Group #35215506 — subgroups

Subgroups — #35215506

Order 20864.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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×#19104646	C₂	1	1	yes	maximal, characteristic
2	#23781535	C₂	1	1	yes	maximal, characteristic
3	#23781427	C₂	1	4	yes	maximal
4	C₂×#125×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₆₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₆₃	—	2	2	no	—
7	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
8	#19104646	—	2	2	no	—
9	#125×C₁₆₃	—	2	2	no	—
10	#125×C₁₆₃	C₄	1	4	yes	complemented
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	4	no	—
19	C₄⋊Dic₁₆₃	—	4	4	no	—
20	C₂²×C₄×C₁₆₃	—	2	2	no	—
21	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
22	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₆₃	—	2	8	no	—
24	C₄⋊C₄×C₁₆₃	—	2	2	no	—
25	C₄⋊C₄×C₁₆₃	C₂×C₄	1	4	yes	—
26	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
27	C₄²×C₁₆₃	D₄	1	2	yes	—
28	C₂×Dic₁₆₃	—	4	2	no	—
29	C₂×Dic₁₆₃	—	4	4	no	—
30	C₁₆₃⋊C₈	—	2	4	no	—
31	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₆₃	—	2	2	no	—
33	C₂×C₄×C₁₆₃	—	2	2	no	—
34	C₂×C₄×C₁₆₃	—	2	4	no	—
35	C₂×C₄×C₁₆₃	—	4	2	no	—
36	C₂×C₄×C₁₆₃	—	4	2	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	—	4	4	no	—
49	C₄×C₁₆₃	C₄×Q₈	1	2	yes	—
50	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
51	Dic₁₆₃	—	8	2	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	D₁₆₃	1	1	yes	characteristic
60	C₂×#97	—	163	1	no	—
61	#384	—	163	1	no	—
62	#340	—	163	4	no	—
63	C₂²×C₈	—	163	2	no	—
64	#125	—	2	2	no	—
65	#125	Dic₁₆₃	1	4	yes	complemented
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	8	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	326	4	no	—
82	C₄⋊C₄	—	652	4	no	—
83	C₄⋊C₄	C₂×Dic₁₆₃	1	4	yes	—
84	C₄²	C₂×Dic₁₆₃	1	2	yes	—
85	C₄²	C₂²⋊D₁₆₃	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	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	652	2	no	—
93	C₂×C₄	—	652	4	no	—
94	C₂×C₄	C₂²⋊Dic₁₆₃	1	4	yes	—
95	C₂×C₄	C₂²×Dic₁₆₃	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₁₆₃	1	2	yes	—
98	C₂×C₄	#14579700	1	2	yes	—
99	C₈	—	326	4	no	—
100	C₂²	C₂×C₂²⋊Dic₁₆₃	1	1	yes	characteristic
101	C₂²	#19104724	1	1	yes	characteristic
102	C₂²	#19104733	1	1	yes	characteristic
103	C₂²	#19104760	1	2	yes	—
104	C₂²	#19104765	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	1304	2	no	—
108	C₄	Q₈×Dic₁₆₃	1	2	yes	—
109	C₄	#19104711	1	2	yes	—
110	C₂	#23781572	1	1	yes	characteristic
111	C₂	#23781666	1	2	yes	—
112	C₂	#23782031	1	1	yes	characteristic
113	C₂	#23782035	1	1	yes	characteristic
114	C₂	#23782075	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104646

C₂

yes

maximal, characteristic

#23781535

C₂

yes

maximal, characteristic

#23781427

C₂

yes

maximal

C₂×#125×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₆₃

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

#19104646

—

#125×C₁₆₃

—

#125×C₁₆₃

C₄

yes

complemented

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

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

C₂×C₄

yes

—

C₄²×C₁₆₃

C₂×C₄

yes

—

C₄²×C₁₆₃

D₄

yes

—

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

—

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

D₁₆₃

yes

characteristic

C₂×#97

—

163

—

#384

—

163

—

#340

—

163

—

C₂²×C₈

—

163

—

#125

—

#125

Dic₁₆₃

yes

complemented

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

—

652

—

C₄⋊C₄

C₂×Dic₁₆₃

yes

—

C₄²

C₂×Dic₁₆₃

yes

—

C₄²

C₂²⋊D₁₆₃

yes

—

C₂³

C₂×C₂²⋊D₁₆₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

652

—

C₂×C₄

—

652

—

C₂×C₄

C₂²⋊Dic₁₆₃

yes

—

C₂×C₄

C₂²×Dic₁₆₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

C₂×C₄

Q₈×D₁₆₃

yes

—

C₂×C₄

#14579700

yes

—

C₈

—

326

—

100

C₂²

C₂×C₂²⋊Dic₁₆₃

yes

characteristic

101

C₂²

#19104724

yes

characteristic

102

C₂²

#19104733

yes

characteristic

103

C₂²

#19104760

yes

—

104

C₂²

#19104765

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

1304

—

108

C₄

Q₈×Dic₁₆₃

yes

—

109

C₄

#19104711

yes

—

110

C₂

#23781572

yes

characteristic

111

C₂

#23781666

yes

—

112

C₂

#23782031

yes

characteristic

113

C₂

#23782035

yes

characteristic

114

C₂

#23782075

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₆₃
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	C₂×#125×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂×D₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₂×#125	D₁₆₃
hypocenter	C₁₆₃	#1992
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₆₃

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

C₂×#125×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 ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₁₆₃	C₄	semidirect
57. C₁₆₃	#1992	semidirect
65. #125	Dic₁₆₃	semidirect

H ≅ G/N

type

10. #125×C₁₆₃

C₄

semidirect

57. C₁₆₃

#1992

semidirect

65. #125

Dic₁₆₃

semidirect

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

subgroup	index
C₂×#19104646	2
#23781535	2
#23781427	2
C₂×#125×C₁₆₃	2
#1992	163

subgroup

index

C₂×#125×C₁₆₃

163

Subgroups — #35215506

Conjugacy classes of subgroups

Complements

Maximal subgroups