Group #14517801 — subgroups

Subgroups — C₃×D₄×C₉⋊C₉ #14517801

Order 1944.

720 conjugacy classes of subgroups; only the 432 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 432 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	C₂²×C₃×C₉⋊C₉	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×C₉⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×D₄×C₉	C₃	1	3	yes	maximal
4	C₃²×D₄×C₉	C₃	1	1	yes	maximal, characteristic
5	D₄×C₉⋊C₉	C₃	1	9	yes	maximal, direct product factor
6	C₂×C₃×C₉⋊C₉	—	2	2	no	—
7	C₂×C₃×C₉⋊C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×C₉	C₂×C₃	1	6	yes	—
9	C₂²×C₃²×C₉	C₂×C₃	1	2	yes	—
10	C₃²×C₄×C₉	C₂×C₃	1	3	yes	—
11	C₃²×C₄×C₉	C₂×C₃	1	1	yes	characteristic
12	C₂²×C₉⋊C₉	C₂×C₃	1	18	yes	complemented
13	C₄×C₉⋊C₉	C₂×C₃	1	9	yes	complemented
14	C₃×C₉⋊C₉	D₄	1	1	yes	characteristic, direct product factor, Sylow
15	C₃³×D₄	C₃²	1	1	yes	characteristic
16	C₃×D₄×C₉	—	3	9	no	—
17	C₃×D₄×C₉	C₉	1	9	yes	complemented
18	C₃×D₄×C₉	C₃²	1	9	yes	—
19	C₃×D₄×C₉	C₃²	1	3	yes	—
20	C₂×C₃²×C₉	—	2	2	no	—
21	C₂×C₃²×C₉	—	2	6	no	—
22	C₂×C₃²×C₉	C₂²×C₃	1	3	yes	—
23	C₂×C₃²×C₉	C₂²×C₃	1	1	yes	characteristic
24	C₂×C₉⋊C₉	—	2	18	no	—
25	C₂×C₉⋊C₉	C₂²×C₃	1	9	yes	—
26	C₂²×C₃³	C₂×C₃²	1	2	yes	—
27	C₃³×C₄	C₂×C₃²	1	1	yes	characteristic
28	C₂²×C₃×C₉	—	3	18	no	—
29	C₂²×C₃×C₉	C₂×C₉	1	18	yes	complemented
30	C₂²×C₃×C₉	C₂×C₃²	1	18	yes	—
31	C₂²×C₃×C₉	C₂×C₃²	1	6	yes	—
32	C₃×C₄×C₉	—	3	9	no	—
33	C₃×C₄×C₉	C₂×C₉	1	9	yes	complemented
34	C₃×C₄×C₉	C₂×C₃²	1	9	yes	—
35	C₃×C₄×C₉	C₂×C₃²	1	3	yes	—
36	C₃²×C₉	C₃×D₄	1	3	yes	—
37	C₃²×C₉	C₃×D₄	1	1	yes	characteristic
38	C₉⋊C₉	C₃×D₄	1	9	yes	direct product factor
39	C₃²×D₄	C₃×C₉	1	3	yes	—
40	C₃²×D₄	3₋¹⁺²	1	9	yes	—
41	C₃²×D₄	C₃³	1	1	yes	characteristic
42	D₄×C₉	—	3	9	no	—
43	D₄×C₉	C₃×C₉	1	9	yes	complemented
44	C₂×C₃³	—	2	2	no	—
45	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
46	C₂×C₃×C₉	—	2	18	no	—
47	C₂×C₃×C₉	—	2	6	no	—
48	C₂×C₃×C₉	—	2	18	no	—
49	C₂×C₃×C₉	—	3	9	no	—
50	C₂×C₃×C₉	—	6	18	no	—
51	C₂×C₃×C₉	C₂²×C₉	1	9	yes	—
52	C₂×C₃×C₉	C₂²×C₃²	1	9	yes	—
53	C₂×C₃×C₉	C₂²×C₃²	1	3	yes	—
54	C₂²×C₃²	C₂×C₃×C₉	1	6	yes	—
55	C₂²×C₃²	C₂×3₋¹⁺²	1	18	yes	—
56	C₂²×C₃²	C₂×C₃³	1	2	yes	—
57	C₃²×C₄	C₂×C₃×C₉	1	3	yes	—
58	C₃²×C₄	C₂×3₋¹⁺²	1	9	yes	—
59	C₃²×C₄	C₂×C₃³	1	1	yes	characteristic
60	C₂²×C₉	—	3	18	no	—
61	C₂²×C₉	C₂×C₃×C₉	1	18	yes	complemented
62	C₄×C₉	—	3	9	no	—
63	C₄×C₉	C₂×C₃×C₉	1	9	yes	complemented
64	C₃³	C₃²×D₄	1	1	yes	characteristic
65	C₃×C₉	—	3	9	no	—
66	C₃×C₉	D₄×C₉	1	9	yes	complemented
67	C₃×C₉	C₃²×D₄	1	9	yes	—
68	C₃×C₉	C₃²×D₄	1	3	yes	—
69	C₃×D₄	C₉⋊C₉	1	9	yes	direct product factor
70	C₃×D₄	C₃²×C₉	1	1	yes	characteristic
71	C₃×D₄	C₃×3₋¹⁺²	1	3	yes	—
72	C₂×C₃²	—	2	6	no	—
73	C₂×C₃²	—	2	18	no	—
74	C₂×C₃²	—	2	2	no	—
75	C₂×C₃²	C₂²×C₃×C₉	1	3	yes	—
76	C₂×C₃²	C₂²×3₋¹⁺²	1	9	yes	—
77	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
78	C₂×C₉	—	2	18	no	—
79	C₂×C₉	—	3	9	no	—
80	C₂×C₉	—	6	18	no	—
81	C₂×C₉	C₂²×C₃×C₉	1	9	yes	—
82	C₂²×C₃	C₂×C₉⋊C₉	1	18	yes	complemented
83	C₂²×C₃	C₂×C₃²×C₉	1	2	yes	—
84	C₂²×C₃	C₂×C₃×3₋¹⁺²	1	6	yes	—
85	C₃×C₄	C₂×C₉⋊C₉	1	9	yes	complemented
86	C₃×C₄	C₂×C₃²×C₉	1	1	yes	characteristic
87	C₃×C₄	C₂×C₃×3₋¹⁺²	1	3	yes	—
88	C₃²	C₃×D₄×C₉	1	3	yes	—
89	C₃²	D₄×3₋¹⁺²	1	9	yes	—
90	C₃²	C₃³×D₄	1	1	yes	characteristic
91	C₉	—	3	9	no	—
92	C₉	C₃×D₄×C₉	1	9	yes	complemented
93	D₄	C₃×C₉⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
94	C₂×C₃	—	2	18	no	—
95	C₂×C₃	—	2	2	no	—
96	C₂×C₃	—	2	6	no	—
97	C₂×C₃	C₂²×C₉⋊C₉	1	9	yes	—
98	C₂×C₃	C₂²×C₃²×C₉	1	1	yes	characteristic
99	C₂×C₃	C₂²×C₃×3₋¹⁺²	1	3	yes	—
100	C₂²	C₂×C₃×C₉⋊C₉	1	2	yes	complemented
101	C₄	C₂×C₃×C₉⋊C₉	1	1	yes	characteristic, complemented
102	C₃	D₄×C₉⋊C₉	1	9	yes	direct product factor
103	C₃	C₃²×D₄×C₉	1	1	yes	characteristic
104	C₃	C₃×D₄×3₋¹⁺²	1	3	yes	—
105	C₂	—	2	2	no	—
106	C₂	C₂²×C₃×C₉⋊C₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃×C₉⋊C₉

C₂

yes

maximal, complemented

C₃×C₄×C₉⋊C₉

C₂

yes

maximal, characteristic, complemented

C₃²×D₄×C₉

C₃

yes

maximal

C₃²×D₄×C₉

C₃

yes

maximal, characteristic

D₄×C₉⋊C₉

C₃

yes

maximal, direct product factor

C₂×C₃×C₉⋊C₉

—

C₂×C₃×C₉⋊C₉

C₂²

yes

characteristic

C₂²×C₃²×C₉

C₂×C₃

yes

—

C₂²×C₃²×C₉

C₂×C₃

yes

—

C₃²×C₄×C₉

C₂×C₃

yes

—

C₃²×C₄×C₉

C₂×C₃

yes

characteristic

C₂²×C₉⋊C₉

C₂×C₃

yes

complemented

C₄×C₉⋊C₉

C₂×C₃

yes

complemented

C₃×C₉⋊C₉

D₄

yes

characteristic, direct product factor, Sylow

C₃³×D₄

C₃²

yes

characteristic

C₃×D₄×C₉

—

C₃×D₄×C₉

C₉

yes

complemented

C₃×D₄×C₉

C₃²

yes

—

C₃×D₄×C₉

C₃²

yes

—

C₂×C₃²×C₉

—

C₂×C₃²×C₉

—

C₂×C₃²×C₉

C₂²×C₃

yes

—

C₂×C₃²×C₉

C₂²×C₃

yes

characteristic

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

C₂²×C₃

yes

—

C₂²×C₃³

C₂×C₃²

yes

—

C₃³×C₄

C₂×C₃²

yes

characteristic

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×C₉

yes

complemented

C₂²×C₃×C₉

C₂×C₃²

yes

—

C₂²×C₃×C₉

C₂×C₃²

yes

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×C₉

yes

complemented

C₃×C₄×C₉

C₂×C₃²

yes

—

C₃×C₄×C₉

C₂×C₃²

yes

—

C₃²×C₉

C₃×D₄

yes

—

C₃²×C₉

C₃×D₄

yes

characteristic

C₉⋊C₉

C₃×D₄

yes

direct product factor

C₃²×D₄

C₃×C₉

yes

—

C₃²×D₄

3₋¹⁺²

yes

—

C₃²×D₄

C₃³

yes

characteristic

D₄×C₉

—

D₄×C₉

C₃×C₉

yes

complemented

C₂×C₃³

—

C₂×C₃³

C₂²×C₃²

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂²×C₉

yes

—

C₂×C₃×C₉

C₂²×C₃²

yes

—

C₂×C₃×C₉

C₂²×C₃²

yes

—

C₂²×C₃²

C₂×C₃×C₉

yes

—

C₂²×C₃²

C₂×3₋¹⁺²

yes

—

C₂²×C₃²

C₂×C₃³

yes

—

C₃²×C₄

C₂×C₃×C₉

yes

—

C₃²×C₄

C₂×3₋¹⁺²

yes

—

C₃²×C₄

C₂×C₃³

yes

characteristic

C₂²×C₉

—

C₂²×C₉

C₂×C₃×C₉

yes

complemented

C₄×C₉

—

C₄×C₉

C₂×C₃×C₉

yes

complemented

C₃³

C₃²×D₄

yes

characteristic

C₃×C₉

—

C₃×C₉

D₄×C₉

yes

complemented

C₃×C₉

C₃²×D₄

yes

—

C₃×C₉

C₃²×D₄

yes

—

C₃×D₄

C₉⋊C₉

yes

direct product factor

C₃×D₄

C₃²×C₉

yes

characteristic

C₃×D₄

C₃×3₋¹⁺²

yes

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂²×C₃×C₉

yes

—

C₂×C₃²

C₂²×3₋¹⁺²

yes

—

C₂×C₃²

C₂²×C₃³

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂²×C₃×C₉

yes

—

C₂²×C₃

C₂×C₉⋊C₉

yes

complemented

C₂²×C₃

C₂×C₃²×C₉

yes

—

C₂²×C₃

C₂×C₃×3₋¹⁺²

yes

—

C₃×C₄

C₂×C₉⋊C₉

yes

complemented

C₃×C₄

C₂×C₃²×C₉

yes

characteristic

C₃×C₄

C₂×C₃×3₋¹⁺²

yes

—

C₃²

C₃×D₄×C₉

yes

—

C₃²

D₄×3₋¹⁺²

yes

—

C₃²

C₃³×D₄

yes

characteristic

C₉

—

C₉

C₃×D₄×C₉

yes

complemented

D₄

C₃×C₉⋊C₉

yes

characteristic, direct product factor, Sylow

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₂²×C₉⋊C₉

yes

—

C₂×C₃

C₂²×C₃²×C₉

yes

characteristic

C₂×C₃

C₂²×C₃×3₋¹⁺²

yes

—

100

C₂²

C₂×C₃×C₉⋊C₉

yes

complemented

101

C₄

C₂×C₃×C₉⋊C₉

yes

characteristic, complemented

102

C₃

D₄×C₉⋊C₉

yes

direct product factor

103

C₃

C₃²×D₄×C₉

yes

characteristic

104

C₃

C₃×D₄×3₋¹⁺²

yes

—

105

C₂

—

106

C₂

C₂²×C₃×C₉⋊C₉

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃³	C₂²×C₃²
commutator	C₂×C₃	C₂²×C₃²×C₉
Frattini, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₃³×D₄	C₃²
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃³ ◁ G

subgroup

G/H

center, socle

C₂×C₃³

C₂²×C₃²

commutator

C₂×C₃

C₂²×C₃²×C₉

Frattini, cosocle

C₂×C₃²

C₂²×C₃³

prime radical

C₃³×D₄

C₃²

LCS

G ▷ C₂×C₃ ▷ C₁

UCS

C₁ ◁ C₂×C₃³ ◁ G

N	H ≅ G/N	type
1. C₂²×C₃×C₉⋊C₉	C₂	semidirect
2. C₃×C₄×C₉⋊C₉	C₂	semidirect
5. D₄×C₉⋊C₉	C₃	direct (N × H)
12. C₂²×C₉⋊C₉	C₂×C₃	semidirect
13. C₄×C₉⋊C₉	C₂×C₃	semidirect
14. C₃×C₉⋊C₉	D₄	direct (N × H)
17. C₃×D₄×C₉	C₉	semidirect
29. C₂²×C₃×C₉	C₂×C₉	semidirect
33. C₃×C₄×C₉	C₂×C₉	semidirect
38. C₉⋊C₉	C₃×D₄	direct (N × H)
43. D₄×C₉	C₃×C₉	semidirect
61. C₂²×C₉	C₂×C₃×C₉	semidirect
63. C₄×C₉	C₂×C₃×C₉	semidirect
66. C₃×C₉	D₄×C₉	semidirect
69. C₃×D₄	C₉⋊C₉	direct (N × H)
82. C₂²×C₃	C₂×C₉⋊C₉	semidirect
85. C₃×C₄	C₂×C₉⋊C₉	semidirect
92. C₉	C₃×D₄×C₉	semidirect
93. D₄	C₃×C₉⋊C₉	direct (N × H)
100. C₂²	C₂×C₃×C₉⋊C₉	semidirect
101. C₄	C₂×C₃×C₉⋊C₉	semidirect
102. C₃	D₄×C₉⋊C₉	direct (N × H)

H ≅ G/N

type

1. C₂²×C₃×C₉⋊C₉

C₂

semidirect

2. C₃×C₄×C₉⋊C₉

C₂

semidirect

5. D₄×C₉⋊C₉

C₃

direct (N × H)

12. C₂²×C₉⋊C₉

C₂×C₃

semidirect

13. C₄×C₉⋊C₉

C₂×C₃

semidirect

14. C₃×C₉⋊C₉

D₄

direct (N × H)

17. C₃×D₄×C₉

C₉

semidirect

29. C₂²×C₃×C₉

C₂×C₉

semidirect

33. C₃×C₄×C₉

C₂×C₉

semidirect

38. C₉⋊C₉

C₃×D₄

direct (N × H)

43. D₄×C₉

C₃×C₉

semidirect

61. C₂²×C₉

C₂×C₃×C₉

semidirect

63. C₄×C₉

C₂×C₃×C₉

semidirect

66. C₃×C₉

D₄×C₉

semidirect

69. C₃×D₄

C₉⋊C₉

direct (N × H)

82. C₂²×C₃

C₂×C₉⋊C₉

semidirect

85. C₃×C₄

C₂×C₉⋊C₉

semidirect

92. C₉

C₃×D₄×C₉

semidirect

93. D₄

C₃×C₉⋊C₉

direct (N × H)

100. C₂²

C₂×C₃×C₉⋊C₉

semidirect

101. C₄

C₂×C₃×C₉⋊C₉

semidirect

102. C₃

D₄×C₉⋊C₉

direct (N × H)

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

subgroup	index
C₂²×C₃×C₉⋊C₉	2
C₃×C₄×C₉⋊C₉	2
C₃²×D₄×C₉	3
D₄×C₉⋊C₉	3

subgroup

index

C₂²×C₃×C₉⋊C₉

C₃×C₄×C₉⋊C₉

C₃²×D₄×C₉

D₄×C₉⋊C₉

Subgroups — C₃×D₄×C₉⋊C₉ #14517801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×D4×C9⋊C9 #14517801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×C₉⋊C₉ #14517801