Group #35214672 — subgroups

Subgroups — #35214672

Order 20864.

188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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	#23781688	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₆₃	C₂	1	1	yes	maximal, characteristic
3	#23781430	C₂	1	1	yes	maximal, characteristic
4	#23781429	C₂	1	2	yes	maximal
5	#341×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579683	C₂²	1	2	yes	—
7	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₆₃	C₂²	1	1	yes	characteristic
9	Dic₁₆₃⋊C₈	—	2	4	no	—
10	C₈×Dic₁₆₃	C₄	1	2	yes	—
11	C₈×Dic₁₆₃	C₄	1	2	yes	—
12	C₂×M₁₆×C₁₆₃	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₆₃	Q₈	1	2	yes	—
15	#14579683	—	2	2	no	—
16	#14579683	—	2	2	no	—
17	C₂×C₁₆₃⋊C₈	—	2	2	no	—
18	C₂×C₁₆₃⋊C₈	—	2	2	no	—
19	C₂×C₁₆₃⋊C₈	C₂×C₄	1	2	yes	—
20	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₆₃	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₁₆₃	D₄	1	2	yes	—
23	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₆₃	—	2	2	no	—
25	M₁₆×C₁₆₃	—	2	2	no	—
26	C₂×C₈×C₁₆₃	—	2	2	no	—
27	C₂×C₈×C₁₆₃	—	2	2	no	—
28	C₂×C₈×C₁₆₃	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₆₃	—	2	4	no	—
30	C₂×Dic₁₆₃	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₁₆₃	C₄⋊C₄	1	2	yes	—
32	C₁₆₃⋊C₈	—	2	2	no	—
33	C₁₆₃⋊C₈	—	4	2	no	—
34	C₂³×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₆₃	#41	1	2	yes	—
38	C₂×C₄×C₁₆₃	#41	1	2	yes	—
39	C₈×C₁₆₃	—	2	2	no	—
40	C₈×C₁₆₃	—	4	2	no	—
41	C₂²×C₁₆₃	—	2	2	no	—
42	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₆₃	#125	1	2	yes	—
44	C₄×C₁₆₃	#117	1	1	yes	characteristic
45	C₄×C₁₆₃	#117	1	1	yes	characteristic
46	C₄×C₁₆₃	#124	1	2	yes	—
47	Dic₁₆₃	—	2	2	no	—
48	Dic₁₆₃	#108	1	4	yes	—
49	C₂×C₁₆₃	—	2	2	no	—
50	C₂×C₁₆₃	#382	1	1	yes	characteristic
51	C₂×C₁₆₃	C₂×#108	1	2	yes	—
52	C₁₆₃	#1837	1	1	yes	characteristic, complemented, Sylow
53	#1837	—	163	1	no	maximal, Sylow
54	#423	—	163	2	no	—
55	C₂×C₄×C₈	—	163	1	no	—
56	#341	—	163	2	no	—
57	#341	—	163	1	no	—
58	#341	D₁₆₃	1	1	yes	characteristic
59	C₂×M₁₆	—	163	2	no	—
60	C₂×M₁₆	C₂×D₁₆₃	1	2	yes	—
61	C₂²×C₈	—	163	1	no	—
62	C₂²×C₈	C₂×D₁₆₃	1	1	yes	characteristic
63	C₂×C₄²	—	163	1	no	—
64	C₄⋊C₈	—	326	4	no	—
65	C₄×C₈	—	163	2	no	—
66	C₄×C₈	—	163	2	no	—
67	C₂²×C₄	—	163	2	no	—
68	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	326	2	no	—
72	M₁₆	—	326	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	326	2	no	—
76	C₂×C₈	—	326	2	no	—
77	C₂×C₈	—	163	2	no	—
78	C₂×C₈	C₄×D₁₆₃	1	2	yes	—
79	C₄²	—	163	1	no	—
80	C₄²	—	163	1	no	—
81	C₄²	—	163	2	no	—
82	C₂³	Q₈×D₁₆₃	1	1	yes	characteristic
83	C₂×C₄	—	326	4	no	—
84	C₂×C₄	—	163	2	no	—
85	C₂×C₄	—	163	2	no	—
86	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
87	C₂×C₄	D₄×D₁₆₃	1	1	yes	characteristic
88	C₂×C₄	#14579698	1	2	yes	—
89	C₂×C₄	#14579699	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	326	2	no	—
93	C₈	—	652	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₄×D₁₆₃	1	1	yes	characteristic
96	C₂²	#19104725	1	2	yes	—
97	C₄	—	326	2	no	—
98	C₄	—	163	4	no	—
99	C₄	#19104691	1	1	yes	characteristic
100	C₄	#19104692	1	1	yes	characteristic
101	C₄	#19104721	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#23781529	1	1	yes	characteristic
104	C₂	#108×D₁₆₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23781688

C₂

yes

maximal

C₂×C₈×Dic₁₆₃

C₂

yes

maximal, characteristic

#23781430

C₂

yes

maximal, characteristic

#23781429

C₂

yes

maximal

#341×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

C₂²×C₁₆₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₆₃

C₂²

yes

characteristic

Dic₁₆₃⋊C₈

—

C₈×Dic₁₆₃

C₄

yes

—

C₈×Dic₁₆₃

C₄

yes

—

C₂×M₁₆×C₁₆₃

C₂²

yes

—

C₂²×C₈×C₁₆₃

C₂²

yes

characteristic

C₂²×Dic₁₆₃

Q₈

yes

—

#14579683

—

#14579683

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₆₃

C₂×C₄

yes

characteristic

C₄×Dic₁₆₃

C₂×C₄

yes

characteristic

C₄×Dic₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

M₁₆×C₁₆₃

—

M₁₆×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

C₂×C₄

yes

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

C₄⋊C₄

yes

—

C₂×Dic₁₆₃

C₄⋊C₄

yes

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₈×C₁₆₃

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₆₃

#125

yes

—

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

—

Dic₁₆₃

—

Dic₁₆₃

#108

yes

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#382

yes

characteristic

C₂×C₁₆₃

C₂×#108

yes

—

C₁₆₃

#1837

yes

characteristic, complemented, Sylow

#1837

—

163

maximal, Sylow

#423

—

163

—

C₂×C₄×C₈

—

163

—

#341

—

163

—

#341

—

163

—

#341

D₁₆₃

yes

characteristic

C₂×M₁₆

—

163

—

C₂×M₁₆

C₂×D₁₆₃

yes

—

C₂²×C₈

—

163

—

C₂²×C₈

C₂×D₁₆₃

yes

characteristic

C₂×C₄²

—

163

—

C₄⋊C₈

—

326

—

C₄×C₈

—

163

—

C₄×C₈

—

163

—

C₂²×C₄

—

163

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

326

—

M₁₆

—

326

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

163

—

C₂×C₈

C₄×D₁₆₃

yes

—

C₄²

—

163

—

C₄²

—

163

—

C₄²

—

163

—

C₂³

Q₈×D₁₆₃

yes

characteristic

C₂×C₄

—

326

—

C₂×C₄

—

163

—

C₂×C₄

—

163

—

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

C₂×C₄

D₄×D₁₆₃

yes

characteristic

C₂×C₄

#14579698

yes

—

C₂×C₄

#14579699

yes

—

C₈

—

C₈

—

C₈

—

326

—

C₈

—

652

—

C₂²

—

C₂²

C₄⋊C₄×D₁₆₃

yes

characteristic

C₂²

#19104725

yes

—

C₄

—

326

—

C₄

—

163

—

C₄

#19104691

yes

characteristic

100

C₄

#19104692

yes

characteristic

101

C₄

#19104721

yes

—

102

C₂

—

103

C₂

#23781529

yes

characteristic

104

C₂

#108×D₁₆₃

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₆₃
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#341×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂×C₄⋊C₄
prime radical	C₂³×C₁₆₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	#341	D₁₆₃
hypocenter	C₁₆₃	#1837
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₆₃

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#341×C₁₆₃

C₂

socle

C₂²×C₁₆₃

C₂×C₄⋊C₄

prime radical

C₂³×C₁₆₃

C₂×Q₈

cosocle

C₂²×C₄×C₁₆₃

C₂³

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
52. C₁₆₃	#1837	semidirect

H ≅ G/N

type

52. C₁₆₃

#1837

semidirect

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

subgroup	index
#23781688	2
C₂×C₈×Dic₁₆₃	2
#23781430	2
#23781429	2
#341×C₁₆₃	2
#1837	163

subgroup

index

C₂×C₈×Dic₁₆₃

#341×C₁₆₃

163

Subgroups — #35214672

Conjugacy classes of subgroups

Complements

Maximal subgroups