Group #35216068 — subgroups

Subgroups — #35216068

Order 20864.

290 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 95 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×#19104667	C₂	1	2	yes	maximal, complemented
2	C₂×#19104657	C₂	1	2	yes	maximal
3	#23781567	C₂	1	1	yes	maximal, characteristic
4	#23781535	C₂	1	1	yes	maximal, characteristic, complemented
5	#336×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₅₂	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₆₃	—	2	2	no	—
8	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
11	#19104667	—	2	4	no	—
12	#19104657	—	2	4	no	—
13	C₂²×C₈×C₁₆₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₂³×D₁₆₃	—	2	1	no	—
16	C₂×D₆₅₂	—	2	2	no	—
17	C₂×D₆₅₂	—	2	2	no	—
18	C₂×D₆₅₂	—	2	1	no	—
19	C₂×D₆₅₂	Q₈	1	2	yes	—
20	C₂²×Dic₁₆₃	—	2	1	no	—
21	C₂²×Dic₁₆₃	—	2	2	no	—
22	C₄⋊Dic₁₆₃	—	2	2	no	—
23	C₄⋊Dic₁₆₃	—	2	2	no	—
24	C₄⋊Dic₁₆₃	—	2	2	no	—
25	C₄⋊Dic₁₆₃	—	4	4	no	—
26	(C₂×D₁₆₃)⋊C₄	—	4	4	no	—
27	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
28	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₁₆₃	—	2	2	no	—
30	C₂×C₈×C₁₆₃	—	2	2	no	—
31	C₂×C₈×C₁₆₃	D₄	1	4	yes	—
32	C₄²×C₁₆₃	—	2	2	no	—
33	C₂²×D₁₆₃	—	2	2	no	—
34	C₂²×D₁₆₃	—	4	1	no	—
35	C₂²×D₁₆₃	—	4	1	no	—
36	C₂²×D₁₆₃	—	4	2	no	—
37	C₂²×D₁₆₃	—	4	2	no	—
38	D₆₅₂	—	2	4	no	—
39	D₆₅₂	—	4	2	no	—
40	C₂×Dic₁₆₃	—	4	1	no	—
41	C₂×Dic₁₆₃	—	4	2	no	—
42	C₂×Dic₁₆₃	—	4	2	no	—
43	C₂×Dic₁₆₃	—	4	4	no	—
44	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₆₃	—	2	2	no	—
46	C₂×C₄×C₁₆₃	—	2	2	no	—
47	C₂×C₄×C₁₆₃	SD₁₆	1	4	yes	—
48	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₆₃	#41	1	2	yes	—
51	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
52	C₈×C₁₆₃	—	2	4	no	—
53	C₂²×C₁₆₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₆₃	#123	1	1	yes	characteristic
55	C₂²×C₁₆₃	#127	1	1	yes	characteristic
56	C₂²×C₁₆₃	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₁₆₃	AGL(1,ℤ/8)	1	2	yes	—
58	C₂×D₁₆₃	—	4	2	no	—
59	C₂×D₁₆₃	—	4	4	no	—
60	C₂×D₁₆₃	—	8	2	no	—
61	C₂×D₁₆₃	—	8	1	no	—
62	C₂×D₁₆₃	—	8	1	no	—
63	C₄×C₁₆₃	—	2	4	no	—
64	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₆₃	#123	1	2	yes	—
66	Dic₁₆₃	—	8	1	no	—
67	Dic₁₆₃	—	8	2	no	—
68	C₂×C₁₆₃	#399	1	1	yes	characteristic
69	C₂×C₁₆₃	#476	1	2	yes	—
70	C₂×C₁₆₃	#481	1	2	yes	—
71	C₂×C₁₆₃	#492	1	1	yes	characteristic
72	C₂×C₁₆₃	#496	1	1	yes	characteristic
73	D₁₆₃	—	8	2	no	—
74	C₁₆₃	#2094	1	1	yes	characteristic, complemented, Sylow
75	#2094	—	163	1	no	maximal, Sylow
76	C₂×#106	—	163	2	no	—
77	C₂×#102	—	163	2	no	—
78	#390	—	163	1	no	—
79	#384	—	163	1	no	—
80	#336	D₁₆₃	1	1	yes	characteristic, complemented
81	C₂²×D₄	—	163	1	no	—
82	C₂²×C₈	C₂×D₁₆₃	1	2	yes	—
83	C₂×C₄⋊C₄	—	326	2	no	—
84	C₂×C₄⋊C₄	—	163	2	no	—
85	C₂×C₄⋊C₄	—	163	1	no	—
86	C₂×C₂²⋊C₄	—	326	2	no	—
87	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
88	#106	—	326	4	no	—
89	#102	—	326	4	no	—
90	C₂⁴	—	326	1	no	—
91	C₂×D₄	—	326	2	no	—
92	C₂×D₄	—	326	2	no	—
93	C₂×D₄	—	326	1	no	—
94	C₂×D₄	—	163	2	no	—
95	C₂²×C₄	—	326	1	no	—
96	C₂²×C₄	—	326	2	no	—
97	C₂²×C₄	D₆₅₂	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	D₆₅₂	1	4	yes	—
102	C₄⋊C₄	—	326	2	no	—
103	C₄⋊C₄	—	326	2	no	—
104	C₄⋊C₄	—	326	2	no	—
105	C₄⋊C₄	—	652	4	no	—
106	C₂²⋊C₄	—	652	4	no	—
107	C₄²	—	2	2	no	—
108	C₂³	—	326	2	no	—
109	C₂³	—	652	1	no	—
110	C₂³	—	652	1	no	—
111	C₂³	—	652	2	no	—
112	C₂³	—	652	2	no	—
113	C₂³	C₂×D₆₅₂	1	1	yes	characteristic
114	D₄	—	326	4	no	—
115	D₄	—	652	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	652	1	no	—
119	C₂×C₄	—	652	2	no	—
120	C₂×C₄	—	652	2	no	—
121	C₂×C₄	—	652	4	no	—
122	C₂×C₄	#14579688	1	4	yes	—
123	C₂×C₄	C₂×D₆₅₂	1	2	yes	—
124	C₂×C₄	Q₈×D₁₆₃	1	1	yes	characteristic
125	C₂×C₄	#14579698	1	2	yes	—
126	C₂×C₄	#14579698	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₂²	—	1304	2	no	—
129	C₂²	—	1304	1	no	—
130	C₂²	—	1304	1	no	—
131	C₂²	—	652	4	no	—
132	C₂²	—	652	2	no	—
133	C₂²	#19104715	1	1	yes	characteristic
134	C₂²	#19104719	1	1	yes	characteristic
135	C₂²	#19104732	1	1	yes	characteristic
136	C₂²	C₂×#14579688	1	2	yes	—
137	C₂²	#19104763	1	2	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	1304	1	no	—
140	C₄	—	1304	2	no	—
141	C₄	#19104715	1	2	yes	—
142	C₄	#19104719	1	2	yes	—
143	C₂	—	1304	2	no	—
144	C₂	#23781602	1	1	yes	characteristic
145	C₂	#23781956	1	2	yes	—
146	C₂	#23781984	1	2	yes	—
147	C₂	#23782038	1	1	yes	characteristic
148	C₂	#23782054	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104667

C₂

yes

maximal, complemented

C₂×#19104657

C₂

yes

maximal

#23781567

C₂

yes

maximal, characteristic

#23781535

C₂

yes

maximal, characteristic, complemented

#336×C₁₆₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₅₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₆₃

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

characteristic

C₂×(C₂×D₁₆₃)⋊C₄

—

#19104667

—

#19104657

—

C₂²×C₈×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₂³×D₁₆₃

—

C₂×D₆₅₂

—

C₂×D₆₅₂

—

C₂×D₆₅₂

—

C₂×D₆₅₂

Q₈

yes

—

C₂²×Dic₁₆₃

—

C₂²×Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

(C₂×D₁₆₃)⋊C₄

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

D₄

yes

—

C₄²×C₁₆₃

—

C₂²×D₁₆₃

—

C₂²×D₁₆₃

—

C₂²×D₁₆₃

—

C₂²×D₁₆₃

—

C₂²×D₁₆₃

—

D₆₅₂

—

D₆₅₂

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

SD₁₆

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

#41

yes

characteristic

C₈×C₁₆₃

—

C₂²×C₁₆₃

C₂²⋊Q₈

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

C₂×SD₁₆

yes

—

C₂²×C₁₆₃

AGL(1,ℤ/8)

yes

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂²⋊Q₈

yes

—

C₄×C₁₆₃

#123

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

#399

yes

characteristic

C₂×C₁₆₃

#476

yes

—

C₂×C₁₆₃

#481

yes

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

D₁₆₃

—

C₁₆₃

#2094

yes

characteristic, complemented, Sylow

#2094

—

163

maximal, Sylow

C₂×#106

—

163

—

C₂×#102

—

163

—

#390

—

163

—

#384

—

163

—

#336

D₁₆₃

yes

characteristic, complemented

C₂²×D₄

—

163

—

C₂²×C₈

C₂×D₁₆₃

yes

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

—

163

—

C₂×C₂²⋊C₄

—

326

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#106

—

326

—

#102

—

326

—

C₂⁴

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

163

—

C₂²×C₄

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

D₆₅₂

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

D₆₅₂

yes

—

102

C₄⋊C₄

—

326

—

103

C₄⋊C₄

—

326

—

104

C₄⋊C₄

—

326

—

105

C₄⋊C₄

—

652

—

106

C₂²⋊C₄

—

652

—

107

C₄²

—

108

C₂³

—

326

—

109

C₂³

—

652

—

110

C₂³

—

652

—

111

C₂³

—

652

—

112

C₂³

—

652

—

113

C₂³

C₂×D₆₅₂

yes

characteristic

114

D₄

—

326

—

115

D₄

—

652

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

652

—

119

C₂×C₄

—

652

—

120

C₂×C₄

—

652

—

121

C₂×C₄

—

652

—

122

C₂×C₄

#14579688

yes

—

123

C₂×C₄

C₂×D₆₅₂

yes

—

124

C₂×C₄

Q₈×D₁₆₃

yes

characteristic

125

C₂×C₄

#14579698

yes

—

126

C₂×C₄

#14579698

yes

characteristic

127

C₈

—

128

C₂²

—

1304

—

129

C₂²

—

1304

—

130

C₂²

—

1304

—

131

C₂²

—

652

—

132

C₂²

—

652

—

133

C₂²

#19104715

yes

characteristic

134

C₂²

#19104719

yes

characteristic

135

C₂²

#19104732

yes

characteristic

136

C₂²

C₂×#14579688

yes

—

137

C₂²

#19104763

yes

—

138

C₄

—

139

C₄

—

1304

—

140

C₄

—

1304

—

141

C₄

#19104715

yes

—

142

C₄

#19104719

yes

—

143

C₂

—

1304

—

144

C₂

#23781602

yes

characteristic

145

C₂

#23781956

yes

—

146

C₂

#23781984

yes

—

147

C₂

#23782038

yes

characteristic

148

C₂

#23782054

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₅₂
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#336×C₁₆₃	C₂
socle	C₂³×C₁₆₃	C₂×D₄
prime radical	C₂²×D₆₅₂	C₂²
hypercenter	#336	D₁₆₃
hypocenter	C₁₆₃	#2094
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₆₅₂

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#336×C₁₆₃

C₂

socle

C₂³×C₁₆₃

C₂×D₄

prime radical

C₂²×D₆₅₂

C₂²

D₁₆₃

C₁₆₃

G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
1. C₂×#19104667	C₂	semidirect
4. #23781535	C₂	semidirect
5. #336×C₁₆₃	C₂	semidirect
74. C₁₆₃	#2094	semidirect
80. #336	D₁₆₃	semidirect

H ≅ G/N

type

1. C₂×#19104667

C₂

semidirect

4. #23781535

C₂

semidirect

5. #336×C₁₆₃

C₂

semidirect

74. C₁₆₃

#2094

semidirect

80. #336

D₁₆₃

semidirect

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

subgroup	index
C₂×#19104667	2
C₂×#19104657	2
#23781567	2
#23781535	2
#336×C₁₆₃	2
#2094	163

subgroup

index

#336×C₁₆₃

163

Subgroups — #35216068

Conjugacy classes of subgroups

Complements

Maximal subgroups