Group #13091369 — subgroups

Subgroups — C₃×D₉×3₋¹⁺² #13091369

Order 1458.

458 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 128 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	C₃×C₉×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×D₃×3₋¹⁺²	—	3	1	no	maximal
3	C₃³×D₉	C₃	1	1	yes	maximal, characteristic
4	D₉×3₋¹⁺²	C₃	1	9	yes	maximal, direct product factor
5	C₃×C₉×D₉	C₃	1	3	yes	maximal, complemented
6	C₃²×3₋¹⁺²	D₃	1	1	yes	characteristic
7	C₃³×C₉	C₂×C₃	1	1	yes	characteristic
8	C₉×3₋¹⁺²	—	2	9	no	—
9	C₉×3₋¹⁺²	C₂×C₃	1	9	yes	complemented
10	C₃×C₉⋊C₉	—	2	3	no	—
11	C₃×C₃²⋊C₉	—	2	1	no	—
12	C₃×C₉²	C₂×C₃	1	3	yes	complemented
13	C₃³×D₃	—	3	1	no	—
14	C₂×C₃×3₋¹⁺²	—	9	1	no	—
15	D₃×3₋¹⁺²	—	3	9	no	—
16	C₃×D₃×C₉	—	3	3	no	—
17	C₃²×D₉	—	3	3	no	—
18	C₃²×D₉	C₃²	1	3	yes	—
19	C₃²×D₉	C₃²	1	1	yes	characteristic
20	C₉×D₉	C₃²	1	9	yes	complemented
21	C₃⁴	C₃×D₃	1	1	yes	characteristic
22	C₃×3₋¹⁺²	—	2	1	no	—
23	C₃×3₋¹⁺²	—	2	9	no	—
24	C₃×3₋¹⁺²	—	2	3	no	—
25	C₃×3₋¹⁺²	D₉	1	1	yes	characteristic, direct product factor
26	C₃×3₋¹⁺²	C₃×D₃	1	9	yes	—
27	C₃²×C₉	—	2	1	no	—
28	C₃²×C₉	—	2	3	no	—
29	C₃²×C₉	—	2	3	no	—
30	C₃²×C₉	—	3	3	no	—
31	C₃²×C₉	—	6	3	no	—
32	C₃²×C₉	C₃×D₃	1	3	yes	—
33	C₃²×C₉	C₂×C₃²	1	1	yes	characteristic
34	C₃²×C₉	C₂×C₃²	1	3	yes	—
35	C₉⋊C₉	—	2	27	no	—
36	C₃²⋊C₉	—	2	9	no	—
37	C₉²	—	2	9	no	—
38	C₉²	C₂×C₃²	1	9	yes	complemented
39	C₂×C₃³	—	9	1	no	—
40	C₃²×D₃	—	3	1	no	—
41	C₃²×D₃	—	3	3	no	—
42	C₃²×D₃	—	9	3	no	—
43	C₂×3₋¹⁺²	—	9	9	no	—
44	C₂×C₃×C₉	—	9	3	no	—
45	D₃×C₉	—	3	9	no	—
46	C₃×D₉	—	3	3	no	—
47	C₃×D₉	3₋¹⁺²	1	3	yes	direct product factor
48	C₃×D₉	C₃³	1	1	yes	characteristic
49	C₃³	—	2	3	no	—
50	C₃³	—	2	1	no	—
51	C₃³	—	3	3	no	—
52	C₃³	—	6	3	no	—
53	C₃³	C₃×D₉	1	1	yes	characteristic
54	C₃³	C₃²×D₃	1	1	yes	characteristic
55	C₃³	C₃²×D₃	1	3	yes	—
56	3₋¹⁺²	—	2	9	no	—
57	3₋¹⁺²	—	2	27	no	—
58	3₋¹⁺²	C₃×D₉	1	9	yes	direct product factor
59	C₃×C₉	—	2	3	no	—
60	C₃×C₉	—	2	3	no	—
61	C₃×C₉	—	2	1	no	—
62	C₃×C₉	—	2	9	no	—
63	C₃×C₉	—	2	9	no	—
64	C₃×C₉	—	2	3	no	—
65	C₃×C₉	—	3	3	no	—
66	C₃×C₉	—	6	9	no	—
67	C₃×C₉	—	6	3	no	—
68	C₃×C₉	—	6	9	no	—
69	C₃×C₉	—	6	3	no	—
70	C₃×C₉	C₃×D₉	1	3	yes	complemented
71	C₃×C₉	C₂×3₋¹⁺²	1	3	yes	complemented
72	C₃×C₉	C₃²×D₃	1	9	yes	—
73	C₃×C₉	C₂×C₃³	1	1	yes	characteristic
74	C₂×C₃²	—	9	1	no	—
75	C₂×C₃²	—	9	3	no	—
76	C₂×C₃²	—	27	3	no	—
77	C₃×D₃	—	3	3	no	—
78	C₃×D₃	—	3	1	no	—
79	C₃×D₃	—	9	3	no	—
80	C₂×C₉	—	9	9	no	—
81	D₉	C₃×3₋¹⁺²	1	1	yes	characteristic, direct product factor
82	C₃²	—	2	3	no	—
83	C₃²	—	2	1	no	—
84	C₃²	—	2	3	no	—
85	C₃²	—	3	3	no	—
86	C₃²	—	3	3	no	—
87	C₃²	—	6	3	no	—
88	C₃²	—	6	9	no	—
89	C₃²	—	6	3	no	—
90	C₃²	C₃²×D₉	1	3	yes	—
91	C₃²	C₃²×D₉	1	1	yes	characteristic
92	C₃²	D₃×3₋¹⁺²	1	3	yes	—
93	C₃²	C₃³×D₃	1	1	yes	characteristic
94	C₉	—	2	9	no	—
95	C₉	—	2	3	no	—
96	C₉	—	2	1	no	—
97	C₉	—	6	9	no	—
98	C₉	—	6	3	no	—
99	C₉	C₃²×D₉	1	9	yes	complemented
100	C₉	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic, complemented
101	C₂×C₃	—	9	3	no	—
102	C₂×C₃	—	9	1	no	—
103	C₂×C₃	—	27	3	no	—
104	D₃	—	3	1	no	—
105	C₃	—	2	1	no	—
106	C₃	—	2	3	no	—
107	C₃	—	3	3	no	—
108	C₃	—	6	3	no	—
109	C₃	D₉×3₋¹⁺²	1	3	yes	direct product factor
110	C₃	C₃³×D₉	1	1	yes	characteristic
111	C₃	C₃×D₃×3₋¹⁺²	1	1	yes	characteristic
112	C₂	—	9	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented, Sylow

C₃×D₃×3₋¹⁺²

—

maximal

C₃³×D₉

C₃

yes

maximal, characteristic

D₉×3₋¹⁺²

C₃

yes

maximal, direct product factor

C₃×C₉×D₉

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

D₃

yes

characteristic

C₃³×C₉

C₂×C₃

yes

characteristic

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

C₂×C₃

yes

complemented

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₉²

C₂×C₃

yes

complemented

C₃³×D₃

—

C₂×C₃×3₋¹⁺²

—

D₃×3₋¹⁺²

—

C₃×D₃×C₉

—

C₃²×D₉

—

C₃²×D₉

C₃²

yes

—

C₃²×D₉

C₃²

yes

characteristic

C₉×D₉

C₃²

yes

complemented

C₃⁴

C₃×D₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

D₉

yes

characteristic, direct product factor

C₃×3₋¹⁺²

C₃×D₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×D₃

yes

—

C₃²×C₉

C₂×C₃²

yes

characteristic

C₃²×C₉

C₂×C₃²

yes

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₉²

C₂×C₃²

yes

complemented

C₂×C₃³

—

C₃²×D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₂×C₃×C₉

—

D₃×C₉

—

C₃×D₉

—

C₃×D₉

3₋¹⁺²

yes

direct product factor

C₃×D₉

C₃³

yes

characteristic

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×D₉

yes

characteristic

C₃³

C₃²×D₃

yes

characteristic

C₃³

C₃²×D₃

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×D₉

yes

direct product factor

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×D₉

yes

complemented

C₃×C₉

C₂×3₋¹⁺²

yes

complemented

C₃×C₉

C₃²×D₃

yes

—

C₃×C₉

C₂×C₃³

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

D₉

C₃×3₋¹⁺²

yes

characteristic, direct product factor

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×D₉

yes

—

C₃²

C₃²×D₉

yes

characteristic

C₃²

D₃×3₋¹⁺²

yes

—

C₃²

C₃³×D₃

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

C₃²×D₉

yes

complemented

100

C₉

C₂×C₃×3₋¹⁺²

yes

characteristic, complemented

101

C₂×C₃

—

102

C₂×C₃

—

103

C₂×C₃

—

104

D₃

—

105

C₃

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

D₉×3₋¹⁺²

yes

direct product factor

110

C₃

C₃³×D₉

yes

characteristic

111

C₃

C₃×D₃×3₋¹⁺²

yes

characteristic

112

C₂

—

Sylow

	subgroup	G/H
center	C₃²	C₃²×D₉
commutator, cosocle	C₃×C₉	C₂×C₃³
Frattini	C₃²	C₃³×D₃
Fitting	C₃×C₉×3₋¹⁺²	C₂
socle	C₃³	C₃²×D₃
prime radical	C₃³×D₉	C₃
hypercenter	C₃×3₋¹⁺²	D₉
hypocenter	C₉	C₂×C₃×3₋¹⁺²
LCS	G ▷ C₃×C₉ ▷ C₉
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃²×D₉

commutator, cosocle

C₃×C₉

C₂×C₃³

Frattini

C₃²

C₃³×D₃

Fitting

C₃×C₉×3₋¹⁺²

C₂

socle

C₃³

C₃²×D₃

prime radical

C₃³×D₉

C₃

hypercenter

C₃×3₋¹⁺²

D₉

hypocenter

C₉

C₂×C₃×3₋¹⁺²

LCS

G ▷ C₃×C₉ ▷ C₉

UCS

C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×C₉×3₋¹⁺²	C₂	semidirect
4. D₉×3₋¹⁺²	C₃	direct (N × H)
5. C₃×C₉×D₉	C₃	semidirect
9. C₉×3₋¹⁺²	C₂×C₃	semidirect
12. C₃×C₉²	C₂×C₃	semidirect
20. C₉×D₉	C₃²	semidirect
25. C₃×3₋¹⁺²	D₉	direct (N × H)
38. C₉²	C₂×C₃²	semidirect
47. C₃×D₉	3₋¹⁺²	direct (N × H)
58. 3₋¹⁺²	C₃×D₉	direct (N × H)
70. C₃×C₉	C₃×D₉	semidirect
71. C₃×C₉	C₂×3₋¹⁺²	semidirect
81. D₉	C₃×3₋¹⁺²	direct (N × H)
99. C₉	C₃²×D₉	semidirect
100. C₉	C₂×C₃×3₋¹⁺²	semidirect
109. C₃	D₉×3₋¹⁺²	direct (N × H)

H ≅ G/N

type

1. C₃×C₉×3₋¹⁺²

C₂

semidirect

4. D₉×3₋¹⁺²

C₃

direct (N × H)

5. C₃×C₉×D₉

C₃

semidirect

9. C₉×3₋¹⁺²

C₂×C₃

semidirect

12. C₃×C₉²

C₂×C₃

semidirect

20. C₉×D₉

C₃²

semidirect

25. C₃×3₋¹⁺²

D₉

direct (N × H)

38. C₉²

C₂×C₃²

semidirect

47. C₃×D₉

3₋¹⁺²

direct (N × H)

58. 3₋¹⁺²

C₃×D₉

direct (N × H)

70. C₃×C₉

C₃×D₉

semidirect

71. C₃×C₉

C₂×3₋¹⁺²

semidirect

81. D₉

C₃×3₋¹⁺²

direct (N × H)

99. C₉

C₃²×D₉

semidirect

100. C₉

C₂×C₃×3₋¹⁺²

semidirect

109. C₃

D₉×3₋¹⁺²

direct (N × H)

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

subgroup	index
C₃×C₉×3₋¹⁺²	2
C₃×D₃×3₋¹⁺²	3
C₃³×D₉	3
D₉×3₋¹⁺²	3
C₃×C₉×D₉	3

subgroup

index

C₃×C₉×3₋¹⁺²

C₃×D₃×3₋¹⁺²

C₃³×D₉

D₉×3₋¹⁺²

C₃×C₉×D₉

Subgroups — C₃×D₉×3₋¹⁺² #13091369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×D9×3−1+2 #13091369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₉×3₋¹⁺² #13091369