Group #14547263 — subgroups

Subgroups — #14547263

Order 2187.

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

normal subgroups only

This group has 420 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×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	#10624812	C₃	1	2	yes	maximal, complemented
3	#10624779	C₃	1	1	yes	maximal, characteristic
4	#10624777	C₃	1	1	yes	maximal, characteristic
5	#10624741	C₃	1	1	yes	maximal, characteristic
6	C₃×3₋¹⁺²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624614	C₃	1	2	yes	maximal, complemented
8	#10624610	C₃	1	2	yes	maximal, complemented
9	#10624604	C₃	1	2	yes	maximal, complemented
10	C₃²×3₋¹⁺²	—	3	1	no	—
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	C₉⋊H(3,𝔽₃)	—	3	3	no	—
16	C₉⋊H(3,𝔽₃)	—	3	6	no	—
17	#6891	C₃²	1	2	yes	—
18	#6891	C₃²	1	2	yes	—
19	#6891	C₃²	1	2	yes	—
20	C₃×C₃²⋊C₉	—	3	2	no	—
21	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
22	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
23	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
24	3₋¹⁺²⋊C₉	—	3	2	no	—
25	3₋¹⁺²⋊C₉	—	3	4	no	—
26	3₋¹⁺²⋊C₉	—	3	1	no	—
27	3₋¹⁺²⋊C₉	—	3	2	no	—
28	H(3,𝔽₃)⋊C₉	—	3	3	no	—
29	H(3,𝔽₃)⋊C₉	—	3	3	no	—
30	#6861	—	3	6	no	—
31	#6858	—	3	6	no	—
32	#6858	—	3	6	no	—
33	C₃⁴	—	3	1	no	—
34	C₃⁴	C₃³	1	1	yes	characteristic
35	C₃×3₋¹⁺²	—	3	9	no	—
36	C₃×3₋¹⁺²	—	3	1	no	—
37	C₃×3₋¹⁺²	—	3	3	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	2	no	—
40	C₃×3₋¹⁺²	—	3	2	no	—
41	C₃×3₋¹⁺²	—	3	2	no	—
42	C₃×3₋¹⁺²	—	9	9	no	—
43	C₃×3₋¹⁺²	—	9	2	no	—
44	C₃×H(3,𝔽₃)	—	3	2	no	—
45	C₃×H(3,𝔽₃)	—	9	3	no	—
46	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
47	C₃²×C₉	—	3	2	no	—
48	C₃²×C₉	—	3	1	no	—
49	C₃²×C₉	—	3	2	no	—
50	C₃²×C₉	—	3	1	no	—
51	C₃²×C₉	—	3	1	no	—
52	C₃²×C₉	—	3	1	no	—
53	C₃²×C₉	—	9	3	no	—
54	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
55	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
56	#731	—	9	9	no	—
57	C₃²⋊C₉	—	3	18	no	—
58	C₃²⋊C₉	—	3	2	no	—
59	C₃²⋊C₉	—	3	2	no	—
60	C₃²⋊C₉	—	3	2	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	—	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	1	no	—
70	C₃²⋊C₉	—	3	1	no	—
71	C₃²⋊C₉	—	3	2	no	—
72	C₃²⋊C₉	—	3	2	no	—
73	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
74	C₃³	—	3	2	no	—
75	C₃³	—	3	1	no	—
76	C₃³	—	3	3	no	—
77	C₃³	—	3	3	no	—
78	C₃³	—	9	1	no	—
79	C₃³	—	9	1	no	—
80	C₃³	—	9	3	no	—
81	C₃³	—	9	3	no	—
82	C₃³	—	9	3	no	—
83	C₃³	—	9	2	no	—
84	C₃³	—	9	1	no	—
85	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
86	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
87	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
88	3₋¹⁺²	—	3	3	no	—
89	3₋¹⁺²	—	9	27	no	—
90	3₋¹⁺²	—	9	2	no	—
91	3₋¹⁺²	—	9	6	no	—
92	3₋¹⁺²	—	9	4	no	—
93	3₋¹⁺²	—	9	2	no	—
94	3₋¹⁺²	—	9	6	no	—
95	3₋¹⁺²	—	27	2	no	—
96	H(3,𝔽₃)	—	3	3	no	—
97	H(3,𝔽₃)	—	3	6	no	—
98	H(3,𝔽₃)	—	9	6	no	—
99	C₃×C₉	—	3	1	no	—
100	C₃×C₉	—	3	3	no	—
101	C₃×C₉	—	3	2	no	—
102	C₃×C₉	—	3	6	no	—
103	C₃×C₉	—	9	3	no	—
104	C₃×C₉	—	9	6	no	—
105	C₃×C₉	—	9	6	no	—
106	C₃×C₉	—	9	1	no	—
107	C₃×C₉	—	9	3	no	—
108	C₃×C₉	—	9	2	no	—
109	C₃×C₉	—	9	1	no	—
110	C₃×C₉	—	9	1	no	—
111	C₃×C₉	—	9	2	no	—
112	C₃×C₉	—	9	2	no	—
113	C₃×C₉	—	9	2	no	—
114	C₃×C₉	—	9	3	no	—
115	C₃×C₉	—	27	1	no	—
116	C₃²	—	3	1	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	1	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	1	no	—
122	C₃²	—	3	2	no	—
123	C₃²	—	3	2	no	—
124	C₃²	—	3	2	no	—
125	C₃²	—	3	2	no	—
126	C₃²	—	9	6	no	—
127	C₃²	—	9	3	no	—
128	C₃²	—	9	3	no	—
129	C₃²	—	9	3	no	—
130	C₃²	—	9	3	no	—
131	C₃²	—	9	3	no	—
132	C₃²	—	27	6	no	—
133	C₃²	—	27	3	no	—
134	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
135	C₃²	#6908	1	1	yes	characteristic
136	C₃²	#6910	1	3	yes	—
137	C₃²	#6913	1	2	yes	—
138	C₉	—	3	3	no	—
139	C₉	—	9	3	no	—
140	C₉	—	9	2	no	—
141	C₉	—	9	6	no	—
142	C₉	—	27	1	no	—
143	C₉	—	27	2	no	—
144	C₉	—	27	2	no	—
145	C₉	—	27	1	no	—
146	C₉	—	27	2	no	—
147	C₃	—	3	2	no	—
148	C₃	—	3	1	no	—
149	C₃	—	3	3	no	—
150	C₃	—	9	2	no	—
151	C₃	—	27	3	no	—
152	C₃	#10624736	1	1	yes	characteristic
153	C₃	#10624746	1	1	yes	characteristic
154	C₃	#10624754	1	1	yes	characteristic
155	C₃	#10624758	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

#10624812

C₃

yes

maximal, complemented

#10624779

C₃

yes

maximal, characteristic

#10624777

C₃

yes

maximal, characteristic

#10624741

C₃

yes

maximal, characteristic

C₃×3₋¹⁺²⋊C₉

C₃

yes

maximal, characteristic, complemented

#10624614

C₃

yes

maximal, complemented

#10624610

C₃

yes

maximal, complemented

#10624604

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃×#731

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6861

—

#6858

—

#6858

—

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×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

characteristic

C₃²×C₉

H(3,𝔽₃)

yes

—

#731

—

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

—

C₃²⋊C₉

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

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

C₃²⋊H(3,𝔽₃)

yes

characteristic

135

C₃²

#6908

yes

characteristic

136

C₃²

#6910

yes

—

137

C₃²

#6913

yes

—

138

C₉

—

139

C₉

—

140

C₉

—

141

C₉

—

142

C₉

—

143

C₉

—

144

C₉

—

145

C₉

—

146

C₉

—

147

C₃

—

148

C₃

—

149

C₃

—

150

C₃

—

151

C₃

—

152

C₃

#10624736

yes

characteristic

153

C₃

#10624746

yes

characteristic

154

C₃

#10624754

yes

characteristic

155

C₃

#10624758

yes

characteristic

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

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
2. #10624812	C₃	semidirect
6. C₃×3₋¹⁺²⋊C₉	C₃	semidirect
7. #10624614	C₃	semidirect
8. #10624610	C₃	semidirect
9. #10624604	C₃	semidirect

H ≅ G/N

type

2. #10624812

C₃

semidirect

6. C₃×3₋¹⁺²⋊C₉

C₃

semidirect

7. #10624614

C₃

semidirect

8. #10624610

C₃

semidirect

9. #10624604

C₃

semidirect

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

subgroup	index
C₃×C₉⋊H(3,𝔽₃)	3
#10624812	3
#10624779	3
#10624777	3
#10624741	3
C₃×3₋¹⁺²⋊C₉	3
#10624614	3
#10624610	3
#10624604	3

subgroup

index

C₃×C₉⋊H(3,𝔽₃)

C₃×3₋¹⁺²⋊C₉

Subgroups — #14547263

Conjugacy classes of subgroups

Complements

Maximal subgroups