Group #35214290 — subgroups

Subgroups — #35214290

Order 20864.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#23781623	C₂	1	1	yes	maximal, characteristic
2	#23781621	C₂	1	2	yes	maximal
3	C₁₆₃⋊#338	C₂	1	1	yes	maximal, characteristic
4	#23781421	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579683	C₂²	1	2	yes	—
7	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
8	#19104694	C₂²	1	1	yes	characteristic
9	#19104646	C₄	1	4	yes	complemented
10	#19104645	C₄	1	2	yes	complemented
11	C₄×C₁₆₃⋊C₈	C₄	1	4	yes	complemented
12	C₈×Dic₁₆₃	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
14	#117×C₁₆₃	C₂²	1	2	yes	—
15	#14579683	—	2	2	no	—
16	#14579683	—	2	2	no	—
17	C₂×C₁₆₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₁₆₃⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆₃⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₆₃⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₁₆₃	—	2	1	no	—
23	C₂²⋊Dic₁₆₃	—	2	1	no	—
24	C₄×Dic₁₆₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₆₃	—	2	2	no	—
27	C₂×C₈×C₁₆₃	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₁₆₃	—	2	2	no	—
29	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
30	C₄²×C₁₆₃	C₂×C₄	1	4	yes	—
31	C₂×Dic₁₆₃	—	2	2	no	—
32	C₁₆₃⋊C₈	—	2	4	no	—
33	C₁₆₃⋊C₈	C₄²	1	1	yes	characteristic
34	C₁₆₃⋊C₈	C₄²	1	1	yes	characteristic
35	C₁₆₃⋊C₈	C₄²	1	2	yes	—
36	C₂³×C₁₆₃	#41	1	1	yes	characteristic
37	C₂×C₄×C₁₆₃	—	2	4	no	—
38	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₆₃	#41	1	2	yes	—
41	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
42	C₈×C₁₆₃	—	2	2	no	—
43	C₂²×C₁₆₃	—	4	1	no	—
44	C₂²×C₁₆₃	#117	1	2	yes	—
45	C₂²×C₁₆₃	#117	1	1	yes	characteristic
46	C₄×C₁₆₃	—	4	4	no	—
47	C₄×C₁₆₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₆₃	#117	1	1	yes	characteristic
49	C₄×C₁₆₃	#117	1	2	yes	—
50	Dic₁₆₃	—	4	2	no	—
51	C₂×C₁₆₃	—	2	1	no	—
52	C₂×C₁₆₃	—	2	2	no	—
53	C₂×C₁₆₃	#376	1	1	yes	characteristic
54	C₁₆₃	#1747	1	1	yes	characteristic, complemented, Sylow
55	#1747	—	163	1	no	maximal, Sylow
56	#405	—	163	1	no	—
57	#405	—	163	2	no	—
58	#338	—	163	1	no	—
59	H(3,ℤ/4)	—	163	2	no	—
60	H(3,ℤ/4)	D₁₆₃	1	1	yes	characteristic
61	C₂×M₁₆	—	163	2	no	—
62	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
63	C₂²×C₈	—	163	1	no	—
64	#117	—	163	1	no	—
65	#117	C₂×D₁₆₃	1	2	yes	—
66	#97	—	163	2	no	—
67	#97	—	163	4	no	—
68	C₄×C₈	—	163	2	no	—
69	C₄×C₈	—	163	4	no	—
70	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	326	2	no	—
73	M₁₆	—	326	2	no	—
74	C₂×C₈	—	163	4	no	—
75	C₂×C₈	—	163	1	no	—
76	C₂×C₈	—	163	2	no	—
77	C₂×C₈	—	163	1	no	—
78	C₂×C₈	—	163	2	no	—
79	C₂×C₈	C₄×D₁₆₃	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	326	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	326	1	no	—
84	C₄²	—	163	2	no	—
85	C₄²	C₄×D₁₆₃	1	4	yes	—
86	C₂³	#14579698	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	326	2	no	—
89	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
91	C₂×C₄	#14579698	1	2	yes	—
92	C₂×C₄	#14579700	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	326	4	no	—
95	C₈	—	163	1	no	—
96	C₈	—	163	1	no	—
97	C₈	—	163	2	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#19104691	1	2	yes	—
100	C₂²	#19104693	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	652	2	no	—
103	C₄	C₄²×D₁₆₃	1	1	yes	characteristic
104	C₄	#19104691	1	1	yes	characteristic
105	C₄	#19104693	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#23781510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23781623

C₂

yes

maximal, characteristic

#23781621

C₂

yes

maximal

C₁₆₃⋊#338

C₂

yes

maximal, characteristic

#23781421

C₂

yes

maximal

H(3,ℤ/4)×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

C₂²×C₁₆₃⋊C₈

C₂²

yes

characteristic

#19104694

C₂²

yes

characteristic

#19104646

C₄

yes

complemented

#19104645

C₄

yes

complemented

C₄×C₁₆₃⋊C₈

C₄

yes

complemented

C₈×Dic₁₆₃

C₄

yes

complemented

C₂×M₁₆×C₁₆₃

C₂²

yes

characteristic

#117×C₁₆₃

C₂²

yes

—

#14579683

—

#14579683

—

C₂×C₁₆₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₆₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₆₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₆₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₆₃⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₆₃

—

C₂²⋊Dic₁₆₃

—

C₄×Dic₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

M₁₆×C₁₆₃

—

C₂×C₈×C₁₆₃

C₂×C₄

yes

—

C₄⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂×C₄

yes

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

C₄²

yes

characteristic

C₁₆₃⋊C₈

C₄²

yes

characteristic

C₁₆₃⋊C₈

C₄²

yes

—

C₂³×C₁₆₃

#41

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

#41

yes

characteristic

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#117

yes

—

C₂²×C₁₆₃

#117

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂×C₄²

yes

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#376

yes

characteristic

C₁₆₃

#1747

yes

characteristic, complemented, Sylow

#1747

—

163

maximal, Sylow

#405

—

163

—

#405

—

163

—

#338

—

163

—

H(3,ℤ/4)

—

163

—

H(3,ℤ/4)

D₁₆₃

yes

characteristic

C₂×M₁₆

—

163

—

C₂×M₁₆

C₂×D₁₆₃

yes

characteristic

C₂²×C₈

—

163

—

#117

—

163

—

#117

C₂×D₁₆₃

yes

—

#97

—

163

—

#97

—

163

—

C₄×C₈

—

163

—

C₄×C₈

—

163

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

M₁₆

—

326

—

M₁₆

—

326

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

C₂×C₈

—

163

—

C₂×C₈

C₄×D₁₆₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₂²⋊C₄

—

C₂²⋊C₄

—

326

—

C₄²

—

163

—

C₄²

C₄×D₁₆₃

yes

—

C₂³

#14579698

yes

characteristic

C₂×C₄

—

C₂×C₄

—

326

—

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

C₂×C₄

C₂×C₄×D₁₆₃

yes

—

C₂×C₄

#14579698

yes

—

C₂×C₄

#14579700

yes

characteristic

C₈

—

C₈

—

326

—

C₈

—

163

—

C₈

—

163

—

C₈

—

163

—

C₂²

—

C₂²

#19104691

yes

—

100

C₂²

#19104693

yes

characteristic

101

C₄

—

102

C₄

—

652

—

103

C₄

C₄²×D₁₆₃

yes

characteristic

104

C₄

#19104691

yes

characteristic

105

C₄

#19104693

yes

—

106

C₂

—

107

C₂

—

108

C₂

#23781510

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₆₃
commutator	C₄×C₁₆₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	H(3,ℤ/4)×C₁₆₃	C₂
socle	C₂×C₁₆₃	#376
prime radical	C₂³×C₁₆₃	#41
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	H(3,ℤ/4)	D₁₆₃
hypocenter	C₁₆₃	#1747
LCS	G ▷ C₄×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₁₆₃

commutator

C₄×C₁₆₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

H(3,ℤ/4)×C₁₆₃

C₂

C₂×C₁₆₃

C₂³×C₁₆₃

C₂²×C₄×C₁₆₃

C₂³

D₁₆₃

C₁₆₃

G ▷ C₄×C₁₆₃ ▷ C₁₆₃

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #19104646	C₄	semidirect
10. #19104645	C₄	semidirect
11. C₄×C₁₆₃⋊C₈	C₄	semidirect
12. C₈×Dic₁₆₃	C₄	semidirect
54. C₁₆₃	#1747	semidirect

H ≅ G/N

type

9. #19104646

C₄

semidirect

10. #19104645

C₄

semidirect

11. C₄×C₁₆₃⋊C₈

C₄

semidirect

12. C₈×Dic₁₆₃

C₄

semidirect

54. C₁₆₃

#1747

semidirect

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

subgroup	index
#23781623	2
#23781621	2
C₁₆₃⋊#338	2
#23781421	2
H(3,ℤ/4)×C₁₆₃	2
#1747	163

subgroup

index

C₁₆₃⋊#338

H(3,ℤ/4)×C₁₆₃

163

Subgroups — #35214290

Conjugacy classes of subgroups

Complements

Maximal subgroups