Group #13090650 — subgroups

Subgroups — #13090650

Order 1458.

245 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 40 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	#10624558	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊D₃×3₋¹⁺²	—	3	1	no	maximal
3	C₃²×C₃⋊D₉	C₃	1	1	yes	maximal, characteristic
4	#92158	C₃	1	3	yes	maximal, complemented
5	C₃²×3₋¹⁺²	D₃	1	1	yes	characteristic
6	C₃³×C₉	C₂×C₃	1	1	yes	characteristic
7	C₃×C₃²⋊C₉	—	2	1	no	—
8	#6855	—	2	3	no	—
9	#6855	C₂×C₃	1	3	yes	complemented
10	C₃²×C₃⋊D₃	—	3	1	no	—
11	C₉×C₃⋊D₃	—	3	3	no	—
12	C₃×C₃⋊D₉	—	3	1	no	—
13	C₃×C₃⋊D₉	C₃²	1	1	yes	characteristic
14	D₃×3₋¹⁺²	—	9	1	no	—
15	D₃×3₋¹⁺²	—	9	2	no	—
16	D₃×3₋¹⁺²	—	9	1	no	—
17	C₃²×D₉	—	9	1	no	—
18	C₃⁴	C₃×D₃	1	1	yes	characteristic
19	C₃×3₋¹⁺²	—	2	1	no	—
20	C₃×3₋¹⁺²	—	2	3	no	—
21	C₃×3₋¹⁺²	—	3	1	no	—
22	C₃×3₋¹⁺²	—	3	2	no	—
23	C₃×3₋¹⁺²	—	6	1	no	—
24	C₃×3₋¹⁺²	—	6	2	no	—
25	C₃×3₋¹⁺²	D₉	1	1	yes	characteristic, complemented
26	C₃²×C₉	—	2	1	no	—
27	C₃²×C₉	—	2	3	no	—
28	C₃²×C₉	—	3	1	no	—
29	C₃²×C₉	—	3	1	no	—
30	C₃²×C₉	—	6	3	no	—
31	C₃²×C₉	—	6	1	no	—
32	C₃²×C₉	—	6	1	no	—
33	C₃²×C₉	C₃×D₃	1	3	yes	—
34	C₃²×C₉	C₂×C₃²	1	1	yes	characteristic
35	C₃²⋊C₉	—	6	2	no	—
36	C₃²⋊C₉	—	6	1	no	—
37	C₃×C₃⋊D₃	—	3	1	no	—
38	C₃×C₃⋊D₃	—	9	1	no	—
39	C₃²×D₃	—	9	1	no	—
40	C₃²×D₃	—	9	1	no	—
41	C₃²×D₃	—	9	2	no	—
42	C₂×3₋¹⁺²	—	27	1	no	—
43	C₃⋊D₉	3₋¹⁺²	1	1	yes	characteristic, complemented
44	D₃×C₉	—	9	3	no	—
45	D₃×C₉	—	9	6	no	—
46	D₃×C₉	—	9	3	no	—
47	C₃×D₉	—	9	1	no	—
48	C₃×D₉	—	9	3	no	—
49	C₃³	—	2	1	no	—
50	C₃³	—	2	1	no	—
51	C₃³	—	2	2	no	—
52	C₃³	—	3	1	no	—
53	C₃³	—	6	1	no	—
54	C₃³	—	6	1	no	—
55	C₃³	—	6	2	no	—
56	C₃³	C₃×D₉	1	1	yes	characteristic
57	C₃³	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
58	C₃³	AGL(1,ℤ/9)	1	2	yes	—
59	C₃³	C₃²×D₃	1	1	yes	characteristic
60	3₋¹⁺²	—	3	1	no	—
61	3₋¹⁺²	—	6	1	no	—
62	3₋¹⁺²	—	6	3	no	—
63	3₋¹⁺²	—	6	6	no	—
64	3₋¹⁺²	—	6	2	no	—
65	3₋¹⁺²	—	6	1	no	—
66	C₃×C₉	—	2	3	no	—
67	C₃×C₉	—	2	3	no	—
68	C₃×C₉	—	2	1	no	—
69	C₃×C₉	—	3	3	no	—
70	C₃×C₉	—	3	1	no	—
71	C₃×C₉	—	3	3	no	—
72	C₃×C₉	—	3	6	no	—
73	C₃×C₉	—	6	3	no	—
74	C₃×C₉	—	6	1	no	—
75	C₃×C₉	—	6	1	no	—
76	C₃×C₉	—	6	3	no	—
77	C₃×C₉	—	6	3	no	—
78	C₃×C₉	—	6	1	no	—
79	C₃×C₉	—	6	1	no	—
80	C₃×C₉	—	6	3	no	—
81	C₃×C₉	—	6	3	no	—
82	C₃×C₉	—	6	6	no	—
83	C₃×C₉	—	6	3	no	—
84	C₃×C₉	C₃×D₉	1	3	yes	complemented
85	C₃×C₉	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
86	C₂×C₃²	—	27	1	no	—
87	C₃⋊D₃	—	3	1	no	—
88	C₃×D₃	—	9	1	no	—
89	C₃×D₃	—	9	1	no	—
90	C₃×D₃	—	9	2	no	—
91	C₃×D₃	—	27	1	no	—
92	C₃×D₃	—	27	1	no	—
93	C₃×D₃	—	27	2	no	—
94	C₂×C₉	—	27	3	no	—
95	D₉	—	9	1	no	—
96	C₃²	—	2	1	no	—
97	C₃²	—	2	1	no	—
98	C₃²	—	2	2	no	—
99	C₃²	—	2	1	no	—
100	C₃²	—	2	1	no	—
101	C₃²	—	2	2	no	—
102	C₃²	—	3	1	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	2	no	—
105	C₃²	—	6	1	no	—
106	C₃²	—	6	1	no	—
107	C₃²	—	6	2	no	—
108	C₃²	—	6	2	no	—
109	C₃²	—	6	2	no	—
110	C₃²	—	6	4	no	—
111	C₃²	—	6	1	no	—
112	C₃²	—	6	1	no	—
113	C₃²	—	6	2	no	—
114	C₃²	#4169	1	1	yes	characteristic
115	C₃²	C₃²×D₉	1	1	yes	characteristic
116	C₃²	C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
117	C₃²	C₃×AGL(1,ℤ/9)	1	2	yes	—
118	C₃²	D₃×3₋¹⁺²	1	1	yes	characteristic
119	C₉	—	3	3	no	—
120	C₉	—	3	1	no	—
121	C₉	—	6	3	no	—
122	C₉	—	6	3	no	—
123	C₉	—	6	1	no	—
124	C₉	—	18	3	no	—
125	C₂×C₃	—	27	1	no	—
126	C₂×C₃	—	81	1	no	—
127	D₃	—	9	1	no	—
128	D₃	—	9	1	no	—
129	D₃	—	9	2	no	—
130	C₃	—	2	1	no	—
131	C₃	—	2	1	no	—
132	C₃	—	2	2	no	—
133	C₃	—	3	1	no	—
134	C₃	—	6	1	no	—
135	C₃	—	6	1	no	—
136	C₃	—	6	2	no	—
137	C₃	C₃×#4169	1	1	yes	characteristic
138	C₃	D₉×3₋¹⁺²	1	1	yes	characteristic
139	C₃	#92259	1	1	yes	characteristic
140	C₃	#6894⋊C₂	1	2	yes	—
141	C₂	—	27	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624558

C₂

yes

maximal, characteristic, complemented, Sylow

C₃⋊D₃×3₋¹⁺²

—

maximal

C₃²×C₃⋊D₉

C₃

yes

maximal, characteristic

#92158

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

D₃

yes

characteristic

C₃³×C₉

C₂×C₃

yes

characteristic

C₃×C₃²⋊C₉

—

#6855

—

#6855

C₂×C₃

yes

complemented

C₃²×C₃⋊D₃

—

C₉×C₃⋊D₃

—

C₃×C₃⋊D₉

—

C₃×C₃⋊D₉

C₃²

yes

characteristic

D₃×3₋¹⁺²

—

D₃×3₋¹⁺²

—

D₃×3₋¹⁺²

—

C₃²×D₉

—

C₃⁴

C₃×D₃

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

D₉

yes

characteristic, complemented

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×D₃

yes

—

C₃²×C₉

C₂×C₃²

yes

characteristic

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₃⋊D₉

3₋¹⁺²

yes

characteristic, complemented

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

C₃×D₉

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×D₉

yes

characteristic

C₃³

(C₃⋊D₃)⋊C₃

yes

characteristic

C₃³

AGL(1,ℤ/9)

yes

—

C₃³

C₃²×D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

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

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×D₉

yes

complemented

C₃×C₉

C₂×3₋¹⁺²

yes

characteristic, complemented

C₂×C₃²

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

D₉

—

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

#4169

yes

characteristic

115

C₃²

C₃²×D₉

yes

characteristic

116

C₃²

C₃×(C₃⋊D₃)⋊C₃

yes

characteristic

117

C₃²

C₃×AGL(1,ℤ/9)

yes

—

118

C₃²

D₃×3₋¹⁺²

yes

characteristic

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₂×C₃

—

126

C₂×C₃

—

127

D₃

—

128

D₃

—

129

D₃

—

130

C₃

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

C₃×#4169

yes

characteristic

138

C₃

D₉×3₋¹⁺²

yes

characteristic

139

C₃

#92259

yes

characteristic

140

C₃

#6894⋊C₂

yes

—

141

C₂

—

Sylow

	subgroup	G/H
center	C₃	C₃×#4169
commutator, cosocle	C₃²×C₉	C₂×C₃²
Frattini, socle	C₃³	C₃²×D₃
Fitting	#10624558	C₂
prime radical	C₃²×C₃⋊D₉	C₃
hypercenter	C₃²	#4169
hypocenter	C₃×C₉	C₂×3₋¹⁺²
LCS	G ▷ C₃²×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₃ ◁ C₃²

subgroup

G/H

center

C₃

C₃×#4169

commutator, cosocle

C₃²×C₉

C₂×C₃²

Frattini, socle

C₃³

C₃²×D₃

Fitting

#10624558

C₂

prime radical

C₃²×C₃⋊D₉

C₃

hypercenter

C₃²

#4169

hypocenter

C₃×C₉

C₂×3₋¹⁺²

LCS

G ▷ C₃²×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₃ ◁ C₃²

N	H ≅ G/N	type
1. #10624558	C₂	semidirect
4. #92158	C₃	semidirect
9. #6855	C₂×C₃	semidirect
25. C₃×3₋¹⁺²	D₉	semidirect
43. C₃⋊D₉	3₋¹⁺²	semidirect
84. C₃×C₉	C₃×D₉	semidirect
85. C₃×C₉	C₂×3₋¹⁺²	semidirect

H ≅ G/N

type

1. #10624558

C₂

semidirect

4. #92158

C₃

semidirect

9. #6855

C₂×C₃

semidirect

25. C₃×3₋¹⁺²

D₉

semidirect

43. C₃⋊D₉

3₋¹⁺²

semidirect

84. C₃×C₉

C₃×D₉

semidirect

85. C₃×C₉

C₂×3₋¹⁺²

semidirect

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

subgroup	index
#10624558	2
C₃⋊D₃×3₋¹⁺²	3
C₃²×C₃⋊D₉	3
#92158	3

subgroup

index

#10624558

C₃⋊D₃×3₋¹⁺²

C₃²×C₃⋊D₉

#92158

Subgroups — #13090650

Conjugacy classes of subgroups

Complements

Maximal subgroups