Group #14550948 — subgroups

Subgroups — #14550948

Order 2187.

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

normal subgroups only

This group has 338 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×#6911	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624771	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624739	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624692	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624478	C₃	1	6	yes	maximal, complemented
6	#10624477	C₃	1	3	yes	maximal, complemented
7	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
8	C₃²×H(3,𝔽₃)	—	3	1	no	—
9	#6911	—	3	9	no	—
10	C₃×#733	C₃²	1	2	yes	complemented
11	C₃×#732	—	3	1	no	—
12	C₃×#732	—	3	1	no	—
13	C₃×#732	C₃²	1	3	yes	—
14	C₃×#732	C₃²	1	1	yes	characteristic, complemented
15	C₃×#731	—	3	1	no	—
16	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
17	#6893	—	3	1	no	—
18	#6891	—	3	1	no	—
19	#6887	—	3	1	no	—
20	#6868	—	3	1	no	—
21	#6867	—	3	1	no	—
22	#6867	—	3	1	no	—
23	#6867	C₃²	1	3	yes	complemented
24	#6867	C₃²	1	3	yes	complemented
25	#6866	—	3	1	no	—
26	C₃⁴	—	9	1	no	—
27	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃×3₋¹⁺²	—	3	2	no	—
29	C₃×3₋¹⁺²	—	3	1	no	—
30	C₃×3₋¹⁺²	—	3	3	no	—
31	C₃×3₋¹⁺²	—	3	2	no	—
32	C₃×3₋¹⁺²	—	9	1	no	—
33	C₃×3₋¹⁺²	—	9	1	no	—
34	C₃×3₋¹⁺²	—	9	1	no	—
35	C₃×H(3,𝔽₃)	—	3	3	no	—
36	C₃×H(3,𝔽₃)	—	3	6	no	—
37	C₃×H(3,𝔽₃)	—	3	3	no	—
38	C₃×H(3,𝔽₃)	—	3	1	no	—
39	C₃×H(3,𝔽₃)	—	9	2	no	—
40	C₃×H(3,𝔽₃)	—	9	1	no	—
41	C₃×H(3,𝔽₃)	—	9	2	no	—
42	C₃×H(3,𝔽₃)	—	9	2	no	—
43	C₃×H(3,𝔽₃)	—	9	1	no	—
44	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
45	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
46	C₃²×C₉	C₃³	1	1	yes	characteristic
47	#733	—	9	2	no	—
48	#732	—	3	9	no	—
49	#732	—	3	9	no	—
50	#732	—	3	9	no	—
51	#732	—	9	1	no	—
52	#731	—	9	3	no	—
53	AΣL(1,𝔽₂₇)	—	9	2	no	—
54	AΣL(1,𝔽₂₇)	—	9	1	no	—
55	C₉⋊C₉	—	9	2	no	—
56	C₃²⋊C₉	—	3	3	no	—
57	C₃²⋊C₉	—	3	3	no	—
58	C₃²⋊C₉	—	9	2	no	—
59	C₃²⋊C₉	—	9	3	no	—
60	C₉²	—	9	1	no	—
61	C₃³	—	3	1	no	—
62	C₃³	—	3	1	no	—
63	C₃³	—	3	1	no	—
64	C₃³	—	9	6	no	—
65	C₃³	—	9	3	no	—
66	C₃³	—	9	3	no	—
67	C₃³	—	9	1	no	—
68	C₃³	—	27	1	no	—
69	C₃³	—	27	9	no	—
70	C₃³	#732	1	2	yes	—
71	C₃³	#732	1	1	yes	characteristic
72	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
73	3₋¹⁺²	—	9	6	no	—
74	3₋¹⁺²	—	9	3	no	—
75	3₋¹⁺²	—	9	2	no	—
76	3₋¹⁺²	—	9	2	no	—
77	3₋¹⁺²	—	9	2	no	—
78	3₋¹⁺²	—	27	3	no	—
79	H(3,𝔽₃)	—	3	9	no	—
80	H(3,𝔽₃)	—	9	6	no	—
81	H(3,𝔽₃)	—	9	6	no	—
82	H(3,𝔽₃)	—	9	6	no	—
83	H(3,𝔽₃)	—	9	3	no	—
84	H(3,𝔽₃)	—	9	3	no	—
85	H(3,𝔽₃)	—	9	1	no	—
86	H(3,𝔽₃)	—	9	1	no	—
87	H(3,𝔽₃)	—	9	1	no	—
88	C₃×C₉	—	3	1	no	—
89	C₃×C₉	—	3	1	no	—
90	C₃×C₉	—	3	1	no	—
91	C₃×C₉	—	3	1	no	—
92	C₃×C₉	—	3	1	no	—
93	C₃×C₉	—	3	1	no	—
94	C₃×C₉	—	3	1	no	—
95	C₃×C₉	—	3	1	no	—
96	C₃×C₉	—	3	1	no	—
97	C₃×C₉	—	3	1	no	—
98	C₃×C₉	—	3	1	no	—
99	C₃×C₉	—	9	2	no	—
100	C₃×C₉	—	9	3	no	—
101	C₃×C₉	—	9	3	no	—
102	C₃×C₉	AΣL(1,𝔽₂₇)	1	3	yes	—
103	C₃²	—	3	2	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	9	1	no	—
106	C₃²	—	9	1	no	—
107	C₃²	—	9	2	no	—
108	C₃²	—	9	3	no	—
109	C₃²	—	9	9	no	—
110	C₃²	—	27	1	no	—
111	C₃²	—	27	1	no	—
112	C₃²	—	27	9	no	—
113	C₃²	—	27	3	no	—
114	C₃²	—	27	9	no	—
115	C₃²	—	27	18	no	—
116	C₃²	—	27	1	no	—
117	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
118	C₃²	C₃×#732	1	1	yes	characteristic
119	C₃²	#6909	1	1	yes	characteristic
120	C₃²	#6911	1	1	yes	characteristic
121	C₉	—	3	3	no	—
122	C₉	—	9	1	no	—
123	C₉	—	9	1	no	—
124	C₉	—	27	3	no	—
125	C₉	—	27	3	no	—
126	C₉	—	27	2	no	—
127	C₃	—	3	1	no	—
128	C₃	—	3	1	no	—
129	C₃	—	3	1	no	—
130	C₃	—	3	1	no	—
131	C₃	—	9	1	no	—
132	C₃	—	9	2	no	—
133	C₃	—	27	9	no	—
134	C₃	—	81	1	no	—
135	C₃	#10624596	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6911

C₃

yes

maximal, characteristic, complemented

#10624771

C₃

yes

maximal, characteristic, complemented

#10624739

C₃

yes

maximal, characteristic, complemented

#10624692

C₃

yes

maximal, characteristic, complemented

#10624478

C₃

yes

maximal, complemented

#10624477

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

—

#6911

—

C₃×#733

C₃²

yes

—

—

C₃²

yes

—

C₃×#732

C₃²

yes

characteristic, complemented

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

—

—

—

—

—

—

C₃²

yes

complemented

#6867

C₃²

yes

complemented

#6866

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

characteristic

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×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

yes

C₃²×C₉

yes

C₃²×C₉

C₃³

yes

—

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

#732

yes

—

C₃³

#732

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

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

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

AΣL(1,𝔽₂₇)

yes

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

118

C₃²

C₃×#732

yes

characteristic

119

C₃²

#6909

yes

characteristic

120

C₃²

#6911

yes

characteristic

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₃

—

128

C₃

—

129

C₃

—

130

C₃

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

—

135

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×#6911	C₃	semidirect
2. #10624771	C₃	semidirect
3. #10624739	C₃	semidirect
4. #10624692	C₃	semidirect
5. #10624478	C₃	semidirect
6. #10624477	C₃	semidirect
10. C₃×#733	C₃²	semidirect
14. C₃×#732	C₃²	semidirect
23. #6867	C₃²	semidirect
24. #6867	C₃²	semidirect

H ≅ G/N

type

1. C₃×#6911

C₃

semidirect

2. #10624771

C₃

semidirect

3. #10624739

C₃

semidirect

4. #10624692

C₃

semidirect

5. #10624478

C₃

semidirect

6. #10624477

C₃

semidirect

10. C₃×#733

C₃²

semidirect

14. C₃×#732

C₃²

semidirect

23. #6867

C₃²

semidirect

24. #6867

C₃²

semidirect

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

subgroup	index
C₃×#6911	3
#10624771	3
#10624739	3
#10624692	3
#10624478	3
#10624477	3

subgroup

index

Subgroups — #14550948

Conjugacy classes of subgroups

Complements

Maximal subgroups