Group #35215950 — subgroups

Subgroups — #35215950

Order 20864.

258 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 79 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×#19104655	C₂	1	2	yes	maximal
2	#23781567	C₂	1	1	yes	maximal, characteristic
3	#23781427	C₂	1	2	yes	maximal, complemented
4	#23781420	C₂	1	1	yes	maximal, characteristic, complemented
5	#382×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₅₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
9	C₂×(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
10	#19104655	—	2	4	no	—
11	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
13	#95×C₁₆₃	—	2	2	no	—
14	C₂³×D₁₆₃	—	2	1	no	—
15	C₂×D₆₅₂	—	2	2	no	—
16	C₂×D₆₅₂	—	2	2	no	—
17	C₂×D₆₅₂	—	2	1	no	—
18	C₂×D₆₅₂	D₄	1	2	yes	—
19	C₂²×Dic₁₆₃	—	2	1	no	—
20	C₂×C₁₆₃⋊C₈	—	2	2	no	—
21	C₂×C₁₆₃⋊C₈	—	2	2	no	—
22	C₂×C₁₆₃⋊C₈	—	2	2	no	—
23	C₄⋊Dic₁₆₃	—	2	2	no	—
24	(C₂×D₁₆₃)⋊C₄	—	4	4	no	—
25	C₂²×C₄×C₁₆₃	—	2	2	no	—
26	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
27	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₆₃	—	2	2	no	—
29	C₄⋊C₄×C₁₆₃	—	2	2	no	—
30	C₄²×C₁₆₃	—	2	2	no	—
31	C₂²×D₁₆₃	—	2	2	no	—
32	C₂²×D₁₆₃	—	4	1	no	—
33	C₂²×D₁₆₃	—	4	1	no	—
34	C₂²×D₁₆₃	—	4	2	no	—
35	C₂²×D₁₆₃	—	4	2	no	—
36	D₆₅₂	—	2	4	no	—
37	D₆₅₂	—	4	2	no	—
38	C₂×Dic₁₆₃	—	4	1	no	—
39	C₂×Dic₁₆₃	—	4	2	no	—
40	C₁₆₃⋊C₈	—	4	2	no	—
41	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₆₃	—	2	2	no	—
43	C₂×C₄×C₁₆₃	—	2	2	no	—
44	C₂×C₄×C₁₆₃	—	2	2	no	—
45	C₂×C₄×C₁₆₃	—	4	2	no	—
46	C₂×C₄×C₁₆₃	—	4	2	no	—
47	C₂×C₄×C₁₆₃	—	4	2	no	—
48	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
50	C₂×C₄×C₁₆₃	#41	1	2	yes	—
51	C₂×C₄×C₁₆₃	#41	1	2	yes	—
52	C₂²×C₁₆₃	#121	1	1	yes	characteristic
53	C₂²×C₁₆₃	#123	1	1	yes	characteristic
54	C₂²×C₁₆₃	#124	1	1	yes	characteristic
55	C₂²×C₁₆₃	#135	1	2	yes	—
56	C₂²×C₁₆₃	AGL(1,ℤ/8)	1	2	yes	—
57	C₂×D₁₆₃	—	4	2	no	—
58	C₂×D₁₆₃	—	4	4	no	—
59	C₂×D₁₆₃	—	8	2	no	—
60	C₂×D₁₆₃	—	8	1	no	—
61	C₂×D₁₆₃	—	8	1	no	—
62	C₄×C₁₆₃	—	4	2	no	—
63	C₄×C₁₆₃	—	8	2	no	—
64	C₄×C₁₆₃	#123	1	2	yes	—
65	C₄×C₁₆₃	#126	1	2	yes	—
66	Dic₁₆₃	—	8	1	no	—
67	C₂×C₁₆₃	#397	1	1	yes	characteristic
68	C₂×C₁₆₃	#463	1	2	yes	—
69	C₂×C₁₆₃	#482	1	2	yes	—
70	C₂×C₁₆₃	#488	1	1	yes	characteristic
71	C₂×C₁₆₃	#490	1	1	yes	characteristic
72	D₁₆₃	—	8	2	no	—
73	C₁₆₃	#2073	1	1	yes	characteristic, complemented, Sylow
74	#2073	—	163	1	no	maximal, Sylow
75	C₂×#102	—	163	2	no	—
76	#390	—	163	1	no	—
77	#382	D₁₆₃	1	1	yes	characteristic, complemented
78	#340	—	163	2	no	—
79	#336	—	163	1	no	—
80	C₂²×D₄	—	163	1	no	—
81	C₂²×C₈	—	163	2	no	—
82	C₂×C₄⋊C₄	—	163	1	no	—
83	C₂×C₄⋊C₄	C₂×D₁₆₃	1	2	yes	—
84	C₂×C₂²⋊C₄	—	326	2	no	—
85	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
86	#102	—	326	4	no	—
87	#95	—	2	2	no	—
88	C₂⁴	—	326	1	no	—
89	C₂×D₄	—	326	1	no	—
90	C₂×D₄	—	326	2	no	—
91	C₂×D₄	—	326	2	no	—
92	C₂×D₄	—	163	2	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	326	1	no	—
95	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
97	C₂×C₈	—	326	2	no	—
98	C₂×C₈	—	326	2	no	—
99	C₂×C₈	—	326	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	326	2	no	—
103	C₂²⋊C₄	—	652	4	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	326	2	no	—
106	C₂³	—	652	1	no	—
107	C₂³	—	652	1	no	—
108	C₂³	—	652	2	no	—
109	C₂³	—	652	2	no	—
110	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
111	D₄	—	326	4	no	—
112	D₄	—	652	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	652	1	no	—
120	C₂×C₄	—	652	2	no	—
121	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
122	C₂×C₄	#14579698	1	1	yes	characteristic
123	C₂×C₄	#14579699	1	2	yes	—
124	C₂×C₄	#14579700	1	2	yes	—
125	C₈	—	652	2	no	—
126	C₂²	—	1304	2	no	—
127	C₂²	—	1304	1	no	—
128	C₂²	—	1304	1	no	—
129	C₂²	—	652	2	no	—
130	C₂²	—	652	4	no	—
131	C₂²	#19104706	1	1	yes	characteristic
132	C₂²	#19104718	1	1	yes	characteristic
133	C₂²	#19104724	1	1	yes	characteristic
134	C₂²	#19104755	1	2	yes	—
135	C₂²	#19104760	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	1304	1	no	—
139	C₄	#19104720	1	2	yes	—
140	C₄	#19104730	1	2	yes	—
141	C₂	—	1304	2	no	—
142	C₂	#23781591	1	1	yes	characteristic
143	C₂	#23781877	1	2	yes	—
144	C₂	#23781989	1	2	yes	—
145	C₂	#23782022	1	1	yes	characteristic
146	C₂	#23782031	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104655

C₂

yes

maximal

#23781567

C₂

yes

maximal, characteristic

#23781427

C₂

yes

maximal, complemented

#23781420

C₂

yes

maximal, characteristic, complemented

#382×C₁₆₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₅₂

C₂²

yes

characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

characteristic

C₂×(C₂×D₁₆₃)⋊C₄

—

#19104655

—

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

#95×C₁₆₃

—

C₂³×D₁₆₃

—

C₂×D₆₅₂

—

C₂×D₆₅₂

—

C₂×D₆₅₂

—

C₂×D₆₅₂

D₄

yes

—

C₂²×Dic₁₆₃

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₄⋊Dic₁₆₃

—

(C₂×D₁₆₃)⋊C₄

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

—

C₂²×D₁₆₃

—

C₂²×D₁₆₃

—

C₂²×D₁₆₃

—

C₂²×D₁₆₃

—

C₂²×D₁₆₃

—

D₆₅₂

—

D₆₅₂

—

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×D₄

yes

C₂×C₄×C₁₆₃

yes

C₂×C₄×C₁₆₃

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

—

C₂²×C₁₆₃

AGL(1,ℤ/8)

yes

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#123

yes

—

C₄×C₁₆₃

#126

yes

—

Dic₁₆₃

—

C₂×C₁₆₃

#397

yes

characteristic

C₂×C₁₆₃

#463

yes

—

C₂×C₁₆₃

#482

yes

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

D₁₆₃

—

C₁₆₃

#2073

yes

characteristic, complemented, Sylow

#2073

—

163

maximal, Sylow

C₂×#102

—

163

—

#390

—

163

—

#382

D₁₆₃

yes

characteristic, complemented

#340

—

163

—

#336

—

163

—

C₂²×D₄

—

163

—

C₂²×C₈

—

163

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

—

C₂×C₂²⋊C₄

—

326

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#102

—

326

—

#95

—

C₂⁴

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

326

—

C₂×D₄

—

163

—

C₂²×C₄

—

C₂²×C₄

—

326

—

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

326

—

103

C₂²⋊C₄

—

652

—

104

C₄²

—

105

C₂³

—

326

—

106

C₂³

—

652

—

107

C₂³

—

652

—

108

C₂³

—

652

—

109

C₂³

—

652

—

110

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

111

D₄

—

326

—

112

D₄

—

652

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

652

—

120

C₂×C₄

—

652

—

121

C₂×C₄

D₄×D₁₆₃

yes

characteristic

122

C₂×C₄

#14579698

yes

characteristic

123

C₂×C₄

#14579699

yes

—

124

C₂×C₄

#14579700

yes

—

125

C₈

—

652

—

126

C₂²

—

1304

—

127

C₂²

—

1304

—

128

C₂²

—

1304

—

129

C₂²

—

652

—

130

C₂²

—

652

—

131

C₂²

#19104706

yes

characteristic

132

C₂²

#19104718

yes

characteristic

133

C₂²

#19104724

yes

characteristic

134

C₂²

#19104755

yes

—

135

C₂²

#19104760

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

1304

—

139

C₄

#19104720

yes

—

140

C₄

#19104730

yes

—

141

C₂

—

1304

—

142

C₂

#23781591

yes

characteristic

143

C₂

#23781877

yes

—

144

C₂

#23781989

yes

—

145

C₂

#23782022

yes

characteristic

146

C₂

#23782031

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₆₃
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#382×C₁₆₃	C₂
socle	C₂³×C₁₆₃	C₂×D₄
prime radical	C₂²×D₆₅₂	C₂²
hypercenter	#382	D₁₆₃
hypocenter	C₁₆₃	#2073
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₆₃

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#382×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 ◁ #382

N	H ≅ G/N	type
3. #23781427	C₂	semidirect
4. #23781420	C₂	semidirect
5. #382×C₁₆₃	C₂	semidirect
73. C₁₆₃	#2073	semidirect
77. #382	D₁₆₃	semidirect

H ≅ G/N

type

3. #23781427

C₂

semidirect

4. #23781420

C₂

semidirect

5. #382×C₁₆₃

C₂

semidirect

73. C₁₆₃

#2073

semidirect

77. #382

D₁₆₃

semidirect

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

subgroup	index
C₂×#19104655	2
#23781567	2
#23781427	2
#23781420	2
#382×C₁₆₃	2
#2073	163

subgroup

index

#382×C₁₆₃

163

Subgroups — #35215950

Conjugacy classes of subgroups

Complements

Maximal subgroups