Group #14554397 — subgroups

Subgroups — #14554397

Order 2187.

1524 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1524 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	C₃×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic
2	#10624883	C₃	1	1	yes	maximal, characteristic
3	#10624883	C₃	1	1	yes	maximal, characteristic
4	C₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	#10624783	C₃	1	9	yes	maximal, complemented
6	#10624782	C₃	1	9	yes	maximal, complemented
7	#10624781	C₃	1	9	yes	maximal, complemented
8	#10624699	C₃	1	6	yes	maximal, complemented
9	C₃×3₋¹⁺²⋊C₉	C₃	1	3	yes	maximal, complemented
10	3₋¹⁺⁴	—	3	27	no	—
11	C₃×#737	—	3	9	no	—
12	C₃²×3₋¹⁺²	C₃²	1	9	yes	—
13	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
14	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
15	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
16	C₃³×C₉	C₃²	1	1	yes	characteristic
17	C₉⋊H(3,𝔽₃)	—	3	9	no	—
18	C₉⋊H(3,𝔽₃)	—	3	9	no	—
19	C₉⋊H(3,𝔽₃)	—	3	9	no	—
20	#6891	C₃²	1	9	yes	—
21	#6891	C₃²	1	9	yes	—
22	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
23	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
24	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
25	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
27	3₋¹⁺²⋊C₉	C₃²	1	9	yes	—
28	3₋¹⁺²⋊C₉	C₃²	1	54	yes	—
29	3₋¹⁺²⋊C₉	C₃²	1	9	yes	—
30	3₋¹⁺²⋊C₉	C₃²	1	9	yes	—
31	C₃⁴	C₃³	1	3	yes	—
32	C₃⁴	C₃³	1	1	yes	characteristic
33	#737	—	3	243	no	—
34	C₃×3₋¹⁺²	—	3	18	no	—
35	C₃×3₋¹⁺²	—	3	6	no	—
36	C₃×3₋¹⁺²	—	3	18	no	—
37	C₃×3₋¹⁺²	—	3	81	no	—
38	C₃×3₋¹⁺²	—	3	9	no	—
39	C₃×3₋¹⁺²	—	3	9	no	—
40	C₃×3₋¹⁺²	—	3	9	no	—
41	C₃×3₋¹⁺²	—	9	9	no	—
42	C₃×3₋¹⁺²	3₋¹⁺²	1	27	yes	complemented
43	C₃×3₋¹⁺²	C₃³	1	9	yes	—
44	C₃×H(3,𝔽₃)	—	3	9	no	—
45	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
46	C₃²×C₉	—	3	9	no	—
47	C₃²×C₉	—	3	3	no	—
48	C₃²×C₉	—	3	3	no	—
49	C₃²×C₉	—	3	3	no	—
50	C₃²×C₉	—	3	3	no	—
51	C₃²×C₉	—	9	3	no	—
52	C₃²×C₉	C₃³	1	1	yes	characteristic
53	C₃²×C₉	C₃³	1	1	yes	characteristic
54	C₃²×C₉	C₃³	1	1	yes	characteristic
55	C₃²×C₉	C₃³	1	6	yes	—
56	C₃²⋊C₉	—	3	9	no	—
57	C₃²⋊C₉	—	3	9	no	—
58	C₃²⋊C₉	—	3	6	no	—
59	C₃²⋊C₉	—	3	6	no	—
60	C₃²⋊C₉	—	3	6	no	—
61	C₃²⋊C₉	—	3	18	no	—
62	C₃²⋊C₉	—	3	18	no	—
63	C₃²⋊C₉	C₃³	1	9	yes	—
64	C₃²⋊C₉	C₃³	1	9	yes	—
65	C₃³	—	3	9	no	—
66	C₃³	—	9	1	no	—
67	C₃³	—	9	1	no	—
68	C₃³	—	9	1	no	—
69	C₃³	—	9	3	no	—
70	C₃³	—	9	3	no	—
71	C₃³	—	9	3	no	—
72	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
73	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
74	C₃³	C₃×3₋¹⁺²	1	9	yes	—
75	C₃³	C₃⁴	1	1	yes	characteristic
76	3₋¹⁺²	—	3	162	no	—
77	3₋¹⁺²	—	3	18	no	—
78	3₋¹⁺²	—	3	54	no	—
79	3₋¹⁺²	—	9	27	no	—
80	3₋¹⁺²	—	27	9	no	—
81	3₋¹⁺²	—	27	9	no	—
82	3₋¹⁺²	C₃×3₋¹⁺²	1	27	yes	—
83	H(3,𝔽₃)	—	3	27	no	—
84	C₃×C₉	—	3	81	no	—
85	C₃×C₉	—	3	9	no	—
86	C₃×C₉	—	3	1	no	—
87	C₃×C₉	—	3	27	no	—
88	C₃×C₉	—	3	3	no	—
89	C₃×C₉	—	3	1	no	—
90	C₃×C₉	—	3	1	no	—
91	C₃×C₉	—	3	6	no	—
92	C₃×C₉	—	3	9	no	—
93	C₃×C₉	—	3	9	no	—
94	C₃×C₉	—	9	9	no	—
95	C₃×C₉	—	9	1	no	—
96	C₃×C₉	—	9	1	no	—
97	C₃×C₉	—	9	6	no	—
98	C₃×C₉	—	27	3	no	—
99	C₃×C₉	—	27	3	no	—
100	C₃²	—	3	1	no	—
101	C₃²	—	3	3	no	—
102	C₃²	—	3	3	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	3	no	—
106	C₃²	—	3	9	no	—
107	C₃²	—	3	9	no	—
108	C₃²	—	3	9	no	—
109	C₃²	—	3	9	no	—
110	C₃²	—	9	9	no	—
111	C₃²	—	27	9	no	—
112	C₃²	#6891	1	9	yes	—
113	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
114	C₃²	C₃²×3₋¹⁺²	1	3	yes	—
115	C₉	—	3	27	no	—
116	C₉	—	3	3	no	—
117	C₉	—	9	1	no	—
118	C₉	—	9	1	no	—
119	C₉	—	9	6	no	—
120	C₉	—	9	9	no	—
121	C₉	—	9	9	no	—
122	C₃	—	3	3	no	—
123	C₃	—	3	9	no	—
124	C₃	—	9	9	no	—
125	C₃	C₃×#6891	1	1	yes	characteristic
126	C₃	#10624930	1	1	yes	characteristic
127	C₃	#10624931	1	1	yes	characteristic
128	C₃	#10624933	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺⁴

C₃

yes

maximal, characteristic

#10624883

C₃

yes

maximal, characteristic

#10624883

C₃

yes

maximal, characteristic

C₃×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

#10624783

C₃

yes

maximal, complemented

#10624782

C₃

yes

maximal, complemented

#10624781

C₃

yes

maximal, complemented

#10624699

C₃

yes

maximal, complemented

C₃×3₋¹⁺²⋊C₉

C₃

yes

maximal, complemented

3₋¹⁺⁴

—

C₃×#737

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

#6891

C₃²

yes

—

#6891

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

—

3₋¹⁺²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

C₃²

yes

—

C₃⁴

C₃³

yes

—

C₃⁴

C₃³

yes

characteristic

#737

—

243

—

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

C₃×3₋¹⁺²

C₃³

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃³

yes

characteristic

C₃²×C₉

C₃³

yes

characteristic

C₃²×C₉

C₃³

yes

characteristic

C₃²×C₉

C₃³

yes

—

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

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

162

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×3₋¹⁺²

yes

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

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

#6891

yes

—

113

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

114

C₃²

C₃²×3₋¹⁺²

yes

—

115

C₉

—

116

C₉

—

117

C₉

—

118

C₉

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₃

—

123

C₃

—

124

C₃

—

125

C₃

C₃×#6891

yes

characteristic

126

C₃

#10624930

yes

characteristic

127

C₃

#10624931

yes

characteristic

128

C₃

#10624933

yes

characteristic

	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 ◁ C₃²×H(3,𝔽₃) ◁ 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 ◁ C₃²×H(3,𝔽₃) ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
5. #10624783	C₃	semidirect
6. #10624782	C₃	semidirect
7. #10624781	C₃	semidirect
8. #10624699	C₃	semidirect
9. C₃×3₋¹⁺²⋊C₉	C₃	semidirect
42. C₃×3₋¹⁺²	3₋¹⁺²	semidirect

H ≅ G/N

type

5. #10624783

C₃

semidirect

6. #10624782

C₃

semidirect

7. #10624781

C₃

semidirect

8. #10624699

C₃

semidirect

9. C₃×3₋¹⁺²⋊C₉

C₃

semidirect

42. 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
C₃×3₋¹⁺⁴	3
#10624883	3
C₃×C₉⋊H(3,𝔽₃)	3
#10624783	3
#10624782	3
#10624781	3
#10624699	3
C₃×3₋¹⁺²⋊C₉	3

subgroup

index

C₃×3₋¹⁺⁴

C₃×C₉⋊H(3,𝔽₃)

C₃×3₋¹⁺²⋊C₉

Subgroups — #14554397

Conjugacy classes of subgroups

Complements

Maximal subgroups