Group #14547004 — subgroups

Subgroups — #14547004

Order 2187.

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

normal subgroups only

This group has 568 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	#10624880	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624811	C₃	1	2	yes	maximal, complemented
3	#10624735	C₃	1	2	yes	maximal, complemented
4	#10624732	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624636	C₃	1	4	yes	maximal, complemented
6	#10624635	C₃	1	2	yes	maximal, complemented
7	#10624555	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×H(3,𝔽₃)	—	3	1	no	—
9	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
10	C₃³×C₉	C₃²	1	1	yes	characteristic, complemented
11	C₃×#732	—	3	3	no	—
12	C₃×AΣL(1,𝔽₂₇)	—	3	4	no	—
13	C₃×AΣL(1,𝔽₂₇)	—	3	2	no	—
14	#6891	C₃²	1	4	yes	complemented
15	C₃²⋊H(3,𝔽₃)	—	3	2	no	—
16	C₃²⋊H(3,𝔽₃)	—	3	6	no	—
17	C₃²⋊H(3,𝔽₃)	—	3	3	no	—
18	C₃²⋊H(3,𝔽₃)	C₃²	1	2	yes	—
19	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
20	C₃²⋊H(3,𝔽₃)	C₃²	1	2	yes	complemented
21	C₃×C₃²⋊C₉	—	3	2	no	—
22	C₃×C₃²⋊C₉	—	3	1	no	—
23	C₃×C₃²⋊C₉	C₃²	1	2	yes	complemented
24	H(3,𝔽₃)⋊C₉	—	3	6	no	—
25	#6859	—	3	6	no	—
26	#6856	—	3	12	no	—
27	C₃⁴	—	3	3	no	—
28	C₃⁴	H(3,𝔽₃)	1	2	yes	—
29	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃⁴	C₃³	1	1	yes	characteristic
31	C₃×3₋¹⁺²	—	3	4	no	—
32	C₃×H(3,𝔽₃)	—	3	6	no	—
33	C₃×H(3,𝔽₃)	—	3	3	no	—
34	C₃×H(3,𝔽₃)	—	3	2	no	—
35	C₃×H(3,𝔽₃)	—	3	1	no	—
36	C₃×H(3,𝔽₃)	—	3	1	no	—
37	C₃×H(3,𝔽₃)	—	3	2	no	—
38	C₃×H(3,𝔽₃)	—	3	2	no	—
39	C₃×H(3,𝔽₃)	—	3	2	no	—
40	C₃×H(3,𝔽₃)	—	3	6	no	—
41	C₃×H(3,𝔽₃)	—	3	12	no	—
42	C₃×H(3,𝔽₃)	—	3	2	no	—
43	C₃×H(3,𝔽₃)	—	3	2	no	—
44	C₃×H(3,𝔽₃)	—	3	2	no	—
45	C₃×H(3,𝔽₃)	—	3	4	no	—
46	C₃×H(3,𝔽₃)	—	9	2	no	—
47	C₃×H(3,𝔽₃)	—	9	6	no	—
48	C₃×H(3,𝔽₃)	—	9	3	no	—
49	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	6	yes	—
50	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
51	C₃²×C₉	—	3	1	no	—
52	C₃²×C₉	—	3	1	no	—
53	C₃²×C₉	—	3	2	no	—
54	C₃²×C₉	—	3	2	no	—
55	C₃²×C₉	—	9	3	no	—
56	#732	—	9	9	no	—
57	AΣL(1,𝔽₂₇)	—	9	12	no	—
58	AΣL(1,𝔽₂₇)	—	9	6	no	—
59	C₃²⋊C₉	—	3	4	no	—
60	C₃²⋊C₉	—	3	4	no	—
61	C₃²⋊C₉	—	3	2	no	—
62	C₃²⋊C₉	—	3	4	no	—
63	C₃²⋊C₉	—	3	4	no	—
64	C₃²⋊C₉	—	9	3	no	—
65	C₃²⋊C₉	—	9	6	no	—
66	C₃³	—	3	1	no	—
67	C₃³	—	3	1	no	—
68	C₃³	—	3	2	no	—
69	C₃³	—	3	2	no	—
70	C₃³	—	3	2	no	—
71	C₃³	—	3	3	no	—
72	C₃³	—	3	6	no	—
73	C₃³	—	3	2	no	—
74	C₃³	—	3	4	no	—
75	C₃³	—	9	1	no	—
76	C₃³	—	9	2	no	—
77	C₃³	—	9	1	no	—
78	C₃³	—	9	2	no	—
79	C₃³	—	9	2	no	—
80	C₃³	—	9	4	no	—
81	C₃³	—	9	9	no	—
82	C₃³	—	9	2	no	—
83	C₃³	—	27	3	no	—
84	C₃³	—	27	6	no	—
85	C₃³	#732	1	3	yes	complemented
86	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
87	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
88	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
89	3₋¹⁺²	—	9	4	no	—
90	3₋¹⁺²	—	9	8	no	—
91	H(3,𝔽₃)	—	9	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	18	no	—
96	H(3,𝔽₃)	—	9	6	no	—
97	H(3,𝔽₃)	—	9	2	no	—
98	H(3,𝔽₃)	—	9	1	no	—
99	H(3,𝔽₃)	—	9	4	no	—
100	H(3,𝔽₃)	—	9	2	no	—
101	H(3,𝔽₃)	—	9	2	no	—
102	H(3,𝔽₃)	—	9	4	no	—
103	H(3,𝔽₃)	—	27	2	no	—
104	H(3,𝔽₃)	—	27	2	no	—
105	H(3,𝔽₃)	—	27	2	no	—
106	H(3,𝔽₃)	—	27	4	no	—
107	H(3,𝔽₃)	—	27	6	no	—
108	H(3,𝔽₃)	—	27	12	no	—
109	C₃×C₉	—	3	3	no	—
110	C₃×C₉	—	3	3	no	—
111	C₃×C₉	—	9	2	no	—
112	C₃×C₉	—	9	2	no	—
113	C₃×C₉	—	9	1	no	—
114	C₃×C₉	—	9	6	no	—
115	C₃×C₉	—	9	2	no	—
116	C₃×C₉	—	9	4	no	—
117	C₃×C₉	—	9	2	no	—
118	C₃×C₉	—	9	4	no	—
119	C₃²	—	3	1	no	—
120	C₃²	—	3	2	no	—
121	C₃²	—	3	3	no	—
122	C₃²	—	3	1	no	—
123	C₃²	—	3	2	no	—
124	C₃²	—	3	1	no	—
125	C₃²	—	3	2	no	—
126	C₃²	—	3	2	no	—
127	C₃²	—	3	2	no	—
128	C₃²	—	3	2	no	—
129	C₃²	—	9	3	no	—
130	C₃²	—	9	3	no	—
131	C₃²	—	9	6	no	—
132	C₃²	—	9	6	no	—
133	C₃²	—	9	1	no	—
134	C₃²	—	9	2	no	—
135	C₃²	—	9	2	no	—
136	C₃²	—	9	3	no	—
137	C₃²	—	9	6	no	—
138	C₃²	—	9	3	no	—
139	C₃²	—	9	2	no	—
140	C₃²	—	9	6	no	—
141	C₃²	—	9	6	no	—
142	C₃²	—	9	4	no	—
143	C₃²	—	9	6	no	—
144	C₃²	—	9	6	no	—
145	C₃²	—	27	2	no	—
146	C₃²	—	27	4	no	—
147	C₃²	—	27	27	no	—
148	C₃²	—	27	2	no	—
149	C₃²	—	27	2	no	—
150	C₃²	—	27	4	no	—
151	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
152	C₃²	C₃×#732	1	1	yes	characteristic
153	C₃²	#6911	1	2	yes	—
154	C₉	—	9	3	no	—
155	C₉	—	27	4	no	—
156	C₉	—	27	2	no	—
157	C₃	—	3	1	no	—
158	C₃	—	3	2	no	—
159	C₃	—	9	1	no	—
160	C₃	—	9	2	no	—
161	C₃	—	9	3	no	—
162	C₃	—	27	2	no	—
163	C₃	—	27	3	no	—
164	C₃	—	27	6	no	—
165	C₃	—	81	2	no	—
166	C₃	#10624740	1	1	yes	characteristic
167	C₃	UT(4,𝔽₃)	1	1	yes	characteristic
168	C₃	#10624748	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624880

C₃

yes

maximal, characteristic, complemented

#10624811

C₃

yes

maximal, complemented

#10624735

C₃

yes

maximal, complemented

#10624732

C₃

yes

maximal, characteristic, complemented

#10624636

C₃

yes

maximal, complemented

#10624635

C₃

yes

maximal, complemented

#10624555

C₃

yes

maximal, characteristic, complemented

C₃²×H(3,𝔽₃)

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic, complemented

C₃×#732

—

C₃×AΣL(1,𝔽₂₇)

—

C₃×AΣL(1,𝔽₂₇)

—

#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

C₃²⋊H(3,𝔽₃)

C₃²

yes

complemented

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

complemented

H(3,𝔽₃)⋊C₉

—

#6859

—

#6856

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

H(3,𝔽₃)

yes

characteristic

C₃⁴

C₃³

yes

characteristic

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)

H(3,𝔽₃)

yes

—

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

#732

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

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

#732

yes

complemented

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

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

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

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

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

147

C₃²

—

148

C₃²

—

149

C₃²

—

150

C₃²

—

151

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

152

C₃²

C₃×#732

yes

characteristic

153

C₃²

#6911

yes

—

154

C₉

—

155

C₉

—

156

C₉

—

157

C₃

—

158

C₃

—

159

C₃

—

160

C₃

—

161

C₃

—

162

C₃

—

163

C₃

—

164

C₃

—

165

C₃

—

166

C₃

#10624740

yes

characteristic

167

C₃

UT(4,𝔽₃)

yes

characteristic

168

C₃

#10624748

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
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

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. #10624880	C₃	semidirect
2. #10624811	C₃	semidirect
3. #10624735	C₃	semidirect
4. #10624732	C₃	semidirect
5. #10624636	C₃	semidirect
6. #10624635	C₃	semidirect
7. #10624555	C₃	semidirect
10. C₃³×C₉	C₃²	semidirect
14. #6891	C₃²	semidirect
20. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
23. C₃×C₃²⋊C₉	C₃²	semidirect
85. C₃³	#732	semidirect

H ≅ G/N

type

1. #10624880

C₃

semidirect

2. #10624811

C₃

semidirect

3. #10624735

C₃

semidirect

4. #10624732

C₃

semidirect

5. #10624636

C₃

semidirect

6. #10624635

C₃

semidirect

7. #10624555

C₃

semidirect

10. C₃³×C₉

C₃²

semidirect

14. #6891

C₃²

semidirect

20. C₃²⋊H(3,𝔽₃)

C₃²

semidirect

23. C₃×C₃²⋊C₉

C₃²

semidirect

85. C₃³

#732

semidirect

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

subgroup	index
#10624880	3
#10624811	3
#10624735	3
#10624732	3
#10624636	3
#10624635	3
#10624555	3

subgroup

index

Subgroups — #14547004

Conjugacy classes of subgroups

Complements

Maximal subgroups