Group #35226340 — subgroups

Subgroups — #35226340

Order 20864.

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

normal subgroups only

This group has 444 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	#23782478	C₂	1	4	yes	maximal, complemented
2	#23781937	C₂	1	4	yes	maximal, complemented
3	#23781847	C₂	1	4	yes	maximal, complemented
4	#23781783	C₂	1	2	yes	maximal, complemented
5	#494×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19104783	C₂²	1	4	yes	—
7	#19104767	—	2	8	no	—
8	#19104766	C₂²	1	8	yes	—
9	#19104758	—	2	4	no	—
10	#19104757	C₂²	1	2	yes	—
11	Q₁₆×D₁₆₃	—	2	4	no	—
12	#19104741	C₂²	1	4	yes	—
13	#19104671	C₂²	1	2	yes	—
14	#19104663	C₂²	1	4	yes	—
15	#19104654	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₆₃	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₆₃	C₂²	1	2	yes	—
18	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
19	#14579700	—	2	8	no	—
20	#14579699	—	4	4	no	—
21	#14579699	C₂³	1	4	yes	—
22	Q₈×D₁₆₃	—	2	8	no	—
23	Dic₆₅₂	D₄	1	2	yes	—
24	#14579689	—	2	8	no	—
25	#14579688	D₄	1	4	yes	—
26	#14579686	—	2	8	no	—
27	D₁₃₀₄	D₄	1	2	yes	—
28	#14579683	C₂³	1	4	yes	—
29	#14579682	—	2	4	no	—
30	C₈×D₁₆₃	—	2	4	no	—
31	C₂×Q₈×C₁₆₃	C₂³	1	4	yes	—
32	Q₁₆×C₁₆₃	—	2	4	no	—
33	Q₁₆×C₁₆₃	—	2	4	no	—
34	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₆₃	—	4	2	no	—
36	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
37	Dic₃₂₆	—	4	4	no	—
38	Dic₃₂₆	C₂×D₄	1	4	yes	—
39	D₆₅₂	—	4	4	no	—
40	D₆₅₂	C₂×D₄	1	4	yes	—
41	C₂²⋊D₁₆₃	—	2	4	no	—
42	C₄×D₁₆₃	—	2	4	no	—
43	C₄×D₁₆₃	—	4	8	no	—
44	C₁₆₃⋊C₈	—	2	4	no	—
45	Q₈×C₁₆₃	—	2	8	no	—
46	C₂×C₄×C₁₆₃	—	2	1	no	—
47	C₂×C₄×C₁₆₃	—	2	4	no	—
48	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₆₃	C₂×D₄	1	2	yes	—
50	C₈×C₁₆₃	C₂×D₄	1	2	yes	—
51	C₂²×C₁₆₃	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₆₃	—	2	4	no	—
53	C₄×C₁₆₃	—	2	2	no	—
54	C₄×C₁₆₃	—	4	4	no	—
55	C₄×C₁₆₃	C₂²×D₄	1	2	yes	—
56	Dic₁₆₃	—	2	4	no	—
57	C₂×C₁₆₃	—	2	1	no	—
58	C₂×C₁₆₃	D₄²	1	1	yes	characteristic
59	D₁₆₃	—	4	4	no	—
60	C₁₆₃	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	163	1	no	maximal, Sylow
62	#578	—	163	4	no	—
63	#494	D₁₆₃	1	1	yes	characteristic, complemented
64	#473	—	163	4	no	—
65	#456	—	163	4	no	—
66	#443	—	163	2	no	—
67	2₋¹⁺⁴	—	163	4	no	—
68	#137	—	326	8	no	—
69	#137	—	163	8	no	—
70	#135	—	326	4	no	—
71	#135	—	163	2	no	—
72	C₂×Q₁₆	—	326	4	no	—
73	C₂×Q₁₆	C₂×D₁₆₃	1	4	yes	—
74	#131	—	163	4	no	—
75	C₄⋊Q₈	C₂×D₁₆₃	1	2	yes	—
76	#108	—	163	2	no	—
77	#104	—	163	4	no	—
78	#101	—	163	4	no	—
79	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
80	#41	—	326	8	no	—
81	#41	—	652	4	no	—
82	#41	—	163	4	no	—
83	C₂×Q₈	—	326	8	no	—
84	C₂×Q₈	C₂²×D₁₆₃	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	326	8	no	—
88	Q₁₆	—	163	2	no	—
89	SD₁₆	—	326	8	no	—
90	SD₁₆	—	163	4	no	—
91	D₈	—	163	2	no	—
92	M₁₆	—	326	4	no	—
93	M₁₆	—	163	4	no	—
94	C₂×C₈	—	326	4	no	—
95	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	652	4	no	—
100	Q₈	—	163	4	no	—
101	D₄	—	326	4	no	—
102	D₄	—	652	4	no	—
103	D₄	—	163	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	326	4	no	—
107	C₂×C₄	—	652	8	no	—
108	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
109	C₈	—	326	4	no	—
110	C₈	D₄×D₁₆₃	1	2	yes	—
111	C₈	D₄×D₁₆₃	1	2	yes	—
112	C₂²	—	326	4	no	—
113	C₂²	#19104780	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	326	4	no	—
117	C₄	C₂×D₄×D₁₆₃	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	652	4	no	—
120	C₂	#23782300	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23782478

C₂

yes

maximal, complemented

#23781937

C₂

yes

maximal, complemented

#23781847

C₂

yes

maximal, complemented

#23781783

C₂

yes

maximal, complemented

#494×C₁₆₃

C₂

yes

maximal, characteristic, complemented

#19104783

C₂²

yes

—

#19104767

—

#19104766

C₂²

yes

—

#19104758

—

#19104757

C₂²

yes

—

Q₁₆×D₁₆₃

—

#19104741

C₂²

yes

—

#19104671

C₂²

yes

—

#19104663

C₂²

yes

—

#19104654

C₂²

yes

—

C₂×Q₁₆×C₁₆₃

C₂²

yes

—

C₄⋊Q₈×C₁₆₃

C₂²

yes

—

C₄×C₈×C₁₆₃

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₆₃

—

Dic₆₅₂

D₄

yes

—

#14579689

—

#14579688

D₄

yes

—

#14579686

—

D₁₃₀₄

D₄

yes

—

#14579683

C₂³

yes

—

#14579682

—

C₈×D₁₆₃

—

C₂×Q₈×C₁₆₃

C₂³

yes

—

Q₁₆×C₁₆₃

—

Q₁₆×C₁₆₃

—

C₂×C₈×C₁₆₃

C₂³

yes

—

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂³

yes

characteristic

Dic₃₂₆

—

Dic₃₂₆

C₂×D₄

yes

—

D₆₅₂

—

D₆₅₂

C₂×D₄

yes

—

C₂²⋊D₁₆₃

—

C₄×D₁₆₃

—

C₄×D₁₆₃

—

C₁₆₃⋊C₈

—

Q₈×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂⁴

yes

characteristic

C₈×C₁₆₃

C₂×D₄

yes

—

C₈×C₁₆₃

C₂×D₄

yes

—

C₂²×C₁₆₃

H(5,𝔽₂)

yes

characteristic

C₂×D₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂²×D₄

yes

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

D₄²

yes

characteristic

D₁₆₃

—

C₁₆₃

#3293

yes

characteristic, complemented, Sylow

#3293

—

163

maximal, Sylow

#578

—

163

—

#494

D₁₆₃

yes

characteristic, complemented

#473

—

163

—

#456

—

163

—

#443

—

163

—

2₋¹⁺⁴

—

163

—

#137

—

326

—

#137

—

163

—

#135

—

326

—

#135

—

163

—

C₂×Q₁₆

—

326

—

C₂×Q₁₆

C₂×D₁₆₃

yes

—

#131

—

163

—

C₄⋊Q₈

C₂×D₁₆₃

yes

—

#108

—

163

—

#104

—

163

—

#101

—

163

—

C₄×C₈

C₂×D₁₆₃

yes

characteristic

#41

—

326

—

#41

—

652

—

#41

—

163

—

C₂×Q₈

—

326

—

C₂×Q₈

C₂²×D₁₆₃

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

326

—

Q₁₆

—

163

—

SD₁₆

—

326

—

SD₁₆

—

163

—

D₈

—

163

—

M₁₆

—

326

—

M₁₆

—

163

—

C₂×C₈

—

326

—

C₂×C₈

C₂²×D₁₆₃

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₆₃

yes

characteristic

Q₈

—

Q₈

—

652

—

100

Q₈

—

163

—

101

D₄

—

326

—

102

D₄

—

652

—

103

D₄

—

163

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

326

—

107

C₂×C₄

—

652

—

108

C₂×C₄

C₂³×D₁₆₃

yes

characteristic

109

C₈

—

326

—

110

C₈

D₄×D₁₆₃

yes

—

111

C₈

D₄×D₁₆₃

yes

—

112

C₂²

—

326

—

113

C₂²

#19104780

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

326

—

117

C₄

C₂×D₄×D₁₆₃

yes

—

118

C₂

—

119

C₂

—

652

—

120

C₂

#23782300

yes

characteristic

	subgroup	G/H
center	C₂	#23782300
commutator, cosocle	C₂×C₄×C₁₆₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	#494×C₁₆₃	C₂
socle	C₂×C₁₆₃	D₄²
hypercenter	#494	D₁₆₃
hypocenter	C₁₆₃	#3293
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#23782300

commutator, cosocle

C₂×C₄×C₁₆₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₆₃

Fitting

#494×C₁₆₃

C₂

socle

C₂×C₁₆₃

D₄²

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #23782478	C₂	semidirect
2. #23781937	C₂	semidirect
3. #23781847	C₂	semidirect
4. #23781783	C₂	semidirect
5. #494×C₁₆₃	C₂	semidirect
60. C₁₆₃	#3293	semidirect
63. #494	D₁₆₃	semidirect

H ≅ G/N

type

1. #23782478

C₂

semidirect

2. #23781937

C₂

semidirect

3. #23781847

C₂

semidirect

4. #23781783

C₂

semidirect

5. #494×C₁₆₃

C₂

semidirect

60. C₁₆₃

#3293

semidirect

63. #494

D₁₆₃

semidirect

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

subgroup	index
#23782478	2
#23781937	2
#23781847	2
#23781783	2
#494×C₁₆₃	2
#3293	163

subgroup

index

#494×C₁₆₃

163

Subgroups — #35226340

Conjugacy classes of subgroups

Complements

Maximal subgroups