Group #13091479 — subgroups

Subgroups — #13091479

Order 1458.

272 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 51 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	#10624884	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×C₃⋊AGL(1,ℤ/9)	C₃	1	1	yes	maximal, characteristic
3	#92294	C₃	1	3	yes	maximal, complemented
4	#92265	—	3	3	no	maximal
5	#92259	—	3	1	no	maximal
6	C₃²×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic
7	#6900	—	2	3	no	—
8	#6900	D₃	1	3	yes	complemented
9	#6899	—	2	6	no	—
10	#6899	—	2	3	no	—
11	#6895	—	2	3	no	—
12	#6891	—	2	1	no	—
13	#6891	D₃	1	1	yes	characteristic
14	#6887	C₂×C₃	1	3	yes	complemented
15	C₃²×C₃⋊D₃	—	3	1	no	—
16	C₃⋊AGL(1,ℤ/9)	—	3	1	no	—
17	C₃×C₃⋊D₉	C₃²	1	1	yes	characteristic
18	D₃×3₋¹⁺²	—	9	1	no	—
19	C₃×AGL(1,ℤ/9)	—	3	3	no	—
20	D₉⋊C₉	—	3	6	no	—
21	(C₃⋊D₃)⋊C₉	—	3	3	no	—
22	C₉×D₉	—	3	3	no	—
23	C₃⁴	C₃×D₃	1	1	yes	characteristic
24	C₃×3₋¹⁺²	—	2	3	no	—
25	C₃×3₋¹⁺²	—	2	1	no	—
26	C₃×3₋¹⁺²	—	2	3	no	—
27	C₃×3₋¹⁺²	—	3	1	no	—
28	C₃×3₋¹⁺²	—	6	3	no	—
29	C₃×3₋¹⁺²	—	6	1	no	—
30	C₃×3₋¹⁺²	C₃×D₃	1	3	yes	—
31	C₃×3₋¹⁺²	C₃⋊D₃	1	1	yes	characteristic
32	C₃²×C₉	—	2	1	no	—
33	C₃²×C₉	C₂×C₃²	1	1	yes	characteristic
34	C₉⋊C₉	—	2	9	no	—
35	C₉⋊C₉	—	2	6	no	—
36	C₉⋊C₉	—	2	3	no	—
37	C₉⋊C₉	—	2	6	no	—
38	C₉⋊C₉	—	2	3	no	—
39	C₉⋊C₉	—	2	6	no	—
40	C₉⋊C₉	C₃×D₃	1	6	yes	complemented
41	C₃²⋊C₉	—	2	2	no	—
42	C₃²⋊C₉	—	2	3	no	—
43	C₃²⋊C₉	—	2	1	no	—
44	C₃²⋊C₉	—	2	3	no	—
45	C₃²⋊C₉	—	2	3	no	—
46	C₃²⋊C₉	C₃×D₃	1	3	yes	—
47	C₉²	—	2	3	no	—
48	C₉²	C₃×D₃	1	3	yes	complemented
49	C₃×C₃⋊D₃	—	3	1	no	—
50	C₃×C₃⋊D₃	—	9	1	no	—
51	C₃²×D₃	—	9	1	no	—
52	C₃²×D₃	—	27	1	no	—
53	C₂×3₋¹⁺²	—	27	1	no	—
54	C₃⋊D₉	3₋¹⁺²	1	1	yes	characteristic, complemented
55	AGL(1,ℤ/9)	—	9	3	no	—
56	D₃×C₉	—	9	3	no	—
57	C₃×D₉	—	3	3	no	—
58	C₃³	—	2	1	no	—
59	C₃³	—	3	1	no	—
60	C₃³	—	6	1	no	—
61	C₃³	—	9	1	no	—
62	C₃³	—	18	1	no	—
63	C₃³	C₃²×D₃	1	1	yes	characteristic
64	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
65	3₋¹⁺²	—	3	3	no	—
66	3₋¹⁺²	—	6	6	no	—
67	3₋¹⁺²	—	6	3	no	—
68	3₋¹⁺²	—	6	3	no	—
69	3₋¹⁺²	—	9	1	no	—
70	3₋¹⁺²	—	18	1	no	—
71	3₋¹⁺²	—	18	3	no	—
72	C₃×C₉	—	2	1	no	—
73	C₃×C₉	—	2	3	no	—
74	C₃×C₉	—	2	6	no	—
75	C₃×C₉	—	2	3	no	—
76	C₃×C₉	—	2	3	no	—
77	C₃×C₉	—	2	3	no	—
78	C₃×C₉	—	6	2	no	—
79	C₃×C₉	—	6	1	no	—
80	C₃×C₉	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
81	C₃×C₉	C₃²×D₃	1	3	yes	—
82	C₃×C₉	C₃×C₃⋊D₃	1	3	yes	—
83	C₂×C₃²	—	27	1	no	—
84	C₃⋊D₃	—	3	1	no	—
85	C₃×D₃	—	9	1	no	—
86	C₃×D₃	—	27	1	no	—
87	C₃×D₃	—	27	1	no	—
88	C₃×D₃	—	27	3	no	—
89	C₂×C₉	—	27	3	no	—
90	D₉	—	3	3	no	—
91	C₃²	—	2	1	no	—
92	C₃²	—	3	1	no	—
93	C₃²	—	3	1	no	—
94	C₃²	—	3	1	no	—
95	C₃²	—	6	1	no	—
96	C₃²	—	6	1	no	—
97	C₃²	—	6	2	no	—
98	C₃²	—	6	1	no	—
99	C₃²	—	6	1	no	—
100	C₃²	—	9	3	no	—
101	C₃²	—	18	3	no	—
102	C₃²	D₃×3₋¹⁺²	1	1	yes	characteristic
103	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
104	C₉	—	2	3	no	—
105	C₉	—	3	3	no	—
106	C₉	—	6	6	no	—
107	C₉	—	6	3	no	—
108	C₉	—	6	3	no	—
109	C₉	—	6	3	no	—
110	C₉	D₃×3₋¹⁺²	1	3	yes	—
111	C₂×C₃	—	27	1	no	—
112	C₂×C₃	—	81	1	no	—
113	D₃	—	9	1	no	—
114	D₃	—	27	1	no	—
115	C₃	—	2	1	no	—
116	C₃	—	3	1	no	—
117	C₃	—	6	1	no	—
118	C₃	—	9	1	no	—
119	C₃	—	18	1	no	—
120	C₃	C₃⋊D₃×3₋¹⁺²	1	1	yes	characteristic
121	C₃	#4207⋊C₃	1	1	yes	characteristic
122	C₂	—	27	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624884

C₂

yes

maximal, characteristic, complemented, Sylow

C₃×C₃⋊AGL(1,ℤ/9)

C₃

yes

maximal, characteristic

#92294

C₃

yes

maximal, complemented

—

—

C₃²×3₋¹⁺²

C₂×C₃

yes

characteristic

#6900

—

#6900

D₃

yes

—

—

—

—

D₃

yes

characteristic

#6887

C₂×C₃

yes

complemented

C₃²×C₃⋊D₃

—

C₃⋊AGL(1,ℤ/9)

—

C₃×C₃⋊D₉

C₃²

yes

characteristic

D₃×3₋¹⁺²

—

C₃×AGL(1,ℤ/9)

—

D₉⋊C₉

—

(C₃⋊D₃)⋊C₉

—

C₉×D₉

—

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×D₃

yes

—

C₃×3₋¹⁺²

C₃⋊D₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₂×C₃²

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×D₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×D₃

yes

—

C₉²

—

C₉²

C₃×D₃

yes

complemented

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

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×D₃

yes

characteristic

C₃³

C₃×C₃⋊D₃

yes

characteristic

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

C₂×3₋¹⁺²

yes

characteristic, complemented

C₃×C₉

C₃²×D₃

yes

—

C₃×C₉

C₃×C₃⋊D₃

yes

—

C₂×C₃²

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

D₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

D₃×3₋¹⁺²

yes

characteristic

103

C₃²

C₃²×C₃⋊D₃

yes

characteristic

104

C₉

—

105

C₉

—

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

D₃×3₋¹⁺²

yes

—

111

C₂×C₃

—

112

C₂×C₃

—

113

D₃

—

114

D₃

—

115

C₃

—

116

C₃

—

117

C₃

—

118

C₃

—

119

C₃

—

120

C₃

C₃⋊D₃×3₋¹⁺²

yes

characteristic

121

C₃

#4207⋊C₃

yes

characteristic

122

C₂

—

Sylow

	subgroup	G/H
center, hypercenter	C₃	#4207⋊C₃
commutator, cosocle	C₃²×C₉	C₂×C₃²
Frattini, socle	C₃²	C₃²×C₃⋊D₃
Fitting	#10624884	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

#4207⋊C₃

commutator, cosocle

C₃²×C₉

C₂×C₃²

Frattini, socle

C₃²

C₃²×C₃⋊D₃

C₂

C₃

C₃×C₉

C₂×3₋¹⁺²

LCS

G ▷ C₃²×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. #10624884	C₂	semidirect
3. #92294	C₃	semidirect
8. #6900	D₃	semidirect
14. #6887	C₂×C₃	semidirect
40. C₉⋊C₉	C₃×D₃	semidirect
48. C₉²	C₃×D₃	semidirect
54. C₃⋊D₉	3₋¹⁺²	semidirect
80. C₃×C₉	C₂×3₋¹⁺²	semidirect

H ≅ G/N

type

1. #10624884

C₂

semidirect

3. #92294

C₃

semidirect

8. #6900

D₃

semidirect

14. #6887

C₂×C₃

semidirect

40. C₉⋊C₉

C₃×D₃

semidirect

48. C₉²

C₃×D₃

semidirect

54. C₃⋊D₉

3₋¹⁺²

semidirect

80. 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
#10624884	2
C₃×C₃⋊AGL(1,ℤ/9)	3
#92294	3
#92265	3
#92259	3

subgroup

index

Subgroups — #13091479

Conjugacy classes of subgroups

Complements

Maximal subgroups