Group #35214417 — subgroups

Subgroups — #35214417

Order 20864.

232 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 117 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×#19104671	C₂	1	1	yes	maximal, characteristic
2	#23781621	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₆₃	C₂	1	1	yes	maximal, characteristic
4	#23781424	C₂	1	2	yes	maximal
5	#427×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579683	C₂²	1	2	yes	—
7	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
9	#19104671	C₄	1	4	yes	complemented
10	#19104646	—	2	2	no	—
11	C₄×C₁₆₃⋊C₈	—	2	2	no	—
12	C₈×Dic₁₆₃	C₄	1	4	yes	complemented
13	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
14	#117×C₁₆₃	C₂²	1	2	yes	—
15	#107×C₁₆₃	C₄	1	2	yes	complemented
16	#106×C₁₆₃	C₄	1	2	yes	complemented
17	C₂²×Dic₁₆₃	—	2	1	no	—
18	#14579683	—	2	2	no	—
19	#14579683	C₂×C₄	1	4	yes	—
20	C₂×C₁₆₃⋊C₈	—	2	2	no	—
21	C₂×C₁₆₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₁₆₃⋊C₈	D₄	1	1	yes	characteristic
23	C₂×C₁₆₃⋊C₈	D₄	1	1	yes	characteristic
24	C₂×C₁₆₃⋊C₈	Q₈	1	2	yes	—
25	C₄×Dic₁₆₃	—	2	1	no	—
26	C₄×Dic₁₆₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₆₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₆₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₄×C₁₆₃	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
34	C₄²×C₁₆₃	—	2	2	no	—
35	C₂×Dic₁₆₃	—	2	4	no	—
36	C₂×Dic₁₆₃	—	2	2	no	—
37	C₁₆₃⋊C₈	—	2	4	no	—
38	C₁₆₃⋊C₈	C₄⋊C₄	1	2	yes	—
39	C₁₆₃⋊C₈	C₄⋊C₄	1	2	yes	—
40	C₂³×C₁₆₃	#41	1	1	yes	characteristic
41	C₂×C₄×C₁₆₃	—	2	4	no	—
42	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
48	C₈×C₁₆₃	C₄²	1	2	yes	—
49	C₈×C₁₆₃	C₄²	1	2	yes	—
50	C₂²×C₁₆₃	—	2	2	no	—
51	C₂²×C₁₆₃	#117	1	1	yes	characteristic
52	C₂²×C₁₆₃	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₁₆₃	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₆₃	—	4	4	no	—
55	C₄×C₁₆₃	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₁₆₃	C₄×D₄	1	1	yes	characteristic
58	C₄×C₁₆₃	C₄×Q₈	1	1	yes	characteristic
59	Dic₁₆₃	—	2	4	no	—
60	C₂×C₁₆₃	—	2	1	no	—
61	C₂×C₁₆₃	—	2	1	no	—
62	C₂×C₁₆₃	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₁₆₃	#443	1	2	yes	—
64	C₁₆₃	#1772	1	1	yes	characteristic, complemented, Sylow
65	#1772	—	163	1	no	maximal, Sylow
66	C₂×#108	—	163	1	no	—
67	#427	D₁₆₃	1	1	yes	characteristic
68	#405	—	163	2	no	—
69	C₂×C₄×C₈	—	163	1	no	—
70	#339	—	163	2	no	—
71	C₂×M₁₆	—	163	2	no	—
72	C₂²×C₈	—	163	1	no	—
73	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
74	#117	C₂×D₁₆₃	1	2	yes	—
75	C₂×C₄²	—	163	1	no	—
76	#108	—	163	4	no	—
77	#107	Dic₁₆₃	1	2	yes	complemented
78	#106	Dic₁₆₃	1	2	yes	complemented
79	#97	—	326	2	no	—
80	C₄×C₈	—	326	2	no	—
81	C₄×C₈	—	163	4	no	—
82	C₂²×C₄	—	326	1	no	—
83	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
84	M₁₆	—	326	2	no	—
85	M₁₆	—	163	4	no	—
86	C₂×C₈	—	326	2	no	—
87	C₂×C₈	—	163	2	no	—
88	C₂×C₈	—	163	1	no	—
89	C₂×C₈	—	163	1	no	—
90	C₂×C₈	—	163	2	no	—
91	C₂×C₈	C₄×D₁₆₃	1	2	yes	—
92	C₂×C₈	C₄×D₁₆₃	1	2	yes	—
93	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
95	C₄⋊C₄	C₂×Dic₁₆₃	1	4	yes	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	326	1	no	—
99	C₄²	—	163	2	no	—
100	C₂³	#14579698	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	326	4	no	—
103	C₂×C₄	—	326	2	no	—
104	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₁₆₃	1	1	yes	characteristic
107	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₁₆₃	1	1	yes	characteristic
109	C₂×C₄	#14579700	1	1	yes	characteristic
110	C₈	—	326	4	no	—
111	C₈	—	163	2	no	—
112	C₈	—	163	2	no	—
113	C₈	C₄×Dic₁₆₃	1	2	yes	—
114	C₈	C₄×Dic₁₆₃	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#19104693	1	1	yes	characteristic
117	C₂²	D₄×Dic₁₆₃	1	1	yes	characteristic
118	C₂²	#19104700	1	1	yes	characteristic
119	C₄	—	4	4	no	—
120	C₄	—	326	4	no	—
121	C₄	C₂×C₄×Dic₁₆₃	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₁₆₃	1	1	yes	characteristic
123	C₄	#19104697	1	1	yes	characteristic
124	C₄	Q₈×Dic₁₆₃	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₁₆₃	1	1	yes	characteristic
128	C₂	#23781782	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104671

C₂

yes

maximal, characteristic

#23781621

C₂

yes

maximal

C₂×C₈×Dic₁₆₃

C₂

yes

maximal, characteristic

#23781424

C₂

yes

maximal

#427×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

C₂²×C₁₆₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₆₃

C₂²

yes

characteristic

#19104671

C₄

yes

complemented

#19104646

—

C₄×C₁₆₃⋊C₈

—

C₈×Dic₁₆₃

C₄

yes

complemented

C₂²×C₈×C₁₆₃

C₂²

yes

characteristic

#117×C₁₆₃

C₂²

yes

—

#107×C₁₆₃

C₄

yes

complemented

#106×C₁₆₃

C₄

yes

complemented

C₂²×Dic₁₆₃

—

#14579683

—

#14579683

C₂×C₄

yes

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₆₃⋊C₈

D₄

yes

characteristic

C₂×C₁₆₃⋊C₈

D₄

yes

characteristic

C₂×C₁₆₃⋊C₈

Q₈

yes

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₈×C₁₆₃

C₂×C₄

yes

—

C₂×C₈×C₁₆₃

C₂×C₄

yes

—

C₂×C₈×C₁₆₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₆₃

C₂×C₄

yes

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

C₄⋊C₄

yes

—

C₁₆₃⋊C₈

C₄⋊C₄

yes

—

C₂³×C₁₆₃

#41

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

characteristic

C₈×C₁₆₃

C₄²

yes

—

C₈×C₁₆₃

C₄²

yes

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#117

yes

characteristic

C₂²×C₁₆₃

C₄×D₄

yes

characteristic

C₂²×C₁₆₃

C₄×Q₈

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂×C₄²

yes

characteristic

C₄×C₁₆₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₆₃

C₄×D₄

yes

characteristic

C₄×C₁₆₃

C₄×Q₈

yes

characteristic

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₆₃

#443

yes

—

C₁₆₃

#1772

yes

characteristic, complemented, Sylow

#1772

—

163

maximal, Sylow

C₂×#108

—

163

—

#427

D₁₆₃

yes

characteristic

#405

—

163

—

C₂×C₄×C₈

—

163

—

#339

—

163

—

C₂×M₁₆

—

163

—

C₂²×C₈

—

163

—

C₂²×C₈

C₂×D₁₆₃

yes

characteristic

#117

C₂×D₁₆₃

yes

—

C₂×C₄²

—

163

—

#108

—

163

—

#107

Dic₁₆₃

yes

complemented

#106

Dic₁₆₃

yes

complemented

#97

—

326

—

C₄×C₈

—

326

—

C₄×C₈

—

163

—

C₂²×C₄

—

326

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

326

—

M₁₆

—

163

—

C₂×C₈

—

326

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

C₂×C₈

C₄×D₁₆₃

yes

—

C₂×C₈

C₄×D₁₆₃

yes

—

C₂×C₈

C₂×Dic₁₆₃

yes

characteristic

C₂×C₈

C₂×Dic₁₆₃

yes

characteristic

C₄⋊C₄

C₂×Dic₁₆₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

326

—

C₄²

—

163

—

100

C₂³

#14579698

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

326

—

103

C₂×C₄

—

326

—

104

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

105

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

106

C₂×C₄

C₂²×Dic₁₆₃

yes

characteristic

107

C₂×C₄

D₄×D₁₆₃

yes

characteristic

108

C₂×C₄

Q₈×D₁₆₃

yes

characteristic

109

C₂×C₄

#14579700

yes

characteristic

110

C₈

—

326

—

111

C₈

—

163

—

112

C₈

—

163

—

113

C₈

C₄×Dic₁₆₃

yes

—

114

C₈

C₄×Dic₁₆₃

yes

—

115

C₂²

—

116

C₂²

#19104693

yes

characteristic

117

C₂²

D₄×Dic₁₆₃

yes

characteristic

118

C₂²

#19104700

yes

characteristic

119

C₄

—

120

C₄

—

326

—

121

C₄

C₂×C₄×Dic₁₆₃

yes

characteristic

122

C₄

C₄⋊C₄×D₁₆₃

yes

characteristic

123

C₄

#19104697

yes

characteristic

124

C₄

Q₈×Dic₁₆₃

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₁₆₃

yes

characteristic

128

C₂

#23781782

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₆₃
commutator	C₄×C₁₆₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#427×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₄×D₄
prime radical	C₂³×C₁₆₃	#41
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	#427	D₁₆₃
hypocenter	C₁₆₃	#1772
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₆₃

commutator

C₄×C₁₆₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#427×C₁₆₃

C₂

socle

C₂²×C₁₆₃

C₄×D₄

prime radical

C₂³×C₁₆₃

#41

cosocle

C₂²×C₄×C₁₆₃

C₂³

D₁₆₃

C₁₆₃

G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
9. #19104671	C₄	semidirect
12. C₈×Dic₁₆₃	C₄	semidirect
15. #107×C₁₆₃	C₄	semidirect
16. #106×C₁₆₃	C₄	semidirect
64. C₁₆₃	#1772	semidirect
77. #107	Dic₁₆₃	semidirect
78. #106	Dic₁₆₃	semidirect

H ≅ G/N

type

9. #19104671

C₄

semidirect

12. C₈×Dic₁₆₃

C₄

semidirect

15. #107×C₁₆₃

C₄

semidirect

16. #106×C₁₆₃

C₄

semidirect

64. C₁₆₃

#1772

semidirect

77. #107

Dic₁₆₃

semidirect

78. #106

Dic₁₆₃

semidirect

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

subgroup	index
C₂×#19104671	2
#23781621	2
C₂×C₈×Dic₁₆₃	2
#23781424	2
#427×C₁₆₃	2
#1772	163

subgroup

index

C₂×C₈×Dic₁₆₃

#427×C₁₆₃

163

Subgroups — #35214417

Conjugacy classes of subgroups

Complements

Maximal subgroups