Group #14518176 — subgroups

Subgroups — D₃×C₉⋊(C₂²⋊C₉) #14518176

Order 1944.

287 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 72 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×C₉⋊(C₂²⋊C₉)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₃×C₉	C₃	1	1	yes	maximal, characteristic
3	C₃×D₃×C₂²⋊C₉	C₃	1	3	yes	maximal
4	C₂×C₉⋊(C₂²⋊C₉)	—	3	1	no	maximal
5	D₃×C₉⋊C₉	—	4	1	no	maximal
6	C₂²×C₃²×C₉	C₂×C₃	1	1	yes	characteristic
7	C₂×C₃×D₃×C₉	—	3	1	no	—
8	C₂×C₃×D₃×C₉	—	3	1	no	—
9	C₃²×C₂²⋊C₉	C₂×C₃	1	3	yes	—
10	C₉⋊(C₂²⋊C₉)	—	2	1	no	—
11	C₉⋊(C₂²⋊C₉)	—	2	3	no	—
12	C₉⋊(C₂²⋊C₉)	D₃	1	1	yes	characteristic, direct product factor
13	C₃×C₉⋊C₉	—	4	1	no	Sylow
14	C₂²×C₃²×D₃	C₃²	1	1	yes	characteristic
15	C₂³×C₃×C₉	—	3	1	no	—
16	C₂²×D₃×C₉	C₉	1	3	yes	complemented
17	C₂×C₃×C₂²⋊C₉	—	3	3	no	—
18	D₃×C₂²⋊C₉	—	3	3	no	—
19	C₂×C₃²×C₉	—	3	1	no	—
20	C₃×D₃×C₉	—	3	1	no	—
21	C₃×D₃×C₉	—	4	3	no	—
22	C₃×D₃×C₉	A₄	1	1	yes	characteristic
23	C₂×C₉⋊C₉	—	12	1	no	—
24	C₂²×C₃³	C₂×C₃²	1	1	yes	characteristic
25	C₂×C₃²×D₃	—	3	1	no	—
26	C₂×C₃²×D₃	—	3	1	no	—
27	C₂²×C₃×C₉	—	2	1	no	—
28	C₂²×C₃×C₉	—	2	3	no	—
29	C₂²×C₃×C₉	—	9	1	no	—
30	C₂²×C₃×C₉	—	9	1	no	—
31	C₂²×C₃×C₉	C₂×C₉	1	3	yes	complemented
32	C₂²×C₃×C₉	C₃×D₃	1	1	yes	characteristic
33	C₂×D₃×C₉	—	3	3	no	—
34	C₂×D₃×C₉	—	3	3	no	—
35	C₃×C₂²⋊C₉	—	2	3	no	—
36	C₃×C₂²⋊C₉	—	3	3	no	—
37	C₃×C₂²⋊C₉	—	6	3	no	—
38	C₃×C₂²⋊C₉	C₃×D₃	1	3	yes	—
39	C₃²×C₉	—	4	3	no	—
40	C₃²×C₉	C₂×A₄	1	1	yes	characteristic
41	C₉⋊C₉	—	4	1	no	—
42	C₉⋊C₉	—	8	3	no	—
43	C₉⋊C₉	—	8	1	no	—
44	C₂³×C₃²	—	3	1	no	—
45	C₂²×C₃×D₃	C₃×C₉	1	1	yes	characteristic
46	C₂²×C₃×D₃	3₋¹⁺²	1	3	yes	—
47	C₂³×C₉	—	3	3	no	—
48	C₂×C₂²⋊C₉	—	9	3	no	—
49	C₂×C₃³	—	3	1	no	—
50	C₃²×D₃	—	3	1	no	—
51	C₃²×D₃	C₃×A₄	1	1	yes	characteristic
52	C₂×C₃×C₉	—	3	1	no	—
53	C₂×C₃×C₉	—	3	1	no	—
54	C₂×C₃×C₉	—	3	3	no	—
55	C₂×C₃×C₉	—	6	3	no	—
56	C₂×C₃×C₉	—	6	1	no	—
57	C₂×C₃×C₉	—	9	1	no	—
58	C₂×C₃×C₉	—	12	3	no	—
59	D₃×C₉	—	3	3	no	—
60	D₃×C₉	—	12	3	no	—
61	D₃×C₉	C₂²⋊C₉	1	3	yes	complemented
62	C₂²×C₃²	—	2	1	no	—
63	C₂²×C₃²	—	2	3	no	—
64	C₂²×C₃²	—	9	1	no	—
65	C₂²×C₃²	—	9	1	no	—
66	C₂²×C₃²	C₂×C₃×C₉	1	1	yes	characteristic
67	C₂²×C₃²	C₂×3₋¹⁺²	1	3	yes	—
68	C₂²×C₃²	C₃²×D₃	1	1	yes	characteristic
69	C₂×C₃×D₃	—	3	3	no	—
70	C₂×C₃×D₃	—	3	1	no	—
71	C₂×C₃×D₃	—	3	1	no	—
72	C₂×C₃×D₃	—	3	3	no	—
73	C₂²×C₉	—	2	3	no	—
74	C₂²×C₉	—	9	3	no	—
75	C₂²×C₉	—	9	3	no	—
76	C₂²×C₉	D₃×C₉	1	3	yes	complemented
77	C₂²⋊C₉	—	3	3	no	—
78	C₂²⋊C₉	—	6	3	no	—
79	C₃³	C₂×C₃×A₄	1	1	yes	characteristic
80	C₃×C₉	—	2	1	no	—
81	C₃×C₉	—	2	3	no	—
82	C₃×C₉	—	4	3	no	—
83	C₃×C₉	—	8	3	no	—
84	C₃×C₉	—	12	3	no	—
85	C₃×C₉	—	24	3	no	—
86	C₃×C₉	C₂×C₂²⋊C₉	1	3	yes	complemented
87	C₃×C₉	D₃×A₄	1	1	yes	characteristic
88	C₂³×C₃	—	3	1	no	—
89	C₂³×C₃	—	3	3	no	—
90	C₂²×D₃	C₉⋊C₉	1	1	yes	characteristic, complemented
91	C₂×C₃²	—	3	1	no	—
92	C₂×C₃²	—	3	1	no	—
93	C₂×C₃²	—	3	3	no	—
94	C₂×C₃²	—	3	1	no	—
95	C₂×C₃²	—	6	1	no	—
96	C₂×C₃²	—	6	3	no	—
97	C₂×C₃²	—	9	1	no	—
98	C₃×D₃	—	3	3	no	—
99	C₃×D₃	—	3	1	no	—
100	C₃×D₃	C₉⋊A₄	1	3	yes	—
101	C₃×D₃	C₃×C₂²⋊C₉	1	1	yes	characteristic
102	C₂×C₉	—	3	3	no	—
103	C₂×C₉	—	3	3	no	—
104	C₂×C₉	—	6	3	no	—
105	C₂×C₉	—	9	3	no	—
106	C₂×C₉	—	36	3	no	—
107	C₂²×C₃	—	2	1	no	—
108	C₂²×C₃	—	2	3	no	—
109	C₂²×C₃	—	9	1	no	—
110	C₂²×C₃	—	9	3	no	—
111	C₂²×C₃	—	9	1	no	—
112	C₂²×C₃	—	9	3	no	—
113	C₂²×C₃	C₂×C₉⋊C₉	1	1	yes	characteristic, complemented
114	C₂²×C₃	C₃×D₃×C₉	1	1	yes	characteristic
115	C₂²×C₃	D₃×3₋¹⁺²	1	3	yes	—
116	C₂×D₃	—	3	1	no	—
117	C₂×D₃	—	3	1	no	—
118	C₃²	—	2	3	no	—
119	C₃²	—	2	1	no	—
120	C₃²	C₂×C₉⋊A₄	1	3	yes	—
121	C₃²	C₂×C₃×C₂²⋊C₉	1	1	yes	characteristic
122	C₃²	C₃×D₃×A₄	1	1	yes	characteristic
123	C₉	—	2	3	no	—
124	C₉	—	12	3	no	—
125	C₉	—	24	3	no	—
126	C₉	D₃×C₂²⋊C₉	1	3	yes	complemented
127	C₂³	—	3	1	no	Sylow
128	C₂×C₃	—	3	1	no	—
129	C₂×C₃	—	3	3	no	—
130	C₂×C₃	—	3	3	no	—
131	C₂×C₃	—	3	1	no	—
132	C₂×C₃	—	3	1	no	—
133	C₂×C₃	—	6	3	no	—
134	C₂×C₃	—	6	1	no	—
135	C₂×C₃	—	9	3	no	—
136	C₂×C₃	—	9	1	no	—
137	D₃	—	3	1	no	—
138	D₃	C₉⋊(C₂²⋊C₉)	1	1	yes	characteristic, direct product factor
139	C₂²	—	9	1	no	—
140	C₂²	—	9	1	no	—
141	C₂²	D₃×C₉⋊C₉	1	1	yes	characteristic, complemented
142	C₃	—	2	3	no	—
143	C₃	—	2	1	no	—
144	C₃	C₂×C₉⋊(C₂²⋊C₉)	1	1	yes	characteristic, complemented
145	C₃	D₃×C₉⋊A₄	1	3	yes	—
146	C₃	C₃×D₃×C₂²⋊C₉	1	1	yes	characteristic
147	C₂	—	3	1	no	—
148	C₂	—	3	1	no	—
149	C₂	—	9	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉⋊(C₂²⋊C₉)

C₂

yes

maximal, characteristic, complemented

C₂²×C₃×D₃×C₉

C₃

yes

maximal, characteristic

C₃×D₃×C₂²⋊C₉

C₃

yes

maximal

C₂×C₉⋊(C₂²⋊C₉)

—

maximal

D₃×C₉⋊C₉

—

maximal

C₂²×C₃²×C₉

C₂×C₃

yes

characteristic

C₂×C₃×D₃×C₉

—

C₂×C₃×D₃×C₉

—

C₃²×C₂²⋊C₉

C₂×C₃

yes

—

C₉⋊(C₂²⋊C₉)

—

C₉⋊(C₂²⋊C₉)

—

C₉⋊(C₂²⋊C₉)

D₃

yes

characteristic, direct product factor

C₃×C₉⋊C₉

—

Sylow

C₂²×C₃²×D₃

C₃²

yes

characteristic

C₂³×C₃×C₉

—

C₂²×D₃×C₉

C₉

yes

complemented

C₂×C₃×C₂²⋊C₉

—

D₃×C₂²⋊C₉

—

C₂×C₃²×C₉

—

C₃×D₃×C₉

—

C₃×D₃×C₉

—

C₃×D₃×C₉

A₄

yes

characteristic

C₂×C₉⋊C₉

—

C₂²×C₃³

C₂×C₃²

yes

characteristic

C₂×C₃²×D₃

—

C₂×C₃²×D₃

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×C₉

yes

complemented

C₂²×C₃×C₉

C₃×D₃

yes

characteristic

C₂×D₃×C₉

—

C₂×D₃×C₉

—

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

C₃×D₃

yes

—

C₃²×C₉

—

C₃²×C₉

C₂×A₄

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₂³×C₃²

—

C₂²×C₃×D₃

C₃×C₉

yes

characteristic

C₂²×C₃×D₃

3₋¹⁺²

yes

—

C₂³×C₉

—

C₂×C₂²⋊C₉

—

C₂×C₃³

—

C₃²×D₃

—

C₃²×D₃

C₃×A₄

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

D₃×C₉

—

D₃×C₉

—

D₃×C₉

C₂²⋊C₉

yes

complemented

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₃×C₉

yes

characteristic

C₂²×C₃²

C₂×3₋¹⁺²

yes

—

C₂²×C₃²

C₃²×D₃

yes

characteristic

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

D₃×C₉

yes

complemented

C₂²⋊C₉

—

C₂²⋊C₉

—

C₃³

C₂×C₃×A₄

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₂×C₂²⋊C₉

yes

complemented

C₃×C₉

D₃×A₄

yes

characteristic

C₂³×C₃

—

C₂³×C₃

—

C₂²×D₃

C₉⋊C₉

yes

characteristic, complemented

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₃×D₃

—

C₃×D₃

—

100

C₃×D₃

C₉⋊A₄

yes

—

101

C₃×D₃

C₃×C₂²⋊C₉

yes

characteristic

102

C₂×C₉

—

103

C₂×C₉

—

104

C₂×C₉

—

105

C₂×C₉

—

106

C₂×C₉

—

107

C₂²×C₃

—

108

C₂²×C₃

—

109

C₂²×C₃

—

110

C₂²×C₃

—

111

C₂²×C₃

—

112

C₂²×C₃

—

113

C₂²×C₃

C₂×C₉⋊C₉

yes

characteristic, complemented

114

C₂²×C₃

C₃×D₃×C₉

yes

characteristic

115

C₂²×C₃

D₃×3₋¹⁺²

yes

—

116

C₂×D₃

—

117

C₂×D₃

—

118

C₃²

—

119

C₃²

—

120

C₃²

C₂×C₉⋊A₄

yes

—

121

C₃²

C₂×C₃×C₂²⋊C₉

yes

characteristic

122

C₃²

C₃×D₃×A₄

yes

characteristic

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

D₃×C₂²⋊C₉

yes

complemented

127

C₂³

—

Sylow

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

—

132

C₂×C₃

—

133

C₂×C₃

—

134

C₂×C₃

—

135

C₂×C₃

—

136

C₂×C₃

—

137

D₃

—

138

D₃

C₉⋊(C₂²⋊C₉)

yes

characteristic, direct product factor

139

C₂²

—

140

C₂²

—

141

C₂²

D₃×C₉⋊C₉

yes

characteristic, complemented

142

C₃

—

143

C₃

—

144

C₃

C₂×C₉⋊(C₂²⋊C₉)

yes

characteristic, complemented

145

C₃

D₃×C₉⋊A₄

yes

—

146

C₃

C₃×D₃×C₂²⋊C₉

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

	subgroup	G/H
center, Frattini	C₃²	C₃×D₃×A₄
commutator	C₂²×C₃²	C₂×C₃×C₉
Fitting	C₂²×C₃²×C₉	C₂×C₃
socle, cosocle	C₂²×C₃³	C₂×C₃²
prime radical	C₂²×C₃²×D₃	C₃²
hypercenter	C₃×C₉	D₃×A₄
hypocenter	C₂²×C₃	C₂×C₉⋊C₉
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃
UCS	C₁ ◁ C₃² ◁ C₃×C₉

subgroup

G/H

center, Frattini

C₃²

C₃×D₃×A₄

commutator

C₂²×C₃²

C₂×C₃×C₉

Fitting

C₂²×C₃²×C₉

C₂×C₃

socle, cosocle

C₂²×C₃³

C₂×C₃²

prime radical

C₂²×C₃²×D₃

C₃²

hypercenter

C₃×C₉

D₃×A₄

hypocenter

C₂²×C₃

C₂×C₉⋊C₉

LCS

G ▷ C₂²×C₃² ▷ C₂²×C₃

UCS

C₁ ◁ C₃² ◁ C₃×C₉

N	H ≅ G/N	type
1. C₃×C₉⋊(C₂²⋊C₉)	C₂	semidirect
12. C₉⋊(C₂²⋊C₉)	D₃	direct (N × H)
16. C₂²×D₃×C₉	C₉	semidirect
31. C₂²×C₃×C₉	C₂×C₉	semidirect
61. D₃×C₉	C₂²⋊C₉	semidirect
76. C₂²×C₉	D₃×C₉	semidirect
86. C₃×C₉	C₂×C₂²⋊C₉	semidirect
90. C₂²×D₃	C₉⋊C₉	semidirect
113. C₂²×C₃	C₂×C₉⋊C₉	semidirect
126. C₉	D₃×C₂²⋊C₉	semidirect
138. D₃	C₉⋊(C₂²⋊C₉)	direct (N × H)
141. C₂²	D₃×C₉⋊C₉	semidirect
144. C₃	C₂×C₉⋊(C₂²⋊C₉)	semidirect

H ≅ G/N

type

1. C₃×C₉⋊(C₂²⋊C₉)

C₂

semidirect

12. C₉⋊(C₂²⋊C₉)

D₃

direct (N × H)

16. C₂²×D₃×C₉

C₉

semidirect

31. C₂²×C₃×C₉

C₂×C₉

semidirect

61. D₃×C₉

C₂²⋊C₉

semidirect

76. C₂²×C₉

D₃×C₉

semidirect

86. C₃×C₉

C₂×C₂²⋊C₉

semidirect

90. C₂²×D₃

C₉⋊C₉

semidirect

113. C₂²×C₃

C₂×C₉⋊C₉

semidirect

126. C₉

D₃×C₂²⋊C₉

semidirect

138. D₃

C₉⋊(C₂²⋊C₉)

direct (N × H)

141. C₂²

D₃×C₉⋊C₉

semidirect

144. C₃

C₂×C₉⋊(C₂²⋊C₉)

semidirect

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

subgroup	index
C₃×C₉⋊(C₂²⋊C₉)	2
C₂²×C₃×D₃×C₉	3
C₃×D₃×C₂²⋊C₉	3
C₂×C₉⋊(C₂²⋊C₉)	3
D₃×C₉⋊C₉	4

subgroup

index

C₃×C₉⋊(C₂²⋊C₉)

C₂²×C₃×D₃×C₉

C₃×D₃×C₂²⋊C₉

C₂×C₉⋊(C₂²⋊C₉)

D₃×C₉⋊C₉

Subgroups — D₃×C₉⋊(C₂²⋊C₉) #14518176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D3×C9⋊(C22⋊C9) #14518176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₉⋊(C₂²⋊C₉) #14518176