Group #14550670 — subgroups

Subgroups — #14550670

Order 2187.

575 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 98 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×#733	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₃×3₋¹⁺²⋊C₉	C₃	1	3	yes	maximal
4	C₃×3₋¹⁺²⋊C₉	C₃	1	3	yes	maximal
5	C₃×#6868	C₃	1	1	yes	maximal, characteristic
6	C₃×#6868	C₃	1	1	yes	maximal, characteristic
7	#10624558	C₃	1	3	yes	maximal
8	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	C₃³×C₉	C₃²	1	1	yes	characteristic
11	C₃³×C₉	C₃²	1	1	yes	characteristic
12	C₃³×C₉	C₃²	1	1	yes	characteristic
13	C₃×#733	—	3	9	no	—
14	C₃×#733	—	3	3	no	—
15	C₃×C₉⋊C₉	—	3	3	no	—
16	C₃×C₉⋊C₉	—	3	3	no	—
17	C₃×C₉⋊C₉	—	3	3	no	—
18	C₃×C₉⋊C₉	—	3	3	no	—
19	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
20	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
21	3₋¹⁺²⋊C₉	—	3	9	no	—
22	3₋¹⁺²⋊C₉	—	3	9	no	—
23	#6868	—	3	3	no	—
24	#6868	—	3	3	no	—
25	#6855	—	3	9	no	—
26	C₃⁴	C₃³	1	1	yes	characteristic
27	C₃×3₋¹⁺²	—	3	6	no	—
28	C₃×3₋¹⁺²	—	3	3	no	—
29	C₃×3₋¹⁺²	—	3	3	no	—
30	C₃×3₋¹⁺²	—	3	3	no	—
31	C₃×3₋¹⁺²	—	3	6	no	—
32	C₃×3₋¹⁺²	—	3	6	no	—
33	C₃×3₋¹⁺²	—	3	6	no	—
34	C₃×3₋¹⁺²	3₋¹⁺²	1	9	yes	—
35	C₃²×C₉	—	3	1	no	—
36	C₃²×C₉	—	3	1	no	—
37	C₃²×C₉	—	3	1	no	—
38	C₃²×C₉	—	3	9	no	—
39	C₃²×C₉	—	3	1	no	—
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	—	3	1	no	—
42	C₃²×C₉	—	3	1	no	—
43	C₃²×C₉	—	3	1	no	—
44	C₃²×C₉	—	3	1	no	—
45	C₃²×C₉	—	3	1	no	—
46	C₃²×C₉	—	3	1	no	—
47	C₃²×C₉	—	3	1	no	—
48	C₃²×C₉	—	3	3	no	—
49	C₃²×C₉	—	3	3	no	—
50	C₃²×C₉	—	3	6	no	—
51	C₃²×C₉	—	3	3	no	—
52	C₃²×C₉	—	3	3	no	—
53	C₃²×C₉	—	9	3	no	—
54	C₃²×C₉	—	9	3	no	—
55	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
56	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
57	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
58	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
59	#733	—	3	27	no	—
60	C₉⋊C₉	—	3	27	no	—
61	C₃²⋊C₉	—	3	3	no	—
62	C₃²⋊C₉	—	3	3	no	—
63	C₃²⋊C₉	—	3	3	no	—
64	C₃²⋊C₉	—	3	3	no	—
65	C₃²⋊C₉	—	3	3	no	—
66	C₃²⋊C₉	—	3	3	no	—
67	C₃³	—	3	3	no	—
68	C₃³	—	3	3	no	—
69	C₃³	—	3	3	no	—
70	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
71	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
72	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
73	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
74	C₃³	C₃×3₋¹⁺²	1	3	yes	—
75	C₃³	#737	1	6	yes	—
76	3₋¹⁺²	—	3	9	no	—
77	3₋¹⁺²	—	3	9	no	—
78	3₋¹⁺²	—	3	9	no	—
79	3₋¹⁺²	—	9	6	no	—
80	3₋¹⁺²	—	9	6	no	—
81	3₋¹⁺²	—	9	6	no	—
82	C₃×C₉	—	3	9	no	—
83	C₃×C₉	—	3	9	no	—
84	C₃×C₉	—	3	3	no	—
85	C₃×C₉	—	3	9	no	—
86	C₃×C₉	—	3	9	no	—
87	C₃×C₉	—	3	9	no	—
88	C₃×C₉	—	3	3	no	—
89	C₃×C₉	—	3	3	no	—
90	C₃×C₉	—	3	3	no	—
91	C₃×C₉	—	3	3	no	—
92	C₃×C₉	—	3	9	no	—
93	C₃×C₉	—	3	6	no	—
94	C₃×C₉	—	9	3	no	—
95	C₃×C₉	—	9	3	no	—
96	C₃×C₉	—	9	3	no	—
97	C₃×C₉	—	9	3	no	—
98	C₃×C₉	—	9	3	no	—
99	C₃×C₉	—	9	3	no	—
100	C₃×C₉	—	9	3	no	—
101	C₃×C₉	—	9	3	no	—
102	C₃×C₉	—	9	3	no	—
103	C₃×C₉	—	9	3	no	—
104	C₃×C₉	—	9	6	no	—
105	C₃×C₉	—	9	3	no	—
106	C₃×C₉	—	9	9	no	—
107	C₃×C₉	—	9	3	no	—
108	C₃×C₉	—	9	3	no	—
109	C₃×C₉	—	9	9	no	—
110	C₃²	—	3	9	no	—
111	C₃²	—	3	18	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	3	1	no	—
115	C₃²	—	3	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	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
122	C₃²	#6891	1	1	yes	characteristic
123	C₃²	C₉⋊H(3,𝔽₃)	1	3	yes	—
124	C₃²	C₉⋊H(3,𝔽₃)	1	3	yes	—
125	C₃²	C₉⋊H(3,𝔽₃)	1	3	yes	—
126	C₃²	#6893	1	2	yes	—
127	C₃²	#6908	1	3	yes	—
128	C₃²	#6912	1	3	yes	—
129	C₃²	#6913	1	3	yes	—
130	C₉	—	3	9	no	—
131	C₉	—	9	3	no	—
132	C₉	—	9	9	no	—
133	C₉	—	9	3	no	—
134	C₉	—	9	3	no	—
135	C₉	—	27	3	no	—
136	C₉	—	27	3	no	—
137	C₃	—	3	1	no	—
138	C₃	—	3	3	no	—
139	C₃	—	3	3	no	—
140	C₃	—	3	2	no	—
141	C₃	#10624563	1	1	yes	characteristic
142	C₃	#10624734	1	1	yes	characteristic
143	C₃	#10624738	1	1	yes	characteristic
144	C₃	#10624739	1	1	yes	characteristic
145	C₃	#10624802	1	9	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#733

C₃

yes

maximal, characteristic

C₃²×C₉⋊C₉

C₃

yes

maximal, characteristic

C₃×3₋¹⁺²⋊C₉

C₃

yes

maximal

C₃×3₋¹⁺²⋊C₉

C₃

yes

maximal

C₃×#6868

C₃

yes

maximal, characteristic

C₃×#6868

C₃

yes

maximal, characteristic

#10624558

C₃

yes

maximal

C₃²×3₋¹⁺²

C₃²

yes

—

C₃³×C₉

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃×#733

—

C₃×#733

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

#6868

—

#6868

—

#6855

—

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

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

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

#733

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

#737

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

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

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

122

C₃²

#6891

yes

characteristic

123

C₃²

C₉⋊H(3,𝔽₃)

yes

—

124

C₃²

C₉⋊H(3,𝔽₃)

yes

—

125

C₃²

C₉⋊H(3,𝔽₃)

yes

—

126

C₃²

#6893

yes

—

127

C₃²

#6908

yes

—

128

C₃²

#6912

yes

—

129

C₃²

#6913

yes

—

130

C₉

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

—

136

C₉

—

137

C₃

—

138

C₃

—

139

C₃

—

140

C₃

—

141

C₃

#10624563

yes

characteristic

142

C₃

#10624734

yes

characteristic

143

C₃

#10624738

yes

characteristic

144

C₃

#10624739

yes

characteristic

145

C₃

#10624802

yes

—

	subgroup	G/H
center, socle	C₃³	C₃×H(3,𝔽₃)
commutator, Frattini, prime radical, cosocle	C₃⁴	C₃³
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃³

C₃×H(3,𝔽₃)

commutator, Frattini, prime radical, cosocle

C₃⁴

C₃³

LCS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

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

subgroup	index
C₃²×#733	3
C₃²×C₉⋊C₉	3
C₃×3₋¹⁺²⋊C₉	3
C₃×#6868	3
#10624558	3

subgroup

index

C₃²×#733

C₃²×C₉⋊C₉

C₃×3₋¹⁺²⋊C₉

C₃×#6868

#10624558

Subgroups — #14550670

Conjugacy classes of subgroups

Maximal subgroups