Group #14554376 — subgroups

Subgroups — #14554376

Order 2187.

2600 conjugacy classes of subgroups; only the 323 normal subgroups are shown.

normal subgroups only

This group has 2600 conjugacy classes of subgroups — too many to draw. Only its 323 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	C₃³×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×#6891	C₃	1	3	yes	maximal, complemented
3	#10624773	C₃	1	9	yes	maximal, complemented
4	#10624769	C₃	1	9	yes	maximal, complemented
5	#10624765	C₃	1	9	yes	maximal, complemented
6	#10624689	C₃	1	9	yes	maximal, complemented
7	C₃⁵	C₃²	1	3	yes	—
8	C₃⁵	C₃²	1	1	yes	characteristic
9	C₃²×3₋¹⁺²	—	3	3	no	—
10	C₃²×H(3,𝔽₃)	—	3	3	no	—
11	C₃²×H(3,𝔽₃)	—	3	6	no	—
12	C₃²×H(3,𝔽₃)	—	3	27	no	—
13	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
14	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
15	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
16	#6891	—	3	6	no	—
17	#6891	—	3	6	no	—
18	#6891	—	3	6	no	—
19	#6891	C₃²	1	9	yes	—
20	#6891	C₃²	1	9	yes	complemented
21	#6891	C₃²	1	9	yes	complemented
22	C₃×C₃²⋊C₉	—	3	6	no	—
23	C₃×C₃²⋊C₉	C₃²	1	9	yes	complemented
24	#6866	C₃²	1	81	yes	complemented
25	C₃⁴	—	3	9	no	—
26	C₃⁴	—	3	9	no	—
27	C₃⁴	—	3	27	no	—
28	C₃⁴	—	9	9	no	—
29	C₃⁴	—	9	27	no	—
30	C₃⁴	H(3,𝔽₃)	1	3	yes	—
31	C₃⁴	H(3,𝔽₃)	1	6	yes	—
32	C₃⁴	C₃³	1	1	yes	characteristic
33	C₃⁴	C₃³	1	1	yes	characteristic
34	C₃⁴	C₃³	1	1	yes	characteristic
35	C₃⁴	C₃³	1	9	yes	—
36	C₃⁴	C₃³	1	1	yes	characteristic
37	C₃×3₋¹⁺²	—	3	9	no	—
38	C₃×3₋¹⁺²	—	3	9	no	—
39	C₃×3₋¹⁺²	—	3	9	no	—
40	C₃×3₋¹⁺²	—	9	27	no	—
41	C₃×H(3,𝔽₃)	—	3	81	no	—
42	C₃×H(3,𝔽₃)	—	3	162	no	—
43	C₃×H(3,𝔽₃)	—	3	6	no	—
44	C₃×H(3,𝔽₃)	—	3	9	no	—
45	C₃×H(3,𝔽₃)	—	3	3	no	—
46	C₃×H(3,𝔽₃)	—	3	9	no	—
47	C₃×H(3,𝔽₃)	—	3	18	no	—
48	C₃×H(3,𝔽₃)	—	3	27	no	—
49	C₃×H(3,𝔽₃)	—	3	27	no	—
50	C₃×H(3,𝔽₃)	3₋¹⁺²	1	27	yes	complemented
51	C₃²×C₉	—	3	9	no	—
52	C₃²⋊C₉	—	3	18	no	—
53	C₃²⋊C₉	—	3	18	no	—
54	C₃²⋊C₉	—	3	18	no	—
55	C₃²⋊C₉	—	3	18	no	—
56	C₃²⋊C₉	C₃³	1	27	yes	—
57	C₃³	—	3	3	no	—
58	C₃³	—	3	9	no	—
59	C₃³	—	3	81	no	—
60	C₃³	—	3	9	no	—
61	C₃³	—	3	3	no	—
62	C₃³	—	3	3	no	—
63	C₃³	—	3	18	no	—
64	C₃³	—	3	9	no	—
65	C₃³	—	3	18	no	—
66	C₃³	—	3	9	no	—
67	C₃³	—	3	18	no	—
68	C₃³	—	9	6	no	—
69	C₃³	—	9	6	no	—
70	C₃³	—	9	6	no	—
71	C₃³	—	9	3	no	—
72	C₃³	—	9	3	no	—
73	C₃³	—	9	3	no	—
74	C₃³	—	9	27	no	—
75	C₃³	—	9	81	no	—
76	C₃³	—	9	243	no	—
77	C₃³	—	9	54	no	—
78	C₃³	—	9	27	no	—
79	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
80	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
81	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
82	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
83	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
84	C₃³	C₃×3₋¹⁺²	1	9	yes	—
85	C₃³	C₃×3₋¹⁺²	1	27	yes	complemented
86	C₃³	C₃⁴	1	1	yes	characteristic
87	3₋¹⁺²	—	9	27	no	—
88	3₋¹⁺²	—	9	27	no	—
89	3₋¹⁺²	—	9	27	no	—
90	H(3,𝔽₃)	—	3	54	no	—
91	H(3,𝔽₃)	—	3	81	no	—
92	H(3,𝔽₃)	—	3	27	no	—
93	H(3,𝔽₃)	—	3	81	no	—
94	C₃×C₉	—	3	27	no	—
95	C₃×C₉	—	9	27	no	—
96	C₃²	—	3	27	no	—
97	C₃²	—	3	3	no	—
98	C₃²	—	3	3	no	—
99	C₃²	—	3	6	no	—
100	C₃²	—	3	3	no	—
101	C₃²	—	3	6	no	—
102	C₃²	—	3	2	no	—
103	C₃²	—	3	3	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	27	no	—
106	C₃²	—	3	3	no	—
107	C₃²	—	3	3	no	—
108	C₃²	—	3	6	no	—
109	C₃²	—	3	6	no	—
110	C₃²	—	9	9	no	—
111	C₃²	—	9	27	no	—
112	C₃²	—	9	54	no	—
113	C₃²	—	9	27	no	—
114	C₃²	—	9	9	no	—
115	C₃²	—	9	9	no	—
116	C₃²	—	9	54	no	—
117	C₃²	—	9	243	no	—
118	C₃²	—	9	27	no	—
119	C₃²	#6891	1	9	yes	—
120	C₃²	#6909	1	9	yes	—
121	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
122	C₃²	C₃²×3₋¹⁺²	1	3	yes	—
123	C₉	—	9	27	no	—
124	C₃	—	3	3	no	—
125	C₃	—	3	9	no	—
126	C₃	—	3	9	no	—
127	C₃	—	9	1	no	—
128	C₃	—	9	3	no	—
129	C₃	—	9	2	no	—
130	C₃	—	9	27	no	—
131	C₃	C₃×#6891	1	1	yes	characteristic
132	C₃	C₃×#6909	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃×#6891

C₃

yes

maximal, complemented

#10624773

C₃

yes

maximal, complemented

#10624769

C₃

yes

maximal, complemented

#10624765

C₃

yes

maximal, complemented

#10624689

C₃

yes

maximal, complemented

C₃⁵

C₃²

yes

—

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

—

C₃²×H(3,𝔽₃)

—

C₃²×H(3,𝔽₃)

—

C₃²×H(3,𝔽₃)

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

—

—

—

—

C₃²

yes

—

#6891

C₃²

yes

complemented

#6891

C₃²

yes

complemented

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

complemented

#6866

C₃²

yes

complemented

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

C₃³

yes

characteristic

C₃⁴

C₃³

yes

characteristic

C₃⁴

C₃³

yes

characteristic

C₃⁴

C₃³

yes

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

162

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

3₋¹⁺²

yes

complemented

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃³

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

243

—

C₃³

—

C₃³

—

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

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

C₃×3₋¹⁺²

yes

complemented

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

—

—

—

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

243

—

118

C₃²

—

119

C₃²

#6891

yes

—

120

C₃²

#6909

yes

—

121

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

122

C₃²

C₃²×3₋¹⁺²

yes

—

123

C₉

—

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

—

128

C₃

—

129

C₃

—

130

C₃

—

131

C₃

C₃×#6891

yes

characteristic

132

C₃

C₃×#6909

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃³×H(3,𝔽₃)	C₃
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

C₃³×H(3,𝔽₃)

C₃

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁵ ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. C₃×#6891	C₃	semidirect
3. #10624773	C₃	semidirect
4. #10624769	C₃	semidirect
5. #10624765	C₃	semidirect
6. #10624689	C₃	semidirect
20. #6891	C₃²	semidirect
21. #6891	C₃²	semidirect
23. C₃×C₃²⋊C₉	C₃²	semidirect
24. #6866	C₃²	semidirect
50. C₃×H(3,𝔽₃)	3₋¹⁺²	semidirect
85. C₃³	C₃×3₋¹⁺²	semidirect

H ≅ G/N

type

2. C₃×#6891

C₃

semidirect

3. #10624773

C₃

semidirect

4. #10624769

C₃

semidirect

5. #10624765

C₃

semidirect

6. #10624689

C₃

semidirect

20. #6891

C₃²

semidirect

21. #6891

C₃²

semidirect

23. C₃×C₃²⋊C₉

C₃²

semidirect

24. #6866

C₃²

semidirect

50. C₃×H(3,𝔽₃)

3₋¹⁺²

semidirect

85. C₃³

C₃×3₋¹⁺²

semidirect

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

subgroup	index
C₃³×H(3,𝔽₃)	3
C₃×#6891	3
#10624773	3
#10624769	3
#10624765	3
#10624689	3

subgroup

index

C₃³×H(3,𝔽₃)

Subgroups — #14554376

Conjugacy classes of subgroups

Complements

Maximal subgroups