Group #35215285 — subgroups

Subgroups — #35215285

Order 20864.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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	#23781692	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	—	2	2	no	—
10	C₄⋊(C₁₆₃⋊C₈)	—	2	4	no	—
11	C₈×Dic₁₆₃	—	2	2	no	—
12	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
13	#117×C₁₆₃	C₂²	1	2	yes	—
14	#107×C₁₆₃	C₄	1	2	yes	complemented
15	#106×C₁₆₃	C₄	1	2	yes	complemented
16	C₂²×Dic₁₆₃	—	2	1	no	—
17	#14579683	—	2	2	no	—
18	#14579683	—	2	2	no	—
19	C₂×C₁₆₃⋊C₈	—	2	1	no	—
20	C₂×C₁₆₃⋊C₈	—	2	2	no	—
21	C₂×C₁₆₃⋊C₈	—	2	1	no	—
22	C₂×C₁₆₃⋊C₈	D₄	1	2	yes	—
23	C₄×Dic₁₆₃	—	2	1	no	—
24	C₄×Dic₁₆₃	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
29	C₂×C₈×C₁₆₃	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₆₃	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
32	C₄²×C₁₆₃	—	2	2	no	—
33	C₂×Dic₁₆₃	—	2	4	no	—
34	C₂×Dic₁₆₃	—	2	2	no	—
35	C₁₆₃⋊C₈	—	2	2	no	—
36	C₁₆₃⋊C₈	—	4	2	no	—
37	C₂³×C₁₆₃	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₆₃	—	2	4	no	—
39	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
45	C₈×C₁₆₃	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₆₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₆₃	—	2	2	no	—
48	C₂²×C₁₆₃	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₆₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₆₃	#125	1	1	yes	characteristic
51	C₄×C₁₆₃	—	4	4	no	—
52	C₄×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₆₃	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₆₃	#121	1	1	yes	characteristic
55	C₄×C₁₆₃	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₆₃	—	2	4	no	—
57	C₂×C₁₆₃	—	2	1	no	—
58	C₂×C₁₆₃	—	2	1	no	—
59	C₂×C₁₆₃	#389	1	1	yes	characteristic
60	C₂×C₁₆₃	#443	1	2	yes	—
61	C₁₆₃	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	163	1	no	maximal, Sylow
63	C₂×#108	—	163	1	no	—
64	#427	D₁₆₃	1	1	yes	characteristic
65	#424	—	163	2	no	—
66	C₂×C₄×C₈	—	163	1	no	—
67	#339	—	163	2	no	—
68	C₂×M₁₆	—	163	2	no	—
69	C₂²×C₈	—	163	1	no	—
70	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
71	#117	C₂×D₁₆₃	1	2	yes	—
72	C₂×C₄²	—	163	1	no	—
73	#108	—	326	2	no	—
74	#107	Dic₁₆₃	1	2	yes	complemented
75	#106	Dic₁₆₃	1	2	yes	complemented
76	C₄⋊C₈	—	326	4	no	—
77	C₄×C₈	—	326	2	no	—
78	C₂²×C₄	—	326	1	no	—
79	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
80	M₁₆	—	326	2	no	—
81	M₁₆	—	326	2	no	—
82	C₂×C₈	—	326	1	no	—
83	C₂×C₈	—	326	1	no	—
84	C₂×C₈	—	326	2	no	—
85	C₂×C₈	—	163	2	no	—
86	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
88	C₂×C₈	D₆₅₂	1	2	yes	—
89	C₂×C₈	Dic₃₂₆	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₁₆₃	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	326	1	no	—
94	C₄²	—	163	2	no	—
95	C₂³	#14579698	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	326	4	no	—
98	C₂×C₄	—	326	2	no	—
99	C₂×C₄	C₂²×Dic₁₆₃	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
101	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₆₃	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₃₂₆	1	1	yes	characteristic
104	C₂×C₄	#14579700	1	1	yes	characteristic
105	C₈	—	326	2	no	—
106	C₈	—	652	2	no	—
107	C₈	C₄⋊Dic₁₆₃	1	2	yes	—
108	C₈	C₄⋊Dic₁₆₃	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₁₆₃	1	1	yes	characteristic
111	C₂²	#19104715	1	1	yes	characteristic
112	C₂²	#19104726	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	326	4	no	—
115	C₄	C₂×C₄⋊Dic₁₆₃	1	1	yes	characteristic
116	C₄	Q₈×Dic₁₆₃	1	1	yes	characteristic
117	C₄	#19104710	1	1	yes	characteristic
118	C₄	#19104734	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#23781564	1	1	yes	characteristic
122	C₂	#23781782	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104671

C₂

yes

maximal, characteristic

#23781692

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⋊(C₁₆₃⋊C₈)

—

C₈×Dic₁₆₃

—

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×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

D₄

yes

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

Q₈

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₈×C₁₆₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₃

D₄

yes

—

C₂×C₈×C₁₆₃

Q₈

yes

—

C₄⋊C₄×C₁₆₃

C₂×C₄

yes

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

#41

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

characteristic

C₈×C₁₆₃

C₄⋊C₄

yes

—

C₈×C₁₆₃

C₄⋊C₄

yes

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₄×D₄

yes

characteristic

C₂²×C₁₆₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₆₃

#125

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₆₃

C₄×Q₈

yes

characteristic

C₄×C₁₆₃

#121

yes

characteristic

C₄×C₁₆₃

C₄⋊Q₈

yes

characteristic

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#389

yes

characteristic

C₂×C₁₆₃

#443

yes

—

C₁₆₃

#1945

yes

characteristic, complemented, Sylow

#1945

—

163

maximal, Sylow

C₂×#108

—

163

—

#427

D₁₆₃

yes

characteristic

#424

—

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

—

326

—

#107

Dic₁₆₃

yes

complemented

#106

Dic₁₆₃

yes

complemented

C₄⋊C₈

—

326

—

C₄×C₈

—

326

—

C₂²×C₄

—

326

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

326

—

M₁₆

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

163

—

C₂×C₈

C₂×Dic₁₆₃

yes

characteristic

C₂×C₈

C₂×Dic₁₆₃

yes

characteristic

C₂×C₈

D₆₅₂

yes

—

C₂×C₈

Dic₃₂₆

yes

—

C₄⋊C₄

C₂×Dic₁₆₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

326

—

C₄²

—

163

—

C₂³

#14579698

yes

characteristic

C₂×C₄

—

C₂×C₄

—

326

—

C₂×C₄

—

326

—

C₂×C₄

C₂²×Dic₁₆₃

yes

characteristic

100

C₂×C₄

D₄×D₁₆₃

yes

characteristic

101

C₂×C₄

C₂×D₆₅₂

yes

characteristic

102

C₂×C₄

Q₈×D₁₆₃

yes

characteristic

103

C₂×C₄

C₂×Dic₃₂₆

yes

characteristic

104

C₂×C₄

#14579700

yes

characteristic

105

C₈

—

326

—

106

C₈

—

652

—

107

C₈

C₄⋊Dic₁₆₃

yes

—

108

C₈

C₄⋊Dic₁₆₃

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₁₆₃

yes

characteristic

111

C₂²

#19104715

yes

characteristic

112

C₂²

#19104726

yes

characteristic

113

C₄

—

114

C₄

—

326

—

115

C₄

C₂×C₄⋊Dic₁₆₃

yes

characteristic

116

C₄

Q₈×Dic₁₆₃

yes

characteristic

117

C₄

#19104710

yes

characteristic

118

C₄

#19104734

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#23781564

yes

characteristic

122

C₂

#23781782

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₆₃
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
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	#1945
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₆₃

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

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 ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₁₆₃	C₄	semidirect
15. #106×C₁₆₃	C₄	semidirect
61. C₁₆₃	#1945	semidirect
74. #107	Dic₁₆₃	semidirect
75. #106	Dic₁₆₃	semidirect

H ≅ G/N

type

14. #107×C₁₆₃

C₄

semidirect

15. #106×C₁₆₃

C₄

semidirect

61. C₁₆₃

#1945

semidirect

74. #107

Dic₁₆₃

semidirect

75. #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
#23781692	2
C₂×C₈×Dic₁₆₃	2
#23781424	2
#427×C₁₆₃	2
#1945	163

subgroup

index

C₂×C₈×Dic₁₆₃

#427×C₁₆₃

163

Subgroups — #35215285

Conjugacy classes of subgroups

Complements

Maximal subgroups