Group #13090669 — subgroups

Subgroups — #13090669

Order 1458.

224 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 35 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	#10624567	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊D₃×3₋¹⁺²	—	3	1	no	maximal
3	C₃×C₃⋊AGL(1,ℤ/9)	C₃	1	1	yes	maximal, characteristic
4	#92158	C₃	1	3	yes	maximal, complemented
5	C₃²×3₋¹⁺²	D₃	1	1	yes	characteristic
6	C₃²×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic
7	C₃×C₃²⋊C₉	—	2	1	no	—
8	#6855	—	2	3	no	—
9	#6855	C₂×C₃	1	3	yes	complemented
10	C₃²×C₃⋊D₃	—	3	1	no	—
11	C₃⋊AGL(1,ℤ/9)	—	3	1	no	—
12	C₉×C₃⋊D₃	—	3	3	no	—
13	C₃×C₃⋊D₉	C₃²	1	1	yes	characteristic
14	D₃×3₋¹⁺²	—	9	1	no	—
15	D₃×3₋¹⁺²	—	9	3	no	—
16	C₃×AGL(1,ℤ/9)	—	9	1	no	—
17	C₃⁴	C₃×D₃	1	1	yes	characteristic
18	C₃×3₋¹⁺²	—	3	1	no	—
19	C₃×3₋¹⁺²	—	3	3	no	—
20	C₃×3₋¹⁺²	—	3	1	no	—
21	C₃×3₋¹⁺²	—	3	1	no	—
22	C₃×3₋¹⁺²	—	6	3	no	—
23	C₃×3₋¹⁺²	—	6	3	no	—
24	C₃×3₋¹⁺²	—	6	1	no	—
25	C₃×3₋¹⁺²	—	6	1	no	—
26	C₃×3₋¹⁺²	—	6	1	no	—
27	C₃²×C₉	—	2	1	no	—
28	C₃²×C₉	—	2	3	no	—
29	C₃²×C₉	C₃×D₃	1	3	yes	—
30	C₃²×C₉	C₂×C₃²	1	1	yes	characteristic
31	C₃²⋊C₉	—	6	2	no	—
32	C₃²⋊C₉	—	6	1	no	—
33	C₃×C₃⋊D₃	—	3	1	no	—
34	C₃×C₃⋊D₃	—	9	1	no	—
35	C₃²×D₃	—	9	3	no	—
36	C₃²×D₃	—	9	1	no	—
37	C₂×3₋¹⁺²	—	27	1	no	—
38	C₃⋊D₉	3₋¹⁺²	1	1	yes	characteristic, complemented
39	AGL(1,ℤ/9)	—	9	3	no	—
40	D₃×C₉	—	9	3	no	—
41	D₃×C₉	—	9	6	no	—
42	D₃×C₉	—	9	3	no	—
43	C₃×D₉	—	9	1	no	—
44	C₃³	—	2	1	no	—
45	C₃³	—	3	1	no	—
46	C₃³	—	3	3	no	—
47	C₃³	—	6	1	no	—
48	C₃³	—	6	3	no	—
49	C₃³	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
50	C₃³	C₃²×D₃	1	1	yes	characteristic
51	3₋¹⁺²	—	3	1	no	—
52	3₋¹⁺²	—	3	3	no	—
53	3₋¹⁺²	—	6	3	no	—
54	3₋¹⁺²	—	6	3	no	—
55	3₋¹⁺²	—	6	3	no	—
56	3₋¹⁺²	—	6	3	no	—
57	3₋¹⁺²	—	6	3	no	—
58	3₋¹⁺²	—	6	1	no	—
59	3₋¹⁺²	—	6	3	no	—
60	3₋¹⁺²	—	6	3	no	—
61	3₋¹⁺²	—	6	3	no	—
62	C₃×C₉	—	2	3	no	—
63	C₃×C₉	—	2	3	no	—
64	C₃×C₉	—	2	1	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	3	no	—
67	C₃×C₉	—	3	6	no	—
68	C₃×C₉	—	6	1	no	—
69	C₃×C₉	—	6	1	no	—
70	C₃×C₉	—	6	1	no	—
71	C₃×C₉	—	6	1	no	—
72	C₃×C₉	—	6	9	no	—
73	C₃×C₉	—	6	3	no	—
74	C₃×C₉	AGL(1,ℤ/9)	1	3	yes	complemented
75	C₃×C₉	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
76	C₂×C₃²	—	27	1	no	—
77	C₃⋊D₃	—	3	1	no	—
78	C₃×D₃	—	9	1	no	—
79	C₃×D₃	—	9	3	no	—
80	C₃×D₃	—	27	1	no	—
81	C₃×D₃	—	27	3	no	—
82	C₂×C₉	—	27	3	no	—
83	D₉	—	9	1	no	—
84	C₃²	—	2	1	no	—
85	C₃²	—	2	3	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	3	1	no	—
88	C₃²	—	6	1	no	—
89	C₃²	—	6	2	no	—
90	C₃²	—	6	1	no	—
91	C₃²	—	6	1	no	—
92	C₃²	—	9	3	no	—
93	C₃²	—	18	3	no	—
94	C₃²	C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
95	C₃²	C₃×AGL(1,ℤ/9)	1	3	yes	—
96	C₃²	D₃×3₋¹⁺²	1	1	yes	characteristic
97	C₉	—	3	1	no	—
98	C₉	—	3	3	no	—
99	C₉	—	6	1	no	—
100	C₉	—	6	3	no	—
101	C₉	—	6	3	no	—
102	C₉	—	18	3	no	—
103	C₂×C₃	—	27	1	no	—
104	C₂×C₃	—	81	1	no	—
105	D₃	—	9	1	no	—
106	D₃	—	9	3	no	—
107	C₃	—	2	1	no	—
108	C₃	—	2	3	no	—
109	C₃	—	9	1	no	—
110	C₃	—	18	1	no	—
111	C₃	#92259	1	1	yes	characteristic
112	C₃	#4169⋊C₃	1	1	yes	characteristic
113	C₃	#92264	1	3	yes	—
114	C₂	—	27	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624567

C₂

yes

maximal, characteristic, complemented, Sylow

C₃⋊D₃×3₋¹⁺²

—

maximal

C₃×C₃⋊AGL(1,ℤ/9)

C₃

yes

maximal, characteristic

#92158

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

D₃

yes

characteristic

C₃²×3₋¹⁺²

C₂×C₃

yes

characteristic

C₃×C₃²⋊C₉

—

#6855

—

#6855

C₂×C₃

yes

complemented

C₃²×C₃⋊D₃

—

C₃⋊AGL(1,ℤ/9)

—

C₉×C₃⋊D₃

—

C₃×C₃⋊D₉

C₃²

yes

characteristic

D₃×3₋¹⁺²

—

D₃×3₋¹⁺²

—

C₃×AGL(1,ℤ/9)

—

C₃⁴

C₃×D₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×D₃

yes

—

C₃²×C₉

C₂×C₃²

yes

characteristic

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₃⋊D₉

3₋¹⁺²

yes

characteristic, complemented

AGL(1,ℤ/9)

—

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

(C₃⋊D₃)⋊C₃

yes

characteristic

C₃³

C₃²×D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

AGL(1,ℤ/9)

yes

complemented

C₃×C₉

C₂×3₋¹⁺²

yes

characteristic, complemented

C₂×C₃²

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

D₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×(C₃⋊D₃)⋊C₃

yes

characteristic

C₃²

C₃×AGL(1,ℤ/9)

yes

—

C₃²

D₃×3₋¹⁺²

yes

characteristic

C₉

—

C₉

—

C₉

—

100

C₉

—

101

C₉

—

102

C₉

—

103

C₂×C₃

—

104

C₂×C₃

—

105

D₃

—

106

D₃

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

#92259

yes

characteristic

112

C₃

#4169⋊C₃

yes

characteristic

113

C₃

#92264

yes

—

114

C₂

—

Sylow

	subgroup	G/H
center, hypercenter	C₃	#4169⋊C₃
commutator, cosocle	C₃²×C₉	C₂×C₃²
Frattini, socle	C₃³	C₃²×D₃
Fitting	#10624567	C₂
prime radical	C₃×C₃⋊AGL(1,ℤ/9)	C₃
hypocenter	C₃×C₉	C₂×3₋¹⁺²
LCS	G ▷ C₃²×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₃

subgroup

G/H

center, hypercenter

C₃

#4169⋊C₃

commutator, cosocle

C₃²×C₉

C₂×C₃²

Frattini, socle

C₃³

C₃²×D₃

C₂

C₃

C₃×C₉

C₂×3₋¹⁺²

LCS

G ▷ C₃²×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. #10624567	C₂	semidirect
4. #92158	C₃	semidirect
9. #6855	C₂×C₃	semidirect
38. C₃⋊D₉	3₋¹⁺²	semidirect
74. C₃×C₉	AGL(1,ℤ/9)	semidirect
75. C₃×C₉	C₂×3₋¹⁺²	semidirect

H ≅ G/N

type

1. #10624567

C₂

semidirect

4. #92158

C₃

semidirect

9. #6855

C₂×C₃

semidirect

38. C₃⋊D₉

3₋¹⁺²

semidirect

74. C₃×C₉

AGL(1,ℤ/9)

semidirect

75. C₃×C₉

C₂×3₋¹⁺²

semidirect

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

subgroup	index
#10624567	2
C₃⋊D₃×3₋¹⁺²	3
C₃×C₃⋊AGL(1,ℤ/9)	3
#92158	3

subgroup

index

#10624567

C₃⋊D₃×3₋¹⁺²

C₃×C₃⋊AGL(1,ℤ/9)

#92158

Subgroups — #13090669

Conjugacy classes of subgroups

Complements

Maximal subgroups