Group #14550934 — subgroups

Subgroups — #14550934

Order 2187.

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

normal subgroups only

This group has 419 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	#10624880	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624773	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624735	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624689	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624475	C₃	1	6	yes	maximal, complemented
6	#10624474	C₃	1	3	yes	maximal, complemented
7	C₃²×H(3,𝔽₃)	—	3	1	no	—
8	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
9	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
10	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
11	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
12	#6891	—	3	1	no	—
13	#6891	C₃²	1	2	yes	complemented
14	C₃²⋊H(3,𝔽₃)	—	3	1	no	—
15	C₃²⋊H(3,𝔽₃)	—	3	1	no	—
16	C₃²⋊H(3,𝔽₃)	—	3	9	no	—
17	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
18	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic, complemented
19	C₃×C₃²⋊C₉	—	3	1	no	—
20	#6866	—	3	1	no	—
21	#6866	—	3	1	no	—
22	#6866	—	3	1	no	—
23	#6866	—	3	1	no	—
24	#6866	C₃²	1	3	yes	complemented
25	#6866	C₃²	1	3	yes	complemented
26	C₃⁴	—	3	3	no	—
27	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃⁴	C₃³	1	1	yes	characteristic
31	C₃×3₋¹⁺²	—	3	2	no	—
32	C₃×3₋¹⁺²	—	9	1	no	—
33	C₃×H(3,𝔽₃)	—	3	6	no	—
34	C₃×H(3,𝔽₃)	—	3	3	no	—
35	C₃×H(3,𝔽₃)	—	3	2	no	—
36	C₃×H(3,𝔽₃)	—	3	1	no	—
37	C₃×H(3,𝔽₃)	—	3	3	no	—
38	C₃×H(3,𝔽₃)	—	3	1	no	—
39	C₃×H(3,𝔽₃)	—	3	9	no	—
40	C₃×H(3,𝔽₃)	—	3	9	no	—
41	C₃×H(3,𝔽₃)	—	3	9	no	—
42	C₃×H(3,𝔽₃)	—	9	1	no	—
43	C₃×H(3,𝔽₃)	—	9	1	no	—
44	C₃×H(3,𝔽₃)	—	9	1	no	—
45	C₃×H(3,𝔽₃)	—	9	2	no	—
46	C₃×H(3,𝔽₃)	—	9	1	no	—
47	C₃×H(3,𝔽₃)	—	9	2	no	—
48	C₃×H(3,𝔽₃)	—	9	2	no	—
49	C₃×H(3,𝔽₃)	—	9	1	no	—
50	C₃×H(3,𝔽₃)	—	9	1	no	—
51	C₃²×C₉	—	9	1	no	—
52	AΣL(1,𝔽₂₇)	—	9	3	no	—
53	AΣL(1,𝔽₂₇)	—	9	3	no	—
54	AΣL(1,𝔽₂₇)	—	9	2	no	—
55	AΣL(1,𝔽₂₇)	—	9	1	no	—
56	C₃²⋊C₉	—	3	3	no	—
57	C₃²⋊C₉	—	3	3	no	—
58	C₃²⋊C₉	—	9	2	no	—
59	C₃²⋊C₉	—	9	2	no	—
60	C₃²⋊C₉	—	9	2	no	—
61	C₃³	—	3	1	no	—
62	C₃³	—	3	1	no	—
63	C₃³	—	3	1	no	—
64	C₃³	—	3	1	no	—
65	C₃³	—	3	1	no	—
66	C₃³	—	3	1	no	—
67	C₃³	—	3	1	no	—
68	C₃³	—	3	1	no	—
69	C₃³	—	3	9	no	—
70	C₃³	—	3	1	no	—
71	C₃³	—	3	1	no	—
72	C₃³	—	3	1	no	—
73	C₃³	—	3	1	no	—
74	C₃³	—	3	1	no	—
75	C₃³	—	3	1	no	—
76	C₃³	—	9	6	no	—
77	C₃³	—	9	3	no	—
78	C₃³	—	9	1	no	—
79	C₃³	—	27	1	no	—
80	C₃³	—	27	1	no	—
81	C₃³	—	27	1	no	—
82	C₃³	—	27	1	no	—
83	C₃³	—	27	9	no	—
84	C₃³	AΣL(1,𝔽₂₇)	1	3	yes	—
85	C₃³	#732	1	2	yes	—
86	C₃³	#732	1	1	yes	characteristic
87	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
88	3₋¹⁺²	—	9	2	no	—
89	3₋¹⁺²	—	9	2	no	—
90	3₋¹⁺²	—	9	2	no	—
91	3₋¹⁺²	—	27	3	no	—
92	H(3,𝔽₃)	—	9	6	no	—
93	H(3,𝔽₃)	—	9	3	no	—
94	H(3,𝔽₃)	—	9	6	no	—
95	H(3,𝔽₃)	—	9	6	no	—
96	H(3,𝔽₃)	—	9	6	no	—
97	H(3,𝔽₃)	—	9	3	no	—
98	H(3,𝔽₃)	—	9	3	no	—
99	H(3,𝔽₃)	—	9	3	no	—
100	H(3,𝔽₃)	—	9	1	no	—
101	H(3,𝔽₃)	—	9	1	no	—
102	H(3,𝔽₃)	—	9	1	no	—
103	H(3,𝔽₃)	—	27	9	no	—
104	H(3,𝔽₃)	—	27	9	no	—
105	H(3,𝔽₃)	—	27	9	no	—
106	H(3,𝔽₃)	—	27	3	no	—
107	C₃×C₉	—	9	3	no	—
108	C₃×C₉	—	9	2	no	—
109	C₃×C₉	—	9	3	no	—
110	C₃×C₉	—	27	3	no	—
111	C₃²	—	3	2	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	3	3	no	—
114	C₃²	—	9	1	no	—
115	C₃²	—	9	3	no	—
116	C₃²	—	9	1	no	—
117	C₃²	—	9	2	no	—
118	C₃²	—	9	1	no	—
119	C₃²	—	9	2	no	—
120	C₃²	—	9	3	no	—
121	C₃²	—	9	3	no	—
122	C₃²	—	9	3	no	—
123	C₃²	—	9	27	no	—
124	C₃²	—	9	1	no	—
125	C₃²	—	9	2	no	—
126	C₃²	—	9	1	no	—
127	C₃²	—	9	1	no	—
128	C₃²	—	9	9	no	—
129	C₃²	—	27	1	no	—
130	C₃²	—	27	1	no	—
131	C₃²	—	27	1	no	—
132	C₃²	—	27	1	no	—
133	C₃²	—	27	1	no	—
134	C₃²	—	27	1	no	—
135	C₃²	—	27	1	no	—
136	C₃²	—	27	1	no	—
137	C₃²	—	27	1	no	—
138	C₃²	—	27	3	no	—
139	C₃²	—	27	9	no	—
140	C₃²	—	27	18	no	—
141	C₃²	—	27	1	no	—
142	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
143	C₃²	C₃×#732	1	1	yes	characteristic
144	C₃²	#6909	1	1	yes	characteristic
145	C₃²	#6911	1	1	yes	characteristic
146	C₉	—	27	3	no	—
147	C₉	—	27	3	no	—
148	C₉	—	27	2	no	—
149	C₃	—	3	1	no	—
150	C₃	—	3	1	no	—
151	C₃	—	3	1	no	—
152	C₃	—	3	1	no	—
153	C₃	—	9	3	no	—
154	C₃	—	9	1	no	—
155	C₃	—	9	2	no	—
156	C₃	—	27	1	no	—
157	C₃	—	27	1	no	—
158	C₃	—	27	9	no	—
159	C₃	—	81	1	no	—
160	C₃	#10624596	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624880

C₃

yes

maximal, characteristic, complemented

#10624773

C₃

yes

maximal, characteristic, complemented

#10624735

C₃

yes

maximal, characteristic, complemented

#10624689

C₃

yes

maximal, characteristic, complemented

#10624475

C₃

yes

maximal, complemented

#10624474

C₃

yes

maximal, complemented

C₃²×H(3,𝔽₃)

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×AΣL(1,𝔽₂₇)

—

C₃×AΣL(1,𝔽₂₇)

—

C₃×AΣL(1,𝔽₂₇)

—

#6891

—

#6891

C₃²

yes

complemented

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

characteristic, complemented

C₃×C₃²⋊C₉

—

—

—

—

—

C₃²

yes

complemented

#6866

C₃²

yes

complemented

C₃⁴

—

C₃⁴

yes

C₃⁴

yes

C₃⁴

yes

C₃⁴

C₃³

yes

characteristic

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)

—

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

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

AΣL(1,𝔽₂₇)

yes

—

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

—

—

—

—

—

—

—

—

—

100

H(3,𝔽₃)

—

101

H(3,𝔽₃)

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

—

104

H(3,𝔽₃)

—

105

H(3,𝔽₃)

—

106

H(3,𝔽₃)

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

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

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

143

C₃²

C₃×#732

yes

characteristic

144

C₃²

#6909

yes

characteristic

145

C₃²

#6911

yes

characteristic

146

C₉

—

147

C₉

—

148

C₉

—

149

C₃

—

150

C₃

—

151

C₃

—

152

C₃

—

153

C₃

—

154

C₃

—

155

C₃

—

156

C₃

—

157

C₃

—

158

C₃

—

159

C₃

—

160

C₃

#10624596

yes

characteristic

	subgroup	G/H
center, socle	C₃	#10624596
commutator, Frattini, cosocle	C₃⁴	C₃³
LCS	G ▷ C₃⁴ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×H(3,𝔽₃) ◁ G
LECS	G ▷ C₃⁴ ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃

#10624596

commutator, Frattini, cosocle

C₃⁴

C₃³

LCS

G ▷ C₃⁴ ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×H(3,𝔽₃) ◁ G

LECS

G ▷ C₃⁴ ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
1. #10624880	C₃	semidirect
2. #10624773	C₃	semidirect
3. #10624735	C₃	semidirect
4. #10624689	C₃	semidirect
5. #10624475	C₃	semidirect
6. #10624474	C₃	semidirect
13. #6891	C₃²	semidirect
18. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
24. #6866	C₃²	semidirect
25. #6866	C₃²	semidirect

H ≅ G/N

type

1. #10624880

C₃

semidirect

2. #10624773

C₃

semidirect

3. #10624735

C₃

semidirect

4. #10624689

C₃

semidirect

5. #10624475

C₃

semidirect

6. #10624474

C₃

semidirect

13. #6891

C₃²

semidirect

18. C₃²⋊H(3,𝔽₃)

C₃²

semidirect

24. #6866

C₃²

semidirect

25. #6866

C₃²

semidirect

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

subgroup	index
#10624880	3
#10624773	3
#10624735	3
#10624689	3
#10624475	3
#10624474	3

subgroup

index

Subgroups — #14550934

Conjugacy classes of subgroups

Complements

Maximal subgroups