Group #35215529 — subgroups

Subgroups — #35215529

Order 20864.

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

normal subgroups only

This group has 228 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	C₂×#19104727	C₂	1	1	yes	maximal, characteristic
2	#23781689	C₂	1	1	yes	maximal, characteristic
3	#23781433	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579683	C₂²	1	2	yes	—
6	#19104727	—	2	1	no	—
7	#19104727	—	2	1	no	—
8	#19104727	—	2	2	no	—
9	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	—
10	C₂×Dic₁₆₃⋊C₄	C₄	1	4	yes	—
11	C₄⋊(C₁₆₃⋊C₈)	—	2	2	no	—
12	C₂×M₁₆×C₁₆₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
14	#97×C₁₆₃	—	2	1	no	—
15	C₄×C₈×C₁₆₃	—	2	1	no	—
16	C₂²×Dic₁₆₃	C₂×C₄	1	4	yes	—
17	#14579683	—	2	4	no	—
18	C₂×C₁₆₃⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₆₃	—	2	2	no	—
20	C₄⋊Dic₁₆₃	—	2	2	no	—
21	Dic₁₆₃⋊C₄	—	2	4	no	—
22	Dic₁₆₃⋊C₄	—	2	4	no	—
23	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₆₃	—	2	2	no	—
26	M₁₆×C₁₆₃	Q₈	1	4	yes	—
27	C₂×C₈×C₁₆₃	—	2	2	no	—
28	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
29	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
30	C₄²×C₁₆₃	D₄	1	2	yes	—
31	C₂×Dic₁₆₃	—	2	4	no	—
32	C₂×Dic₁₆₃	—	4	4	no	—
33	C₁₆₃⋊C₈	—	4	2	no	—
34	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₆₃	—	4	2	no	—
36	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₆₃	C₂×D₄	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	C₈×C₁₆₃	—	2	4	no	—
43	C₂²×C₁₆₃	—	2	2	no	—
44	C₂²×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₆₃	C₄×Q₈	1	2	yes	—
46	C₄×C₁₆₃	—	2	2	no	—
47	C₄×C₁₆₃	—	2	2	no	—
48	C₄×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₁₆₃	#124	1	1	yes	characteristic
50	C₄×C₁₆₃	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₁₆₃	—	4	4	no	—
52	C₂×C₁₆₃	—	2	2	no	—
53	C₂×C₁₆₃	#391	1	1	yes	characteristic
54	C₂×C₁₆₃	#413	1	2	yes	—
55	C₁₆₃	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	163	1	no	maximal, Sylow
57	C₂×#125	—	163	1	no	—
58	#423	—	163	1	no	—
59	C₄×M₁₆	D₁₆₃	1	1	yes	characteristic
60	#343	—	163	4	no	—
61	C₂×M₁₆	—	163	2	no	—
62	C₂×M₁₆	C₂×D₁₆₃	1	2	yes	—
63	#125	—	326	1	no	—
64	#125	—	326	1	no	—
65	#125	—	326	2	no	—
66	C₂×C₄⋊C₄	—	163	4	no	—
67	C₂×C₄⋊C₄	—	163	2	no	—
68	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
69	C₄⋊C₈	—	326	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	—	163	4	no	—
73	C₂²×C₄	C₄×D₁₆₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	326	4	no	—
77	M₁₆	Dic₃₂₆	1	4	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	326	2	no	—
80	C₄⋊C₄	—	326	4	no	—
81	C₄⋊C₄	—	326	4	no	—
82	C₄⋊C₄	—	326	2	no	—
83	C₄⋊C₄	—	326	2	no	—
84	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
85	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
86	C₄²	D₆₅₂	1	2	yes	—
87	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	326	4	no	—
90	C₂×C₄	—	652	4	no	—
91	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	2	yes	—
92	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
94	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
95	C₂×C₄	C₂×Dic₃₂₆	1	2	yes	—
96	C₂×C₄	#14579699	1	2	yes	—
97	C₈	—	2	4	no	—
98	C₈	—	652	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
101	C₂²	C₄×Dic₃₂₆	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	652	4	no	—
105	C₄	#19104714	1	2	yes	—
106	C₄	#19104723	1	1	yes	characteristic
107	C₄	#19104735	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#23781571	1	1	yes	characteristic
110	C₂	#23781647	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19104727

C₂

yes

maximal, characteristic

#23781689

C₂

yes

maximal, characteristic

#23781433

C₂

yes

maximal

C₄×M₁₆×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

#19104727

—

#19104727

—

#19104727

—

C₂×C₄⋊Dic₁₆₃

C₂²

yes

—

C₂×Dic₁₆₃⋊C₄

C₄

yes

—

C₄⋊(C₁₆₃⋊C₈)

—

C₂×M₁₆×C₁₆₃

C₂²

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

#97×C₁₆₃

—

C₄×C₈×C₁₆₃

—

C₂²×Dic₁₆₃

C₂×C₄

yes

—

#14579683

—

C₂×C₁₆₃⋊C₈

—

C₄⋊Dic₁₆₃

—

C₄⋊Dic₁₆₃

—

Dic₁₆₃⋊C₄

—

Dic₁₆₃⋊C₄

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

M₁₆×C₁₆₃

—

M₁₆×C₁₆₃

Q₈

yes

—

C₂×C₈×C₁₆₃

—

C₄²×C₁₆₃

D₄

yes

characteristic

C₄²×C₁₆₃

D₄

yes

characteristic

C₄²×C₁₆₃

D₄

yes

—

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×Q₈

yes

—

C₂×C₄×C₁₆₃

#41

yes

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₆₃

C₄×Q₈

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂²⋊Q₈

yes

—

C₄×C₁₆₃

#124

yes

characteristic

C₄×C₁₆₃

C₄⋊Q₈

yes

characteristic

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#391

yes

characteristic

C₂×C₁₆₃

#413

yes

—

C₁₆₃

#1998

yes

characteristic, complemented, Sylow

#1998

—

163

maximal, Sylow

C₂×#125

—

163

—

#423

—

163

—

C₄×M₁₆

D₁₆₃

yes

characteristic

#343

—

163

—

C₂×M₁₆

—

163

—

C₂×M₁₆

C₂×D₁₆₃

yes

—

#125

—

326

—

#125

—

326

—

#125

—

326

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄⋊C₄

—

163

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

C₄⋊C₈

—

326

—

#97

—

C₄×C₈

—

C₂²×C₄

—

163

—

C₂²×C₄

C₄×D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

M₁₆

—

326

—

M₁₆

Dic₃₂₆

yes

—

C₂×C₈

—

C₂×C₈

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄²

C₂²⋊D₁₆₃

yes

characteristic

C₄²

C₂²⋊D₁₆₃

yes

characteristic

C₄²

D₆₅₂

yes

—

C₂³

C₂×C₄×D₁₆₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

326

—

C₂×C₄

—

652

—

C₂×C₄

(C₂×D₁₆₃)⋊C₄

yes

—

C₂×C₄

(C₂×D₁₆₃)⋊C₄

yes

—

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

C₂×C₄

C₂×D₆₅₂

yes

characteristic

C₂×C₄

C₂×Dic₃₂₆

yes

—

C₂×C₄

#14579699

yes

—

C₈

—

C₈

—

652

—

C₂²

—

100

C₂²

C₂×(C₂×D₁₆₃)⋊C₄

yes

characteristic

101

C₂²

C₄×Dic₃₂₆

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

652

—

105

C₄

#19104714

yes

—

106

C₄

#19104723

yes

characteristic

107

C₄

#19104735

yes

characteristic

108

C₂

—

109

C₂

#23781571

yes

characteristic

110

C₂

#23781647

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₆₃)⋊C₄
commutator, prime radical	C₂³×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	C₄×M₁₆×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₄×M₁₆	D₁₆₃
hypocenter	C₁₆₃	#1998
LCS	G ▷ C₂³×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₆₃)⋊C₄

commutator, prime radical

C₂³×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

C₄×M₁₆×C₁₆₃

C₂

socle

C₂²×C₁₆₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₆₃

C₂³

hypercenter

C₄×M₁₆

D₁₆₃

hypocenter

C₁₆₃

#1998

LCS

G ▷ C₂³×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
55. C₁₆₃	#1998	semidirect

H ≅ G/N

type

55. C₁₆₃

#1998

semidirect

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

subgroup	index
C₂×#19104727	2
#23781689	2
#23781433	2
C₄×M₁₆×C₁₆₃	2
#1998	163

subgroup

index

C₄×M₁₆×C₁₆₃

163

Subgroups — #35215529

Conjugacy classes of subgroups

Complements

Maximal subgroups