Group #14548268 — subgroups

Subgroups — #14548268

Order 2187.

1125 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1125 conjugacy classes of subgroups — too many to draw. Only its 62 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	C₃×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624824	C₃	1	2	yes	maximal, complemented
3	#10624823	C₃	1	2	yes	maximal, complemented
4	#10624819	C₃	1	2	yes	maximal, complemented
5	#10624782	C₃	1	2	yes	maximal, complemented
6	#10624745	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624632	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624628	C₃	1	1	yes	maximal, characteristic
9	C₃×#6859	C₃	1	1	yes	maximal, characteristic, complemented
10	3₋¹⁺⁴	—	3	27	no	—
11	C₃×#737	—	3	3	no	—
12	C₃×#737	—	3	6	no	—
13	C₃²×3₋¹⁺²	—	3	3	no	—
14	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
15	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
16	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
17	C₃×#731	—	3	3	no	—
18	#6891	C₃²	1	2	yes	—
19	#6891	C₃²	1	2	yes	—
20	#6891	C₃²	1	1	yes	characteristic
21	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
22	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
23	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
24	3₋¹⁺²⋊C₉	—	3	6	no	—
25	3₋¹⁺²⋊C₉	—	3	6	no	—
26	3₋¹⁺²⋊C₉	—	3	6	no	—
27	H(3,𝔽₃)⋊C₉	—	3	6	no	—
28	#6859	—	3	3	no	—
29	#6859	—	3	3	no	—
30	#6856	—	3	3	no	—
31	C₃⁴	—	3	1	no	—
32	C₃⁴	C₃³	1	1	yes	characteristic
33	#737	—	3	81	no	—
34	#737	—	3	162	no	—
35	C₃×3₋¹⁺²	—	3	18	no	—
36	C₃×3₋¹⁺²	—	3	54	no	—
37	C₃×3₋¹⁺²	—	3	2	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	2	no	—
40	C₃×3₋¹⁺²	—	3	2	no	—
41	C₃×3₋¹⁺²	—	3	27	no	—
42	C₃×3₋¹⁺²	—	3	9	no	—
43	C₃×3₋¹⁺²	—	3	18	no	—
44	C₃×3₋¹⁺²	—	3	2	no	—
45	C₃×3₋¹⁺²	—	3	2	no	—
46	C₃×3₋¹⁺²	—	3	1	no	—
47	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
48	C₃×H(3,𝔽₃)	—	3	9	no	—
49	C₃×H(3,𝔽₃)	—	3	2	no	—
50	C₃×H(3,𝔽₃)	—	3	2	no	—
51	C₃×H(3,𝔽₃)	—	3	1	no	—
52	C₃×H(3,𝔽₃)	—	3	1	no	—
53	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
54	C₃²×C₉	—	3	9	no	—
55	C₃²×C₉	—	3	1	no	—
56	C₃²×C₉	—	3	1	no	—
57	C₃²×C₉	—	3	2	no	—
58	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
59	#731	—	9	9	no	—
60	C₃²⋊C₉	—	3	2	no	—
61	C₃²⋊C₉	—	3	2	no	—
62	C₃²⋊C₉	—	3	1	no	—
63	C₃²⋊C₉	—	3	2	no	—
64	C₃²⋊C₉	—	3	1	no	—
65	C₃²⋊C₉	—	3	2	no	—
66	C₃²⋊C₉	—	3	2	no	—
67	C₃²⋊C₉	—	3	2	no	—
68	C₃²⋊C₉	—	3	2	no	—
69	C₃²⋊C₉	—	3	2	no	—
70	C₃²⋊C₉	—	3	2	no	—
71	C₃²⋊C₉	—	3	2	no	—
72	C₃²⋊C₉	—	3	2	no	—
73	C₃³	—	3	3	no	—
74	C₃³	—	3	3	no	—
75	C₃³	—	3	9	no	—
76	C₃³	—	9	2	no	—
77	C₃³	—	9	1	no	—
78	C₃³	—	9	1	no	—
79	C₃³	—	27	2	no	—
80	C₃³	—	27	1	no	—
81	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
82	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
83	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
84	3₋¹⁺²	—	3	18	no	—
85	3₋¹⁺²	—	3	162	no	—
86	3₋¹⁺²	—	3	18	no	—
87	3₋¹⁺²	—	3	18	no	—
88	3₋¹⁺²	—	3	18	no	—
89	3₋¹⁺²	—	3	6	no	—
90	3₋¹⁺²	—	9	6	no	—
91	3₋¹⁺²	—	9	6	no	—
92	3₋¹⁺²	—	9	3	no	—
93	3₋¹⁺²	AΣL(1,𝔽₂₇)	1	9	yes	—
94	H(3,𝔽₃)	—	3	9	no	—
95	H(3,𝔽₃)	—	3	18	no	—
96	H(3,𝔽₃)	—	9	6	no	—
97	H(3,𝔽₃)	—	9	3	no	—
98	H(3,𝔽₃)	—	27	1	no	—
99	C₃×C₉	—	3	9	no	—
100	C₃×C₉	—	3	81	no	—
101	C₃×C₉	—	3	3	no	—
102	C₃×C₉	—	3	6	no	—
103	C₃×C₉	—	3	18	no	—
104	C₃×C₉	—	9	1	no	—
105	C₃×C₉	—	9	2	no	—
106	C₃×C₉	—	9	1	no	—
107	C₃×C₉	—	9	2	no	—
108	C₃×C₉	—	9	2	no	—
109	C₃×C₉	—	9	3	no	—
110	C₃×C₉	—	27	2	no	—
111	C₃²	—	3	2	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	3	2	no	—
115	C₃²	—	3	9	no	—
116	C₃²	—	3	2	no	—
117	C₃²	—	3	2	no	—
118	C₃²	—	3	2	no	—
119	C₃²	—	3	1	no	—
120	C₃²	—	3	2	no	—
121	C₃²	—	9	6	no	—
122	C₃²	—	9	3	no	—
123	C₃²	—	27	3	no	—
124	C₃²	—	27	9	no	—
125	C₃²	—	27	1	no	—
126	C₃²	—	27	2	no	—
127	C₃²	—	27	1	no	—
128	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
129	C₃²	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
130	C₉	—	3	9	no	—
131	C₉	—	3	18	no	—
132	C₉	—	27	1	no	—
133	C₉	—	27	1	no	—
134	C₉	—	27	2	no	—
135	C₉	—	27	2	no	—
136	C₉	—	27	2	no	—
137	C₃	—	3	3	no	—
138	C₃	—	9	2	no	—
139	C₃	—	9	1	no	—
140	C₃	—	9	9	no	—
141	C₃	—	81	1	no	—
142	C₃	#10624730	1	1	yes	characteristic
143	C₃	#10624751	1	2	yes	—
144	C₃	#10624753	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺⁴

C₃

yes

maximal, characteristic, complemented

#10624824

C₃

yes

maximal, complemented

#10624823

C₃

yes

maximal, complemented

#10624819

C₃

yes

maximal, complemented

#10624782

C₃

yes

maximal, complemented

#10624745

C₃

yes

maximal, characteristic, complemented

#10624632

C₃

yes

maximal, characteristic, complemented

#10624628

C₃

yes

maximal, characteristic

C₃×#6859

C₃

yes

maximal, characteristic, complemented

3₋¹⁺⁴

—

C₃×#737

—

C₃×#737

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×#731

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

characteristic

C₃²⋊H(3,𝔽₃)

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6859

—

#6859

—

#6856

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

#737

—

#737

—

162

—

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

H(3,𝔽₃)

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

#731

—

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

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

162

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

AΣL(1,𝔽₂₇)

yes

—

—

—

—

—

—

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

—

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

C₃²⋊H(3,𝔽₃)

yes

characteristic

129

C₃²

C₃×AΣL(1,𝔽₂₇)

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

—

142

C₃

#10624730

yes

characteristic

143

C₃

#10624751

yes

—

144

C₃

#10624753

yes

characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624628	C₃
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

C₃

G ▷ C₃⁴ ▷ C₃² ▷ C₁

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. C₃×3₋¹⁺⁴	C₃	semidirect
2. #10624824	C₃	semidirect
3. #10624823	C₃	semidirect
4. #10624819	C₃	semidirect
5. #10624782	C₃	semidirect
6. #10624745	C₃	semidirect
7. #10624632	C₃	semidirect
9. C₃×#6859	C₃	semidirect

H ≅ G/N

type

1. C₃×3₋¹⁺⁴

C₃

semidirect

2. #10624824

C₃

semidirect

3. #10624823

C₃

semidirect

4. #10624819

C₃

semidirect

5. #10624782

C₃

semidirect

6. #10624745

C₃

semidirect

7. #10624632

C₃

semidirect

9. C₃×#6859

C₃

semidirect

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

subgroup	index
C₃×3₋¹⁺⁴	3
#10624824	3
#10624823	3
#10624819	3
#10624782	3
#10624745	3
#10624632	3
#10624628	3
C₃×#6859	3

subgroup

index

C₃×3₋¹⁺⁴

Subgroups — #14548268

Conjugacy classes of subgroups

Complements

Maximal subgroups