Group #14554477 — subgroups

Subgroups — #14554477

Order 2187.

1142 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 269 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×#6891	C₃	1	2	yes	maximal
2	C₃×#6891	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	#10624807	C₃	1	18	yes	maximal, complemented
5	#10624806	C₃	1	9	yes	maximal, complemented
6	#10624691	C₃	1	9	yes	maximal, complemented
7	C₃⁵	C₃²	1	1	yes	characteristic
8	C₃²×3₋¹⁺²	—	3	2	no	—
9	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
10	C₃³×C₉	—	3	1	no	—
11	#6891	—	3	2	no	—
12	#6891	—	3	2	no	—
13	#6891	—	3	4	no	—
14	#6891	—	3	4	no	—
15	#6891	—	3	1	no	—
16	#6891	—	3	2	no	—
17	#6891	—	3	4	no	—
18	#6891	—	3	2	no	—
19	#6891	C₃²	1	6	yes	—
20	#6891	C₃²	1	12	yes	—
21	C₃×C₃²⋊C₉	—	3	2	no	—
22	C₃×C₃²⋊C₉	—	3	4	no	—
23	C₃×C₃²⋊C₉	—	3	2	no	—
24	C₃×C₃²⋊C₉	—	3	4	no	—
25	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
27	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
28	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
29	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
30	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
31	#6868	C₃²	1	81	yes	complemented
32	C₃⁴	—	3	9	no	—
33	C₃⁴	—	9	6	no	—
34	C₃⁴	—	9	3	no	—
35	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃⁴	H(3,𝔽₃)	1	2	yes	—
37	C₃⁴	H(3,𝔽₃)	1	4	yes	—
38	C₃⁴	H(3,𝔽₃)	1	2	yes	—
39	C₃⁴	C₃³	1	2	yes	—
40	C₃⁴	C₃³	1	1	yes	characteristic
41	C₃⁴	C₃³	1	1	yes	characteristic
42	C₃×3₋¹⁺²	—	3	12	no	—
43	C₃×3₋¹⁺²	—	3	6	no	—
44	C₃×3₋¹⁺²	—	3	6	no	—
45	C₃×3₋¹⁺²	—	3	3	no	—
46	C₃×3₋¹⁺²	—	3	27	no	—
47	C₃×3₋¹⁺²	—	9	18	no	—
48	C₃²×C₉	—	3	6	no	—
49	C₃²×C₉	—	3	3	no	—
50	C₃²×C₉	—	3	3	no	—
51	C₃²×C₉	—	3	6	no	—
52	C₃²×C₉	—	9	9	no	—
53	C₃²×C₉	C₃³	1	9	yes	—
54	C₃²⋊C₉	—	3	6	no	—
55	C₃²⋊C₉	—	3	12	no	—
56	C₃²⋊C₉	—	3	6	no	—
57	C₃²⋊C₉	—	3	12	no	—
58	C₃²⋊C₉	—	3	12	no	—
59	C₃²⋊C₉	—	3	12	no	—
60	C₃²⋊C₉	—	3	6	no	—
61	C₃²⋊C₉	—	3	12	no	—
62	C₃²⋊C₉	—	3	6	no	—
63	C₃²⋊C₉	—	3	3	no	—
64	C₃²⋊C₉	—	3	6	no	—
65	C₃²⋊C₉	—	3	6	no	—
66	C₃²⋊C₉	C₃³	1	9	yes	—
67	C₃²⋊C₉	C₃³	1	18	yes	—
68	C₃³	—	3	6	no	—
69	C₃³	—	3	3	no	—
70	C₃³	—	3	3	no	—
71	C₃³	—	3	12	no	—
72	C₃³	—	3	6	no	—
73	C₃³	—	3	6	no	—
74	C₃³	—	3	3	no	—
75	C₃³	—	3	6	no	—
76	C₃³	—	3	12	no	—
77	C₃³	—	3	6	no	—
78	C₃³	—	3	6	no	—
79	C₃³	—	3	3	no	—
80	C₃³	—	9	4	no	—
81	C₃³	—	9	2	no	—
82	C₃³	—	9	4	no	—
83	C₃³	—	9	4	no	—
84	C₃³	—	9	4	no	—
85	C₃³	—	9	1	no	—
86	C₃³	—	9	2	no	—
87	C₃³	—	9	2	no	—
88	C₃³	—	9	2	no	—
89	C₃³	—	9	2	no	—
90	C₃³	—	9	81	no	—
91	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
92	C₃³	C₃×H(3,𝔽₃)	1	4	yes	—
93	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
94	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
95	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
96	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
97	C₃³	#737	1	9	yes	—
98	C₃³	C₃⁴	1	1	yes	characteristic
99	3₋¹⁺²	—	9	36	no	—
100	3₋¹⁺²	—	9	9	no	—
101	3₋¹⁺²	—	9	18	no	—
102	3₋¹⁺²	—	9	18	no	—
103	C₃×C₉	—	3	9	no	—
104	C₃×C₉	—	3	9	no	—
105	C₃×C₉	—	3	18	no	—
106	C₃×C₉	—	3	27	no	—
107	C₃×C₉	—	9	18	no	—
108	C₃×C₉	—	9	9	no	—
109	C₃×C₉	—	9	18	no	—
110	C₃×C₉	—	9	9	no	—
111	C₃²	—	3	27	no	—
112	C₃²	—	3	2	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	3	2	no	—
115	C₃²	—	3	1	no	—
116	C₃²	—	3	2	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	2	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	2	no	—
121	C₃²	—	3	4	no	—
122	C₃²	—	3	2	no	—
123	C₃²	—	3	2	no	—
124	C₃²	—	3	4	no	—
125	C₃²	—	3	4	no	—
126	C₃²	—	3	2	no	—
127	C₃²	—	3	4	no	—
128	C₃²	—	3	4	no	—
129	C₃²	—	3	4	no	—
130	C₃²	—	9	9	no	—
131	C₃²	—	9	18	no	—
132	C₃²	—	9	36	no	—
133	C₃²	—	9	9	no	—
134	C₃²	—	9	18	no	—
135	C₃²	—	9	18	no	—
136	C₃²	#6893	1	3	yes	—
137	C₃²	#6893	1	6	yes	—
138	C₃²	#6911	1	3	yes	—
139	C₃²	#6912	1	6	yes	—
140	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
141	C₃²	C₃×#737	1	3	yes	—
142	C₉	—	9	9	no	—
143	C₉	—	9	9	no	—
144	C₉	—	9	18	no	—
145	C₃	—	3	6	no	—
146	C₃	—	3	3	no	—
147	C₃	—	3	3	no	—
148	C₃	—	3	3	no	—
149	C₃	—	3	6	no	—
150	C₃	—	9	4	no	—
151	C₃	—	9	2	no	—
152	C₃	C₃×#6893	1	1	yes	characteristic
153	C₃	C₃×#6911	1	1	yes	characteristic
154	C₃	C₃×#6912	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6891

C₃

yes

maximal

C₃×#6891

C₃

yes

maximal, characteristic

C₃²×C₃²⋊C₉

C₃

yes

maximal, characteristic

#10624807

C₃

yes

maximal, complemented

#10624806

C₃

yes

maximal, complemented

#10624691

C₃

yes

maximal, complemented

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃³×C₉

—

—

—

—

—

—

—

—

—

C₃²

yes

—

#6891

C₃²

yes

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

#6868

C₃²

yes

complemented

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

characteristic

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

C₃³

yes

—

C₃⁴

C₃³

yes

characteristic

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

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃³

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

C₃³

yes

—

C₃²⋊C₉

C₃³

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

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

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

#737

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

—

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

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

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

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

#6893

yes

—

137

C₃²

#6893

yes

—

138

C₃²

#6911

yes

—

139

C₃²

#6912

yes

—

140

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

141

C₃²

C₃×#737

yes

—

142

C₉

—

143

C₉

—

144

C₉

—

145

C₃

—

146

C₃

—

147

C₃

—

148

C₃

—

149

C₃

—

150

C₃

—

151

C₃

—

152

C₃

C₃×#6893

yes

characteristic

153

C₃

C₃×#6911

yes

characteristic

154

C₃

C₃×#6912

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁵ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃⁴

prime radical

C₃⁵

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁵ ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
4. #10624807	C₃	semidirect
5. #10624806	C₃	semidirect
6. #10624691	C₃	semidirect
31. #6868	C₃²	semidirect

H ≅ G/N

type

4. #10624807

C₃

semidirect

5. #10624806

C₃

semidirect

6. #10624691

C₃

semidirect

31. #6868

C₃²

semidirect

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

subgroup	index
C₃×#6891	3
C₃²×C₃²⋊C₉	3
#10624807	3
#10624806	3
#10624691	3

subgroup

index

C₃²×C₃²⋊C₉

Subgroups — #14554477

Conjugacy classes of subgroups

Complements

Maximal subgroups