Group #35213859 — subgroups

Subgroups — #35213859

Order 20864.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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	#23782035	C₂	1	1	yes	maximal, characteristic, complemented
2	#23782024	C₂	1	2	yes	maximal, complemented
3	#23781407	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781405	C₂	1	2	yes	maximal, complemented
5	#562×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	#19104735	C₂²	1	1	yes	characteristic
7	#19104659	—	2	2	no	—
8	#19104658	—	2	2	no	—
9	#19104656	—	2	2	no	—
10	#19104646	C₂²	1	1	yes	characteristic
11	#19104646	C₂²	1	2	yes	—
12	#125×C₁₆₃	C₂²	1	1	yes	characteristic
13	#124×C₁₆₃	C₂²	1	2	yes	—
14	#123×C₁₆₃	—	2	2	no	—
15	C₂²⋊Q₈×C₁₆₃	—	2	2	no	—
16	#121×C₁₆₃	—	2	2	no	—
17	C₂×Dic₃₂₆	—	2	1	no	—
18	C₂×C₁₆₃⋊C₈	—	2	2	no	—
19	C₂×C₁₆₃⋊C₈	—	2	1	no	—
20	C₄⋊Dic₁₆₃	—	2	2	no	—
21	C₂×Q₈×C₁₆₃	D₄	1	2	yes	—
22	C₂×D₄×C₁₆₃	—	2	2	no	—
23	C₂×D₄×C₁₆₃	D₄	1	2	yes	—
24	C₂²×C₄×C₁₆₃	—	2	2	no	—
25	C₄⋊C₄×C₁₆₃	—	2	2	no	—
26	C₄⋊C₄×C₁₆₃	—	2	2	no	—
27	C₄⋊C₄×C₁₆₃	D₄	1	2	yes	—
28	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
29	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
30	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
31	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
32	Dic₃₂₆	—	4	2	no	—
33	C₂×Dic₁₆₃	—	4	1	no	—
34	C₁₆₃⋊C₈	—	4	1	no	—
35	C₁₆₃⋊C₈	—	4	2	no	—
36	C₂³×C₁₆₃	—	2	2	no	—
37	Q₈×C₁₆₃	—	4	2	no	—
38	D₄×C₁₆₃	—	4	2	no	—
39	D₄×C₁₆₃	—	8	2	no	—
40	C₂×C₄×C₁₆₃	—	2	2	no	—
41	C₂×C₄×C₁₆₃	—	2	2	no	—
42	C₂×C₄×C₁₆₃	—	8	2	no	—
43	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
45	C₂²×C₁₆₃	—	4	2	no	—
46	C₂²×C₁₆₃	—	4	2	no	—
47	C₂²×C₁₆₃	—	4	2	no	—
48	C₂²×C₁₆₃	#120	1	1	yes	characteristic
49	C₄×C₁₆₃	—	2	1	no	—
50	C₄×C₁₆₃	—	2	2	no	—
51	C₄×C₁₆₃	—	4	2	no	—
52	C₄×C₁₆₃	—	4	2	no	—
53	Dic₁₆₃	—	8	1	no	—
54	C₂×C₁₆₃	—	8	2	no	—
55	C₂×C₁₆₃	#454	1	1	yes	characteristic
56	C₂×C₁₆₃	#455	1	2	yes	—
57	C₁₆₃	#1662	1	1	yes	characteristic, complemented, Sylow
58	#1662	—	163	1	no	maximal, Sylow
59	#562	D₁₆₃	1	1	yes	characteristic
60	#491	—	163	1	no	—
61	#489	—	163	2	no	—
62	#330	—	163	1	no	—
63	#329	—	163	2	no	—
64	C₄⋊Q₈	—	163	1	no	—
65	#125	C₂×D₁₆₃	1	1	yes	characteristic
66	#124	C₂×D₁₆₃	1	2	yes	—
67	#123	—	2	2	no	—
68	C₂²⋊Q₈	—	2	2	no	—
69	#121	—	2	2	no	—
70	#103	—	326	2	no	—
71	#103	—	326	2	no	—
72	#102	—	326	2	no	—
73	#97	—	163	1	no	—
74	#97	—	163	2	no	—
75	C₂×Q₈	—	326	1	no	—
76	C₂×Q₈	C₂²⋊D₁₆₃	1	2	yes	—
77	C₂×D₄	—	2	2	no	—
78	C₂×D₄	C₂²⋊D₁₆₃	1	2	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂×C₈	—	326	2	no	—
81	C₂×C₈	—	326	1	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	326	2	no	—
85	C₄⋊C₄	C₂²⋊D₁₆₃	1	2	yes	—
86	C₂²⋊C₄	—	2	2	no	—
87	C₂²⋊C₄	—	2	2	no	—
88	C₂²⋊C₄	—	2	2	no	—
89	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
90	C₂³	—	2	2	no	—
91	Q₈	—	4	2	no	—
92	Q₈	—	652	2	no	—
93	D₄	—	4	2	no	—
94	D₄	—	8	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	652	1	no	—
99	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
101	C₈	—	652	1	no	—
102	C₈	—	652	2	no	—
103	C₂²	—	4	2	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	#19104704	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	2	no	—
109	C₄	—	4	2	no	—
110	C₄	—	4	2	no	—
111	C₄	—	1304	1	no	—
112	C₂	—	8	2	no	—
113	C₂	#23781838	1	1	yes	characteristic
114	C₂	#23781843	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23782035

C₂

yes

maximal, characteristic, complemented

#23782024

C₂

yes

maximal, complemented

#23781407

C₂

yes

maximal, characteristic, complemented

#23781405

C₂

yes

maximal, complemented

#562×C₁₆₃

C₂

yes

maximal, characteristic

#19104735

C₂²

yes

—

—

—

C₂²

yes

characteristic

#19104646

C₂²

yes

—

#125×C₁₆₃

C₂²

yes

characteristic

#124×C₁₆₃

C₂²

yes

—

#123×C₁₆₃

—

C₂²⋊Q₈×C₁₆₃

—

#121×C₁₆₃

—

C₂×Dic₃₂₆

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₄⋊Dic₁₆₃

—

C₂×Q₈×C₁₆₃

D₄

yes

—

C₂×D₄×C₁₆₃

—

C₂×D₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

D₄

yes

—

C₂²⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂³

yes

characteristic

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

—

Q₈×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

#120

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#454

yes

characteristic

C₂×C₁₆₃

#455

yes

—

C₁₆₃

#1662

yes

characteristic, complemented, Sylow

#1662

—

163

maximal, Sylow

#562

D₁₆₃

yes

characteristic

#491

—

163

—

#489

—

163

—

#330

—

163

—

#329

—

163

—

C₄⋊Q₈

—

163

—

#125

C₂×D₁₆₃

yes

characteristic

#124

C₂×D₁₆₃

yes

—

#123

—

C₂²⋊Q₈

—

#121

—

#103

—

326

—

#103

—

326

—

#102

—

326

—

#97

—

163

—

#97

—

163

—

C₂×Q₈

—

326

—

C₂×Q₈

C₂²⋊D₁₆₃

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₄⋊C₄

C₂²⋊D₁₆₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²×D₁₆₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

652

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

652

—

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

101

C₈

—

652

—

102

C₈

—

652

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

#19104704

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

1304

—

112

C₂

—

113

C₂

#23781838

yes

characteristic

114

C₂

#23781843

yes

—

	subgroup	G/H
center	C₂²	#19104704
commutator, cosocle	C₄²×C₁₆₃	C₂³
Frattini	C₄²	C₂²×D₁₆₃
Fitting, prime radical	#562×C₁₆₃	C₂
socle	C₂²×C₁₆₃	#120
hypercenter	#562	D₁₆₃
hypocenter	C₁₆₃	#1662
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #562

subgroup

G/H

center

C₂²

#19104704

commutator, cosocle

C₄²×C₁₆₃

C₂³

Frattini

C₄²

C₂²×D₁₆₃

Fitting, prime radical

#562×C₁₆₃

C₂

C₂²×C₁₆₃

D₁₆₃

C₁₆₃

G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #562

N	H ≅ G/N	type
1. #23782035	C₂	semidirect
2. #23782024	C₂	semidirect
3. #23781407	C₂	semidirect
4. #23781405	C₂	semidirect
57. C₁₆₃	#1662	semidirect

H ≅ G/N

type

1. #23782035

C₂

semidirect

2. #23782024

C₂

semidirect

3. #23781407

C₂

semidirect

4. #23781405

C₂

semidirect

57. C₁₆₃

#1662

semidirect

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

subgroup	index
#23782035	2
#23782024	2
#23781407	2
#23781405	2
#562×C₁₆₃	2
#1662	163

subgroup

index

#562×C₁₆₃

163

Subgroups — #35213859

Conjugacy classes of subgroups

Complements

Maximal subgroups