Group #14547216 — subgroups

Subgroups — #14547216

Order 2187.

644 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 59 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×#6891	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624783	C₃	1	2	yes	maximal, complemented
3	#10624765	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624745	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624730	C₃	1	1	yes	maximal, characteristic
6	C₃×#6866	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624610	C₃	1	2	yes	maximal, complemented
8	#10624609	C₃	1	2	yes	maximal, complemented
9	#10624592	C₃	1	2	yes	maximal, complemented
10	C₃⁵	C₃²	1	1	yes	characteristic
11	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
13	C₃×#731	—	3	3	no	—
14	C₃×AΣL(1,𝔽₂₇)	—	3	2	no	—
15	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
16	#6891	—	3	3	no	—
17	#6891	—	3	6	no	—
18	#6891	C₃²	1	2	yes	—
19	#6891	C₃²	1	2	yes	—
20	#6891	C₃²	1	1	yes	characteristic
21	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
22	C₃×C₃²⋊C₉	—	3	2	no	—
23	C₃×C₃²⋊C₉	—	3	1	no	—
24	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
25	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	3₋¹⁺²⋊C₉	—	3	6	no	—
27	#6866	—	3	2	no	—
28	#6866	—	3	1	no	—
29	#6866	—	3	2	no	—
30	#6866	—	3	1	no	—
31	#6858	—	3	6	no	—
32	#6858	—	3	6	no	—
33	#6857	—	3	6	no	—
34	C₃⁴	—	3	3	no	—
35	C₃⁴	—	3	2	no	—
36	C₃⁴	—	3	1	no	—
37	C₃⁴	—	9	2	no	—
38	C₃⁴	—	9	6	no	—
39	C₃⁴	—	9	3	no	—
40	C₃⁴	H(3,𝔽₃)	1	2	yes	—
41	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
42	C₃⁴	C₃³	1	1	yes	characteristic
43	C₃×3₋¹⁺²	—	3	2	no	—
44	C₃×3₋¹⁺²	—	3	2	no	—
45	C₃×3₋¹⁺²	—	3	2	no	—
46	C₃×3₋¹⁺²	—	3	2	no	—
47	C₃×3₋¹⁺²	—	3	1	no	—
48	C₃×3₋¹⁺²	—	3	2	no	—
49	C₃×3₋¹⁺²	—	9	3	no	—
50	C₃×3₋¹⁺²	—	9	6	no	—
51	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
52	C₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
53	C₃×H(3,𝔽₃)	—	3	1	no	—
54	C₃×H(3,𝔽₃)	—	3	1	no	—
55	C₃×H(3,𝔽₃)	—	3	2	no	—
56	C₃²×C₉	—	3	1	no	—
57	C₃²×C₉	—	3	2	no	—
58	C₃²×C₉	—	3	1	no	—
59	C₃²×C₉	—	3	2	no	—
60	#731	—	9	9	no	—
61	AΣL(1,𝔽₂₇)	—	9	6	no	—
62	AΣL(1,𝔽₂₇)	—	9	3	no	—
63	C₃²⋊C₉	—	3	9	no	—
64	C₃²⋊C₉	—	3	18	no	—
65	C₃²⋊C₉	—	3	2	no	—
66	C₃²⋊C₉	—	3	2	no	—
67	C₃²⋊C₉	—	3	2	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	1	no	—
74	C₃²⋊C₉	—	3	2	no	—
75	C₃²⋊C₉	—	3	1	no	—
76	C₃²⋊C₉	—	3	2	no	—
77	C₃²⋊C₉	—	3	2	no	—
78	C₃³	—	3	6	no	—
79	C₃³	—	3	1	no	—
80	C₃³	—	3	2	no	—
81	C₃³	—	3	1	no	—
82	C₃³	—	3	3	no	—
83	C₃³	—	3	2	no	—
84	C₃³	—	3	2	no	—
85	C₃³	—	3	2	no	—
86	C₃³	—	3	2	no	—
87	C₃³	—	3	2	no	—
88	C₃³	—	9	2	no	—
89	C₃³	—	9	3	no	—
90	C₃³	—	9	6	no	—
91	C₃³	—	9	6	no	—
92	C₃³	—	9	6	no	—
93	C₃³	—	9	6	no	—
94	C₃³	—	9	1	no	—
95	C₃³	—	9	2	no	—
96	C₃³	—	9	2	no	—
97	C₃³	—	9	2	no	—
98	C₃³	—	9	6	no	—
99	C₃³	—	9	27	no	—
100	C₃³	—	9	54	no	—
101	C₃³	—	9	3	no	—
102	C₃³	—	9	1	no	—
103	C₃³	#731	1	3	yes	complemented
104	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
105	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
106	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
107	3₋¹⁺²	—	3	3	no	—
108	3₋¹⁺²	—	3	6	no	—
109	3₋¹⁺²	—	9	6	no	—
110	3₋¹⁺²	—	9	6	no	—
111	3₋¹⁺²	—	9	6	no	—
112	3₋¹⁺²	—	9	6	no	—
113	3₋¹⁺²	—	9	2	no	—
114	3₋¹⁺²	—	9	3	no	—
115	3₋¹⁺²	—	9	4	no	—
116	3₋¹⁺²	—	27	2	no	—
117	H(3,𝔽₃)	—	9	1	no	—
118	H(3,𝔽₃)	—	9	3	no	—
119	H(3,𝔽₃)	—	9	2	no	—
120	H(3,𝔽₃)	—	27	2	no	—
121	C₃×C₉	—	3	3	no	—
122	C₃×C₉	—	3	6	no	—
123	C₃×C₉	—	9	3	no	—
124	C₃×C₉	—	9	6	no	—
125	C₃×C₉	—	9	1	no	—
126	C₃×C₉	—	9	2	no	—
127	C₃×C₉	—	9	1	no	—
128	C₃×C₉	—	9	2	no	—
129	C₃×C₉	—	9	2	no	—
130	C₃×C₉	—	9	3	no	—
131	C₃×C₉	—	27	2	no	—
132	C₃²	—	3	1	no	—
133	C₃²	—	3	1	no	—
134	C₃²	—	3	2	no	—
135	C₃²	—	3	2	no	—
136	C₃²	—	3	2	no	—
137	C₃²	—	3	3	no	—
138	C₃²	—	3	2	no	—
139	C₃²	—	3	2	no	—
140	C₃²	—	3	2	no	—
141	C₃²	—	3	6	no	—
142	C₃²	—	9	3	no	—
143	C₃²	—	9	6	no	—
144	C₃²	—	9	3	no	—
145	C₃²	—	9	6	no	—
146	C₃²	—	9	6	no	—
147	C₃²	—	9	6	no	—
148	C₃²	—	9	6	no	—
149	C₃²	—	9	27	no	—
150	C₃²	—	9	54	no	—
151	C₃²	—	9	2	no	—
152	C₃²	—	9	1	no	—
153	C₃²	—	9	2	no	—
154	C₃²	—	9	2	no	—
155	C₃²	—	9	2	no	—
156	C₃²	—	27	1	no	—
157	C₃²	—	27	1	no	—
158	C₃²	—	27	2	no	—
159	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
160	C₃²	C₃×#731	1	1	yes	characteristic
161	C₃²	#6909	1	3	yes	—
162	C₃²	#6910	1	2	yes	—
163	C₉	—	9	3	no	—
164	C₉	—	9	6	no	—
165	C₉	—	27	1	no	—
166	C₉	—	27	1	no	—
167	C₉	—	27	2	no	—
168	C₉	—	27	2	no	—
169	C₉	—	27	2	no	—
170	C₃	—	3	1	no	—
171	C₃	—	3	3	no	—
172	C₃	—	3	2	no	—
173	C₃	—	9	2	no	—
174	C₃	—	9	6	no	—
175	C₃	—	9	3	no	—
176	C₃	—	81	1	no	—
177	C₃	#10624735	1	1	yes	characteristic
178	C₃	#10624745	1	1	yes	characteristic
179	C₃	#10624755	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6891

C₃

yes

maximal, characteristic, complemented

#10624783

C₃

yes

maximal, complemented

#10624765

C₃

yes

maximal, characteristic, complemented

#10624745

C₃

yes

maximal, characteristic, complemented

#10624730

C₃

yes

maximal, characteristic

C₃×#6866

C₃

yes

maximal, characteristic, complemented

#10624610

C₃

yes

maximal, complemented

#10624609

C₃

yes

maximal, complemented

#10624592

C₃

yes

maximal, complemented

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

C₃×AΣL(1,𝔽₂₇)

—

#6891

—

#6891

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

characteristic

C₃²⋊H(3,𝔽₃)

C₃²

yes

characteristic

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

3₋¹⁺²⋊C₉

—

—

—

—

—

—

—

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

H(3,𝔽₃)

yes

characteristic

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

H(3,𝔽₃)

yes

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

#731

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

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

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

100

C₃³

—

101

C₃³

—

102

C₃³

—

103

C₃³

#731

yes

complemented

104

C₃³

C₃×H(3,𝔽₃)

yes

—

105

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

106

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

110

3₋¹⁺²

—

111

3₋¹⁺²

—

112

3₋¹⁺²

—

113

3₋¹⁺²

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

H(3,𝔽₃)

—

118

H(3,𝔽₃)

—

119

H(3,𝔽₃)

—

120

H(3,𝔽₃)

—

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

—

149

C₃²

—

150

C₃²

—

151

C₃²

—

152

C₃²

—

153

C₃²

—

154

C₃²

—

155

C₃²

—

156

C₃²

—

157

C₃²

—

158

C₃²

—

159

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

160

C₃²

C₃×#731

yes

characteristic

161

C₃²

#6909

yes

—

162

C₃²

#6910

yes

—

163

C₉

—

164

C₉

—

165

C₉

—

166

C₉

—

167

C₉

—

168

C₉

—

169

C₉

—

170

C₃

—

171

C₃

—

172

C₃

—

173

C₃

—

174

C₃

—

175

C₃

—

176

C₃

—

177

C₃

#10624735

yes

characteristic

178

C₃

#10624745

yes

characteristic

179

C₃

#10624755

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624730	C₃
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

C₃

G ▷ C₃⁴ ▷ C₃² ▷ C₁

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. C₃×#6891	C₃	semidirect
2. #10624783	C₃	semidirect
3. #10624765	C₃	semidirect
4. #10624745	C₃	semidirect
6. C₃×#6866	C₃	semidirect
7. #10624610	C₃	semidirect
8. #10624609	C₃	semidirect
9. #10624592	C₃	semidirect
103. C₃³	#731	semidirect

H ≅ G/N

type

1. C₃×#6891

C₃

semidirect

2. #10624783

C₃

semidirect

3. #10624765

C₃

semidirect

4. #10624745

C₃

semidirect

6. C₃×#6866

C₃

semidirect

7. #10624610

C₃

semidirect

8. #10624609

C₃

semidirect

9. #10624592

C₃

semidirect

103. C₃³

#731

semidirect

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

subgroup	index
C₃×#6891	3
#10624783	3
#10624765	3
#10624745	3
#10624730	3
C₃×#6866	3
#10624610	3
#10624609	3
#10624592	3

subgroup

index

Subgroups — #14547216

Conjugacy classes of subgroups

Complements

Maximal subgroups