Group #35213593 — subgroups

Subgroups — #35213593

Order 20864.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#23782401	C₂	1	1	yes	maximal, characteristic, complemented
2	#23781862	C₂	1	2	yes	maximal, complemented
3	#23781409	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781397	C₂	1	2	yes	maximal, complemented
5	#493×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579687	—	2	2	no	—
7	#19104722	—	2	2	no	—
8	#19104716	—	2	2	no	—
9	C₄×Dic₃₂₆	C₂²	1	2	yes	—
10	D₄×Dic₁₆₃	—	2	2	no	—
11	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
12	#19104656	—	2	2	no	—
13	C₂×D₈×C₁₆₃	—	2	2	no	—
14	#127×C₁₆₃	C₂²	1	1	yes	characteristic
15	#127×C₁₆₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₂₆	D₄	1	2	yes	—
18	C₂²×Dic₁₆₃	—	2	2	no	—
19	#14579687	—	4	4	no	—
20	C₂×C₁₆₃⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₆₃	D₄	1	2	yes	—
22	Dic₁₆₃⋊C₄	—	2	2	no	—
23	C₂²⋊Dic₁₆₃	—	2	4	no	—
24	C₂²⋊Dic₁₆₃	—	2	2	no	—
25	C₄×Dic₁₆₃	—	2	2	no	—
26	C₂×D₄×C₁₆₃	—	2	2	no	—
27	C₂×D₄×C₁₆₃	—	2	1	no	—
28	C₂×D₄×C₁₆₃	—	2	2	no	—
29	C₂×D₄×C₁₆₃	D₄	1	2	yes	—
30	D₈×C₁₆₃	—	4	4	no	—
31	C₂×C₈×C₁₆₃	—	2	1	no	—
32	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
33	Dic₃₂₆	—	2	2	no	—
34	Dic₃₂₆	—	2	2	no	—
35	C₂×Dic₁₆₃	—	2	2	no	—
36	C₂×Dic₁₆₃	—	2	2	no	—
37	C₂×Dic₁₆₃	—	8	2	no	—
38	C₁₆₃⋊C₈	—	4	2	no	—
39	C₂³×C₁₆₃	—	2	2	no	—
40	C₂³×C₁₆₃	—	4	1	no	—
41	D₄×C₁₆₃	—	4	2	no	—
42	D₄×C₁₆₃	—	4	2	no	—
43	D₄×C₁₆₃	—	4	2	no	—
44	D₄×C₁₆₃	—	4	2	no	—
45	D₄×C₁₆₃	—	4	2	no	—
46	D₄×C₁₆₃	—	4	2	no	—
47	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₆₃	—	2	2	no	—
50	C₂²×C₁₆₃	—	4	4	no	—
51	C₂²×C₁₆₃	—	4	2	no	—
52	C₂²×C₁₆₃	—	8	2	no	—
53	C₂²×C₁₆₃	—	8	1	no	—
54	C₂²×C₁₆₃	#120	1	1	yes	characteristic
55	C₄×C₁₆₃	—	2	1	no	—
56	C₄×C₁₆₃	#135	1	2	yes	—
57	C₄×C₁₆₃	AGL(1,ℤ/8)	1	2	yes	—
58	Dic₁₆₃	—	4	2	no	—
59	Dic₁₆₃	—	4	2	no	—
60	C₂×C₁₆₃	—	8	2	no	—
61	C₂×C₁₆₃	—	16	1	no	—
62	C₂×C₁₆₃	#449	1	2	yes	—
63	C₂×C₁₆₃	#453	1	1	yes	characteristic
64	C₁₆₃	#1620	1	1	yes	characteristic, complemented, Sylow
65	#1620	—	163	1	no	maximal, Sylow
66	#559	—	163	1	no	—
67	#493	D₁₆₃	1	1	yes	characteristic
68	#460	—	163	2	no	—
69	#331	—	163	1	no	—
70	Q₈⋊C₈	—	163	2	no	—
71	C₂×SD₁₆	—	326	2	no	—
72	C₂×D₈	—	2	2	no	—
73	#127	C₂×D₁₆₃	1	1	yes	characteristic
74	#127	C₂×D₁₆₃	1	1	yes	characteristic
75	#124	—	326	2	no	—
76	#123	—	326	2	no	—
77	C₄×Q₈	—	163	2	no	—
78	C₄×D₄	—	326	2	no	—
79	C₄⋊C₈	—	163	2	no	—
80	#102	—	326	2	no	—
81	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
82	C₂×Q₈	—	163	2	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	C₂²⋊D₁₆₃	1	2	yes	—
87	C₂²×C₄	—	326	2	no	—
88	SD₁₆	—	652	4	no	—
89	D₈	—	4	4	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	326	2	no	—
92	C₄⋊C₄	—	326	2	no	—
93	C₄⋊C₄	—	163	2	no	—
94	C₂²⋊C₄	—	326	4	no	—
95	C₂²⋊C₄	—	326	2	no	—
96	C₄²	—	326	2	no	—
97	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	4	1	no	—
100	Q₈	—	326	2	no	—
101	Q₈	—	326	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	C₂×C₄	—	1304	2	no	—
109	C₂×C₄	—	326	2	no	—
110	C₂×C₄	—	326	2	no	—
111	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	652	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	8	2	no	—
118	C₂²	—	8	1	no	—
119	C₂²	#19104703	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	652	2	no	—
122	C₄	—	652	2	no	—
123	C₄	#19104754	1	2	yes	—
124	C₄	#19104759	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	16	1	no	—
127	C₂	#23781807	1	2	yes	—
128	C₂	#23781832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23782401

C₂

yes

maximal, characteristic, complemented

#23781862

C₂

yes

maximal, complemented

#23781409

C₂

yes

maximal, characteristic, complemented

#23781397

C₂

yes

maximal, complemented

#493×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579687

—

#19104722

—

#19104716

—

C₄×Dic₃₂₆

C₂²

yes

—

D₄×Dic₁₆₃

—

C₄⋊(C₁₆₃⋊C₈)

C₂²

yes

—

#19104656

—

C₂×D₈×C₁₆₃

—

#127×C₁₆₃

C₂²

yes

characteristic

#127×C₁₆₃

C₂²

yes

characteristic

C₄×C₈×C₁₆₃

C₂²

yes

characteristic

C₂×Dic₃₂₆

D₄

yes

—

C₂²×Dic₁₆₃

—

#14579687

—

C₂×C₁₆₃⋊C₈

—

C₄⋊Dic₁₆₃

D₄

yes

—

Dic₁₆₃⋊C₄

—

C₂²⋊Dic₁₆₃

—

C₂²⋊Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

D₄

yes

—

D₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₄²×C₁₆₃

C₂³

yes

characteristic

Dic₃₂₆

—

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

—

C₂³×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#120

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

#135

yes

—

C₄×C₁₆₃

AGL(1,ℤ/8)

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#449

yes

—

C₂×C₁₆₃

#453

yes

characteristic

C₁₆₃

#1620

yes

characteristic, complemented, Sylow

#1620

—

163

maximal, Sylow

#559

—

163

—

#493

D₁₆₃

yes

characteristic

#460

—

163

—

#331

—

163

—

Q₈⋊C₈

—

163

—

C₂×SD₁₆

—

326

—

C₂×D₈

—

#127

C₂×D₁₆₃

yes

characteristic

#127

C₂×D₁₆₃

yes

characteristic

#124

—

326

—

#123

—

326

—

C₄×Q₈

—

163

—

C₄×D₄

—

326

—

C₄⋊C₈

—

163

—

#102

—

326

—

C₄×C₈

C₂×D₁₆₃

yes

characteristic

C₂×Q₈

—

163

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

—

326

—

SD₁₆

—

652

—

D₈

—

C₂×C₈

—

C₂×C₈

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

163

—

C₂²⋊C₄

—

326

—

C₂²⋊C₄

—

326

—

C₄²

—

326

—

C₄²

C₂²×D₁₆₃

yes

characteristic

C₂³

—

C₂³

—

100

Q₈

—

326

—

101

Q₈

—

326

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

1304

—

109

C₂×C₄

—

326

—

110

C₂×C₄

—

326

—

111

C₂×C₄

D₄×D₁₆₃

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

113

C₈

—

114

C₈

—

652

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#19104703

yes

characteristic

120

C₄

—

121

C₄

—

652

—

122

C₄

—

652

—

123

C₄

#19104754

yes

—

124

C₄

#19104759

yes

—

125

C₂

—

126

C₂

—

127

C₂

#23781807

yes

—

128

C₂

#23781832

yes

characteristic

	subgroup	G/H
center	C₂²	#19104703
commutator, cosocle	C₄²×C₁₆₃	C₂³
Frattini	C₄²	C₂²×D₁₆₃
Fitting, prime radical	#493×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#120
hypercenter	#493	D₁₆₃
hypocenter	C₁₆₃	#1620
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#19104703

commutator, cosocle

C₄²×C₁₆₃

C₂³

Frattini

C₄²

C₂²×D₁₆₃

Fitting, prime radical

#493×C₁₆₃

C₂

C₂²×C₁₆₃

D₁₆₃

C₁₆₃

G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #23782401	C₂	semidirect
2. #23781862	C₂	semidirect
3. #23781409	C₂	semidirect
4. #23781397	C₂	semidirect
64. C₁₆₃	#1620	semidirect

H ≅ G/N

type

1. #23782401

C₂

semidirect

2. #23781862

C₂

semidirect

3. #23781409

C₂

semidirect

4. #23781397

C₂

semidirect

64. C₁₆₃

#1620

semidirect

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

subgroup	index
#23782401	2
#23781862	2
#23781409	2
#23781397	2
#493×C₁₆₃	2
#1620	163

subgroup

index

#493×C₁₆₃

163

Subgroups — #35213593

Conjugacy classes of subgroups

Complements

Maximal subgroups