Group #14550945 — subgroups

Subgroups — #14550945

Order 2187.

324 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 47 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	#10624737	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624692	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624565	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624481	C₃	1	2	yes	maximal, complemented
6	#10624481	C₃	1	2	yes	maximal, complemented
7	#10624481	C₃	1	2	yes	maximal, complemented
8	#10624480	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624480	C₃	1	2	yes	maximal, complemented
10	C₃×#737	—	3	1	no	—
11	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	#6913	—	3	9	no	—
13	C₃×#733	C₃²	1	2	yes	—
14	C₃×#732	—	3	1	no	—
15	C₃×#732	—	3	1	no	—
16	C₃×#732	C₃²	1	1	yes	characteristic
17	C₃×#732	C₃²	1	1	yes	characteristic
18	C₃×#732	C₃²	1	2	yes	—
19	C₃×#731	—	3	1	no	—
20	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
21	C₃²⋊H(3,𝔽₃)	—	3	1	no	—
22	#6887	—	3	1	no	—
23	C₃×C₉⋊C₉	—	3	1	no	—
24	C₃×C₃²⋊C₉	—	3	1	no	—
25	#6868	C₃²	1	1	yes	characteristic
26	#6868	C₃²	1	2	yes	—
27	#6867	—	3	1	no	—
28	#6866	—	3	1	no	—
29	#6855	—	3	1	no	—
30	#6855	C₃²	1	1	yes	characteristic
31	#6855	C₃²	1	2	yes	—
32	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
33	#737	—	3	9	no	—
34	#737	—	9	3	no	—
35	#737	—	9	3	no	—
36	C₃×3₋¹⁺²	—	3	3	no	—
37	C₃×3₋¹⁺²	—	3	3	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	9	1	no	—
40	C₃×3₋¹⁺²	—	9	1	no	—
41	C₃×3₋¹⁺²	—	9	2	no	—
42	C₃×H(3,𝔽₃)	—	3	1	no	—
43	C₃×H(3,𝔽₃)	—	3	1	no	—
44	C₃×H(3,𝔽₃)	—	3	2	no	—
45	C₃×H(3,𝔽₃)	—	9	1	no	—
46	C₃×H(3,𝔽₃)	—	9	2	no	—
47	C₃²×C₉	—	3	1	no	—
48	C₃²×C₉	—	3	2	no	—
49	C₃²×C₉	—	9	1	no	—
50	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
51	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
52	C₃²×C₉	C₃³	1	1	yes	characteristic
53	#733	—	3	6	no	—
54	#732	—	3	3	no	—
55	#732	—	9	1	no	—
56	#732	—	9	2	no	—
57	#732	—	9	3	no	—
58	#732	—	9	3	no	—
59	#731	—	3	9	no	—
60	AΣL(1,𝔽₂₇)	—	3	9	no	—
61	C₉⋊C₉	—	9	2	no	—
62	C₉⋊C₉	—	9	3	no	—
63	C₃²⋊C₉	—	3	1	no	—
64	C₃²⋊C₉	—	3	2	no	—
65	C₃²⋊C₉	—	9	3	no	—
66	C₉²	—	9	1	no	—
67	C₃³	—	3	1	no	—
68	C₃³	—	3	1	no	—
69	C₃³	—	3	1	no	—
70	C₃³	—	3	1	no	—
71	C₃³	—	9	1	no	—
72	C₃³	—	9	2	no	—
73	C₃³	—	9	1	no	—
74	C₃³	—	9	1	no	—
75	C₃³	—	9	2	no	—
76	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
77	3₋¹⁺²	—	3	6	no	—
78	3₋¹⁺²	—	9	9	no	—
79	3₋¹⁺²	—	9	18	no	—
80	3₋¹⁺²	—	9	2	no	—
81	3₋¹⁺²	—	9	2	no	—
82	H(3,𝔽₃)	—	3	3	no	—
83	H(3,𝔽₃)	—	9	1	no	—
84	H(3,𝔽₃)	—	9	1	no	—
85	H(3,𝔽₃)	—	9	1	no	—
86	H(3,𝔽₃)	—	9	2	no	—
87	H(3,𝔽₃)	—	9	1	no	—
88	H(3,𝔽₃)	—	9	2	no	—
89	H(3,𝔽₃)	—	9	1	no	—
90	H(3,𝔽₃)	—	9	2	no	—
91	H(3,𝔽₃)	—	27	3	no	—
92	H(3,𝔽₃)	—	27	6	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	—	3	3	no	—
100	C₃×C₉	—	3	1	no	—
101	C₃×C₉	—	3	1	no	—
102	C₃×C₉	—	3	1	no	—
103	C₃×C₉	—	3	1	no	—
104	C₃×C₉	—	9	9	no	—
105	C₃×C₉	—	9	2	no	—
106	C₃×C₉	—	9	1	no	—
107	C₃×C₉	—	9	2	no	—
108	C₃×C₉	—	9	2	no	—
109	C₃×C₉	—	9	4	no	—
110	C₃×C₉	—	9	2	no	—
111	C₃×C₉	—	9	1	no	—
112	C₃×C₉	—	9	2	no	—
113	C₃×C₉	AΣL(1,𝔽₂₇)	1	3	yes	—
114	C₃×C₉	#732	1	3	yes	—
115	C₃²	—	3	1	no	—
116	C₃²	—	3	2	no	—
117	C₃²	—	9	3	no	—
118	C₃²	—	9	1	no	—
119	C₃²	—	9	3	no	—
120	C₃²	—	9	1	no	—
121	C₃²	—	9	2	no	—
122	C₃²	—	27	1	no	—
123	C₃²	—	27	1	no	—
124	C₃²	—	27	1	no	—
125	C₃²	—	27	2	no	—
126	C₃²	—	27	1	no	—
127	C₃²	—	27	2	no	—
128	C₃²	—	27	1	no	—
129	C₃²	—	27	2	no	—
130	C₃²	—	27	2	no	—
131	C₃²	—	27	4	no	—
132	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
133	C₃²	C₃×#732	1	1	yes	characteristic
134	C₃²	#6909	1	1	yes	characteristic
135	C₃²	#6911	1	1	yes	characteristic
136	C₉	—	3	3	no	—
137	C₉	—	3	3	no	—
138	C₉	—	9	3	no	—
139	C₉	—	9	1	no	—
140	C₉	—	9	6	no	—
141	C₉	—	27	2	no	—
142	C₉	—	27	1	no	—
143	C₉	—	27	2	no	—
144	C₃	—	3	1	no	—
145	C₃	—	3	1	no	—
146	C₃	—	3	1	no	—
147	C₃	—	27	1	no	—
148	C₃	—	27	3	no	—
149	C₃	—	81	1	no	—
150	C₃	—	81	2	no	—
151	C₃	#10624596	1	1	yes	characteristic
152	C₃	#10624851	1	1	yes	characteristic
153	C₃	#10624853	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6913

C₃

yes

maximal, characteristic, complemented

#10624737

C₃

yes

maximal, characteristic, complemented

#10624692

C₃

yes

maximal, characteristic, complemented

#10624565

C₃

yes

maximal, characteristic, complemented

#10624481

C₃

yes

maximal, complemented

#10624481

C₃

yes

maximal, complemented

#10624481

C₃

yes

maximal, complemented

#10624480

C₃

yes

maximal, characteristic, complemented

#10624480

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

characteristic

C₃×#732

C₃²

yes

—

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

C₃²⋊H(3,𝔽₃)

—

#6887

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

#6868

C₃²

yes

characteristic

#6868

C₃²

yes

—

—

—

—

C₃²

yes

characteristic

#6855

C₃²

yes

—

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)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

yes

C₃²×C₉

yes

C₃²×C₉

C₃³

yes

—

—

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

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

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

AΣL(1,𝔽₂₇)

yes

—

114

C₃×C₉

#732

yes

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

133

C₃²

C₃×#732

yes

characteristic

134

C₃²

#6909

yes

characteristic

135

C₃²

#6911

yes

characteristic

136

C₉

—

137

C₉

—

138

C₉

—

139

C₉

—

140

C₉

—

141

C₉

—

142

C₉

—

143

C₉

—

144

C₃

—

145

C₃

—

146

C₃

—

147

C₃

—

148

C₃

—

149

C₃

—

150

C₃

—

151

C₃

#10624596

yes

characteristic

152

C₃

#10624851

yes

characteristic

153

C₃

#10624853

yes

—

	subgroup	G/H
center, socle	C₃²	#6911
commutator, Frattini, cosocle	C₃²×C₉	C₃³
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×3₋¹⁺² ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

#6911

commutator, Frattini, cosocle

C₃²×C₉

C₃³

LCS

G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×3₋¹⁺² ◁ G

LECS

G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
1. C₃×#6913	C₃	semidirect
2. #10624737	C₃	semidirect
3. #10624692	C₃	semidirect
4. #10624565	C₃	semidirect
5. #10624481	C₃	semidirect
6. #10624481	C₃	semidirect
7. #10624481	C₃	semidirect
8. #10624480	C₃	semidirect
9. #10624480	C₃	semidirect

H ≅ G/N

type

1. C₃×#6913

C₃

semidirect

2. #10624737

C₃

semidirect

3. #10624692

C₃

semidirect

4. #10624565

C₃

semidirect

5. #10624481

C₃

semidirect

6. #10624481

C₃

semidirect

7. #10624481

C₃

semidirect

8. #10624480

C₃

semidirect

9. #10624480

C₃

semidirect

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

subgroup	index
C₃×#6913	3
#10624737	3
#10624692	3
#10624565	3
#10624481	3
#10624480	3

subgroup

index

Subgroups — #14550945

Conjugacy classes of subgroups

Complements

Maximal subgroups