

—

C₂×C₄×C₁₆₃

Q₁₆

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₆₃

C₂×D₈

yes

characteristic

C₂²×C₁₆₃

C₂×Q₁₆

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂²⋊Q₈

yes

—

C₄×C₁₆₃

#125

yes

—

Dic₁₆₃

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, complemented, Sylow

#2083

—

163

maximal, Sylow

C₂×#107

—

163

—

#389

D₁₆₃

yes

characteristic

#384

—

163

—

#340

—

163

—

#336

—

163

—

C₂²×C₈

—

163

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

characteristic

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#107

—

326

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

326

—

C₂²×C₄

C₂²⋊D₁₆₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₆₃

yes

characteristic

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

326

—

100

C₂×C₈

—

326

—

101

C₂×C₈

—

163

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

326

—

106

C₄⋊C₄

—

326

—

107

C₄⋊C₄

—

652

—

108

C₄⋊C₄

—

652

—

109

C₄⋊C₄

D₆₅₂

yes

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

652

—

123

C₂×C₄

—

652

—

124

C₂×C₄

—

652

—

125

C₂×C₄

#14579684

yes

—

126

C₂×C₄

#14579689

yes

—

127

C₂×C₄

C₂×D₆₅₂

yes

characteristic

128

C₂×C₄

Q₈×D₁₆₃

yes

—

129

C₂×C₄

#14579698

yes

—

130

C₂×C₄

#14579699

yes

characteristic

131

C₈

—

326

—

132

C₂²

#19104709

yes

characteristic

133

C₂²

#19104716

yes

characteristic

134

C₂²

#19104733

yes

characteristic

135

C₂²

C₂×#14579684

yes

characteristic

136

C₂²

C₂×#14579689

yes

characteristic

137

C₂²

#19104760

yes

characteristic

138

C₂²

#19104765

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

1304

—

144

C₄

#19104715

yes

—

145

C₄

#19104725

yes

—

146

C₂

#23781595

yes

characteristic

147

C₂

#23781860

yes

characteristic

148

C₂

#23781875

yes

characteristic

149

C₂

#23781976

yes

characteristic

150

C₂

#23781998

yes

characteristic

151

C₂

#23782068

yes

characteristic

152

C₂

#23782075

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₆₃
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#389×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂×D₄
hypercenter	#389	D₁₆₃
hypocenter	C₁₆₃	#2083
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₆₃

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#389×C₁₆₃

C₂

socle, prime radical

C₂³×C₁₆₃

C₂×D₄

D₁₆₃

C₁₆₃

G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
76. C₁₆₃	#2083	semidirect

H ≅ G/N

type

76. C₁₆₃

#2083

semidirect

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

subgroup	index
C₂×#19104668	2
#23781535	2
#23781427	2
#23781420	2
#389×C₁₆₃	2
#2083	163

subgroup

index

#389×C₁₆₃

163

Subgroups — #35216007

Conjugacy classes of subgroups

Complements

Maximal subgroups