Group #14554263 — subgroups

Subgroups — #14554263

Order 2187.

2063 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2063 conjugacy classes of subgroups — too many to draw. Only its 314 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	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#6908	C₃	1	9	yes	maximal, complemented
3	#10624869	C₃	1	3	yes	maximal, complemented
4	#10624739	C₃	1	18	yes	maximal, complemented
5	#10624738	C₃	1	2	yes	maximal, complemented
6	#10624737	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624736	C₃	1	3	yes	maximal, complemented
8	#10624735	C₃	1	3	yes	maximal, complemented
9	C₃×#737	—	3	6	no	—
10	C₃×#737	—	3	3	no	—
11	C₃×#737	—	3	27	no	—
12	C₃×#737	C₃²	1	9	yes	—
13	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
14	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
15	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
16	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
17	C₃³×C₉	C₃²	1	3	yes	—
18	C₃³×C₉	C₃²	1	1	yes	characteristic
19	#6908	—	3	81	no	—
20	C₃×#733	C₃²	1	18	yes	—
21	C₃×#732	C₃²	1	9	yes	—
22	C₃×#731	C₃²	1	27	yes	—
23	C₃×AΣL(1,𝔽₂₇)	C₃²	1	27	yes	—
24	#6893	C₃²	1	18	yes	—
25	C₉⋊H(3,𝔽₃)	—	3	18	no	—
26	#6891	C₃²	1	6	yes	—
27	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
28	C₉×H(3,𝔽₃)	—	3	9	no	—
29	C₃⁴	C₃³	1	3	yes	—
30	C₃⁴	C₃³	1	1	yes	characteristic
31	#737	—	3	81	no	—
32	#737	—	3	162	no	—
33	#737	—	3	81	no	—
34	C₃×3₋¹⁺²	—	3	3	no	—
35	C₃×3₋¹⁺²	—	3	3	no	—
36	C₃×3₋¹⁺²	—	3	18	no	—
37	C₃×3₋¹⁺²	—	3	18	no	—
38	C₃×3₋¹⁺²	—	3	9	no	—
39	C₃×3₋¹⁺²	—	3	18	no	—
40	C₃×3₋¹⁺²	—	3	9	no	—
41	C₃×3₋¹⁺²	—	3	18	no	—
42	C₃×3₋¹⁺²	—	3	18	no	—
43	C₃×3₋¹⁺²	C₃³	1	18	yes	—
44	C₃×H(3,𝔽₃)	—	3	3	no	—
45	C₃×H(3,𝔽₃)	—	3	9	no	—
46	C₃×H(3,𝔽₃)	—	3	9	no	—
47	C₃×H(3,𝔽₃)	C₃³	1	9	yes	—
48	C₃²×C₉	—	3	6	no	—
49	C₃²×C₉	—	3	18	no	—
50	C₃²×C₉	—	3	9	no	—
51	C₃²×C₉	—	3	3	no	—
52	C₃²×C₉	—	3	9	no	—
53	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
54	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
55	C₃²×C₉	H(3,𝔽₃)	1	27	yes	complemented
56	C₃²×C₉	C₃³	1	3	yes	—
57	C₃²×C₉	C₃³	1	3	yes	—
58	C₃²×C₉	C₃³	1	2	yes	—
59	C₃²×C₉	C₃³	1	1	yes	characteristic
60	#733	—	3	54	no	—
61	#732	—	3	27	no	—
62	#731	—	3	81	no	—
63	AΣL(1,𝔽₂₇)	—	3	81	no	—
64	C₃²⋊C₉	—	3	18	no	—
65	C₃²⋊C₉	—	3	54	no	—
66	C₃³	—	3	6	no	—
67	C₃³	—	3	3	no	—
68	C₃³	—	3	3	no	—
69	C₃³	—	3	9	no	—
70	C₃³	—	9	3	no	—
71	C₃³	—	9	9	no	—
72	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
73	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
74	C₃³	C₃⁴	1	1	yes	characteristic
75	3₋¹⁺²	—	3	162	no	—
76	3₋¹⁺²	—	3	27	no	—
77	3₋¹⁺²	—	3	27	no	—
78	3₋¹⁺²	—	3	54	no	—
79	3₋¹⁺²	—	9	54	no	—
80	H(3,𝔽₃)	—	3	27	no	—
81	H(3,𝔽₃)	—	3	27	no	—
82	H(3,𝔽₃)	—	9	27	no	—
83	C₃×C₉	—	3	6	no	—
84	C₃×C₉	—	3	6	no	—
85	C₃×C₉	—	3	6	no	—
86	C₃×C₉	—	3	3	no	—
87	C₃×C₉	—	3	3	no	—
88	C₃×C₉	—	3	9	no	—
89	C₃×C₉	—	3	18	no	—
90	C₃×C₉	—	3	27	no	—
91	C₃×C₉	—	3	54	no	—
92	C₃×C₉	—	3	18	no	—
93	C₃×C₉	—	9	27	no	—
94	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
95	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
96	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
97	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
98	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
99	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
100	C₃²	—	3	3	no	—
101	C₃²	—	3	3	no	—
102	C₃²	—	3	9	no	—
103	C₃²	—	3	9	no	—
104	C₃²	—	9	9	no	—
105	C₃²	—	9	9	no	—
106	C₃²	—	9	18	no	—
107	C₃²	—	9	9	no	—
108	C₃²	—	9	9	no	—
109	C₃²	—	9	27	no	—
110	C₃²	C₃²×H(3,𝔽₃)	1	3	yes	—
111	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
112	C₉	—	3	6	no	—
113	C₉	—	3	9	no	—
114	C₉	—	3	9	no	—
115	C₉	—	9	18	no	—
116	C₉	C₃²⋊H(3,𝔽₃)	1	3	yes	—
117	C₉	C₃²⋊H(3,𝔽₃)	1	6	yes	—
118	C₃	—	3	3	no	—
119	C₃	—	9	3	no	—
120	C₃	—	9	9	no	—
121	C₃	—	27	9	no	—
122	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
123	C₃	#10624929	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#737

C₃

yes

maximal, characteristic, complemented

C₃×#6908

C₃

yes

maximal, complemented

#10624869

C₃

yes

maximal, complemented

#10624739

C₃

yes

maximal, complemented

#10624738

C₃

yes

maximal, complemented

#10624737

C₃

yes

maximal, characteristic, complemented

#10624736

C₃

yes

maximal, complemented

#10624735

C₃

yes

maximal, complemented

—

—

—

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

C₃³×C₉

C₃²

yes

characteristic

#6908

—

C₃×#733

C₃²

yes

—

C₃×#732

C₃²

yes

—

C₃×#731

C₃²

yes

—

C₃×AΣL(1,𝔽₂₇)

C₃²

yes

—

#6893

C₃²

yes

—

C₉⋊H(3,𝔽₃)

—

#6891

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

—

C₉×H(3,𝔽₃)

—

C₃⁴

C₃³

yes

—

C₃⁴

C₃³

yes

characteristic

#737

—

#737

—

162

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃³

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃³

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

complemented

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

—

—

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

162

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

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×H(3,𝔽₃)

yes

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

characteristic

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

C₃²×H(3,𝔽₃)

yes

—

111

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

C₃²⋊H(3,𝔽₃)

yes

—

117

C₉

C₃²⋊H(3,𝔽₃)

yes

—

118

C₃

—

119

C₃

—

120

C₃

—

121

C₃

—

122

C₃

C₃×C₃²⋊H(3,𝔽₃)

yes

characteristic

123

C₃

#10624929

yes

—

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

subgroup

G/H

center

C₃×C₉

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃⁴

socle

C₃²

C₃²×H(3,𝔽₃)

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃×C₉ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
1. C₃²×#737	C₃	semidirect
2. C₃×#6908	C₃	semidirect
3. #10624869	C₃	semidirect
4. #10624739	C₃	semidirect
5. #10624738	C₃	semidirect
6. #10624737	C₃	semidirect
7. #10624736	C₃	semidirect
8. #10624735	C₃	semidirect
55. C₃²×C₉	H(3,𝔽₃)	semidirect

H ≅ G/N

type

1. C₃²×#737

C₃

semidirect

2. C₃×#6908

C₃

semidirect

3. #10624869

C₃

semidirect

4. #10624739

C₃

semidirect

5. #10624738

C₃

semidirect

6. #10624737

C₃

semidirect

7. #10624736

C₃

semidirect

8. #10624735

C₃

semidirect

55. C₃²×C₉

H(3,𝔽₃)

semidirect

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

subgroup	index
C₃²×#737	3
C₃×#6908	3
#10624869	3
#10624739	3
#10624738	3
#10624737	3
#10624736	3
#10624735	3

subgroup

index

C₃²×#737

Subgroups — #14554263

Conjugacy classes of subgroups

Complements

Maximal subgroups