Group #14550951 — subgroups

Subgroups — #14550951

Order 2187.

307 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 44 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	C₃×#6913	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624774	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624737	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624690	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624478	C₃	1	6	yes	maximal, complemented
6	#10624477	C₃	1	3	yes	maximal, complemented
7	C₃×#737	—	3	1	no	—
8	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
9	#6913	—	3	9	no	—
10	C₃×#733	C₃²	1	2	yes	—
11	C₃×#732	—	3	1	no	—
12	C₃×#732	—	3	1	no	—
13	C₃×#732	C₃²	1	1	yes	characteristic
14	C₃×#732	C₃²	1	3	yes	—
15	C₃×#731	—	3	1	no	—
16	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
17	#6895	—	3	1	no	—
18	C₃²⋊H(3,𝔽₃)	—	3	1	no	—
19	C₃×C₃²⋊C₉	—	3	1	no	—
20	#6868	—	3	1	no	—
21	#6867	—	3	1	no	—
22	#6867	—	3	1	no	—
23	#6867	C₃²	1	3	yes	—
24	#6867	C₃²	1	3	yes	—
25	#6866	—	3	1	no	—
26	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
27	#737	—	3	9	no	—
28	#737	—	9	3	no	—
29	#737	—	9	3	no	—
30	C₃×3₋¹⁺²	—	3	3	no	—
31	C₃×3₋¹⁺²	—	3	3	no	—
32	C₃×3₋¹⁺²	—	3	2	no	—
33	C₃×3₋¹⁺²	—	9	1	no	—
34	C₃×3₋¹⁺²	—	9	1	no	—
35	C₃×3₋¹⁺²	—	9	2	no	—
36	C₃×H(3,𝔽₃)	—	3	1	no	—
37	C₃×H(3,𝔽₃)	—	3	3	no	—
38	C₃×H(3,𝔽₃)	—	9	3	no	—
39	C₃²×C₉	—	9	1	no	—
40	C₃²×C₉	—	9	1	no	—
41	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
42	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
43	C₃²×C₉	C₃³	1	1	yes	characteristic
44	#733	—	3	6	no	—
45	#732	—	3	3	no	—
46	#732	—	9	3	no	—
47	#732	—	9	3	no	—
48	#732	—	9	3	no	—
49	#731	—	3	9	no	—
50	AΣL(1,𝔽₂₇)	—	3	9	no	—
51	C₉⋊C₉	—	9	3	no	—
52	C₃²⋊C₉	—	3	3	no	—
53	C₃²⋊C₉	—	3	3	no	—
54	C₃²⋊C₉	—	9	2	no	—
55	C₃³	—	3	1	no	—
56	C₃³	—	3	1	no	—
57	C₃³	—	3	1	no	—
58	C₃³	—	9	1	no	—
59	C₃³	—	9	3	no	—
60	C₃³	—	27	1	no	—
61	C₃³	AΣL(1,𝔽₂₇)	1	3	yes	—
62	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
63	3₋¹⁺²	—	3	6	no	—
64	3₋¹⁺²	—	9	9	no	—
65	3₋¹⁺²	—	9	18	no	—
66	3₋¹⁺²	—	9	2	no	—
67	3₋¹⁺²	—	9	2	no	—
68	H(3,𝔽₃)	—	3	3	no	—
69	H(3,𝔽₃)	—	9	1	no	—
70	H(3,𝔽₃)	—	9	1	no	—
71	H(3,𝔽₃)	—	9	3	no	—
72	H(3,𝔽₃)	—	9	3	no	—
73	H(3,𝔽₃)	—	9	3	no	—
74	H(3,𝔽₃)	—	27	9	no	—
75	C₃×C₉	—	3	1	no	—
76	C₃×C₉	—	3	1	no	—
77	C₃×C₉	—	3	1	no	—
78	C₃×C₉	—	3	1	no	—
79	C₃×C₉	—	3	1	no	—
80	C₃×C₉	—	3	1	no	—
81	C₃×C₉	—	3	1	no	—
82	C₃×C₉	—	3	1	no	—
83	C₃×C₉	—	3	1	no	—
84	C₃×C₉	—	3	1	no	—
85	C₃×C₉	—	3	1	no	—
86	C₃×C₉	—	9	9	no	—
87	C₃×C₉	—	9	2	no	—
88	C₃×C₉	—	9	3	no	—
89	C₃×C₉	—	9	3	no	—
90	C₃×C₉	—	27	3	no	—
91	C₃×C₉	#732	1	3	yes	—
92	C₃²	—	3	3	no	—
93	C₃²	—	9	1	no	—
94	C₃²	—	9	3	no	—
95	C₃²	—	9	1	no	—
96	C₃²	—	9	2	no	—
97	C₃²	—	9	3	no	—
98	C₃²	—	27	1	no	—
99	C₃²	—	27	1	no	—
100	C₃²	—	27	3	no	—
101	C₃²	—	27	9	no	—
102	C₃²	—	27	3	no	—
103	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
104	C₃²	C₃×#732	1	1	yes	characteristic
105	C₃²	#6909	1	1	yes	characteristic
106	C₃²	#6911	1	1	yes	characteristic
107	C₉	—	3	3	no	—
108	C₉	—	9	1	no	—
109	C₉	—	9	1	no	—
110	C₉	—	9	6	no	—
111	C₉	—	27	3	no	—
112	C₉	—	27	3	no	—
113	C₃	—	3	1	no	—
114	C₃	—	3	1	no	—
115	C₃	—	3	1	no	—
116	C₃	—	3	1	no	—
117	C₃	—	9	3	no	—
118	C₃	—	27	3	no	—
119	C₃	—	81	3	no	—
120	C₃	#10624596	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6913

C₃

yes

maximal, characteristic, complemented

#10624774

C₃

yes

maximal, characteristic, complemented

#10624737

C₃

yes

maximal, characteristic, complemented

#10624690

C₃

yes

maximal, characteristic, complemented

#10624478

C₃

yes

maximal, complemented

#10624477

C₃

yes

maximal, complemented

C₃×#737

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

#6913

—

C₃×#733

C₃²

yes

—

C₃×#732

—

C₃×#732

—

C₃×#732

C₃²

yes

characteristic

C₃×#732

C₃²

yes

—

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

#6895

—

C₃²⋊H(3,𝔽₃)

—

C₃×C₃²⋊C₉

—

—

—

—

C₃²

yes

—

#6867

C₃²

yes

—

#6866

—

C₃⁴

yes

—

—

—

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×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

yes

C₃²×C₉

yes

C₃²×C₉

C₃³

yes

—

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

AΣL(1,𝔽₂₇)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

—

—

—

—

—

—

—

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×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

#732

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

104

C₃²

C₃×#732

yes

characteristic

105

C₃²

#6909

yes

characteristic

106

C₃²

#6911

yes

characteristic

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

—

117

C₃

—

118

C₃

—

119

C₃

—

120

C₃

#10624596

yes

characteristic

	subgroup	G/H
center, socle	C₃	#10624596
commutator, Frattini, cosocle	C₃²×C₉	C₃³
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×3₋¹⁺² ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃

#10624596

commutator, Frattini, cosocle

C₃²×C₉

C₃³

LCS

G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×3₋¹⁺² ◁ G

LECS

G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
1. C₃×#6913	C₃	semidirect
2. #10624774	C₃	semidirect
3. #10624737	C₃	semidirect
4. #10624690	C₃	semidirect
5. #10624478	C₃	semidirect
6. #10624477	C₃	semidirect

H ≅ G/N

type

1. C₃×#6913

C₃

semidirect

2. #10624774

C₃

semidirect

3. #10624737

C₃

semidirect

4. #10624690

C₃

semidirect

5. #10624478

C₃

semidirect

6. #10624477

C₃

semidirect

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

subgroup	index
C₃×#6913	3
#10624774	3
#10624737	3
#10624690	3
#10624478	3
#10624477	3

subgroup

index

Subgroups — #14550951

Conjugacy classes of subgroups

Complements

Maximal subgroups