Group #14547314 — subgroups

Subgroups — #14547314

Order 2187.

483 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 56 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×#6910	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624885	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624824	C₃	1	2	yes	maximal, complemented
4	#10624776	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624742	C₃	1	1	yes	maximal, characteristic
6	#10624594	C₃	1	2	yes	maximal, complemented
7	#10624591	C₃	1	2	yes	maximal, complemented
8	#10624590	C₃	1	2	yes	maximal, complemented
9	#10624563	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
11	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	#6910	—	3	9	no	—
15	C₃×#731	—	3	2	no	—
16	C₃×#731	—	3	1	no	—
17	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
18	#6896	—	3	6	no	—
19	#6896	—	3	3	no	—
20	#6895	—	3	2	no	—
21	#6891	C₃²	1	1	yes	characteristic
22	#6891	C₃²	1	2	yes	—
23	#6891	C₃²	1	2	yes	—
24	#6891	C₃²	1	1	yes	characteristic
25	#6887	—	3	1	no	—
26	C₃×C₃²⋊C₉	—	3	2	no	—
27	C₃×C₃²⋊C₉	—	3	1	no	—
28	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
29	3₋¹⁺²⋊C₉	—	3	6	no	—
30	#6868	—	3	2	no	—
31	#6868	—	3	1	no	—
32	#6860	—	3	6	no	—
33	#6857	—	3	6	no	—
34	#6857	—	3	6	no	—
35	C₃⁴	—	3	1	no	—
36	C₃⁴	—	3	1	no	—
37	C₃⁴	C₃³	1	1	yes	characteristic
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	1	no	—
40	C₃×3₋¹⁺²	—	3	3	no	—
41	C₃×3₋¹⁺²	—	3	2	no	—
42	C₃×3₋¹⁺²	—	3	6	no	—
43	C₃×3₋¹⁺²	—	3	2	no	—
44	C₃×3₋¹⁺²	—	3	2	no	—
45	C₃×3₋¹⁺²	—	3	1	no	—
46	C₃×3₋¹⁺²	—	3	2	no	—
47	C₃×3₋¹⁺²	—	9	2	no	—
48	C₃×3₋¹⁺²	—	9	4	no	—
49	C₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
50	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
51	C₃×H(3,𝔽₃)	—	3	1	no	—
52	C₃×H(3,𝔽₃)	—	3	1	no	—
53	C₃×H(3,𝔽₃)	—	3	3	no	—
54	C₃×H(3,𝔽₃)	—	3	2	no	—
55	C₃×H(3,𝔽₃)	—	3	2	no	—
56	C₃×H(3,𝔽₃)	—	9	3	no	—
57	C₃×H(3,𝔽₃)	—	9	2	no	—
58	C₃²×C₉	—	3	2	no	—
59	C₃²×C₉	—	3	2	no	—
60	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
61	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
62	#731	—	3	18	no	—
63	#731	—	3	9	no	—
64	AΣL(1,𝔽₂₇)	—	9	9	no	—
65	C₉⋊C₉	—	3	6	no	—
66	C₉⋊C₉	—	3	18	no	—
67	C₃²⋊C₉	—	3	2	no	—
68	C₃²⋊C₉	—	3	1	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	1	no	—
77	C₃²⋊C₉	—	3	2	no	—
78	C₃²⋊C₉	—	3	2	no	—
79	C₃²⋊C₉	—	3	2	no	—
80	C₃²⋊C₉	—	9	6	no	—
81	C₃²⋊C₉	—	9	3	no	—
82	C₉²	—	3	3	no	—
83	C₃³	—	3	1	no	—
84	C₃³	—	3	2	no	—
85	C₃³	—	3	3	no	—
86	C₃³	—	9	1	no	—
87	C₃³	—	9	2	no	—
88	C₃³	—	9	2	no	—
89	C₃³	—	9	2	no	—
90	C₃³	—	9	4	no	—
91	C₃³	—	9	3	no	—
92	C₃³	—	9	3	no	—
93	C₃³	—	9	1	no	—
94	C₃³	—	9	1	no	—
95	C₃³	—	9	1	no	—
96	C₃³	—	9	2	no	—
97	C₃³	—	27	2	no	—
98	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
99	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
100	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
101	3₋¹⁺²	—	3	6	no	—
102	3₋¹⁺²	—	9	6	no	—
103	3₋¹⁺²	—	9	3	no	—
104	3₋¹⁺²	—	9	4	no	—
105	3₋¹⁺²	—	9	6	no	—
106	3₋¹⁺²	—	9	6	no	—
107	3₋¹⁺²	—	9	3	no	—
108	3₋¹⁺²	—	9	6	no	—
109	3₋¹⁺²	—	9	3	no	—
110	3₋¹⁺²	—	27	2	no	—
111	H(3,𝔽₃)	—	3	3	no	—
112	H(3,𝔽₃)	—	3	3	no	—
113	H(3,𝔽₃)	—	9	6	no	—
114	H(3,𝔽₃)	—	9	6	no	—
115	H(3,𝔽₃)	—	9	2	no	—
116	H(3,𝔽₃)	—	9	2	no	—
117	C₃×C₉	—	3	1	no	—
118	C₃×C₉	—	3	2	no	—
119	C₃×C₉	—	3	3	no	—
120	C₃×C₉	—	3	6	no	—
121	C₃×C₉	—	3	2	no	—
122	C₃×C₉	—	3	2	no	—
123	C₃×C₉	—	3	2	no	—
124	C₃×C₉	—	3	2	no	—
125	C₃×C₉	—	9	2	no	—
126	C₃×C₉	—	9	6	no	—
127	C₃×C₉	—	9	2	no	—
128	C₃×C₉	—	9	1	no	—
129	C₃×C₉	—	9	2	no	—
130	C₃×C₉	—	9	2	no	—
131	C₃×C₉	—	9	4	no	—
132	C₃×C₉	AΣL(1,𝔽₂₇)	1	3	yes	complemented
133	C₃²	—	3	1	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	—	3	2	no	—
140	C₃²	—	3	2	no	—
141	C₃²	—	3	2	no	—
142	C₃²	—	9	3	no	—
143	C₃²	—	9	6	no	—
144	C₃²	—	9	3	no	—
145	C₃²	—	9	3	no	—
146	C₃²	—	27	1	no	—
147	C₃²	—	27	2	no	—
148	C₃²	—	27	2	no	—
149	C₃²	—	27	4	no	—
150	C₃²	—	27	9	no	—
151	C₃²	—	27	1	no	—
152	C₃²	—	27	1	no	—
153	C₃²	—	27	2	no	—
154	C₃²	—	27	2	no	—
155	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
156	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
157	C₃²	#6909	1	2	yes	—
158	C₉	—	3	3	no	—
159	C₉	—	9	2	no	—
160	C₉	—	9	6	no	—
161	C₉	—	9	3	no	—
162	C₉	—	9	6	no	—
163	C₉	—	27	2	no	—
164	C₉	—	27	1	no	—
165	C₉	—	27	2	no	—
166	C₃	—	3	1	no	—
167	C₃	—	3	2	no	—
168	C₃	—	9	2	no	—
169	C₃	—	9	1	no	—
170	C₃	—	27	3	no	—
171	C₃	—	27	3	no	—
172	C₃	#10624743	1	1	yes	characteristic
173	C₃	#10624749	1	1	yes	characteristic
174	C₃	#10624750	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6910

C₃

yes

maximal, characteristic, complemented

#10624885

C₃

yes

maximal, characteristic, complemented

#10624824

C₃

yes

maximal, complemented

#10624776

C₃

yes

maximal, characteristic, complemented

#10624742

C₃

yes

maximal, characteristic

#10624594

C₃

yes

maximal, complemented

#10624591

C₃

yes

maximal, complemented

#10624590

C₃

yes

maximal, complemented

#10624563

C₃

yes

maximal, characteristic, complemented

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

—

—

—

C₃×AΣL(1,𝔽₂₇)

—

—

—

—

C₃²

yes

characteristic

#6891

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

characteristic

#6887

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

—

—

—

—

—

C₃⁴

—

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

—

C₃×3₋¹⁺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)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

yes

—

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

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

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

100

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

110

3₋¹⁺²

—

111

H(3,𝔽₃)

—

112

H(3,𝔽₃)

—

113

H(3,𝔽₃)

—

114

H(3,𝔽₃)

—

115

H(3,𝔽₃)

—

116

H(3,𝔽₃)

—

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

C₃×C₉

AΣL(1,𝔽₂₇)

yes

complemented

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

C₃²⋊H(3,𝔽₃)

yes

characteristic

156

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

157

C₃²

#6909

yes

—

158

C₉

—

159

C₉

—

160

C₉

—

161

C₉

—

162

C₉

—

163

C₉

—

164

C₉

—

165

C₉

—

166

C₃

—

167

C₃

—

168

C₃

—

169

C₃

—

170

C₃

—

171

C₃

—

172

C₃

#10624743

yes

characteristic

173

C₃

#10624749

yes

characteristic

174

C₃

#10624750

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624742	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×#6910	C₃	semidirect
2. #10624885	C₃	semidirect
3. #10624824	C₃	semidirect
4. #10624776	C₃	semidirect
6. #10624594	C₃	semidirect
7. #10624591	C₃	semidirect
8. #10624590	C₃	semidirect
9. #10624563	C₃	semidirect
132. C₃×C₉	AΣL(1,𝔽₂₇)	semidirect

H ≅ G/N

type

1. C₃×#6910

C₃

semidirect

2. #10624885

C₃

semidirect

3. #10624824

C₃

semidirect

4. #10624776

C₃

semidirect

6. #10624594

C₃

semidirect

7. #10624591

C₃

semidirect

8. #10624590

C₃

semidirect

9. #10624563

C₃

semidirect

132. C₃×C₉

AΣL(1,𝔽₂₇)

semidirect

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

subgroup	index
C₃×#6910	3
#10624885	3
#10624824	3
#10624776	3
#10624742	3
#10624594	3
#10624591	3
#10624590	3
#10624563	3

subgroup

index

Subgroups — #14547314

Conjugacy classes of subgroups

Complements

Maximal subgroups