Group #35214582 — subgroups

Subgroups — #35214582

Order 20864.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	#23781703	C₂	1	2	yes	maximal, complemented
2	#23781639	C₂	1	2	yes	maximal
3	#23781422	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	#19104694	C₂²	1	4	yes	—
6	#19104669	—	2	2	no	—
7	#19104667	—	2	2	no	—
8	#19104650	—	2	4	no	—
9	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
10	#131×C₁₆₃	—	2	2	no	—
11	#131×C₁₆₃	—	2	2	no	—
12	C₂×M₁₆×C₁₆₃	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₆₃	C₄	1	2	yes	complemented
15	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₆₃	—	2	4	no	—
17	C₂²⋊Dic₁₆₃	—	2	4	no	—
18	C₄×Dic₁₆₃	—	2	4	no	—
19	#41×C₁₆₃	—	2	2	no	—
20	#41×C₁₆₃	—	2	2	no	—
21	C₂×Q₈×C₁₆₃	Q₈	1	1	yes	characteristic
22	C₂×D₄×C₁₆₃	D₄	1	2	yes	—
23	C₂×D₄×C₁₆₃	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₆₃	—	2	2	no	—
27	M₁₆×C₁₆₃	—	2	4	no	—
28	C₂×C₈×C₁₆₃	—	2	2	no	—
29	C₂×C₈×C₁₆₃	—	2	2	no	—
30	C₂×C₈×C₁₆₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
34	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
35	C₂×Dic₁₆₃	—	4	4	no	—
36	C₂³×C₁₆₃	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₁₆₃	—	2	2	no	—
39	D₄×C₁₆₃	—	2	4	no	—
40	D₄×C₁₆₃	—	2	2	no	—
41	C₂×C₄×C₁₆₃	—	2	2	no	—
42	C₂×C₄×C₁₆₃	—	2	2	no	—
43	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₆₃	#41	1	2	yes	—
48	C₈×C₁₆₃	—	2	2	no	—
49	C₈×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₈×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₈×C₁₆₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₁₆₃	—	2	2	no	—
53	C₂²×C₁₆₃	—	4	1	no	—
54	C₂²×C₁₆₃	—	4	2	no	—
55	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₆₃	#121	1	2	yes	—
57	C₄×C₁₆₃	—	2	2	no	—
58	C₄×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₁₆₃	#117	1	1	yes	characteristic
60	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
61	Dic₁₆₃	—	8	4	no	—
62	C₂×C₁₆₃	—	2	2	no	—
63	C₂×C₁₆₃	—	2	1	no	—
64	C₂×C₁₆₃	—	4	2	no	—
65	C₂×C₁₆₃	#380	1	1	yes	characteristic
66	C₁₆₃	#1812	1	1	yes	characteristic, complemented, Sylow
67	#1812	—	163	1	no	maximal, Sylow
68	C₂×#131	D₁₆₃	1	1	yes	characteristic
69	#427	—	163	2	no	—
70	#410	—	163	2	no	—
71	H(3,ℤ/4)	—	163	2	no	—
72	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
73	#131	—	2	2	no	—
74	#131	—	2	2	no	—
75	C₂×M₁₆	Dic₁₆₃	1	2	yes	complemented
76	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
77	C₂²×C₈	Dic₁₆₃	1	2	yes	complemented
78	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
79	#117	—	163	4	no	—
80	#107	—	326	2	no	—
81	#106	—	326	2	no	—
82	#99	—	326	4	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	Dic₃₂₆	1	1	yes	characteristic
86	C₂×D₄	D₆₅₂	1	2	yes	—
87	C₂×D₄	Dic₃₂₆	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₆₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	C₂×Dic₁₆₃	1	2	yes	—
95	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
98	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
99	C₄⋊C₄	—	326	4	no	—
100	C₂²⋊C₄	—	326	4	no	—
101	C₄²	—	326	4	no	—
102	C₂³	C₄⋊Dic₁₆₃	1	2	yes	—
103	C₂³	C₂×D₆₅₂	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	652	4	no	—
110	C₂×C₄	C₄⋊Dic₁₆₃	1	2	yes	—
111	C₂×C₄	C₂²×Dic₁₆₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
113	C₂×C₄	C₂×Dic₃₂₆	1	1	yes	characteristic
114	C₂×C₄	#14579699	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₈	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
117	C₈	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
118	C₈	C₂²⋊Dic₁₆₃	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	2	no	—
122	C₂²	C₂×C₄⋊Dic₁₆₃	1	1	yes	characteristic
123	C₂²	#19104709	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	—	1304	4	no	—
126	C₄	C₂×C₂²⋊Dic₁₆₃	1	1	yes	characteristic
127	C₄	#19104694	1	1	yes	characteristic
128	C₄	#19104714	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	2	1	no	—
131	C₂	—	4	2	no	—
132	C₂	#23781525	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23781703

C₂

yes

maximal, complemented

#23781639

C₂

yes

maximal

#23781422

C₂

yes

maximal, complemented

C₂×#131×C₁₆₃

C₂

yes

maximal, characteristic

#19104694

C₂²

yes

—

#19104669

—

#19104667

—

#19104650

—

C₂×#41×C₁₆₃

C₂²

yes

characteristic

#131×C₁₆₃

—

#131×C₁₆₃

—

C₂×M₁₆×C₁₆₃

C₄

yes

complemented

C₂×M₁₆×C₁₆₃

C₂²

yes

characteristic

C₂²×C₈×C₁₆₃

C₄

yes

complemented

C₂²×C₈×C₁₆₃

C₂²

yes

characteristic

C₄⋊Dic₁₆₃

—

C₂²⋊Dic₁₆₃

—

C₄×Dic₁₆₃

—

#41×C₁₆₃

—

#41×C₁₆₃

—

C₂×Q₈×C₁₆₃

Q₈

yes

characteristic

C₂×D₄×C₁₆₃

D₄

yes

—

C₂×D₄×C₁₆₃

Q₈

yes

characteristic

C₂²×C₄×C₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

M₁₆×C₁₆₃

—

M₁₆×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

C₂×C₄

yes

—

C₂×C₈×C₁₆₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₃

D₄

yes

—

C₂×C₈×C₁₆₃

D₄

yes

—

C₂×Dic₁₆₃

—

C₂³×C₁₆₃

C₄⋊C₄

yes

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

Q₈×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₈×C₁₆₃

—

C₈×C₁₆₃

C₂²⋊C₄

yes

characteristic

C₈×C₁₆₃

C₂²⋊C₄

yes

characteristic

C₈×C₁₆₃

C₂²⋊C₄

yes

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₆₃

#121

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₆₃

#117

yes

characteristic

C₄×C₁₆₃

C₂²⋊Q₈

yes

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#380

yes

characteristic

C₁₆₃

#1812

yes

characteristic, complemented, Sylow

#1812

—

163

maximal, Sylow

C₂×#131

D₁₆₃

yes

characteristic

#427

—

163

—

#410

—

163

—

H(3,ℤ/4)

—

163

—

C₂×#41

C₂×D₁₆₃

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₁₆₃

yes

complemented

C₂×M₁₆

C₂×D₁₆₃

yes

characteristic

C₂²×C₈

Dic₁₆₃

yes

complemented

C₂²×C₈

C₂×D₁₆₃

yes

characteristic

#117

—

163

—

#107

—

326

—

#106

—

326

—

#99

—

326

—

#41

—

#41

—

C₂×Q₈

Dic₃₂₆

yes

characteristic

C₂×D₄

D₆₅₂

yes

—

C₂×D₄

Dic₃₂₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₆₃

yes

—

C₂×C₈

C₂×Dic₁₆₃

yes

characteristic

C₂×C₈

C₂×Dic₁₆₃

yes

characteristic

C₂×C₈

C₂²⋊D₁₆₃

yes

—

C₂×C₈

C₂²⋊D₁₆₃

yes

—

C₄⋊C₄

—

326

—

100

C₂²⋊C₄

—

326

—

101

C₄²

—

326

—

102

C₂³

C₄⋊Dic₁₆₃

yes

—

103

C₂³

C₂×D₆₅₂

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

652

—

110

C₂×C₄

C₄⋊Dic₁₆₃

yes

—

111

C₂×C₄

C₂²×Dic₁₆₃

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

113

C₂×C₄

C₂×Dic₃₂₆

yes

characteristic

114

C₂×C₄

#14579699

yes

—

115

C₈

—

116

C₈

C₂²⋊Dic₁₆₃

yes

characteristic

117

C₈

C₂²⋊Dic₁₆₃

yes

characteristic

118

C₈

C₂²⋊Dic₁₆₃

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₄⋊Dic₁₆₃

yes

characteristic

123

C₂²

#19104709

yes

—

124

C₄

—

125

C₄

—

1304

—

126

C₄

C₂×C₂²⋊Dic₁₆₃

yes

characteristic

127

C₄

#19104694

yes

characteristic

128

C₄

#19104714

yes

—

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

#23781525

yes

characteristic

	subgroup	G/H
center	C₄	#19104694
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	C₂×#131×C₁₆₃	C₂
socle	C₂×C₁₆₃	#380
prime radical	C₂×#41×C₁₆₃	C₂²
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₂×#131	D₁₆₃
hypocenter	C₁₆₃	#1812
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#19104694

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

C₂×#131×C₁₆₃

C₂

socle

C₂×C₁₆₃

#380

prime radical

C₂×#41×C₁₆₃

C₂²

cosocle

C₂²×C₄×C₁₆₃

C₂³

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. #23781703	C₂	semidirect
3. #23781422	C₂	semidirect
12. C₂×M₁₆×C₁₆₃	C₄	semidirect
14. C₂²×C₈×C₁₆₃	C₄	semidirect
66. C₁₆₃	#1812	semidirect
75. C₂×M₁₆	Dic₁₆₃	semidirect
77. C₂²×C₈	Dic₁₆₃	semidirect

H ≅ G/N

type

1. #23781703

C₂

semidirect

3. #23781422

C₂

semidirect

12. C₂×M₁₆×C₁₆₃

C₄

semidirect

14. C₂²×C₈×C₁₆₃

C₄

semidirect

66. C₁₆₃

#1812

semidirect

75. C₂×M₁₆

Dic₁₆₃

semidirect

77. C₂²×C₈

Dic₁₆₃

semidirect

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

subgroup	index
#23781703	2
#23781639	2
#23781422	2
C₂×#131×C₁₆₃	2
#1812	163

subgroup

index

C₂×#131×C₁₆₃

163

Subgroups — #35214582

Conjugacy classes of subgroups

Complements

Maximal subgroups