Group #13091702 — subgroups

Subgroups — C₃×C₃⋊#4208 #13091702

Order 1458.

1811 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1811 conjugacy classes of subgroups — too many to draw. Only its 182 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	C₃²×#737	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊#4208	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₃⋊#272	C₃	1	1	yes	maximal, characteristic
4	C₃×#4208	—	3	9	no	maximal
5	C₃×C₉×C₃⋊D₃	—	3	4	no	maximal
6	C₃×#737	—	2	27	no	—
7	C₃×#737	—	2	9	no	—
8	C₃×#737	—	2	12	no	—
9	C₃×#737	D₃	1	9	yes	complemented
10	C₃×#737	C₂×C₃	1	3	yes	complemented
11	C₃²×3₋¹⁺²	—	2	4	no	—
12	C₃²×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
13	C₃³×C₉	D₃	1	4	yes	complemented
14	C₃²×C₃⋊D₃	—	3	4	no	—
15	C₃⋊#272	C₃²	1	1	yes	characteristic
16	#4208	—	3	27	no	—
17	C₃×#272	—	3	9	no	—
18	C₉×C₃⋊D₃	—	3	12	no	—
19	C₃×D₃×C₉	—	9	1	no	—
20	C₃×D₃×C₉	—	9	12	no	—
21	C₃⁴	C₃×D₃	1	4	yes	—
22	#737	—	2	216	no	—
23	#737	—	2	27	no	—
24	#737	—	2	108	no	—
25	#737	C₃×D₃	1	27	yes	complemented
26	C₃×3₋¹⁺²	—	2	72	no	—
27	C₃×3₋¹⁺²	—	2	24	no	—
28	C₃×3₋¹⁺²	—	2	12	no	—
29	C₃×3₋¹⁺²	—	2	36	no	—
30	C₃×H(3,𝔽₃)	—	2	4	no	—
31	C₃×H(3,𝔽₃)	—	2	9	no	—
32	C₃×H(3,𝔽₃)	C₃×D₃	1	9	yes	—
33	C₃×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
34	C₃²×C₉	—	2	4	no	—
35	C₃²×C₉	—	2	12	no	—
36	C₃²×C₉	—	2	12	no	—
37	C₃²×C₉	—	2	36	no	—
38	C₃²×C₉	C₃×D₃	1	12	yes	complemented
39	C₃²×C₉	C₃⋊D₃	1	1	yes	characteristic
40	C₃²×C₉	C₃⋊D₃	1	12	yes	complemented
41	C₃×C₃⋊D₃	—	3	12	no	—
42	C₃×C₃⋊D₃	—	3	4	no	—
43	C₃²×D₃	—	9	12	no	—
44	C₃²×D₃	—	9	1	no	—
45	C₂×C₃×C₉	—	27	1	no	—
46	#272	—	3	9	no	—
47	D₃×C₉	—	9	3	no	—
48	D₃×C₉	—	9	36	no	—
49	C₃³	—	2	12	no	—
50	C₃³	—	2	4	no	—
51	C₃³	—	3	12	no	—
52	C₃³	—	6	12	no	—
53	C₃³	C₃²×D₃	1	4	yes	—
54	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
55	C₃³	C₃×C₃⋊D₃	1	12	yes	—
56	3₋¹⁺²	—	2	216	no	—
57	3₋¹⁺²	—	2	108	no	—
58	H(3,𝔽₃)	—	2	36	no	—
59	H(3,𝔽₃)	C₃²×D₃	1	9	yes	—
60	C₃×C₉	—	2	24	no	—
61	C₃×C₉	—	2	72	no	—
62	C₃×C₉	—	2	12	no	—
63	C₃×C₉	—	2	36	no	—
64	C₃×C₉	—	2	1	no	—
65	C₃×C₉	—	2	12	no	—
66	C₃×C₉	—	2	3	no	—
67	C₃×C₉	—	2	36	no	—
68	C₃×C₉	C₃×C₃⋊D₃	1	3	yes	—
69	C₃×C₉	C₃×C₃⋊D₃	1	36	yes	complemented
70	C₃×C₉	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
71	C₂×C₃²	—	27	1	no	—
72	C₃⋊D₃	—	3	4	no	—
73	C₃×D₃	—	9	36	no	—
74	C₃×D₃	—	9	3	no	—
75	C₃×D₃	—	9	12	no	—
76	C₃×D₃	—	9	1	no	—
77	C₂×C₉	—	27	3	no	—
78	C₃²	—	2	3	no	—
79	C₃²	—	2	1	no	—
80	C₃²	—	2	12	no	—
81	C₃²	—	3	4	no	—
82	C₃²	—	3	36	no	—
83	C₃²	—	6	4	no	—
84	C₃²	—	6	36	no	—
85	C₃²	—	6	8	no	—
86	C₃²	—	6	4	no	—
87	C₃²	#4208	1	3	yes	complemented
88	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
89	C₃²	C₃²×C₃⋊D₃	1	12	yes	—
90	C₃²	C₃×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
91	C₉	—	2	3	no	—
92	C₉	—	2	36	no	—
93	C₉	C₃×C₃⋊(C₃⋊D₃)	1	3	yes	—
94	C₂×C₃	—	27	3	no	—
95	C₂×C₃	—	27	1	no	—
96	D₃	—	9	12	no	—
97	D₃	—	9	1	no	—
98	C₃	—	2	3	no	—
99	C₃	—	2	1	no	—
100	C₃	—	3	12	no	—
101	C₃	—	6	12	no	—
102	C₃	C₃×#4208	1	1	yes	characteristic, complemented
103	C₃	C₃⋊#4208	1	3	yes	direct product factor
104	C₃	C₃²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
105	C₂	—	27	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#737

C₂

yes

maximal, characteristic, complemented, Sylow

C₃⋊#4208

C₃

yes

maximal, direct product factor

C₃×C₃⋊#272

C₃

yes

maximal, characteristic

C₃×#4208

—

maximal

C₃×C₉×C₃⋊D₃

—

—

—

—

D₃

yes

complemented

C₃×#737

C₂×C₃

yes

complemented

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

C₂×C₃

yes

characteristic

C₃³×C₉

D₃

yes

complemented

C₃²×C₃⋊D₃

—

C₃⋊#272

C₃²

yes

characteristic

#4208

—

C₃×#272

—

C₉×C₃⋊D₃

—

C₃×D₃×C₉

—

C₃×D₃×C₉

—

C₃⁴

C₃×D₃

yes

—

#737

—

216

—

#737

—

#737

—

108

—

#737

C₃×D₃

yes

complemented

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×D₃

yes

—

C₃×H(3,𝔽₃)

C₂×C₃²

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×D₃

yes

complemented

C₃²×C₉

C₃⋊D₃

yes

characteristic

C₃²×C₉

C₃⋊D₃

yes

complemented

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₂×C₃×C₉

—

#272

—

D₃×C₉

—

D₃×C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×D₃

yes

—

C₃³

C₃×C₃⋊D₃

yes

characteristic

C₃³

C₃×C₃⋊D₃

yes

—

3₋¹⁺²

—

216

—

3₋¹⁺²

—

108

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃²×D₃

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×C₃⋊D₃

yes

—

C₃×C₉

C₃×C₃⋊D₃

yes

complemented

C₃×C₉

C₃⋊(C₃⋊D₃)

yes

characteristic

C₂×C₃²

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

#4208

yes

complemented

C₃²

C₃²×C₃⋊D₃

yes

characteristic

C₃²

C₃²×C₃⋊D₃

yes

—

C₃²

C₃×C₃⋊(C₃⋊D₃)

yes

characteristic

C₉

—

C₉

—

C₉

C₃×C₃⋊(C₃⋊D₃)

yes

—

C₂×C₃

—

C₂×C₃

—

D₃

—

D₃

—

C₃

—

C₃

—

100

C₃

—

101

C₃

—

102

C₃

C₃×#4208

yes

characteristic, complemented

103

C₃

C₃⋊#4208

yes

direct product factor

104

C₃

C₃²×C₃⋊(C₃⋊D₃)

yes

characteristic

105

C₂

—

Sylow

	subgroup	G/H
center, hypercenter	C₃×C₉	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃×H(3,𝔽₃)	C₂×C₃²
Frattini	C₃	C₃²×C₃⋊(C₃⋊D₃)
Fitting	C₃²×#737	C₂
socle	C₃³	C₃×C₃⋊D₃
prime radical	C₃×C₃⋊#272	C₃
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₉

subgroup

G/H

center, hypercenter

C₃×C₉

C₃⋊(C₃⋊D₃)

commutator, cosocle, hypocenter

C₃×H(3,𝔽₃)

C₂×C₃²

Frattini

C₃

C₃²×C₃⋊(C₃⋊D₃)

Fitting

C₃²×#737

C₂

socle

C₃³

C₃×C₃⋊D₃

prime radical

C₃×C₃⋊#272

C₃

LCS

G ▷ C₃×H(3,𝔽₃)

UCS

C₁ ◁ C₃×C₉

N	H ≅ G/N	type
1. C₃²×#737	C₂	semidirect
2. C₃⋊#4208	C₃	direct (N × H)
9. C₃×#737	D₃	semidirect
10. C₃×#737	C₂×C₃	semidirect
13. C₃³×C₉	D₃	semidirect
25. #737	C₃×D₃	semidirect
38. C₃²×C₉	C₃×D₃	semidirect
40. C₃²×C₉	C₃⋊D₃	semidirect
69. C₃×C₉	C₃×C₃⋊D₃	semidirect
87. C₃²	#4208	semidirect
102. C₃	C₃×#4208	semidirect
103. C₃	C₃⋊#4208	direct (N × H)

H ≅ G/N

type

1. C₃²×#737

C₂

semidirect

2. C₃⋊#4208

C₃

direct (N × H)

9. C₃×#737

D₃

semidirect

10. C₃×#737

C₂×C₃

semidirect

13. C₃³×C₉

D₃

semidirect

25. #737

C₃×D₃

semidirect

38. C₃²×C₉

C₃×D₃

semidirect

40. C₃²×C₉

C₃⋊D₃

semidirect

69. C₃×C₉

C₃×C₃⋊D₃

semidirect

87. C₃²

#4208

semidirect

102. C₃

C₃×#4208

semidirect

103. C₃

C₃⋊#4208

direct (N × H)

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

subgroup	index
C₃²×#737	2
C₃⋊#4208	3
C₃×C₃⋊#272	3
C₃×#4208	3
C₃×C₉×C₃⋊D₃	3

subgroup

index

C₃²×#737

C₃⋊#4208

C₃×C₃⋊#272

C₃×#4208

C₃×C₉×C₃⋊D₃

Subgroups — C₃×C₃⋊#4208 #13091702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×C3⋊#4208 #13091702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃⋊#4208 #13091702