Group #14554138 — subgroups

Subgroups — #14554138

Order 2187.

1862 conjugacy classes of subgroups; only the 584 normal subgroups are shown.

normal subgroups only

This group has 1862 conjugacy classes of subgroups — too many to draw. Only its 584 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	#10624863	C₃	1	3	yes	maximal, complemented
3	#10624692	C₃	1	18	yes	maximal, complemented
4	#10624691	C₃	1	3	yes	maximal, complemented
5	#10624690	C₃	1	3	yes	maximal, complemented
6	#10624688	C₃	1	9	yes	maximal, complemented
7	C₃×#6866	C₃	1	3	yes	maximal
8	C₃⁵	C₃²	1	1	yes	characteristic
9	C₃²×3₋¹⁺²	—	3	9	no	—
10	C₃²×3₋¹⁺²	—	3	18	no	—
11	C₃²×3₋¹⁺²	C₉	1	27	yes	complemented
12	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
13	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
14	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
15	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
16	C₃³×C₉	C₃²	1	3	yes	—
17	C₉×3₋¹⁺²	—	3	27	no	—
18	#6887	C₃²	1	27	yes	complemented
19	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
20	#6868	C₃²	1	27	yes	complemented
21	#6867	C₃²	1	27	yes	complemented
22	#6866	C₃²	1	9	yes	—
23	#6866	C₃²	1	18	yes	—
24	C₃⁴	—	3	3	no	—
25	C₃⁴	—	3	6	no	—
26	C₃⁴	—	9	9	no	—
27	C₃⁴	C₃×C₉	1	9	yes	—
28	C₃⁴	C₃³	1	1	yes	characteristic
29	C₃⁴	C₃³	1	2	yes	—
30	C₃⁴	C₃³	1	1	yes	characteristic
31	C₃×3₋¹⁺²	—	3	3	no	—
32	C₃×3₋¹⁺²	—	3	3	no	—
33	C₃×3₋¹⁺²	—	3	6	no	—
34	C₃×3₋¹⁺²	—	3	6	no	—
35	C₃×3₋¹⁺²	—	3	243	no	—
36	C₃×3₋¹⁺²	—	3	18	no	—
37	C₃×3₋¹⁺²	—	3	36	no	—
38	C₃×3₋¹⁺²	C₃×C₉	1	9	yes	—
39	C₃×3₋¹⁺²	C₃×C₉	1	9	yes	—
40	C₃×3₋¹⁺²	C₃×C₉	1	9	yes	—
41	C₃×3₋¹⁺²	C₃×C₉	1	54	yes	complemented
42	C₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
43	C₃×3₋¹⁺²	3₋¹⁺²	1	18	yes	—
44	C₃²×C₉	—	3	9	no	—
45	C₃²×C₉	—	3	18	no	—
46	C₃²×C₉	—	3	9	no	—
47	C₃²×C₉	C₃×C₉	1	27	yes	complemented
48	C₃²×C₉	C₃³	1	3	yes	—
49	C₃²×C₉	C₃³	1	3	yes	—
50	C₃²×C₉	C₃³	1	3	yes	—
51	C₉⋊C₉	—	3	54	no	—
52	C₃²⋊C₉	—	3	18	no	—
53	C₃²⋊C₉	C₃³	1	27	yes	—
54	C₉²	—	3	27	no	—
55	C₃³	—	3	9	no	—
56	C₃³	—	3	1	no	—
57	C₃³	—	3	2	no	—
58	C₃³	—	3	2	no	—
59	C₃³	—	3	4	no	—
60	C₃³	—	3	27	no	—
61	C₃³	—	9	3	no	—
62	C₃³	—	9	27	no	—
63	C₃³	—	9	27	no	—
64	C₃³	—	9	6	no	—
65	C₃³	—	9	54	no	—
66	C₃³	C₃²×C₉	1	3	yes	—
67	C₃³	C₃²×C₉	1	6	yes	—
68	C₃³	C₃²×C₉	1	3	yes	—
69	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
70	C₃³	C₃×3₋¹⁺²	1	6	yes	—
71	C₃³	C₃⁴	1	1	yes	characteristic
72	3₋¹⁺²	—	3	27	no	—
73	3₋¹⁺²	—	3	27	no	—
74	3₋¹⁺²	—	3	162	no	—
75	3₋¹⁺²	C₃²⋊C₉	1	81	yes	—
76	C₃×C₉	—	3	3	no	—
77	C₃×C₉	—	3	3	no	—
78	C₃×C₉	—	3	6	no	—
79	C₃×C₉	—	3	6	no	—
80	C₃×C₉	—	3	81	no	—
81	C₃×C₉	—	3	27	no	—
82	C₃×C₉	—	27	9	no	—
83	C₃×C₉	C₃²×C₉	1	9	yes	—
84	C₃×C₉	C₃²×C₉	1	9	yes	—
85	C₃×C₉	C₃²×C₉	1	9	yes	—
86	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
87	C₃×C₉	C₃×3₋¹⁺²	1	18	yes	—
88	C₃²	—	3	9	no	—
89	C₃²	—	3	18	no	—
90	C₃²	—	3	3	no	—
91	C₃²	—	3	9	no	—
92	C₃²	—	3	9	no	—
93	C₃²	—	3	3	no	—
94	C₃²	—	3	6	no	—
95	C₃²	—	9	3	no	—
96	C₃²	—	9	6	no	—
97	C₃²	—	9	81	no	—
98	C₃²	—	9	6	no	—
99	C₃²	—	9	12	no	—
100	C₃²	—	9	6	no	—
101	C₃²	C₃×C₃²⋊C₉	1	9	yes	—
102	C₃²	C₃³×C₉	1	1	yes	characteristic
103	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
104	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
105	C₉	—	3	9	no	—
106	C₉	—	3	9	no	—
107	C₉	—	9	27	no	—
108	C₉	C₃×C₃²⋊C₉	1	27	yes	—
109	C₃	—	3	9	no	—
110	C₃	—	3	1	no	—
111	C₃	—	3	2	no	—
112	C₃	—	3	9	no	—
113	C₃	—	9	6	no	—
114	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic
115	C₃	#10624934	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×3₋¹⁺²

C₃

yes

maximal, characteristic

#10624863

C₃

yes

maximal, complemented

#10624692

C₃

yes

maximal, complemented

#10624691

C₃

yes

maximal, complemented

#10624690

C₃

yes

maximal, complemented

#10624688

C₃

yes

maximal, complemented

C₃×#6866

C₃

yes

maximal

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

characteristic

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

C₉×3₋¹⁺²

—

#6887

C₃²

yes

complemented

C₃×C₃²⋊C₉

C₃²

yes

—

#6868

C₃²

yes

complemented

#6867

C₃²

yes

complemented

#6866

C₃²

yes

—

#6866

C₃²

yes

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃×C₉

yes

—

C₃⁴

C₃³

yes

characteristic

C₃⁴

C₃³

yes

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

243

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₉

yes

—

C₃×3₋¹⁺²

C₃×C₉

yes

—

C₃×3₋¹⁺²

C₃×C₉

yes

—

C₃×3₋¹⁺²

C₃×C₉

yes

complemented

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃×3₋¹⁺²

3₋¹⁺²

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₉

yes

complemented

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃³

yes

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₉

yes

—

C₃³

C₃²×C₉

yes

—

C₃³

C₃²×C₉

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

162

—

3₋¹⁺²

C₃²⋊C₉

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃²×C₉

yes

—

C₃×C₉

C₃²×C₉

yes

—

C₃×C₉

C₃²×C₉

yes

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

C₃×C₃²⋊C₉

yes

—

102

C₃²

C₃³×C₉

yes

characteristic

103

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

104

C₃²

C₃²×3₋¹⁺²

yes

—

105

C₉

—

106

C₉

—

107

C₉

—

108

C₉

C₃×C₃²⋊C₉

yes

—

109

C₃

—

110

C₃

—

111

C₃

—

112

C₃

—

113

C₃

—

114

C₃

C₃²×C₃²⋊C₉

yes

characteristic

115

C₃

#10624934

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator	C₃²	C₃³×C₉
Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺² ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²×H(3,𝔽₃)

commutator

C₃²

C₃³×C₉

Frattini, cosocle

C₃³

C₃⁴

prime radical

C₃⁵

C₃²

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃²×3₋¹⁺² ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624863	C₃	semidirect
3. #10624692	C₃	semidirect
4. #10624691	C₃	semidirect
5. #10624690	C₃	semidirect
6. #10624688	C₃	semidirect
11. C₃²×3₋¹⁺²	C₉	semidirect
18. #6887	C₃²	semidirect
20. #6868	C₃²	semidirect
21. #6867	C₃²	semidirect
41. C₃×3₋¹⁺²	C₃×C₉	semidirect
47. C₃²×C₉	C₃×C₉	semidirect

H ≅ G/N

type

2. #10624863

C₃

semidirect

3. #10624692

C₃

semidirect

4. #10624691

C₃

semidirect

5. #10624690

C₃

semidirect

6. #10624688

C₃

semidirect

11. C₃²×3₋¹⁺²

C₉

semidirect

18. #6887

C₃²

semidirect

20. #6868

C₃²

semidirect

21. #6867

C₃²

semidirect

41. C₃×3₋¹⁺²

C₃×C₉

semidirect

47. C₃²×C₉

C₃×C₉

semidirect

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

subgroup	index
C₃³×3₋¹⁺²	3
#10624863	3
#10624692	3
#10624691	3
#10624690	3
#10624688	3
C₃×#6866	3

subgroup

index

C₃³×3₋¹⁺²

Subgroups — #14554138

Conjugacy classes of subgroups

Complements

Maximal subgroups