Group #35216090 — subgroups

Subgroups — #35216090

Order 20864.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#23781714	C₂	1	1	yes	maximal, characteristic, complemented
2	#23781692	C₂	1	2	yes	maximal, complemented
3	#23781683	C₂	1	2	yes	maximal
4	#23781633	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14579699	C₂²	1	1	yes	characteristic
7	C₂×#14579683	C₂²	1	2	yes	—
8	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
9	#19104671	—	2	2	no	—
10	C₄⋊(C₁₆₃⋊C₈)	—	2	4	no	—
11	#19104663	—	2	4	no	—
12	(C₂×D₁₆₃)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
14	#117×C₁₆₃	C₂²	1	2	yes	—
15	#14579699	—	2	2	no	—
16	#14579699	—	2	2	no	—
17	C₂×Dic₃₂₆	Q₈	1	1	yes	characteristic
18	C₂×D₆₅₂	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₆₃	—	2	1	no	—
20	C₂×C₄×D₁₆₃	—	2	1	no	—
21	#14579683	—	2	2	no	—
22	#14579683	—	2	2	no	—
23	C₂×C₁₆₃⋊C₈	—	2	1	no	—
24	C₂×C₁₆₃⋊C₈	—	2	2	no	—
25	C₂×C₁₆₃⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₆₃	—	2	2	no	—
28	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₆₃	—	2	2	no	—
30	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
31	C₄²×C₁₆₃	D₄	1	4	yes	—
32	C₂²×D₁₆₃	—	2	1	no	—
33	Dic₃₂₆	—	2	2	no	—
34	D₆₅₂	—	2	2	no	—
35	C₂²⋊D₁₆₃	—	4	2	no	—
36	C₂×Dic₁₆₃	—	2	1	no	—
37	C₄×D₁₆₃	—	4	1	no	—
38	C₄×D₁₆₃	—	4	1	no	—
39	C₁₆₃⋊C₈	—	4	1	no	—
40	C₁₆₃⋊C₈	—	4	2	no	—
41	C₂³×C₁₆₃	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₆₃	—	2	4	no	—
43	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₆₃	#41	1	2	yes	—
47	C₈×C₁₆₃	—	2	2	no	—
48	C₂²×C₁₆₃	—	4	1	no	—
49	C₂²×C₁₆₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₆₃	#123	1	2	yes	—
51	C₂×D₁₆₃	—	4	2	no	—
52	C₂×D₁₆₃	—	4	1	no	—
53	C₄×C₁₆₃	—	4	4	no	—
54	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₆₃	#123	1	1	yes	characteristic
56	C₄×C₁₆₃	#127	1	1	yes	characteristic
57	Dic₁₆₃	—	4	1	no	—
58	C₂×C₁₆₃	—	2	2	no	—
59	C₂×C₁₆₃	—	2	1	no	—
60	C₂×C₁₆₃	#399	1	1	yes	characteristic
61	D₁₆₃	—	8	1	no	—
62	C₁₆₃	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	163	1	no	maximal, Sylow
64	#430	—	163	1	no	—
65	#424	—	163	2	no	—
66	#421	—	163	2	no	—
67	#408	—	163	1	no	—
68	H(3,ℤ/4)	D₁₆₃	1	1	yes	characteristic, complemented
69	C₂×#41	—	163	1	no	—
70	C₂×M₁₆	—	163	2	no	—
71	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
72	C₂²×C₈	—	163	1	no	—
73	#117	C₂×D₁₆₃	1	2	yes	—
74	#108	—	326	2	no	—
75	C₄⋊C₈	—	326	4	no	—
76	#104	—	326	4	no	—
77	C₂²⋊C₈	—	326	2	no	—
78	#41	—	326	2	no	—
79	#41	—	326	2	no	—
80	C₂×Q₈	—	163	1	no	—
81	C₂×D₄	—	326	1	no	—
82	C₂×D₄	—	163	1	no	—
83	C₂²×C₄	—	326	1	no	—
84	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	326	2	no	—
87	M₁₆	—	326	2	no	—
88	C₂×C₈	—	326	1	no	—
89	C₂×C₈	—	326	2	no	—
90	C₂×C₈	—	326	2	no	—
91	C₂×C₈	D₆₅₂	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₆₅₂	1	4	yes	—
95	C₂³	—	326	1	no	—
96	C₂³	#14579698	1	1	yes	characteristic
97	Q₈	—	326	2	no	—
98	D₄	—	326	2	no	—
99	D₄	—	652	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	326	1	no	—
102	C₂×C₄	—	652	1	no	—
103	C₂×C₄	—	652	1	no	—
104	C₂×C₄	C₂×D₆₅₂	1	2	yes	—
105	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₆₃	1	1	yes	characteristic
107	C₂×C₄	#14579698	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	652	1	no	—
110	C₈	—	652	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	652	2	no	—
113	C₂²	—	652	1	no	—
114	C₂²	#19104715	1	1	yes	characteristic
115	C₂²	#19104719	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	652	1	no	—
118	C₄	#19104715	1	2	yes	—
119	C₄	#19104719	1	1	yes	characteristic
120	C₄	#19104732	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1304	1	no	—
124	C₂	#23781602	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23781714

C₂

yes

maximal, characteristic, complemented

#23781692

C₂

yes

maximal, complemented

#23781683

C₂

yes

maximal

#23781633

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₆₃

C₂

yes

maximal, characteristic, complemented

C₂×#14579699

C₂²

yes

characteristic

C₂×#14579683

C₂²

yes

—

C₂²×C₁₆₃⋊C₈

C₂²

yes

characteristic

#19104671

—

C₄⋊(C₁₆₃⋊C₈)

—

#19104663

—

(C₂×D₁₆₃)⋊C₈

—

C₂×M₁₆×C₁₆₃

C₂²

yes

characteristic

#117×C₁₆₃

C₂²

yes

—

#14579699

—

#14579699

—

C₂×Dic₃₂₆

Q₈

yes

characteristic

C₂×D₆₅₂

Q₈

yes

characteristic

C₂×C₂²⋊D₁₆₃

—

C₂×C₄×D₁₆₃

—

#14579683

—

#14579683

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

M₁₆×C₁₆₃

—

C₂×C₈×C₁₆₃

D₄

yes

—

C₄⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

D₄

yes

—

C₂²×D₁₆₃

—

Dic₃₂₆

—

D₆₅₂

—

C₂²⋊D₁₆₃

—

C₂×Dic₁₆₃

—

C₄×D₁₆₃

—

C₄×D₁₆₃

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

#41

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₆₃

#123

yes

—

C₂×D₁₆₃

—

C₂×D₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂²⋊Q₈

yes

—

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#399

yes

characteristic

D₁₆₃

—

C₁₆₃

#2098

yes

characteristic, complemented, Sylow

#2098

—

163

maximal, Sylow

#430

—

163

—

#424

—

163

—

#421

—

163

—

#408

—

163

—

H(3,ℤ/4)

D₁₆₃

yes

characteristic, complemented

C₂×#41

—

163

—

C₂×M₁₆

—

163

—

C₂×M₁₆

C₂×D₁₆₃

yes

characteristic

C₂²×C₈

—

163

—

#117

C₂×D₁₆₃

yes

—

#108

—

326

—

C₄⋊C₈

—

326

—

#104

—

326

—

C₂²⋊C₈

—

326

—

#41

—

326

—

#41

—

326

—

C₂×Q₈

—

163

—

C₂×D₄

—

326

—

C₂×D₄

—

163

—

C₂²×C₄

—

326

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

M₁₆

—

326

—

M₁₆

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

D₆₅₂

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₆₅₂

yes

—

C₂³

—

326

—

C₂³

#14579698

yes

characteristic

Q₈

—

326

—

D₄

—

326

—

D₄

—

652

—

100

C₂×C₄

—

101

C₂×C₄

—

326

—

102

C₂×C₄

—

652

—

103

C₂×C₄

—

652

—

104

C₂×C₄

C₂×D₆₅₂

yes

—

105

C₂×C₄

C₂×D₆₅₂

yes

characteristic

106

C₂×C₄

Q₈×D₁₆₃

yes

characteristic

107

C₂×C₄

#14579698

yes

—

108

C₈

—

109

C₈

—

652

—

110

C₈

—

652

—

111

C₂²

—

112

C₂²

—

652

—

113

C₂²

—

652

—

114

C₂²

#19104715

yes

characteristic

115

C₂²

#19104719

yes

—

116

C₄

—

117

C₄

—

652

—

118

C₄

#19104715

yes

—

119

C₄

#19104719

yes

characteristic

120

C₄

#19104732

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1304

—

124

C₂

#23781602

yes

characteristic

	subgroup	G/H
center	C₄	#19104732
commutator, cosocle	C₂²×C₄×C₁₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	H(3,ℤ/4)×C₁₆₃	C₂
socle	C₂×C₁₆₃	#399
prime radical	C₂×#14579699	C₂²
hypercenter	H(3,ℤ/4)	D₁₆₃
hypocenter	C₁₆₃	#2098
LCS	G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#19104732

commutator, cosocle

C₂²×C₄×C₁₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

H(3,ℤ/4)×C₁₆₃

C₂

C₂×C₁₆₃

C₂²

D₁₆₃

C₁₆₃

G ▷ C₂²×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #23781714	C₂	semidirect
2. #23781692	C₂	semidirect
5. H(3,ℤ/4)×C₁₆₃	C₂	semidirect
62. C₁₆₃	#2098	semidirect
68. H(3,ℤ/4)	D₁₆₃	semidirect

H ≅ G/N

type

1. #23781714

C₂

semidirect

2. #23781692

C₂

semidirect

5. H(3,ℤ/4)×C₁₆₃

C₂

semidirect

62. C₁₆₃

#2098

semidirect

68. H(3,ℤ/4)

D₁₆₃

semidirect

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

subgroup	index
#23781714	2
#23781692	2
#23781683	2
#23781633	2
H(3,ℤ/4)×C₁₆₃	2
#2098	163

subgroup

index

H(3,ℤ/4)×C₁₆₃

163

Subgroups — #35216090

Conjugacy classes of subgroups

Complements

Maximal subgroups