Group #14554381 — subgroups

Subgroups — #14554381

Order 2187.

1460 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1460 conjugacy classes of subgroups — too many to draw. Only its 314 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	#10624872	C₃	1	3	yes	maximal, complemented
3	#10624776	C₃	1	2	yes	maximal, complemented
4	#10624774	C₃	1	9	yes	maximal, complemented
5	#10624772	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624771	C₃	1	3	yes	maximal, complemented
7	#10624769	C₃	1	3	yes	maximal
8	#10624766	C₃	1	9	yes	maximal, complemented
9	#10624692	C₃	1	9	yes	maximal, complemented
10	C₃×#737	—	3	27	no	—
11	C₃×#737	—	3	6	no	—
12	C₃×#737	—	3	3	no	—
13	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
14	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
15	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
16	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
17	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
18	C₃³×C₉	C₃²	1	3	yes	—
19	C₃³×C₉	C₃²	1	1	yes	characteristic
20	#6895	C₃²	1	9	yes	—
21	#6895	C₃²	1	9	yes	—
22	#6894	—	3	6	no	—
23	#6894	—	3	6	no	—
24	#6894	—	3	6	no	—
25	#6891	C₃²	1	3	yes	—
26	#6891	C₃²	1	6	yes	—
27	C₉×3₋¹⁺²	—	3	6	no	—
28	C₉×3₋¹⁺²	—	3	3	no	—
29	#6887	C₃²	1	9	yes	—
30	#6868	C₃²	1	9	yes	—
31	#6868	C₃²	1	18	yes	—
32	#6867	C₃²	1	27	yes	—
33	#6866	C₃²	1	27	yes	—
34	C₃⁴	C₃³	1	1	yes	characteristic
35	C₃⁴	C₃³	1	3	yes	—
36	#737	—	3	162	no	—
37	#737	—	3	81	no	—
38	C₃×3₋¹⁺²	—	3	3	no	—
39	C₃×3₋¹⁺²	—	3	3	no	—
40	C₃×3₋¹⁺²	—	3	9	no	—
41	C₃×3₋¹⁺²	—	3	3	no	—
42	C₃×3₋¹⁺²	—	3	6	no	—
43	C₃×3₋¹⁺²	—	3	9	no	—
44	C₃×3₋¹⁺²	—	3	9	no	—
45	C₃×3₋¹⁺²	—	3	9	no	—
46	C₃×3₋¹⁺²	—	3	9	no	—
47	C₃×3₋¹⁺²	—	3	9	no	—
48	C₃×3₋¹⁺²	—	3	9	no	—
49	C₃×3₋¹⁺²	—	3	9	no	—
50	C₃×3₋¹⁺²	—	3	9	no	—
51	C₃×3₋¹⁺²	3₋¹⁺²	1	27	yes	complemented
52	C₃×H(3,𝔽₃)	—	3	3	no	—
53	C₃×H(3,𝔽₃)	—	3	9	no	—
54	C₃²×C₉	—	3	9	no	—
55	C₃²×C₉	—	3	6	no	—
56	C₃²×C₉	—	3	3	no	—
57	C₃²×C₉	—	3	9	no	—
58	C₃²×C₉	—	3	9	no	—
59	C₃²×C₉	—	3	9	no	—
60	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
61	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
62	C₃²×C₉	C₃³	1	1	yes	characteristic
63	C₃²×C₉	C₃³	1	1	yes	characteristic
64	C₃²×C₉	C₃³	1	3	yes	—
65	C₃²×C₉	C₃³	1	3	yes	—
66	C₃²×C₉	C₃³	1	1	yes	characteristic
67	C₉⋊C₉	—	3	18	no	—
68	C₉⋊C₉	—	3	18	no	—
69	C₉⋊C₉	—	3	9	no	—
70	C₉⋊C₉	—	3	18	no	—
71	C₉⋊C₉	—	3	9	no	—
72	C₃²⋊C₉	—	3	6	no	—
73	C₃²⋊C₉	—	3	6	no	—
74	C₃²⋊C₉	—	3	6	no	—
75	C₃²⋊C₉	C₃³	1	9	yes	—
76	C₃²⋊C₉	C₃³	1	18	yes	—
77	C₉²	—	3	9	no	—
78	C₃³	—	3	3	no	—
79	C₃³	—	3	3	no	—
80	C₃³	—	3	3	no	—
81	C₃³	—	3	3	no	—
82	C₃³	—	9	1	no	—
83	C₃³	—	9	2	no	—
84	C₃³	—	9	3	no	—
85	C₃³	—	9	6	no	—
86	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
87	C₃³	C₃×3₋¹⁺²	1	9	yes	—
88	C₃³	C₃⁴	1	1	yes	characteristic
89	3₋¹⁺²	—	3	81	no	—
90	3₋¹⁺²	—	3	27	no	—
91	3₋¹⁺²	—	3	27	no	—
92	3₋¹⁺²	—	3	81	no	—
93	3₋¹⁺²	—	9	9	no	—
94	3₋¹⁺²	—	9	18	no	—
95	H(3,𝔽₃)	—	3	27	no	—
96	C₃×C₉	—	3	54	no	—
97	C₃×C₉	—	3	27	no	—
98	C₃×C₉	—	3	3	no	—
99	C₃×C₉	—	3	3	no	—
100	C₃×C₉	—	3	3	no	—
101	C₃×C₉	—	3	3	no	—
102	C₃×C₉	—	3	3	no	—
103	C₃×C₉	—	3	3	no	—
104	C₃×C₉	—	3	3	no	—
105	C₃×C₉	—	3	3	no	—
106	C₃×C₉	—	3	18	no	—
107	C₃×C₉	—	3	9	no	—
108	C₃×C₉	—	3	9	no	—
109	C₃×C₉	—	3	18	no	—
110	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
111	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
112	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
113	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
114	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	—
115	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	—
116	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	2	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	3	9	no	—
121	C₃²	—	3	3	no	—
122	C₃²	—	3	6	no	—
123	C₃²	—	9	9	no	—
124	C₃²	—	9	9	no	—
125	C₃²	—	9	9	no	—
126	C₃²	—	9	9	no	—
127	C₃²	—	9	3	no	—
128	C₃²	—	9	6	no	—
129	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
130	C₃²	C₃²×3₋¹⁺²	1	3	yes	—
131	C₉	—	3	2	no	—
132	C₉	—	3	1	no	—
133	C₉	—	3	3	no	—
134	C₉	—	3	9	no	—
135	C₉	—	3	9	no	—
136	C₉	—	9	9	no	—
137	C₉	—	9	18	no	—
138	C₉	#6891	1	9	yes	—
139	C₃	—	3	3	no	—
140	C₃	—	9	3	no	—
141	C₃	—	9	9	no	—
142	C₃	C₃×#6891	1	1	yes	characteristic
143	C₃	#10624929	1	1	yes	characteristic
144	C₃	#10624929	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#737

C₃

yes

maximal, characteristic

#10624872

C₃

yes

maximal, complemented

#10624776

C₃

yes

maximal, complemented

#10624774

C₃

yes

maximal, complemented

#10624772

C₃

yes

maximal, characteristic, complemented

#10624771

C₃

yes

maximal, complemented

#10624769

C₃

yes

maximal

#10624766

C₃

yes

maximal, complemented

#10624692

C₃

yes

maximal, complemented

C₃×#737

—

C₃×#737

—

C₃×#737

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

C₃³×C₉

C₃²

yes

characteristic

#6895

C₃²

yes

—

#6895

C₃²

yes

—

—

—

—

C₃²

yes

—

#6891

C₃²

yes

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

#6887

C₃²

yes

—

#6868

C₃²

yes

—

#6868

C₃²

yes

—

#6867

C₃²

yes

—

#6866

C₃²

yes

—

C₃⁴

C₃³

yes

characteristic

C₃⁴

C₃³

yes

—

#737

—

162

—

#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

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

3₋¹⁺²

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

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

C₃³

yes

characteristic

C₃²×C₉

C₃³

yes

characteristic

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃³

yes

—

C₃²⋊C₉

C₃³

yes

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

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

C₃×H(3,𝔽₃)

yes

—

111

C₃×C₉

C₃×H(3,𝔽₃)

yes

characteristic

112

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

113

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

114

C₃×C₉

C₃×3₋¹⁺²

yes

—

115

C₃×C₉

C₃×3₋¹⁺²

yes

—

116

C₃×C₉

C₃×3₋¹⁺²

yes

—

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

C₃²×H(3,𝔽₃)

yes

characteristic

130

C₃²

C₃²×3₋¹⁺²

yes

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

—

136

C₉

—

137

C₉

—

138

C₉

#6891

yes

—

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

C₃×#6891

yes

characteristic

143

C₃

#10624929

yes

characteristic

144

C₃

#10624929

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃²×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃⁴

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624872	C₃	semidirect
3. #10624776	C₃	semidirect
4. #10624774	C₃	semidirect
5. #10624772	C₃	semidirect
6. #10624771	C₃	semidirect
8. #10624766	C₃	semidirect
9. #10624692	C₃	semidirect
51. C₃×3₋¹⁺²	3₋¹⁺²	semidirect

H ≅ G/N

type

2. #10624872

C₃

semidirect

3. #10624776

C₃

semidirect

4. #10624774

C₃

semidirect

5. #10624772

C₃

semidirect

6. #10624771

C₃

semidirect

8. #10624766

C₃

semidirect

9. #10624692

C₃

semidirect

51. C₃×3₋¹⁺²

3₋¹⁺²

semidirect

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

subgroup	index
C₃²×#737	3
#10624872	3
#10624776	3
#10624774	3
#10624772	3
#10624771	3
#10624769	3
#10624766	3
#10624692	3

subgroup

index

C₃²×#737

Subgroups — #14554381

Conjugacy classes of subgroups

Complements

Maximal subgroups