Group #35216029 — subgroups

Subgroups — #35216029

Order 20864.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#19104669	C₂	1	1	yes	maximal, characteristic
2	#23781563	C₂	1	2	yes	maximal
3	#23781428	C₂	1	2	yes	maximal
4	#23781428	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
8	C₂×Dic₁₆₃⋊C₄	—	2	2	no	—
9	#19104669	—	2	2	no	—
10	C₁₆₃⋊#95	—	2	2	no	—
11	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₆₃	—	2	2	no	—
15	C₂²×Dic₁₆₃	—	2	2	no	—
16	C₂²×Dic₁₆₃	—	2	2	no	—
17	C₄⋊Dic₁₆₃	—	2	2	no	—
18	C₄⋊Dic₁₆₃	—	2	2	no	—
19	C₄⋊Dic₁₆₃	—	2	2	no	—
20	C₄⋊Dic₁₆₃	Q₈	1	4	yes	—
21	Dic₁₆₃⋊C₄	—	4	4	no	—
22	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₆₃	—	2	2	no	—
26	C₂×C₈×C₁₆₃	—	2	1	no	—
27	C₂×C₈×C₁₆₃	—	2	2	no	—
28	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
29	C₄²×C₁₆₃	—	2	1	no	—
30	C₄²×C₁₆₃	—	2	1	no	—
31	C₂×Dic₁₆₃	—	2	4	no	—
32	C₂×Dic₁₆₃	—	4	2	no	—
33	C₂×Dic₁₆₃	—	4	2	no	—
34	C₂×Dic₁₆₃	—	4	2	no	—
35	C₂×Dic₁₆₃	—	4	2	no	—
36	C₂×Dic₁₆₃	—	4	2	no	—
37	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₆₃	—	2	1	no	—
39	C₂×C₄×C₁₆₃	—	2	1	no	—
40	C₂×C₄×C₁₆₃	—	2	1	no	—
41	C₂×C₄×C₁₆₃	—	2	2	no	—
42	C₂×C₄×C₁₆₃	Q₁₆	1	2	yes	—
43	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₆₃	#41	1	2	yes	—
46	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
47	C₈×C₁₆₃	—	2	2	no	—
48	C₈×C₁₆₃	—	4	1	no	—
49	C₂²×C₁₆₃	#121	1	1	yes	characteristic
50	C₂²×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₁₆₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₂²×C₁₆₃	#135	1	1	yes	characteristic
53	C₂²×C₁₆₃	AGL(1,ℤ/8)	1	2	yes	—
54	C₄×C₁₆₃	—	2	2	no	—
55	C₄×C₁₆₃	—	4	1	no	—
56	C₄×C₁₆₃	#123	1	1	yes	characteristic
57	C₄×C₁₆₃	#125	1	2	yes	—
58	C₄×C₁₆₃	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₁₆₃	—	4	4	no	—
60	Dic₁₆₃	—	8	2	no	—
61	C₂×C₁₆₃	#398	1	1	yes	characteristic
62	C₂×C₁₆₃	#467	1	1	yes	characteristic
63	C₂×C₁₆₃	#469	1	1	yes	characteristic
64	C₂×C₁₆₃	#477	1	2	yes	—
65	C₂×C₁₆₃	#478	1	2	yes	—
66	C₁₆₃	#2087	1	1	yes	characteristic, complemented, Sylow
67	#2087	—	163	1	no	maximal, Sylow
68	C₂×#107	—	163	1	no	—
69	C₂×C₄⋊C₈	D₁₆₃	1	1	yes	characteristic
70	#389	—	163	2	no	—
71	#340	—	163	2	no	—
72	#340	—	163	1	no	—
73	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	326	2	no	—
76	C₂×C₄⋊C₄	—	163	2	no	—
77	C₂×C₄⋊C₄	—	163	2	no	—
78	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
79	#107	—	326	2	no	—
80	C₄⋊C₈	—	2	2	no	—
81	#95	—	326	2	no	—
82	C₂²×C₄	—	326	2	no	—
83	C₂²×C₄	—	326	2	no	—
84	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
85	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
91	C₄⋊C₄	—	326	2	no	—
92	C₄⋊C₄	—	326	2	no	—
93	C₄⋊C₄	—	326	2	no	—
94	C₄⋊C₄	—	652	4	no	—
95	C₄⋊C₄	—	163	4	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	C₂×D₆₅₂	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	326	4	no	—
104	C₂×C₄	—	652	2	no	—
105	C₂×C₄	—	652	2	no	—
106	C₂×C₄	—	652	2	no	—
107	C₂×C₄	—	652	2	no	—
108	C₂×C₄	—	652	2	no	—
109	C₂×C₄	Dic₆₅₂	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₁₆₃	1	2	yes	—
112	C₂×C₄	#14579698	1	2	yes	—
113	C₂×C₄	#14579699	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	1	no	—
116	C₂²	#19104709	1	1	yes	characteristic
117	C₂²	#19104715	1	2	yes	—
118	C₂²	C₂×Dic₆₅₂	1	1	yes	characteristic
119	C₂²	#19104757	1	1	yes	characteristic
120	C₂²	#19104763	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	1304	2	no	—
124	C₄	—	652	4	no	—
125	C₄	#19104716	1	1	yes	characteristic
126	C₄	#19104725	1	2	yes	—
127	C₄	#19104733	1	1	yes	characteristic
128	C₂	#23781595	1	1	yes	characteristic
129	C₂	#23781906	1	1	yes	characteristic
130	C₂	#23781918	1	1	yes	characteristic
131	C₂	#23781961	1	2	yes	—
132	C₂	#23781966	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104669

C₂

yes

maximal, characteristic

#23781563

C₂

yes

maximal

#23781428

C₂

yes

maximal

#23781428

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₆₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

C₂×Dic₁₆₃⋊C₄

—

#19104669

—

C₁₆₃⋊#95

—

C₂²×C₈×C₁₆₃

C₂²

yes

characteristic

C₂²×C₈×C₁₆₃

C₂²

yes

characteristic

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₃

—

C₂²×Dic₁₆₃

—

C₂²×Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

Q₈

yes

—

Dic₁₆₃⋊C₄

—

C₂²×C₄×C₁₆₃

D₄

yes

characteristic

C₂²×C₄×C₁₆₃

D₄

yes

characteristic

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

D₄

yes

—

C₄²×C₁₆₃

—

C₄²×C₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

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

#41

yes

characteristic

C₈×C₁₆₃

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

#121

yes

characteristic

C₂²×C₁₆₃

C₂²⋊Q₈

yes

—

C₂²×C₁₆₃

C₂×Q₁₆

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#123

yes

characteristic

C₄×C₁₆₃

#125

yes

—

C₄×C₁₆₃

C₄⋊Q₈

yes

characteristic

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

—

C₂×C₁₆₃

#478

yes

—

C₁₆₃

#2087

yes

characteristic, complemented, Sylow

#2087

—

163

maximal, Sylow

C₂×#107

—

163

—

C₂×C₄⋊C₈

D₁₆₃

yes

characteristic

#389

—

163

—

#340

—

163

—

#340

—

163

—

C₂²×C₈

C₂×D₁₆₃

yes

characteristic

C₂²×C₈

C₂×D₁₆₃

yes

characteristic

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#107

—

326

—

C₄⋊C₈

—

#95

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

D₆₅₂

yes

characteristic

C₂²×C₄

D₆₅₂

yes

characteristic

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₆₃

yes

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

652

—

C₄⋊C₄

—

163

—

C₄²

—

C₄²

—

C₂³

C₂×D₆₅₂

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

326

—

104

C₂×C₄

—

652

—

105

C₂×C₄

—

652

—

106

C₂×C₄

—

652

—

107

C₂×C₄

—

652

—

108

C₂×C₄

—

652

—

109

C₂×C₄

Dic₆₅₂

yes

—

110

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

111

C₂×C₄

Q₈×D₁₆₃

yes

—

112

C₂×C₄

#14579698

yes

—

113

C₂×C₄

#14579699

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

#19104709

yes

characteristic

117

C₂²

#19104715

yes

—

118

C₂²

C₂×Dic₆₅₂

yes

characteristic

119

C₂²

#19104757

yes

characteristic

120

C₂²

#19104763

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

1304

—

124

C₄

—

652

—

125

C₄

#19104716

yes

characteristic

126

C₄

#19104725

yes

—

127

C₄

#19104733

yes

characteristic

128

C₂

#23781595

yes

characteristic

129

C₂

#23781906

yes

characteristic

130

C₂

#23781918

yes

characteristic

131

C₂

#23781961

yes

—

132

C₂

#23781966

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₅₂
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	C₂×C₄⋊C₈×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₁₆₃
hypocenter	C₁₆₃	#2087
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₆₅₂

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

C₂×C₄⋊C₈×C₁₆₃

C₂

socle, prime radical

C₂³×C₁₆₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₁₆₃

hypocenter

C₁₆₃

#2087

LCS

G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
66. C₁₆₃	#2087	semidirect

H ≅ G/N

type

66. C₁₆₃

#2087

semidirect

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

subgroup	index
C₂×#19104669	2
#23781563	2
#23781428	2
C₂×C₄⋊C₈×C₁₆₃	2
#2087	163

subgroup

index

C₂×C₄⋊C₈×C₁₆₃

163

Subgroups — #35216029

Conjugacy classes of subgroups

Complements

Maximal subgroups