Group #35212366 — subgroups

Subgroups — #35212366

Order 20864.

152 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 77 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	#23781718	C₂	1	1	yes	maximal, characteristic
2	C₂×#19104646	C₂	1	1	yes	maximal, characteristic
3	#431×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆₃⋊C₈	C₄	1	2	yes	—
5	C₄⋊(C₁₆₃⋊C₈)	—	2	1	no	—
6	C₂²⋊(C₁₆₃⋊C₈)	—	2	1	no	—
7	#19104646	—	2	1	no	—
8	#19104646	—	2	1	no	—
9	C₄×C₁₆₃⋊C₈	C₄	1	2	yes	—
10	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₆₃	—	2	1	no	—
12	C₂²⋊C₈×C₁₆₃	—	2	1	no	—
13	C₄×C₈×C₁₆₃	C₄	1	2	yes	—
14	C₂×C₁₆₃⋊C₈	—	2	2	no	—
15	C₂×C₁₆₃⋊C₈	—	2	2	no	—
16	C₂×C₁₆₃⋊C₈	—	2	2	no	—
17	C₂×C₁₆₃⋊C₈	C₈	1	4	yes	complemented
18	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₆₃	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₁₆₃	—	2	2	no	—
22	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
25	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
26	C₁₆₃⋊C₈	—	2	4	no	—
27	C₁₆₃⋊C₈	—	4	2	no	—
28	C₂³×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₆₃	—	4	1	no	—
30	C₂×C₄×C₁₆₃	—	4	1	no	—
31	C₂×C₄×C₁₆₃	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₆₃	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₁₆₃	M₁₆	1	2	yes	—
39	C₈×C₁₆₃	—	4	2	no	—
40	C₂²×C₁₆₃	—	2	1	no	—
41	C₂²×C₁₆₃	—	2	1	no	—
42	C₂²×C₁₆₃	#95	1	1	yes	characteristic
43	C₂²×C₁₆₃	C₄⋊C₈	1	2	yes	—
44	C₄×C₁₆₃	—	2	1	no	—
45	C₄×C₁₆₃	—	2	1	no	—
46	C₄×C₁₆₃	—	2	1	no	—
47	C₄×C₁₆₃	—	2	1	no	—
48	C₄×C₁₆₃	C₄×C₈	1	1	yes	characteristic
49	C₄×C₁₆₃	#97	1	1	yes	characteristic
50	C₄×C₁₆₃	C₂²⋊C₈	1	2	yes	—
51	C₂×C₁₆₃	—	2	2	no	—
52	C₂×C₁₆₃	#336	1	1	yes	characteristic
53	C₂×C₁₆₃	H(3,ℤ/4)	1	1	yes	characteristic
54	C₂×C₁₆₃	#338	1	1	yes	characteristic
55	C₁₆₃	#1275	1	1	yes	characteristic, complemented, Sylow
56	#1275	—	163	1	no	maximal, Sylow
57	#431	—	163	1	no	—
58	#431	D₁₆₃	1	1	yes	characteristic
59	C₂×#97	—	163	1	no	—
60	C₂²×C₈	—	163	2	no	—
61	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	326	1	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	C₂²⋊C₈	—	326	1	no	—
66	#97	—	326	1	no	—
67	#97	—	326	1	no	—
68	C₄×C₈	—	163	2	no	—
69	C₄×C₈	Dic₁₆₃	1	2	yes	—
70	C₂²×C₄	C₄×D₁₆₃	1	1	yes	characteristic
71	C₂²×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
72	C₂²×C₄	Dic₃₂₆	1	1	yes	characteristic
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	326	2	no	—
75	C₂×C₈	—	326	2	no	—
76	C₂×C₈	—	326	2	no	—
77	C₂×C₈	—	163	4	no	—
78	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
79	C₄²	C₂×Dic₁₆₃	1	1	yes	characteristic
80	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
81	C₄²	D₆₅₂	1	1	yes	characteristic
82	C₂³	Dic₁₆₃⋊C₄	1	1	yes	characteristic
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	C₄×Dic₁₆₃	1	1	yes	characteristic
86	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
88	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
89	C₂×C₄	Dic₁₆₃⋊C₄	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₁₆₃	1	1	yes	characteristic
91	C₂×C₄	C₈×D₁₆₃	1	2	yes	—
92	C₂×C₄	#14579682	1	2	yes	—
93	C₈	—	4	2	no	—
94	C₈	—	326	4	no	—
95	C₈	—	652	2	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	C₁₆₃⋊#95	1	1	yes	characteristic
99	C₂²	Dic₁₆₃⋊C₈	1	2	yes	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₁₆₃	1	1	yes	characteristic
105	C₄	#19104645	1	1	yes	characteristic
106	C₄	(C₂×D₁₆₃)⋊C₈	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	#23781419	1	1	yes	characteristic
109	C₂	#23781421	1	1	yes	characteristic
110	C₂	C₁₆₃⋊#338	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23781718

C₂

yes

maximal, characteristic

C₂×#19104646

C₂

yes

maximal, characteristic

#431×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

C₄

yes

—

C₄⋊(C₁₆₃⋊C₈)

—

C₂²⋊(C₁₆₃⋊C₈)

—

#19104646

—

#19104646

—

C₄×C₁₆₃⋊C₈

C₄

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₃

—

C₂²⋊C₈×C₁₆₃

—

C₄×C₈×C₁₆₃

C₄

yes

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

C₈

yes

complemented

C₂²×C₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₃

D₄

yes

characteristic

C₂²×C₄×C₁₆₃

Q₈

yes

characteristic

C₂×C₈×C₁₆₃

—

C₄²×C₁₆₃

C₂×C₄

yes

characteristic

C₄²×C₁₆₃

C₂×C₄

yes

characteristic

C₄²×C₁₆₃

D₄

yes

characteristic

C₄²×C₁₆₃

D₄

yes

characteristic

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₄²

yes

characteristic

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×C₈

yes

—

C₂×C₄×C₁₆₃

M₁₆

yes

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#95

yes

characteristic

C₂²×C₁₆₃

C₄⋊C₈

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄×C₈

yes

characteristic

C₄×C₁₆₃

#97

yes

characteristic

C₄×C₁₆₃

C₂²⋊C₈

yes

—

C₂×C₁₆₃

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, complemented, Sylow

#1275

—

163

maximal, Sylow

#431

—

163

—

#431

D₁₆₃

yes

characteristic

C₂×#97

—

163

—

C₂²×C₈

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

326

—

C₂²⋊C₈

—

C₂²⋊C₈

—

326

—

#97

—

326

—

#97

—

326

—

C₄×C₈

—

163

—

C₄×C₈

Dic₁₆₃

yes

—

C₂²×C₄

C₄×D₁₆₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₆₃

yes

characteristic

C₂²×C₄

Dic₃₂₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

163

—

C₄²

C₄×D₁₆₃

yes

characteristic

C₄²

C₂×Dic₁₆₃

yes

characteristic

C₄²

C₂²⋊D₁₆₃

yes

characteristic

C₄²

D₆₅₂

yes

characteristic

C₂³

Dic₁₆₃⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₆₃

yes

characteristic

C₂×C₄

(C₂×D₁₆₃)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₁₆₃)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₁₆₃

yes

characteristic

C₂×C₄

Dic₁₆₃⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₁₆₃

yes

characteristic

C₂×C₄

C₈×D₁₆₃

yes

—

C₂×C₄

#14579682

yes

—

C₈

—

C₈

—

326

—

C₈

—

652

—

C₂²

—

C₂²

—

C₂²

C₁₆₃⋊#95

yes

characteristic

C₂²

Dic₁₆₃⋊C₈

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₁₆₃

yes

characteristic

105

C₄

#19104645

yes

characteristic

106

C₄

(C₂×D₁₆₃)⋊C₈

yes

—

107

C₂

—

108

C₂

#23781419

yes

characteristic

109

C₂

#23781421

yes

characteristic

110

C₂

C₁₆₃⋊#338

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₆₃
commutator	C₄×C₁₆₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₆₃
Fitting	#431×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#95
prime radical	C₂³×C₁₆₃	C₄⋊C₄
cosocle	C₂×C₄²×C₁₆₃	C₂²
hypercenter	#431	D₁₆₃
hypocenter	C₁₆₃	#1275
LCS	G ▷ C₄×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center

C₂×C₄

C₄×Dic₁₆₃

commutator

C₄×C₁₆₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₆₃

Fitting

#431×C₁₆₃

C₂

socle

C₂²×C₁₆₃

#95

prime radical

C₂³×C₁₆₃

C₄⋊C₄

cosocle

C₂×C₄²×C₁₆₃

C₂²

D₁₆₃

C₁₆₃

G ▷ C₄×C₁₆₃ ▷ C₁₆₃

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
17. C₂×C₁₆₃⋊C₈	C₈	semidirect
55. C₁₆₃	#1275	semidirect

H ≅ G/N

type

17. C₂×C₁₆₃⋊C₈

C₈

semidirect

55. C₁₆₃

#1275

semidirect

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

subgroup	index
#23781718	2
C₂×#19104646	2
#431×C₁₆₃	2
#1275	163

subgroup

index

#23781718

C₂×#19104646

#431×C₁₆₃

#1275

163

Subgroups — #35212366

Conjugacy classes of subgroups

Complements

Maximal subgroups