Group #13091494 — subgroups

Subgroups — #13091494

Order 1458.

263 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 48 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	#10624890	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	#92311	C₃	1	3	yes	maximal, complemented
3	#92302	C₃	1	1	yes	maximal, characteristic
4	C₉×AGL(1,ℤ/9)	—	3	3	no	maximal
5	C₃×(C₃⋊D₃)⋊C₉	—	3	1	no	maximal
6	#6899	—	2	6	no	—
7	#6899	—	2	3	no	—
8	#6898	—	2	3	no	—
9	#6898	C₂×C₃	1	3	yes	complemented
10	#6894	—	2	3	no	—
11	#6891	—	2	1	no	—
12	#6891	C₂×C₃	1	1	yes	characteristic
13	C₉×3₋¹⁺²	D₃	1	3	yes	complemented
14	C₃×C₃²⋊C₉	D₃	1	1	yes	characteristic
15	C₃²×C₃⋊D₃	—	3	1	no	—
16	C₃×AGL(1,ℤ/9)	—	3	3	no	—
17	C₃×D₃×C₉	—	9	1	no	—
18	#4181	C₃²	1	1	yes	characteristic
19	D₉⋊C₉	—	3	6	no	—
20	(C₃⋊D₃)⋊C₉	—	3	3	no	—
21	C₉×D₉	—	3	3	no	—
22	C₃⁴	C₃×D₃	1	1	yes	characteristic
23	C₃×3₋¹⁺²	—	2	2	no	—
24	C₃×3₋¹⁺²	—	2	1	no	—
25	C₃×3₋¹⁺²	C₃×D₃	1	3	yes	—
26	C₃²×C₉	—	2	1	no	—
27	C₃²×C₉	C₃⋊D₃	1	1	yes	characteristic
28	C₉⋊C₉	—	2	9	no	—
29	C₉⋊C₉	—	2	6	no	—
30	C₉⋊C₉	—	2	3	no	—
31	C₉⋊C₉	—	2	3	no	—
32	C₉⋊C₉	—	2	6	no	—
33	C₉⋊C₉	—	2	6	no	—
34	C₉⋊C₉	C₃×D₃	1	6	yes	complemented
35	C₃²⋊C₉	—	2	3	no	—
36	C₃²⋊C₉	—	2	3	no	—
37	C₃²⋊C₉	—	2	1	no	—
38	C₃²⋊C₉	—	2	3	no	—
39	C₃²⋊C₉	—	2	3	no	—
40	C₃²⋊C₉	—	2	3	no	—
41	C₃²⋊C₉	C₃×D₃	1	3	yes	—
42	C₃²⋊C₉	C₂×C₃²	1	1	yes	characteristic
43	C₉²	—	2	3	no	—
44	C₉²	C₃×D₃	1	3	yes	complemented
45	C₃×C₃⋊D₃	—	3	3	no	—
46	C₃×C₃⋊D₃	—	3	1	no	—
47	C₃²×D₃	—	9	1	no	—
48	C₃²×D₃	—	27	1	no	—
49	C₂×C₃×C₉	—	27	1	no	—
50	AGL(1,ℤ/9)	—	3	9	no	—
51	D₃×C₉	—	9	3	no	—
52	C₃×D₉	—	3	3	no	—
53	C₃³	—	2	1	no	—
54	C₃³	—	3	3	no	—
55	C₃³	—	9	1	no	—
56	C₃³	—	18	1	no	—
57	C₃³	C₃²×D₃	1	1	yes	characteristic
58	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
59	3₋¹⁺²	—	3	9	no	—
60	3₋¹⁺²	—	18	2	no	—
61	3₋¹⁺²	—	18	1	no	—
62	C₃×C₉	—	2	3	no	—
63	C₃×C₉	—	2	6	no	—
64	C₃×C₉	—	2	3	no	—
65	C₃×C₉	—	2	3	no	—
66	C₃×C₉	—	9	1	no	—
67	C₃×C₉	—	18	1	no	—
68	C₃×C₉	C₃²×D₃	1	3	yes	—
69	C₃×C₉	C₃×C₃⋊D₃	1	3	yes	—
70	C₂×C₃²	—	27	1	no	—
71	C₃⋊D₃	—	3	1	no	—
72	C₃×D₃	—	9	3	no	—
73	C₃×D₃	—	9	1	no	—
74	C₃×D₃	—	27	1	no	—
75	C₃×D₃	—	27	3	no	—
76	C₂×C₉	—	27	3	no	—
77	D₉	—	3	3	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	1	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	6	1	no	—
82	C₃²	—	6	1	no	—
83	C₃²	—	6	1	no	—
84	C₃²	—	6	1	no	—
85	C₃²	—	6	1	no	—
86	C₃²	—	6	1	no	—
87	C₃²	—	9	3	no	—
88	C₃²	—	18	3	no	—
89	C₃²	#4208	1	3	yes	—
90	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
91	C₉	—	3	3	no	—
92	C₉	—	3	3	no	—
93	C₉	—	6	6	no	—
94	C₉	—	6	3	no	—
95	C₉	—	6	3	no	—
96	C₉	—	6	3	no	—
97	C₂×C₃	—	27	3	no	—
98	C₂×C₃	—	27	1	no	—
99	D₃	—	9	1	no	—
100	D₃	—	27	1	no	—
101	C₃	—	2	1	no	—
102	C₃	—	3	3	no	—
103	C₃	—	9	1	no	—
104	C₃	—	18	1	no	—
105	C₃	C₃×#4208	1	1	yes	characteristic
106	C₃	#4207⋊C₃	1	1	yes	characteristic
107	C₂	—	27	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624890

C₂

yes

maximal, characteristic, complemented, Sylow

#92311

C₃

yes

maximal, complemented

#92302

C₃

yes

maximal, characteristic

C₉×AGL(1,ℤ/9)

—

maximal

C₃×(C₃⋊D₃)⋊C₉

—

—

—

—

C₂×C₃

yes

—

—

C₂×C₃

yes

characteristic

C₉×3₋¹⁺²

D₃

yes

complemented

C₃×C₃²⋊C₉

D₃

yes

characteristic

C₃²×C₃⋊D₃

—

C₃×AGL(1,ℤ/9)

—

C₃×D₃×C₉

—

#4181

C₃²

yes

characteristic

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×D₃

yes

—

C₃²×C₉

—

C₃²×C₉

C₃⋊D₃

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

—

C₃²⋊C₉

C₃×D₃

yes

—

C₃²⋊C₉

C₂×C₃²

yes

characteristic

C₉²

—

C₉²

C₃×D₃

yes

complemented

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₂×C₃×C₉

—

AGL(1,ℤ/9)

—

D₃×C₉

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×D₃

yes

characteristic

C₃³

C₃×C₃⋊D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

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

—

C₃²

—

C₃²

—

C₃²

#4208

yes

—

C₃²

C₃²×C₃⋊D₃

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₂×C₃

—

C₂×C₃

—

D₃

—

100

D₃

—

101

C₃

—

102

C₃

—

103

C₃

—

104

C₃

—

105

C₃

C₃×#4208

yes

characteristic

106

C₃

#4207⋊C₃

yes

characteristic

107

C₂

—

Sylow

	subgroup	G/H
center, hypercenter	C₃	#4207⋊C₃
commutator, cosocle, hypocenter	C₃²⋊C₉	C₂×C₃²
Frattini, socle	C₃²	C₃²×C₃⋊D₃
Fitting	#10624890	C₂
prime radical	#92302	C₃
LCS	G ▷ C₃²⋊C₉
UCS	C₁ ◁ C₃

subgroup

G/H

center, hypercenter

C₃

#4207⋊C₃

commutator, cosocle, hypocenter

C₃²⋊C₉

C₂×C₃²

Frattini, socle

C₃²

C₃²×C₃⋊D₃

C₂

C₃

G ▷ C₃²⋊C₉

C₁ ◁ C₃

N	H ≅ G/N	type
1. #10624890	C₂	semidirect
2. #92311	C₃	semidirect
9. #6898	C₂×C₃	semidirect
13. C₉×3₋¹⁺²	D₃	semidirect
34. C₉⋊C₉	C₃×D₃	semidirect
44. C₉²	C₃×D₃	semidirect

H ≅ G/N

type

1. #10624890

C₂

semidirect

2. #92311

C₃

semidirect

9. #6898

C₂×C₃

semidirect

13. C₉×3₋¹⁺²

D₃

semidirect

34. C₉⋊C₉

C₃×D₃

semidirect

44. C₉²

C₃×D₃

semidirect

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

subgroup	index
#10624890	2
#92311	3
#92302	3
C₉×AGL(1,ℤ/9)	3
C₃×(C₃⋊D₃)⋊C₉	3

subgroup

index

C₃×(C₃⋊D₃)⋊C₉

Subgroups — #13091494

Conjugacy classes of subgroups

Complements

Maximal subgroups