Group #14550288 — subgroups

Subgroups — #14550288

Order 2187.

1295 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1295 conjugacy classes of subgroups — too many to draw. Only its 170 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×#737	C₃	1	1	yes	maximal, characteristic
2	C₃×3₋¹⁺²⋊C₉	C₃	1	6	yes	maximal, complemented
3	C₃×H(3,𝔽₃)⋊C₉	C₃	1	3	yes	maximal
4	#10624549	C₃	1	2	yes	maximal, complemented
5	C₃×#6855	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#737	—	3	6	no	—
7	C₃×#737	—	3	12	no	—
8	C₃×#737	—	3	3	no	—
9	C₃×#737	—	3	6	no	—
10	C₃×#737	—	3	6	no	—
11	C₃×#737	C₉	1	9	yes	complemented
12	C₃²×3₋¹⁺²	—	3	2	no	—
13	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
14	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₃³×C₉	—	3	1	no	—
16	C₃³×C₉	C₃²	1	3	yes	—
17	C₃³×C₉	C₃²	1	1	yes	characteristic
18	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
19	3₋¹⁺²⋊C₉	—	3	18	no	—
20	H(3,𝔽₃)⋊C₉	—	3	9	no	—
21	#6855	—	3	1	no	—
22	#6855	—	3	2	no	—
23	#6855	—	3	2	no	—
24	#6855	—	3	4	no	—
25	C₃⁴	—	3	1	no	—
26	C₃⁴	C₃³	1	1	yes	characteristic
27	#737	—	3	54	no	—
28	#737	—	3	108	no	—
29	#737	—	3	54	no	—
30	#737	—	3	27	no	—
31	C₃×3₋¹⁺²	—	3	18	no	—
32	C₃×3₋¹⁺²	—	3	36	no	—
33	C₃×3₋¹⁺²	—	3	2	no	—
34	C₃×3₋¹⁺²	—	3	2	no	—
35	C₃×3₋¹⁺²	—	3	4	no	—
36	C₃×3₋¹⁺²	—	3	4	no	—
37	C₃×3₋¹⁺²	—	3	4	no	—
38	C₃×3₋¹⁺²	—	3	18	no	—
39	C₃×3₋¹⁺²	C₃×C₉	1	6	yes	—
40	C₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
41	C₃×3₋¹⁺²	3₋¹⁺²	1	12	yes	—
42	C₃×H(3,𝔽₃)	—	3	2	no	—
43	C₃×H(3,𝔽₃)	—	3	4	no	—
44	C₃×H(3,𝔽₃)	—	3	2	no	—
45	C₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
46	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
47	C₃×H(3,𝔽₃)	3₋¹⁺²	1	6	yes	—
48	C₃²×C₉	—	3	9	no	—
49	C₃²×C₉	—	3	18	no	—
50	C₃²×C₉	—	3	2	no	—
51	C₃²×C₉	—	3	1	no	—
52	C₃²×C₉	—	3	2	no	—
53	C₃²×C₉	—	3	4	no	—
54	C₃²×C₉	—	3	2	no	—
55	C₃²×C₉	—	3	9	no	—
56	C₃²×C₉	—	3	3	no	—
57	C₃²×C₉	—	3	9	no	—
58	C₃²×C₉	—	3	3	no	—
59	C₃²×C₉	—	3	6	no	—
60	C₃²×C₉	—	3	6	no	—
61	C₃²×C₉	—	9	6	no	—
62	C₃²×C₉	C₃×C₉	1	3	yes	—
63	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
64	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
65	C₃²×C₉	3₋¹⁺²	1	6	yes	—
66	C₃²⋊C₉	—	3	6	no	—
67	C₃²⋊C₉	—	3	12	no	—
68	C₃³	—	3	6	no	—
69	C₃³	—	3	3	no	—
70	C₃³	—	3	3	no	—
71	C₃³	—	3	3	no	—
72	C₃³	—	3	6	no	—
73	C₃³	—	9	6	no	—
74	C₃³	—	9	3	no	—
75	C₃³	C₃²×C₉	1	1	yes	characteristic
76	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
77	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
78	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
79	C₃³	C₃×3₋¹⁺²	1	4	yes	—
80	C₃³	C₃×3₋¹⁺²	1	2	yes	—
81	C₃³	C₃×3₋¹⁺²	1	2	yes	—
82	3₋¹⁺²	—	3	162	no	—
83	3₋¹⁺²	—	3	6	no	—
84	3₋¹⁺²	—	3	6	no	—
85	3₋¹⁺²	—	3	12	no	—
86	3₋¹⁺²	—	3	12	no	—
87	3₋¹⁺²	—	3	12	no	—
88	3₋¹⁺²	C₃²⋊C₉	1	18	yes	—
89	H(3,𝔽₃)	—	3	6	no	—
90	H(3,𝔽₃)	—	3	6	no	—
91	H(3,𝔽₃)	—	3	12	no	—
92	H(3,𝔽₃)	C₃²⋊C₉	1	9	yes	—
93	C₃×C₉	—	3	81	no	—
94	C₃×C₉	—	3	6	no	—
95	C₃×C₉	—	3	6	no	—
96	C₃×C₉	—	3	3	no	—
97	C₃×C₉	—	3	1	no	—
98	C₃×C₉	—	3	6	no	—
99	C₃×C₉	—	3	2	no	—
100	C₃×C₉	—	3	2	no	—
101	C₃×C₉	—	3	12	no	—
102	C₃×C₉	—	3	4	no	—
103	C₃×C₉	—	3	6	no	—
104	C₃×C₉	—	3	12	no	—
105	C₃×C₉	—	3	9	no	—
106	C₃×C₉	—	9	6	no	—
107	C₃×C₉	—	9	3	no	—
108	C₃×C₉	—	9	9	no	—
109	C₃×C₉	—	9	18	no	—
110	C₃×C₉	—	9	12	no	—
111	C₃×C₉	—	9	6	no	—
112	C₃×C₉	—	9	6	no	—
113	C₃×C₉	C₃²⋊C₉	1	3	yes	—
114	C₃×C₉	C₃²⋊C₉	1	6	yes	—
115	C₃²	—	3	12	no	—
116	C₃²	—	3	6	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	2	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	4	no	—
121	C₃²	—	3	6	no	—
122	C₃²	—	3	3	no	—
123	C₃²	—	3	9	no	—
124	C₃²	—	9	18	no	—
125	C₃²	—	9	3	no	—
126	C₃²	—	9	6	no	—
127	C₃²	—	9	9	no	—
128	C₃²	C₃×C₃²⋊C₉	1	2	yes	—
129	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
130	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
131	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
132	C₃²	#6891	1	2	yes	—
133	C₃²	#6891	1	4	yes	—
134	C₃²	#6891	1	2	yes	—
135	C₃²	#6908	1	3	yes	—
136	C₃²	#6913	1	6	yes	—
137	C₉	—	3	3	no	—
138	C₉	—	3	18	no	—
139	C₉	—	3	6	no	—
140	C₉	—	9	9	no	—
141	C₉	—	27	6	no	—
142	C₃	—	3	2	no	—
143	C₃	—	3	1	no	—
144	C₃	—	3	2	no	—
145	C₃	—	3	4	no	—
146	C₃	—	9	9	no	—
147	C₃	#10624547	1	1	yes	characteristic
148	C₃	#10624697	1	1	yes	characteristic
149	C₃	#10624700	1	2	yes	—
150	C₃	#10624746	1	3	yes	—
151	C₃	#10624780	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#737

C₃

yes

maximal, characteristic

C₃×3₋¹⁺²⋊C₉

C₃

yes

maximal, complemented

C₃×H(3,𝔽₃)⋊C₉

C₃

yes

maximal

#10624549

C₃

yes

maximal, complemented

C₃×#6855

C₃

yes

maximal, characteristic, complemented

—

—

—

—

—

C₉

yes

complemented

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

—

C₃³×C₉

C₃²

yes

—

C₃³×C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

—

—

—

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

#737

—

#737

—

108

—

#737

—

#737

—

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

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×C₉

yes

—

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃×H(3,𝔽₃)

3₋¹⁺²

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

—

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

characteristic

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₉

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

3₋¹⁺²

—

162

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃²⋊C₉

yes

—

—

—

—

C₃²⋊C₉

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

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

C₃²⋊C₉

yes

—

114

C₃×C₉

C₃²⋊C₉

yes

—

115

C₃²

—

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

C₃×C₃²⋊C₉

yes

—

129

C₃²

C₃×C₃²⋊C₉

yes

characteristic

130

C₃²

C₃×C₃²⋊C₉

yes

characteristic

131

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

132

C₃²

#6891

yes

—

133

C₃²

#6891

yes

—

134

C₃²

#6891

yes

—

135

C₃²

#6908

yes

—

136

C₃²

#6913

yes

—

137

C₉

—

138

C₉

—

139

C₉

—

140

C₉

—

141

C₉

—

142

C₃

—

143

C₃

—

144

C₃

—

145

C₃

—

146

C₃

—

147

C₃

#10624547

yes

characteristic

148

C₃

#10624697

yes

characteristic

149

C₃

#10624700

yes

—

150

C₃

#10624746

yes

—

151

C₃

#10624780

yes

—

	subgroup	G/H
center, socle	C₃³	C₃×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 ◁ C₃³ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃³

C₃×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 ◁ C₃³ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. C₃×3₋¹⁺²⋊C₉	C₃	semidirect
4. #10624549	C₃	semidirect
5. C₃×#6855	C₃	semidirect
11. C₃×#737	C₉	semidirect

H ≅ G/N

type

2. C₃×3₋¹⁺²⋊C₉

C₃

semidirect

4. #10624549

C₃

semidirect

5. C₃×#6855

C₃

semidirect

11. C₃×#737

C₉

semidirect

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

subgroup	index
C₃²×#737	3
C₃×3₋¹⁺²⋊C₉	3
C₃×H(3,𝔽₃)⋊C₉	3
#10624549	3
C₃×#6855	3

subgroup

index

C₃²×#737

C₃×3₋¹⁺²⋊C₉

C₃×H(3,𝔽₃)⋊C₉

#10624549

C₃×#6855

Subgroups — #14550288

Conjugacy classes of subgroups

Complements

Maximal subgroups