Group #14554121 — subgroups

Subgroups — C₃²×3₋¹⁺²⋊C₉ #14554121

Order 2187.

3302 conjugacy classes of subgroups; only the 764 normal subgroups are shown.

normal subgroups only

This group has 3302 conjugacy classes of subgroups — too many to draw. Only its 764 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	C₃³×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃⁴×C₉	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₃²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
5	C₃×3₋¹⁺²⋊C₉	C₃	1	36	yes	maximal, direct product factor
6	C₃⁵	C₃²	1	1	yes	characteristic
7	C₃²×3₋¹⁺²	—	3	8	no	—
8	C₃²×3₋¹⁺²	—	3	18	no	—
9	C₃²×3₋¹⁺²	C₉	1	27	yes	complemented
10	C₃²×3₋¹⁺²	C₃²	1	12	yes	—
11	C₃³×C₉	—	3	8	no	—
12	C₃³×C₉	—	3	1	no	—
13	C₃³×C₉	—	3	1	no	—
14	C₃³×C₉	—	3	27	no	—
15	C₃³×C₉	—	3	1	no	—
16	C₃³×C₉	—	3	1	no	—
17	C₃³×C₉	C₃²	1	12	yes	—
18	C₃×C₃²⋊C₉	—	3	8	no	—
19	C₃×C₃²⋊C₉	—	3	8	no	—
20	C₃×C₃²⋊C₉	C₃²	1	12	yes	—
21	C₃×C₃²⋊C₉	C₃²	1	12	yes	—
22	3₋¹⁺²⋊C₉	C₃²	1	81	yes	direct product factor
23	C₃⁴	—	3	9	no	—
24	C₃⁴	—	3	9	no	—
25	C₃⁴	—	3	9	no	—
26	C₃⁴	C₃×C₉	1	9	yes	—
27	C₃⁴	H(3,𝔽₃)	1	8	yes	—
28	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃⁴	3₋¹⁺²	1	18	yes	—
30	C₃⁴	C₃³	1	4	yes	—
31	C₃×3₋¹⁺²	—	3	216	no	—
32	C₃×3₋¹⁺²	—	3	24	no	—
33	C₃×3₋¹⁺²	—	3	72	no	—
34	C₃×3₋¹⁺²	C₃×C₉	1	108	yes	complemented
35	C₃×3₋¹⁺²	C₃³	1	9	yes	—
36	C₃²×C₉	—	3	216	no	—
37	C₃²×C₉	—	3	27	no	—
38	C₃²×C₉	—	3	27	no	—
39	C₃²×C₉	—	3	24	no	—
40	C₃²×C₉	—	3	12	no	—
41	C₃²×C₉	—	3	12	no	—
42	C₃²×C₉	—	3	108	no	—
43	C₃²×C₉	—	3	12	no	—
44	C₃²×C₉	—	3	12	no	—
45	C₃²×C₉	—	9	9	no	—
46	C₃²×C₉	—	9	9	no	—
47	C₃²×C₉	C₃³	1	9	yes	—
48	C₃²⋊C₉	—	3	24	no	—
49	C₃²⋊C₉	—	3	24	no	—
50	C₃²⋊C₉	C₃³	1	9	yes	—
51	C₃²⋊C₉	C₃³	1	9	yes	—
52	C₃³	—	3	81	no	—
53	C₃³	—	3	162	no	—
54	C₃³	—	3	24	no	—
55	C₃³	—	3	12	no	—
56	C₃³	—	3	24	no	—
57	C₃³	—	3	12	no	—
58	C₃³	—	3	24	no	—
59	C₃³	—	3	12	no	—
60	C₃³	C₃²⋊C₉	1	72	yes	—
61	C₃³	C₃²⋊C₉	1	9	yes	—
62	C₃³	AΣL(1,𝔽₂₇)	1	9	yes	—
63	C₃³	#731	1	9	yes	—
64	C₃³	#733	1	9	yes	—
65	C₃³	C₃²×C₉	1	12	yes	—
66	C₃³	C₃×H(3,𝔽₃)	1	8	yes	—
67	C₃³	C₃×H(3,𝔽₃)	1	4	yes	—
68	C₃³	C₃×3₋¹⁺²	1	24	yes	—
69	C₃³	C₃⁴	1	1	yes	characteristic
70	3₋¹⁺²	—	3	18	no	—
71	3₋¹⁺²	—	3	216	no	—
72	3₋¹⁺²	C₃²×C₉	1	27	yes	complemented
73	C₃×C₉	—	3	216	no	—
74	C₃×C₉	—	3	9	no	—
75	C₃×C₉	—	3	9	no	—
76	C₃×C₉	—	3	108	no	—
77	C₃×C₉	—	3	108	no	—
78	C₃×C₉	—	3	27	no	—
79	C₃×C₉	—	3	9	no	—
80	C₃×C₉	—	3	9	no	—
81	C₃×C₉	—	9	36	no	—
82	C₃×C₉	—	9	36	no	—
83	C₃²	—	3	8	no	—
84	C₃²	—	3	1	no	—
85	C₃²	—	3	8	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	3	8	no	—
88	C₃²	—	3	1	no	—
89	C₃²	—	3	216	no	—
90	C₃²	—	3	36	no	—
91	C₃²	—	3	72	no	—
92	C₃²	3₋¹⁺²⋊C₉	1	81	yes	direct product factor
93	C₃²	C₃×C₃²⋊C₉	1	24	yes	—
94	C₃²	C₃×C₃²⋊C₉	1	12	yes	—
95	C₃²	C₃×AΣL(1,𝔽₂₇)	1	12	yes	—
96	C₃²	C₃×#731	1	12	yes	—
97	C₃²	C₃×#733	1	12	yes	—
98	C₃²	C₃³×C₉	1	1	yes	characteristic
99	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
100	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
101	C₉	—	3	27	no	—
102	C₉	—	3	27	no	—
103	C₉	—	9	9	no	—
104	C₉	—	9	9	no	—
105	C₃	—	3	1	no	—
106	C₃	—	3	2	no	—
107	C₃	—	3	24	no	—
108	C₃	C₃×3₋¹⁺²⋊C₉	1	36	yes	direct product factor
109	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic
110	C₃	C₃²×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
111	C₃	C₃²×#731	1	1	yes	characteristic
112	C₃	C₃²×#733	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃⁴×C₉

C₃

yes

maximal, characteristic

C₃²×C₃²⋊C₉

C₃

yes

maximal, characteristic

C₃²×C₃²⋊C₉

C₃

yes

maximal, characteristic

C₃×3₋¹⁺²⋊C₉

C₃

yes

maximal, direct product factor

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₉

yes

complemented

C₃²×3₋¹⁺²

C₃²

yes

—

C₃³×C₉

—

C₃³×C₉

—

C₃³×C₉

—

C₃³×C₉

—

C₃³×C₉

—

C₃³×C₉

—

C₃³×C₉

C₃²

yes

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

C₃²

yes

direct product factor

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃×C₉

yes

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

H(3,𝔽₃)

yes

characteristic

C₃⁴

3₋¹⁺²

yes

—

C₃⁴

C₃³

yes

—

C₃×3₋¹⁺²

—

216

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₉

108

yes

complemented

C₃×3₋¹⁺²

C₃³

yes

—

C₃²×C₉

—

216

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

108

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃³

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃³

yes

—

C₃²⋊C₉

C₃³

yes

—

C₃³

—

C₃³

—

162

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊C₉

yes

—

C₃³

C₃²⋊C₉

yes

—

C₃³

AΣL(1,𝔽₂₇)

yes

—

C₃³

#731

yes

—

C₃³

#733

yes

—

C₃³

C₃²×C₉

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

216

—

3₋¹⁺²

C₃²×C₉

yes

complemented

C₃×C₉

—

216

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

108

—

C₃×C₉

—

108

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

216

—

C₃²

—

C₃²

—

C₃²

3₋¹⁺²⋊C₉

yes

direct product factor

C₃²

C₃×C₃²⋊C₉

yes

—

C₃²

C₃×C₃²⋊C₉

yes

—

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

—

C₃²

C₃×#731

yes

—

C₃²

C₃×#733

yes

—

C₃²

C₃³×C₉

yes

characteristic

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

100

C₃²

C₃²×3₋¹⁺²

yes

—

101

C₉

—

102

C₉

—

103

C₉

—

104

C₉

—

105

C₃

—

106

C₃

—

107

C₃

—

108

C₃

C₃×3₋¹⁺²⋊C₉

yes

direct product factor

109

C₃

C₃²×C₃²⋊C₉

yes

characteristic

110

C₃

C₃²×AΣL(1,𝔽₂₇)

yes

characteristic

111

C₃

C₃²×#731

yes

characteristic

112

C₃

C₃²×#733

yes

characteristic

	subgroup	G/H
center, socle	C₃⁴	H(3,𝔽₃)
commutator	C₃²	C₃³×C₉
Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃⁴ ◁ C₃⁵ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃⁴

H(3,𝔽₃)

commutator

C₃²

C₃³×C₉

Frattini, cosocle

C₃³

C₃⁴

prime radical

C₃⁵

C₃²

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃⁴ ◁ C₃⁵ ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
5. C₃×3₋¹⁺²⋊C₉	C₃	direct (N × H)
9. C₃²×3₋¹⁺²	C₉	semidirect
22. 3₋¹⁺²⋊C₉	C₃²	direct (N × H)
34. C₃×3₋¹⁺²	C₃×C₉	semidirect
72. 3₋¹⁺²	C₃²×C₉	semidirect
92. C₃²	3₋¹⁺²⋊C₉	direct (N × H)
108. C₃	C₃×3₋¹⁺²⋊C₉	direct (N × H)

H ≅ G/N

type

5. C₃×3₋¹⁺²⋊C₉

C₃

direct (N × H)

9. C₃²×3₋¹⁺²

C₉

semidirect

22. 3₋¹⁺²⋊C₉

C₃²

direct (N × H)

34. C₃×3₋¹⁺²

C₃×C₉

semidirect

72. 3₋¹⁺²

C₃²×C₉

semidirect

92. C₃²

3₋¹⁺²⋊C₉

direct (N × H)

108. C₃

C₃×3₋¹⁺²⋊C₉

direct (N × H)

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

subgroup	index
C₃³×3₋¹⁺²	3
C₃⁴×C₉	3
C₃²×C₃²⋊C₉	3
C₃×3₋¹⁺²⋊C₉	3

subgroup

index

C₃³×3₋¹⁺²

C₃⁴×C₉

C₃²×C₃²⋊C₉

C₃×3₋¹⁺²⋊C₉

Subgroups — C₃²×3₋¹⁺²⋊C₉ #14554121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C32×3−1+2⋊C9 #14554121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×3₋¹⁺²⋊C₉ #14554121