Group #14553746 — subgroups

Subgroups — #14553746

Order 2187.

618 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 242 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	3₋¹⁺²²	C₃	1	3	yes	maximal
2	#10624885	C₃	1	1	yes	maximal, characteristic
3	#10624674	C₃	1	6	yes	maximal, complemented
4	#10624668	C₃	1	3	yes	maximal
5	#10624655	C₃	1	6	yes	maximal, complemented
6	#10624649	C₃	1	2	yes	maximal, complemented
7	#10624631	C₃	1	3	yes	maximal
8	#10624626	C₃	1	6	yes	maximal, complemented
9	#10624618	C₃	1	6	yes	maximal, complemented
10	#10624607	C₃	1	3	yes	maximal
11	#10624588	C₃	1	1	yes	maximal, characteristic, complemented
12	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
13	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
14	#6900	—	3	3	no	—
15	#6900	—	3	6	no	—
16	#6896	—	3	3	no	—
17	#6896	—	3	6	no	—
18	#6896	—	3	3	no	—
19	#6896	—	3	6	no	—
20	#6895	C₃²	1	6	yes	—
21	#6894	—	3	6	no	—
22	#6894	C₃²	1	6	yes	—
23	#6894	C₃²	1	6	yes	—
24	#6894	C₃²	1	6	yes	—
25	#6891	C₃²	1	3	yes	—
26	#6891	C₃²	1	3	yes	—
27	#6891	C₃²	1	3	yes	—
28	C₉×3₋¹⁺²	—	3	3	no	—
29	C₉×3₋¹⁺²	C₃²	1	3	yes	—
30	C₉×3₋¹⁺²	C₃²	1	6	yes	—
31	#6887	C₃²	1	1	yes	characteristic
32	#6887	C₃²	1	2	yes	—
33	#6861	C₃²	1	3	yes	—
34	#6861	C₃²	1	12	yes	—
35	#6861	C₃²	1	6	yes	—
36	#6861	C₃²	1	6	yes	—
37	#6860	C₃²	1	3	yes	—
38	#6860	C₃²	1	12	yes	—
39	#6860	C₃²	1	6	yes	—
40	#6860	C₃²	1	6	yes	—
41	#6858	C₃²	1	3	yes	—
42	#6858	C₃²	1	6	yes	—
43	#6858	C₃²	1	12	yes	—
44	#6858	C₃²	1	6	yes	—
45	C₃⁴	C₃³	1	1	yes	characteristic
46	C₃×3₋¹⁺²	—	3	3	no	—
47	C₃×3₋¹⁺²	—	3	3	no	—
48	C₃×3₋¹⁺²	—	3	3	no	—
49	C₃×3₋¹⁺²	—	3	3	no	—
50	C₃×3₋¹⁺²	—	3	3	no	—
51	C₃×3₋¹⁺²	—	9	3	no	—
52	C₃×3₋¹⁺²	—	9	9	no	—
53	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
54	C₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
55	C₃×3₋¹⁺²	C₃³	1	3	yes	—
56	C₃×3₋¹⁺²	C₃³	1	6	yes	—
57	C₃²×C₉	—	3	3	no	—
58	C₃²×C₉	C₃³	1	1	yes	characteristic
59	C₃²×C₉	C₃³	1	2	yes	—
60	C₉⋊C₉	—	3	2	no	—
61	C₉⋊C₉	—	3	2	no	—
62	C₉⋊C₉	—	3	2	no	—
63	C₉⋊C₉	—	3	6	no	—
64	C₉⋊C₉	—	3	6	no	—
65	C₉⋊C₉	—	3	6	no	—
66	C₉⋊C₉	—	3	6	no	—
67	C₉⋊C₉	—	3	6	no	—
68	C₉⋊C₉	—	3	6	no	—
69	C₉⋊C₉	—	3	12	no	—
70	C₉⋊C₉	—	3	6	no	—
71	C₉⋊C₉	—	3	6	no	—
72	C₉⋊C₉	—	3	6	no	—
73	C₉⋊C₉	—	3	6	no	—
74	C₉⋊C₉	—	3	18	no	—
75	C₃²⋊C₉	—	3	3	no	—
76	C₃²⋊C₉	—	3	3	no	—
77	C₃²⋊C₉	—	3	6	no	—
78	C₃²⋊C₉	—	3	3	no	—
79	C₃²⋊C₉	—	3	3	no	—
80	C₃²⋊C₉	C₃³	1	3	yes	—
81	C₃²⋊C₉	C₃³	1	6	yes	—
82	C₃²⋊C₉	C₃³	1	3	yes	—
83	C₃²⋊C₉	C₃³	1	6	yes	—
84	C₃²⋊C₉	C₃³	1	3	yes	—
85	C₃²⋊C₉	C₃³	1	6	yes	—
86	C₉²	—	3	2	no	—
87	C₉²	—	3	1	no	—
88	C₉²	—	3	6	no	—
89	C₉²	—	3	3	no	—
90	C₃³	—	9	1	no	—
91	C₃³	—	9	3	no	—
92	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
93	C₃³	C₃⁴	1	1	yes	characteristic
94	3₋¹⁺²	—	3	9	no	—
95	3₋¹⁺²	—	3	9	no	—
96	3₋¹⁺²	—	9	6	no	—
97	3₋¹⁺²	—	9	9	no	—
98	3₋¹⁺²	—	9	9	no	—
99	3₋¹⁺²	—	9	6	no	—
100	3₋¹⁺²	—	27	3	no	—
101	3₋¹⁺²	—	27	3	no	—
102	3₋¹⁺²	—	27	3	no	—
103	C₃×C₉	—	3	3	no	—
104	C₃×C₉	—	3	3	no	—
105	C₃×C₉	—	3	6	no	—
106	C₃×C₉	—	3	3	no	—
107	C₃×C₉	—	3	6	no	—
108	C₃×C₉	—	3	3	no	—
109	C₃×C₉	—	3	6	no	—
110	C₃×C₉	—	3	3	no	—
111	C₃×C₉	—	3	6	no	—
112	C₃×C₉	—	9	2	no	—
113	C₃×C₉	—	9	9	no	—
114	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
115	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
116	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
117	C₃²	—	3	1	no	—
118	C₃²	—	3	3	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	3	3	no	—
121	C₃²	—	3	3	no	—
122	C₃²	—	3	3	no	—
123	C₃²	—	27	3	no	—
124	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
125	C₉	—	3	2	no	—
126	C₉	—	3	6	no	—
127	C₉	—	3	9	no	—
128	C₉	—	9	6	no	—
129	C₉	—	9	6	no	—
130	C₉	—	9	3	no	—
131	C₉	—	9	6	no	—
132	C₉	—	9	3	no	—
133	C₉	—	9	6	no	—
134	C₉	—	9	3	no	—
135	C₉	#6912	1	3	yes	—
136	C₃	—	9	1	no	—
137	C₃	—	9	3	no	—
138	C₃	C₃×#6912	1	1	yes	characteristic
139	C₃	#10624934	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²²

C₃

yes

maximal

#10624885

C₃

yes

maximal, characteristic

#10624674

C₃

yes

maximal, complemented

#10624668

C₃

yes

maximal

#10624655

C₃

yes

maximal, complemented

#10624649

C₃

yes

maximal, complemented

#10624631

C₃

yes

maximal

#10624626

C₃

yes

maximal, complemented

#10624618

C₃

yes

maximal, complemented

#10624607

C₃

yes

maximal

#10624588

C₃

yes

maximal, characteristic, complemented

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

—

—

—

—

—

C₃²

yes

—

#6894

—

#6894

C₃²

yes

—

#6894

C₃²

yes

—

#6894

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

C₃²

yes

—

C₉×3₋¹⁺²

C₃²

yes

—

#6887

C₃²

yes

characteristic

#6887

C₃²

yes

—

#6861

C₃²

yes

—

#6861

C₃²

yes

—

#6861

C₃²

yes

—

#6861

C₃²

yes

—

#6860

C₃²

yes

—

#6860

C₃²

yes

—

#6860

C₃²

yes

—

#6860

C₃²

yes

—

#6858

C₃²

yes

—

#6858

C₃²

yes

—

#6858

C₃²

yes

—

#6858

C₃²

yes

—

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

H(3,𝔽₃)

yes

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃×3₋¹⁺²

C₃³

yes

—

C₃×3₋¹⁺²

C₃³

yes

—

C₃²×C₉

—

C₃²×C₉

C₃³

yes

characteristic

C₃²×C₉

C₃³

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃³

yes

—

C₃²⋊C₉

C₃³

yes

—

C₃²⋊C₉

C₃³

yes

—

C₃²⋊C₉

C₃³

yes

—

C₃²⋊C₉

C₃³

yes

—

C₃²⋊C₉

C₃³

yes

—

C₉²

—

C₉²

—

C₉²

—

C₉²

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

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

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

115

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

116

C₃×C₉

C₃×H(3,𝔽₃)

yes

characteristic

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

125

C₉

—

126

C₉

—

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

#6912

yes

—

136

C₃

—

137

C₃

—

138

C₃

C₃×#6912

yes

characteristic

139

C₃

#10624934

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃⁴	C₃³
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺² ◁ G
LECS	G ▷ C₃³ ▷ C₃² ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃⁴

prime radical

C₃⁴

C₃³

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃²×3₋¹⁺² ◁ G

LECS

G ▷ C₃³ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
3. #10624674	C₃	semidirect
5. #10624655	C₃	semidirect
6. #10624649	C₃	semidirect
8. #10624626	C₃	semidirect
9. #10624618	C₃	semidirect
11. #10624588	C₃	semidirect

H ≅ G/N

type

3. #10624674

C₃

semidirect

5. #10624655

C₃

semidirect

6. #10624649

C₃

semidirect

8. #10624626

C₃

semidirect

9. #10624618

C₃

semidirect

11. #10624588

C₃

semidirect

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

subgroup	index
3₋¹⁺²²	3
#10624885	3
#10624674	3
#10624668	3
#10624655	3
#10624649	3
#10624631	3
#10624626	3
#10624618	3
#10624607	3
#10624588	3

subgroup

index

3₋¹⁺²²

Subgroups — #14553746

Conjugacy classes of subgroups

Complements

Maximal subgroups