Group #35215504 — subgroups

Subgroups — #35215504

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	C₂×#19104727	C₂	1	1	yes	maximal, characteristic
2	C₂×#19104646	C₂	1	1	yes	maximal, characteristic
3	#23781427	C₂	1	4	yes	maximal
4	#384×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
6	#19104727	—	2	2	no	—
7	#19104727	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
9	C₂×Dic₁₆₃⋊C₄	—	2	2	no	—
10	#19104646	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₆₃	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₆₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₆₃	—	2	2	no	—
15	C₂×C₁₆₃⋊C₈	—	2	2	no	—
16	C₂×C₁₆₃⋊C₈	—	2	2	no	—
17	C₂×C₁₆₃⋊C₈	Q₈	1	4	yes	—
18	C₄⋊Dic₁₆₃	—	2	2	no	—
19	C₄⋊Dic₁₆₃	C₂×C₄	1	4	yes	—
20	Dic₁₆₃⋊C₄	—	2	8	no	—
21	C₂²×C₄×C₁₆₃	—	2	2	no	—
22	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
23	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₆₃	—	2	4	no	—
25	C₄⋊C₄×C₁₆₃	—	4	4	no	—
26	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
27	C₄²×C₁₆₃	D₄	1	2	yes	—
28	C₂×Dic₁₆₃	—	2	4	no	—
29	C₂×Dic₁₆₃	—	4	2	no	—
30	C₂×Dic₁₆₃	—	4	2	no	—
31	C₁₆₃⋊C₈	—	2	4	no	—
32	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₆₃	—	2	2	no	—
34	C₂×C₄×C₁₆₃	—	2	2	no	—
35	C₂×C₄×C₁₆₃	—	4	2	no	—
36	C₂×C₄×C₁₆₃	—	4	4	no	—
37	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₆₃	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₆₃	#41	1	2	yes	—
42	C₂²×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₆₃	#124	1	1	yes	characteristic
44	C₂²×C₁₆₃	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₁₆₃	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₁₆₃	#137	1	2	yes	—
47	C₄×C₁₆₃	—	2	4	no	—
48	C₄×C₁₆₃	—	8	2	no	—
49	C₄×C₁₆₃	C₄×Q₈	1	2	yes	—
50	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
51	Dic₁₆₃	—	4	4	no	—
52	C₂×C₁₆₃	#391	1	1	yes	characteristic
53	C₂×C₁₆₃	#417	1	2	yes	—
54	C₂×C₁₆₃	#490	1	1	yes	characteristic
55	C₂×C₁₆₃	#491	1	1	yes	characteristic
56	C₂×C₁₆₃	#501	1	2	yes	—
57	C₁₆₃	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	163	1	no	maximal, Sylow
59	C₂×#125	—	163	1	no	—
60	C₂×#97	—	163	1	no	—
61	#384	D₁₆₃	1	1	yes	characteristic
62	#340	—	163	4	no	—
63	C₂²×C₈	—	163	2	no	—
64	#125	—	326	2	no	—
65	#125	—	163	4	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	326	2	no	—
68	C₂×C₄⋊C₄	—	163	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₆₃	1	2	yes	—
70	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
71	#97	—	326	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	326	2	no	—
74	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
76	C₂×C₈	—	326	2	no	—
77	C₂×C₈	—	326	2	no	—
78	C₂×C₈	—	163	4	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	4	4	no	—
81	C₄⋊C₄	—	326	8	no	—
82	C₄⋊C₄	—	326	2	no	—
83	C₄⋊C₄	—	163	4	no	—
84	C₄²	C₄×D₁₆₃	1	2	yes	—
85	C₄²	D₆₅₂	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	—	326	4	no	—
92	C₂×C₄	—	652	2	no	—
93	C₂×C₄	—	652	2	no	—
94	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
96	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₁₆₃	1	2	yes	—
98	C₂×C₄	#14579698	1	2	yes	—
99	C₈	—	326	4	no	—
100	C₂²	C₂×(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
101	C₂²	#19104722	1	1	yes	characteristic
102	C₂²	#19104733	1	1	yes	characteristic
103	C₂²	#19104761	1	2	yes	—
104	C₂²	#19104766	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	8	2	no	—
107	C₄	—	652	4	no	—
108	C₄	#19104700	1	2	yes	—
109	C₄	#19104715	1	2	yes	—
110	C₂	#23781570	1	1	yes	characteristic
111	C₂	#23781664	1	2	yes	—
112	C₂	#23782029	1	1	yes	characteristic
113	C₂	#23782033	1	1	yes	characteristic
114	C₂	#23782071	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104727

C₂

yes

maximal, characteristic

C₂×#19104646

C₂

yes

maximal, characteristic

#23781427

C₂

yes

maximal

#384×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

—

#19104727

—

#19104727

C₄

yes

complemented

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

C₂×Dic₁₆₃⋊C₄

—

#19104646

—

C₂×C₄⋊C₄×C₁₆₃

—

C₂×C₄⋊C₄×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₂²×Dic₁₆₃

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

Q₈

yes

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

C₂×C₄

yes

—

Dic₁₆₃⋊C₄

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂×C₄

yes

—

C₄²×C₁₆₃

D₄

yes

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₂²×C₁₆₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₆₃

#124

yes

characteristic

C₂²×C₁₆₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₆₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₆₃

#137

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄×Q₈

yes

—

C₄×C₁₆₃

C₂²⋊Q₈

yes

—

Dic₁₆₃

—

C₂×C₁₆₃

#391

yes

characteristic

C₂×C₁₆₃

#417

yes

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

—

C₁₆₃

#1992

yes

characteristic, complemented, Sylow

#1992

—

163

maximal, Sylow

C₂×#125

—

163

—

C₂×#97

—

163

—

#384

D₁₆₃

yes

characteristic

#340

—

163

—

C₂²×C₈

—

163

—

#125

—

326

—

#125

—

163

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

C₂×D₁₆₃

yes

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

#97

—

326

—

C₂²×C₄

—

C₂²×C₄

—

326

—

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

163

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

163

—

C₄²

C₄×D₁₆₃

yes

—

C₄²

D₆₅₂

yes

—

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

326

—

C₂×C₄

—

652

—

C₂×C₄

—

652

—

C₂×C₄

(C₂×D₁₆₃)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

C₂×C₄

C₂×D₆₅₂

yes

characteristic

C₂×C₄

Q₈×D₁₆₃

yes

—

C₂×C₄

#14579698

yes

—

C₈

—

326

—

100

C₂²

C₂×(C₂×D₁₆₃)⋊C₄

yes

characteristic

101

C₂²

#19104722

yes

characteristic

102

C₂²

#19104733

yes

characteristic

103

C₂²

#19104761

yes

—

104

C₂²

#19104766

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

652

—

108

C₄

#19104700

yes

—

109

C₄

#19104715

yes

—

110

C₂

#23781570

yes

characteristic

111

C₂

#23781664

yes

—

112

C₂

#23782029

yes

characteristic

113

C₂

#23782033

yes

characteristic

114

C₂

#23782071

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₆₃
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#384×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂×D₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	#384	D₁₆₃
hypocenter	C₁₆₃	#1992
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₆₃

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#384×C₁₆₃

C₂

socle, prime radical

C₂³×C₁₆₃

C₂×D₄

cosocle

C₂²×C₄×C₁₆₃

C₂³

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #19104727	C₄	semidirect
57. C₁₆₃	#1992	semidirect

H ≅ G/N

type

7. #19104727

C₄

semidirect

57. C₁₆₃

#1992

semidirect

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

subgroup	index
C₂×#19104727	2
C₂×#19104646	2
#23781427	2
#384×C₁₆₃	2
#1992	163

subgroup

index

#384×C₁₆₃

163

Subgroups — #35215504

Conjugacy classes of subgroups

Complements

Maximal subgroups