Group #14550437 — subgroups

Subgroups — #14550437

Order 2187.

503 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 107 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×#731	C₃	1	1	yes	maximal, characteristic
2	#10624863	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624699	C₃	1	2	yes	maximal, complemented
4	#10624699	C₃	1	2	yes	maximal, complemented
5	#10624698	C₃	1	2	yes	maximal
6	#10624690	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624689	C₃	1	1	yes	maximal, characteristic
8	C₃×#6868	C₃	1	2	yes	maximal, complemented
9	C₃×#6867	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
11	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃³×C₉	C₃²	1	1	yes	characteristic
14	C₃×#731	—	3	3	no	—
15	C₃×#731	—	3	6	no	—
16	C₃×#731	C₉	1	9	yes	complemented
17	C₉×3₋¹⁺²	—	3	3	no	—
18	C₉×3₋¹⁺²	—	3	6	no	—
19	#6887	—	3	3	no	—
20	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
21	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
23	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
24	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
25	3₋¹⁺²⋊C₉	—	3	6	no	—
26	3₋¹⁺²⋊C₉	—	3	6	no	—
27	H(3,𝔽₃)⋊C₉	—	3	6	no	—
28	#6868	—	3	6	no	—
29	#6867	—	3	3	no	—
30	#6867	—	3	3	no	—
31	#6866	—	3	3	no	—
32	C₃⁴	—	3	1	no	—
33	C₃⁴	C₃³	1	1	yes	characteristic
34	C₃×3₋¹⁺²	—	3	2	no	—
35	C₃×3₋¹⁺²	—	3	2	no	—
36	C₃×3₋¹⁺²	—	3	4	no	—
37	C₃×3₋¹⁺²	—	3	2	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	1	no	—
40	C₃×3₋¹⁺²	—	3	2	no	—
41	C₃×3₋¹⁺²	—	3	4	no	—
42	C₃×3₋¹⁺²	C₃×C₉	1	3	yes	—
43	C₃×3₋¹⁺²	C₃×C₉	1	3	yes	—
44	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
45	C₃×3₋¹⁺²	3₋¹⁺²	1	6	yes	—
46	C₃×H(3,𝔽₃)	—	3	2	no	—
47	C₃×H(3,𝔽₃)	—	3	1	no	—
48	C₃×H(3,𝔽₃)	—	3	2	no	—
49	C₃×H(3,𝔽₃)	—	3	4	no	—
50	C₃×H(3,𝔽₃)	—	3	2	no	—
51	C₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
52	C₃²×C₉	—	3	1	no	—
53	C₃²×C₉	—	3	1	no	—
54	C₃²×C₉	—	3	2	no	—
55	C₃²×C₉	—	3	1	no	—
56	C₃²×C₉	—	3	1	no	—
57	C₃²×C₉	—	3	2	no	—
58	C₃²×C₉	—	3	2	no	—
59	C₃²×C₉	—	3	4	no	—
60	C₃²×C₉	—	3	1	no	—
61	C₃²×C₉	—	3	2	no	—
62	C₃²×C₉	—	3	2	no	—
63	C₃²×C₉	—	3	2	no	—
64	C₃²×C₉	—	3	2	no	—
65	C₃²×C₉	C₃×C₉	1	3	yes	—
66	#731	—	3	27	no	—
67	C₉⋊C₉	—	3	18	no	—
68	C₃²⋊C₉	—	3	2	no	—
69	C₃²⋊C₉	—	3	2	no	—
70	C₃²⋊C₉	—	3	2	no	—
71	C₃²⋊C₉	—	3	2	no	—
72	C₃²⋊C₉	—	3	2	no	—
73	C₃²⋊C₉	—	3	2	no	—
74	C₃²⋊C₉	—	3	2	no	—
75	C₃²⋊C₉	—	3	2	no	—
76	C₃²⋊C₉	—	3	2	no	—
77	C₃²⋊C₉	—	3	2	no	—
78	C₃²⋊C₉	—	3	2	no	—
79	C₃²⋊C₉	—	3	2	no	—
80	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
81	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
82	C₉²	—	3	9	no	—
83	C₃³	—	3	3	no	—
84	C₃³	—	9	2	no	—
85	C₃³	—	9	1	no	—
86	C₃³	—	9	1	no	—
87	C₃³	—	9	2	no	—
88	C₃³	—	27	2	no	—
89	C₃³	—	27	1	no	—
90	C₃³	C₃²×C₉	1	1	yes	characteristic
91	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
92	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
93	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
94	C₃³	C₃×3₋¹⁺²	1	2	yes	—
95	C₃³	#737	1	2	yes	—
96	C₃³	#737	1	4	yes	—
97	3₋¹⁺²	—	3	6	no	—
98	3₋¹⁺²	—	3	12	no	—
99	3₋¹⁺²	—	3	6	no	—
100	3₋¹⁺²	—	3	3	no	—
101	3₋¹⁺²	—	3	6	no	—
102	3₋¹⁺²	—	9	4	no	—
103	3₋¹⁺²	—	9	2	no	—
104	3₋¹⁺²	C₃²⋊C₉	1	9	yes	—
105	H(3,𝔽₃)	—	3	3	no	—
106	H(3,𝔽₃)	—	3	6	no	—
107	H(3,𝔽₃)	—	9	4	no	—
108	H(3,𝔽₃)	—	9	2	no	—
109	C₃×C₉	—	3	3	no	—
110	C₃×C₉	—	3	3	no	—
111	C₃×C₉	—	3	6	no	—
112	C₃×C₉	—	3	3	no	—
113	C₃×C₉	—	3	3	no	—
114	C₃×C₉	—	3	6	no	—
115	C₃×C₉	—	3	3	no	—
116	C₃×C₉	—	3	6	no	—
117	C₃×C₉	—	3	3	no	—
118	C₃×C₉	—	9	1	no	—
119	C₃×C₉	—	9	1	no	—
120	C₃×C₉	—	9	2	no	—
121	C₃×C₉	—	9	4	no	—
122	C₃×C₉	—	9	2	no	—
123	C₃×C₉	—	9	2	no	—
124	C₃×C₉	—	9	2	no	—
125	C₃×C₉	—	9	2	no	—
126	C₃×C₉	—	9	2	no	—
127	C₃×C₉	—	27	2	no	—
128	C₃×C₉	—	27	1	no	—
129	C₃×C₉	—	27	2	no	—
130	C₃×C₉	—	27	2	no	—
131	C₃×C₉	—	27	2	no	—
132	C₃²	—	3	2	no	—
133	C₃²	—	3	2	no	—
134	C₃²	—	3	2	no	—
135	C₃²	—	3	2	no	—
136	C₃²	—	3	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	2	no	—
139	C₃²	—	9	1	no	—
140	C₃²	—	9	2	no	—
141	C₃²	—	9	2	no	—
142	C₃²	—	9	4	no	—
143	C₃²	—	9	3	no	—
144	C₃²	—	27	3	no	—
145	C₃²	—	27	6	no	—
146	C₃²	—	27	2	no	—
147	C₃²	—	27	1	no	—
148	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
149	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
150	C₃²	C₉×H(3,𝔽₃)	1	2	yes	—
151	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
152	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
153	C₃²	C₉⋊H(3,𝔽₃)	1	4	yes	—
154	C₃²	#6893	1	2	yes	—
155	C₉	—	3	9	no	—
156	C₉	—	9	3	no	—
157	C₉	—	9	3	no	—
158	C₉	—	9	6	no	—
159	C₉	—	9	3	no	—
160	C₉	—	27	2	no	—
161	C₉	—	27	2	no	—
162	C₉	—	27	2	no	—
163	C₃	—	3	1	no	—
164	C₃	—	3	2	no	—
165	C₃	—	3	1	no	—
166	C₃	—	3	2	no	—
167	C₃	—	3	4	no	—
168	C₃	—	3	2	no	—
169	C₃	—	27	3	no	—
170	C₃	#10624555	1	1	yes	characteristic
171	C₃	#10624763	1	2	yes	—
172	C₃	#10624764	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#731

C₃

yes

maximal, characteristic

#10624863

C₃

yes

maximal, characteristic, complemented

#10624699

C₃

yes

maximal, complemented

#10624699

C₃

yes

maximal, complemented

#10624698

C₃

yes

maximal

#10624690

C₃

yes

maximal, characteristic, complemented

#10624689

C₃

yes

maximal, characteristic

C₃×#6868

C₃

yes

maximal, complemented

C₃×#6867

C₃

yes

maximal, characteristic, complemented

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

—

C₉

yes

complemented

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

#6887

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

—

—

—

—

C₃⁴

—

C₃⁴

C₃³

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

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₉

yes

—

C₃×3₋¹⁺²

C₃×C₉

yes

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃×3₋¹⁺²

3₋¹⁺²

yes

—

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×C₉

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₉

yes

—

#731

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₉

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

#737

yes

—

C₃³

#737

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

C₃²⋊C₉

yes

—

105

H(3,𝔽₃)

—

106

H(3,𝔽₃)

—

107

H(3,𝔽₃)

—

108

H(3,𝔽₃)

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₃×C₉

—

125

C₃×C₉

—

126

C₃×C₉

—

127

C₃×C₉

—

128

C₃×C₉

—

129

C₃×C₉

—

130

C₃×C₉

—

131

C₃×C₉

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

147

C₃²

—

148

C₃²

C₃×C₃²⋊C₉

yes

characteristic

149

C₃²

C₉×H(3,𝔽₃)

yes

characteristic

150

C₃²

C₉×H(3,𝔽₃)

yes

—

151

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

152

C₃²

C₉⋊H(3,𝔽₃)

yes

—

153

C₃²

C₉⋊H(3,𝔽₃)

yes

—

154

C₃²

#6893

yes

—

155

C₉

—

156

C₉

—

157

C₉

—

158

C₉

—

159

C₉

—

160

C₉

—

161

C₉

—

162

C₉

—

163

C₃

—

164

C₃

—

165

C₃

—

166

C₃

—

167

C₃

—

168

C₃

—

169

C₃

—

170

C₃

#10624555

yes

characteristic

171

C₃

#10624763

yes

—

172

C₃

#10624764

yes

characteristic

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

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator

C₃³

C₃²×C₉

Frattini, cosocle

C₃⁴

C₃³

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624863	C₃	semidirect
3. #10624699	C₃	semidirect
4. #10624699	C₃	semidirect
6. #10624690	C₃	semidirect
8. C₃×#6868	C₃	semidirect
9. C₃×#6867	C₃	semidirect
16. C₃×#731	C₉	semidirect

H ≅ G/N

type

2. #10624863

C₃

semidirect

3. #10624699

C₃

semidirect

4. #10624699

C₃

semidirect

6. #10624690

C₃

semidirect

8. C₃×#6868

C₃

semidirect

9. C₃×#6867

C₃

semidirect

16. C₃×#731

C₉

semidirect

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

subgroup	index
C₃²×#731	3
#10624863	3
#10624699	3
#10624698	3
#10624690	3
#10624689	3
C₃×#6868	3
C₃×#6867	3

subgroup

index

C₃²×#731

Subgroups — #14550437

Conjugacy classes of subgroups

Complements

Maximal subgroups