Group #35215381 — subgroups

Subgroups — #35215381

Order 20864.

292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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	#23781689	C₂	1	1	yes	maximal, characteristic
2	C₂×#19104654	C₂	1	4	yes	maximal
3	C₄×#14579683	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579683	C₂²	1	2	yes	—
6	C₂×#14579683	C₂²	1	2	yes	—
7	C₄⋊(C₁₆₃⋊C₈)	—	2	2	no	—
8	#19104654	—	2	8	no	—
9	#19104646	—	2	1	no	—
10	C₄×C₁₆₃⋊C₈	—	2	1	no	—
11	C₂²×Q₈×C₁₆₃	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₆₃	—	2	1	no	—
13	C₄⋊Q₈×C₁₆₃	—	2	1	no	—
14	C₄⋊Q₈×C₁₆₃	—	2	2	no	—
15	C₂×C₄⋊C₄×C₁₆₃	C₄	1	4	yes	—
16	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
17	#14579683	—	2	4	no	—
18	#14579683	—	2	2	no	—
19	#14579683	D₄	1	4	yes	—
20	C₂×C₁₆₃⋊C₈	—	2	2	no	—
21	C₂×C₁₆₃⋊C₈	—	2	2	no	—
22	C₂×Q₈×C₁₆₃	—	2	2	no	—
23	C₂×Q₈×C₁₆₃	—	2	2	no	—
24	C₂×Q₈×C₁₆₃	—	2	4	no	—
25	C₂×Q₈×C₁₆₃	—	2	4	no	—
26	C₂²×C₄×C₁₆₃	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₆₃	—	2	4	no	—
30	C₄⋊C₄×C₁₆₃	—	2	4	no	—
31	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
32	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
33	C₄²×C₁₆₃	D₄	1	2	yes	—
34	C₁₆₃⋊C₈	—	2	4	no	—
35	C₁₆₃⋊C₈	—	4	2	no	—
36	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₁₆₃	—	4	4	no	—
38	Q₈×C₁₆₃	—	4	2	no	—
39	Q₈×C₁₆₃	—	4	2	no	—
40	C₂×C₄×C₁₆₃	—	2	4	no	—
41	C₂×C₄×C₁₆₃	—	2	4	no	—
42	C₂×C₄×C₁₆₃	—	2	8	no	—
43	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₆₃	#41	1	2	yes	—
49	C₂²×C₁₆₃	—	2	2	no	—
50	C₂²×C₁₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₆₃	C₄×D₄	1	2	yes	—
52	C₄×C₁₆₃	—	2	2	no	—
53	C₄×C₁₆₃	—	4	4	no	—
54	C₄×C₁₆₃	#101	1	4	yes	complemented
55	C₄×C₁₆₃	#121	1	2	yes	—
56	C₄×C₁₆₃	#124	1	1	yes	characteristic
57	C₄×C₁₆₃	#127	1	1	yes	characteristic
58	C₂×C₁₆₃	—	2	2	no	—
59	C₂×C₁₆₃	#390	1	1	yes	characteristic
60	C₂×C₁₆₃	C₂×#101	1	2	yes	—
61	C₁₆₃	#1966	1	1	yes	characteristic, complemented, Sylow
62	#1966	—	163	1	no	maximal, Sylow
63	C₂×C₄⋊Q₈	D₁₆₃	1	1	yes	characteristic
64	#423	—	163	1	no	—
65	C₂×#101	—	163	4	no	—
66	C₄×M₁₆	—	163	1	no	—
67	C₂²×Q₈	C₂×D₁₆₃	1	2	yes	—
68	C₂×M₁₆	—	163	2	no	—
69	C₂×M₁₆	—	163	2	no	—
70	C₄⋊Q₈	—	2	1	no	—
71	C₄⋊Q₈	—	2	1	no	—
72	C₄⋊Q₈	—	2	2	no	—
73	C₂×C₄⋊C₄	Dic₁₆₃	1	4	yes	—
74	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
75	C₄⋊C₈	—	326	2	no	—
76	#101	—	326	8	no	—
77	#97	—	326	1	no	—
78	C₄×C₈	—	326	1	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂²×C₄	C₂×Dic₁₆₃	1	4	yes	—
84	C₂²×C₄	C₂×Dic₁₆₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
86	M₁₆	—	326	2	no	—
87	M₁₆	—	326	4	no	—
88	M₁₆	—	163	4	no	—
89	C₂×C₈	—	326	2	no	—
90	C₂×C₈	—	326	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
94	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
95	C₄²	C₂²⋊D₁₆₃	1	2	yes	—
96	C₂³	C₂²×Dic₁₆₃	1	1	yes	characteristic
97	Q₈	—	4	4	no	—
98	Q₈	—	4	2	no	—
99	Q₈	—	4	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	C₂²⋊Dic₁₆₃	1	2	yes	—
104	C₂×C₄	C₂²⋊Dic₁₆₃	1	2	yes	—
105	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
108	C₂×C₄	#14579698	1	2	yes	—
109	C₈	—	326	4	no	—
110	C₈	—	652	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×C₂²⋊Dic₁₆₃	1	1	yes	characteristic
113	C₂²	D₄×Dic₁₆₃	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	#19104654	1	4	yes	complemented
117	C₄	#19104706	1	2	yes	—
118	C₄	#19104722	1	1	yes	characteristic
119	C₄	#19104731	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#23781568	1	1	yes	characteristic
122	C₂	C₂×#19104654	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23781689

C₂

yes

maximal, characteristic

C₂×#19104654

C₂

yes

maximal

C₄×#14579683

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

—

C₂×#14579683

C₂²

yes

—

C₄⋊(C₁₆₃⋊C₈)

—

#19104654

—

#19104646

—

C₄×C₁₆₃⋊C₈

—

C₂²×Q₈×C₁₆₃

C₂²

yes

—

C₄⋊Q₈×C₁₆₃

—

C₄⋊Q₈×C₁₆₃

—

C₄⋊Q₈×C₁₆₃

—

C₂×C₄⋊C₄×C₁₆₃

C₄

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

—

—

D₄

yes

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

D₄

yes

characteristic

C₄²×C₁₆₃

D₄

yes

characteristic

C₄²×C₁₆₃

D₄

yes

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂²×C₄

yes

characteristic

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₆₃

C₄×D₄

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#101

yes

complemented

C₄×C₁₆₃

#121

yes

—

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

C₂×C₁₆₃

—

C₂×C₁₆₃

#390

yes

characteristic

C₂×C₁₆₃

C₂×#101

yes

—

C₁₆₃

#1966

yes

characteristic, complemented, Sylow

#1966

—

163

maximal, Sylow

C₂×C₄⋊Q₈

D₁₆₃

yes

characteristic

#423

—

163

—

C₂×#101

—

163

—

C₄×M₁₆

—

163

—

C₂²×Q₈

C₂×D₁₆₃

yes

—

C₂×M₁₆

—

163

—

C₂×M₁₆

—

163

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₁₆₃

yes

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

C₄⋊C₈

—

326

—

#101

—

326

—

#97

—

326

—

C₄×C₈

—

326

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₁₆₃

yes

—

C₂²×C₄

C₂×Dic₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

326

—

M₁₆

—

326

—

M₁₆

—

163

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₁₆₃

yes

characteristic

C₄²

C₂²⋊D₁₆₃

yes

characteristic

C₄²

C₂²⋊D₁₆₃

yes

—

C₂³

C₂²×Dic₁₆₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²⋊Dic₁₆₃

yes

—

104

C₂×C₄

C₂²⋊Dic₁₆₃

yes

—

105

C₂×C₄

D₄×D₁₆₃

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

108

C₂×C₄

#14579698

yes

—

109

C₈

—

326

—

110

C₈

—

652

—

111

C₂²

—

112

C₂²

C₂×C₂²⋊Dic₁₆₃

yes

characteristic

113

C₂²

D₄×Dic₁₆₃

yes

—

114

C₄

—

115

C₄

—

116

C₄

#19104654

yes

complemented

117

C₄

#19104706

yes

—

118

C₄

#19104722

yes

characteristic

119

C₄

#19104731

yes

characteristic

120

C₂

—

121

C₂

#23781568

yes

characteristic

122

C₂

C₂×#19104654

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₆₃
commutator, prime radical	C₂³×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	C₂×C₄⋊Q₈×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₆₃
hypocenter	C₁₆₃	#1966
LCS	G ▷ C₂³×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₆₃

commutator, prime radical

C₂³×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

C₂×C₄⋊Q₈×C₁₆₃

C₂

socle

C₂²×C₁₆₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₆₃

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₆₃

hypocenter

C₁₆₃

#1966

LCS

G ▷ C₂³×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
54. C₄×C₁₆₃	#101	semidirect
61. C₁₆₃	#1966	semidirect
116. C₄	#19104654	semidirect

H ≅ G/N

type

54. C₄×C₁₆₃

#101

semidirect

61. C₁₆₃

#1966

semidirect

116. C₄

#19104654

semidirect

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

subgroup	index
#23781689	2
C₂×#19104654	2
C₄×#14579683	2
C₂×C₄⋊Q₈×C₁₆₃	2
#1966	163

subgroup

index

C₂×C₄⋊Q₈×C₁₆₃

163

Subgroups — #35215381

Conjugacy classes of subgroups

Complements

Maximal subgroups