Group #35223476 — subgroups

Subgroups — #35223476

Order 20864.

348 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 189 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	#23782222	C₂	1	1	yes	maximal, characteristic
2	#23781793	C₂	1	8	yes	maximal, complemented
3	#23781691	C₂	1	2	yes	maximal
4	#23781689	C₂	1	1	yes	maximal, characteristic
5	#23781620	C₂	1	2	yes	maximal
6	C₂×#97×C₁₆₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14579683	C₂²	1	2	yes	—
8	#19104735	C₄	1	2	yes	—
9	#19104727	C₄	1	2	yes	—
10	#19104714	C₄	1	4	yes	—
11	C₄×Dic₃₂₆	C₂²	1	4	yes	—
12	#19104694	C₂²	1	2	yes	—
13	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	4	yes	—
14	Dic₁₆₃⋊C₈	C₂²	1	8	yes	—
15	#19104645	C₂²	1	4	yes	—
16	C₈×Dic₁₆₃	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
20	#97×C₁₆₃	C₂²	1	4	yes	—
21	C₂×Dic₃₂₆	C₂×C₄	1	4	yes	—
22	#14579683	—	2	4	no	—
23	C₂×C₁₆₃⋊C₈	C₂³	1	4	yes	—
24	C₄⋊Dic₁₆₃	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₆₃	C₂×C₄	1	4	yes	—
26	Dic₁₆₃⋊C₄	C₂×C₄	1	8	yes	—
27	C₂²⋊Dic₁₆₃	C₂×C₄	1	4	yes	—
28	C₄×Dic₁₆₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₆₃	—	2	2	no	—
32	C₂×C₈×C₁₆₃	Q₈	1	4	yes	complemented
33	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
34	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
35	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₆₃	C₂³	1	2	yes	—
38	Dic₃₂₆	—	4	4	no	—
39	C₂×Dic₁₆₃	C₂²×C₄	1	8	yes	—
40	C₁₆₃⋊C₈	—	2	4	no	—
41	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₆₃	—	4	2	no	—
43	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₆₃	#41	1	2	yes	—
48	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₆₃	—	2	2	no	—
50	C₈×C₁₆₃	C₂×Q₈	1	4	yes	—
51	C₂²×C₁₆₃	—	2	2	no	—
52	C₂²×C₁₆₃	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₆₃	—	2	2	no	—
55	C₄×C₁₆₃	—	2	2	no	—
56	C₄×C₁₆₃	C₄×Q₈	1	2	yes	—
57	C₄×C₁₆₃	C₂²×Q₈	1	1	yes	characteristic
58	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
59	Dic₁₆₃	—	2	8	no	—
60	C₂×C₁₆₃	—	2	2	no	—
61	C₂×C₁₆₃	C₂×C₄×Q₈	1	1	yes	characteristic
62	C₂×C₁₆₃	#568	1	2	yes	—
63	C₁₆₃	#2961	1	1	yes	characteristic, complemented, Sylow
64	#2961	—	163	1	no	maximal, Sylow
65	#533	—	163	1	no	—
66	#446	—	163	8	no	—
67	#424	—	163	2	no	—
68	#423	—	163	1	no	—
69	#405	—	163	2	no	—
70	C₂×#97	D₁₆₃	1	1	yes	characteristic
71	C₂×M₁₆	—	163	2	no	—
72	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
74	C₄⋊Q₈	—	163	2	no	—
75	#125	—	163	2	no	—
76	C₂²⋊Q₈	—	163	4	no	—
77	C₄×Q₈	—	163	4	no	—
78	#117	—	163	2	no	—
79	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
80	C₄⋊C₈	—	163	8	no	—
81	C₄⋊C₈	—	163	4	no	—
82	#97	—	163	4	no	—
83	#97	C₂×D₁₆₃	1	4	yes	—
84	C₄×C₈	—	163	4	no	—
85	C₂×Q₈	—	163	4	no	—
86	C₂²×C₄	C₄×D₁₆₃	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
88	M₁₆	—	326	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	163	4	no	—
91	C₂×C₈	Dic₃₂₆	1	4	yes	complemented
92	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
93	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
94	C₄⋊C₄	—	163	4	no	—
95	C₄⋊C₄	—	163	8	no	—
96	C₄⋊C₄	—	163	4	no	—
97	C₂²⋊C₄	—	163	4	no	—
98	C₄²	—	163	4	no	—
99	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
100	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
101	C₄²	C₂²×D₁₆₃	1	2	yes	—
102	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
103	Q₈	—	652	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	163	8	no	—
106	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
107	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₃₂₆	1	2	yes	—
110	C₂×C₄	#14579699	1	2	yes	—
111	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	326	4	no	—
114	C₈	C₂×Dic₃₂₆	1	4	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	C₄×Dic₃₂₆	1	2	yes	—
117	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	326	8	no	—
121	C₄	C₄×Dic₃₂₆	1	2	yes	—
122	C₄	C₂²×Dic₃₂₆	1	1	yes	characteristic
123	C₄	C₂×#14579699	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×C₄×Dic₃₂₆	1	1	yes	characteristic
126	C₂	#23782431	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23782222

C₂

yes

maximal, characteristic

#23781793

C₂

yes

maximal, complemented

#23781691

C₂

yes

maximal

#23781689

C₂

yes

maximal, characteristic

#23781620

C₂

yes

maximal

C₂×#97×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

#19104735

C₄

yes

—

#19104727

C₄

yes

—

#19104714

C₄

yes

—

C₄×Dic₃₂₆

C₂²

yes

—

#19104694

C₂²

yes

—

C₄⋊(C₁₆₃⋊C₈)

C₂²

yes

—

Dic₁₆₃⋊C₈

C₂²

yes

—

#19104645

C₂²

yes

—

C₈×Dic₁₆₃

C₂²

yes

—

C₂²×C₈×C₁₆₃

C₂²

yes

characteristic

C₂²×C₈×C₁₆₃

C₂²

yes

characteristic

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

#97×C₁₆₃

C₂²

yes

—

C₂×Dic₃₂₆

C₂×C₄

yes

—

#14579683

—

C₂×C₁₆₃⋊C₈

C₂³

yes

—

C₄⋊Dic₁₆₃

C₂×C₄

yes

—

C₄⋊Dic₁₆₃

C₂×C₄

yes

—

Dic₁₆₃⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₁₆₃

C₂×C₄

yes

—

C₄×Dic₁₆₃

C₂³

yes

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

Q₈

yes

complemented

C₂×C₈×C₁₆₃

C₂³

yes

—

C₂×C₈×C₁₆₃

C₂³

yes

—

C₄²×C₁₆₃

C₂×C₄

yes

characteristic

C₄²×C₁₆₃

C₂×C₄

yes

characteristic

C₄²×C₁₆₃

C₂³

yes

—

Dic₃₂₆

—

C₂×Dic₁₆₃

C₂²×C₄

yes

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

C₂⁴

yes

characteristic

C₈×C₁₆₃

—

C₈×C₁₆₃

C₂×Q₈

yes

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₄×Q₈

yes

—

C₂²×C₁₆₃

C₂³×C₄

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄×Q₈

yes

—

C₄×C₁₆₃

C₂²×Q₈

yes

characteristic

C₄×C₁₆₃

C₂×#41

yes

characteristic

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₆₃

#568

yes

—

C₁₆₃

#2961

yes

characteristic, complemented, Sylow

#2961

—

163

maximal, Sylow

#533

—

163

—

#446

—

163

—

#424

—

163

—

#423

—

163

—

#405

—

163

—

C₂×#97

D₁₆₃

yes

characteristic

C₂×M₁₆

—

163

—

C₂²×C₈

C₂×D₁₆₃

yes

characteristic

C₂²×C₈

C₂×D₁₆₃

yes

characteristic

C₄⋊Q₈

—

163

—

#125

—

163

—

C₂²⋊Q₈

—

163

—

C₄×Q₈

—

163

—

#117

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

C₄⋊C₈

—

163

—

C₄⋊C₈

—

163

—

#97

—

163

—

#97

C₂×D₁₆₃

yes

—

C₄×C₈

—

163

—

C₂×Q₈

—

163

—

C₂²×C₄

C₄×D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

326

—

C₂×C₈

—

C₂×C₈

—

163

—

C₂×C₈

Dic₃₂₆

yes

complemented

C₂×C₈

C₂²×D₁₆₃

yes

—

C₂×C₈

C₂²×D₁₆₃

yes

—

C₄⋊C₄

—

163

—

C₄⋊C₄

—

163

—

C₄⋊C₄

—

163

—

C₂²⋊C₄

—

163

—

C₄²

—

163

—

C₄²

C₄×D₁₆₃

yes

characteristic

100

C₄²

C₄×D₁₆₃

yes

characteristic

101

C₄²

C₂²×D₁₆₃

yes

—

102

C₂³

C₂×C₄×D₁₆₃

yes

characteristic

103

Q₈

—

652

—

104

C₂×C₄

—

105

C₂×C₄

—

163

—

106

C₂×C₄

C₂×C₄×D₁₆₃

yes

—

107

C₂×C₄

C₂×C₄×D₁₆₃

yes

—

108

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

109

C₂×C₄

C₂×Dic₃₂₆

yes

—

110

C₂×C₄

#14579699

yes

—

111

C₂×C₄

C₂³×D₁₆₃

yes

characteristic

112

C₈

—

113

C₈

—

326

—

114

C₈

C₂×Dic₃₂₆

yes

—

115

C₂²

—

116

C₂²

C₄×Dic₃₂₆

yes

—

117

C₂²

C₂²×C₄×D₁₆₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

326

—

121

C₄

C₄×Dic₃₂₆

yes

—

122

C₄

C₂²×Dic₃₂₆

yes

characteristic

123

C₄

C₂×#14579699

yes

characteristic

124

C₂

—

125

C₂

C₂×C₄×Dic₃₂₆

yes

characteristic

126

C₂

#23782431

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Fitting	C₂×#97×C₁₆₃	C₂
prime radical	C₂³×C₁₆₃	C₂²×C₄
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	C₂×#97	D₁₆₃
hypocenter	C₁₆₃	#2961
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₆₃

commutator, socle

C₂²×C₁₆₃

C₂³×C₄

Fitting

C₂×#97×C₁₆₃

C₂

prime radical

C₂³×C₁₆₃

C₂²×C₄

cosocle

C₂×C₄×C₁₆₃

C₂⁴

hypercenter

C₂×#97

D₁₆₃

C₁₆₃

G ▷ C₂²×C₁₆₃ ▷ C₁₆₃

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
2. #23781793	C₂	semidirect
32. C₂×C₈×C₁₆₃	Q₈	semidirect
63. C₁₆₃	#2961	semidirect
91. C₂×C₈	Dic₃₂₆	semidirect

H ≅ G/N

type

2. #23781793

C₂

semidirect

32. C₂×C₈×C₁₆₃

Q₈

semidirect

63. C₁₆₃

#2961

semidirect

91. C₂×C₈

Dic₃₂₆

semidirect

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

subgroup	index
#23782222	2
#23781793	2
#23781691	2
#23781689	2
#23781620	2
C₂×#97×C₁₆₃	2
#2961	163

subgroup

index

C₂×#97×C₁₆₃

163

Subgroups — #35223476

Conjugacy classes of subgroups

Complements

Maximal subgroups