Group #14517419 — subgroups

Subgroups — C₂×3₋¹⁺²×Dic₉ #14517419

Order 1944.

317 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₂²×C₉×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×Dic₉	C₂	1	2	yes	maximal, direct product factor
3	C₂×Dic₃×3₋¹⁺²	—	3	1	no	maximal
4	C₂×C₃²×Dic₉	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×Dic₉	C₃	1	3	yes	maximal, complemented
6	C₂×C₉×3₋¹⁺²	C₄	1	2	yes	complemented
7	C₂×C₉×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂²×C₃×3₋¹⁺²	D₃	1	1	yes	characteristic
9	C₂²×C₃²×C₉	C₂×C₃	1	1	yes	characteristic
10	Dic₃×3₋¹⁺²	—	3	2	no	—
11	C₃²×Dic₉	C₂×C₃	1	2	yes	—
12	C₂²×C₉⋊C₉	—	2	3	no	—
13	C₂²×C₃²⋊C₉	—	2	1	no	—
14	C₂²×C₉²	C₂×C₃	1	3	yes	—
15	C₉×Dic₉	C₂×C₃	1	6	yes	complemented
16	C₉×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₃²×Dic₃	—	3	1	no	—
18	C₂×C₄×3₋¹⁺²	—	9	1	no	—
19	C₂×C₉×Dic₃	—	3	3	no	—
20	C₂×C₃×Dic₉	—	3	1	no	—
21	C₂×C₃×Dic₉	C₃²	1	1	yes	characteristic
22	C₂×C₃×3₋¹⁺²	Dic₃	1	2	yes	—
23	C₂×C₃×3₋¹⁺²	C₂×D₃	1	1	yes	characteristic
24	C₂×C₃²×C₉	C₃×C₄	1	2	yes	—
25	C₂×C₃²×C₉	C₂²×C₃	1	1	yes	characteristic
26	C₂×C₉⋊C₉	—	2	6	no	—
27	C₂×C₉⋊C₉	—	2	3	no	—
28	C₂×C₃²⋊C₉	—	2	2	no	—
29	C₂×C₃²⋊C₉	—	2	1	no	—
30	C₂×C₉²	C₃×C₄	1	6	yes	complemented
31	C₂×C₉²	C₂²×C₃	1	3	yes	—
32	C₂²×C₃³	C₃×D₃	1	1	yes	characteristic
33	C₃²×Dic₃	—	3	2	no	—
34	C₂²×3₋¹⁺²	—	2	1	no	—
35	C₂²×3₋¹⁺²	—	2	3	no	—
36	C₂²×3₋¹⁺²	D₉	1	1	yes	characteristic
37	C₂²×C₃×C₉	—	2	3	no	—
38	C₂²×C₃×C₉	—	2	1	no	—
39	C₂²×C₃×C₉	—	3	1	no	—
40	C₂²×C₃×C₉	—	6	1	no	—
41	C₂²×C₃×C₉	C₃×D₃	1	3	yes	—
42	C₂²×C₃×C₉	C₂×C₃²	1	1	yes	characteristic
43	C₄×3₋¹⁺²	—	9	2	no	—
44	C₉×Dic₃	—	3	6	no	—
45	C₃×Dic₉	—	3	2	no	—
46	C₃×Dic₉	C₂×C₃²	1	2	yes	—
47	C₃×3₋¹⁺²	C₂×Dic₃	1	1	yes	characteristic
48	C₃²×C₉	C₂×C₃×C₄	1	1	yes	characteristic
49	C₉⋊C₉	—	2	3	no	—
50	C₃²⋊C₉	—	2	1	no	—
51	C₉²	C₂×C₃×C₄	1	3	yes	complemented
52	C₂×C₃²×C₄	—	9	1	no	—
53	C₂×C₃×Dic₃	—	3	1	no	—
54	C₂×C₃×Dic₃	—	9	1	no	—
55	C₂×C₄×C₉	—	9	3	no	—
56	C₂×Dic₉	3₋¹⁺²	1	1	yes	characteristic, direct product factor
57	C₂×C₃³	C₃×Dic₃	1	2	yes	—
58	C₂×C₃³	C₂×C₃×D₃	1	1	yes	characteristic
59	C₂×3₋¹⁺²	—	2	2	no	—
60	C₂×3₋¹⁺²	—	2	1	no	—
61	C₂×3₋¹⁺²	—	2	6	no	—
62	C₂×3₋¹⁺²	—	2	3	no	—
63	C₂×3₋¹⁺²	Dic₉	1	2	yes	direct product factor
64	C₂×3₋¹⁺²	C₂×D₉	1	1	yes	characteristic
65	C₂×C₃×C₉	—	2	6	no	—
66	C₂×C₃×C₉	—	2	2	no	—
67	C₂×C₃×C₉	—	2	3	no	—
68	C₂×C₃×C₉	—	2	1	no	—
69	C₂×C₃×C₉	—	3	2	no	—
70	C₂×C₃×C₉	—	3	1	no	—
71	C₂×C₃×C₉	—	6	2	no	—
72	C₂×C₃×C₉	—	6	1	no	—
73	C₂×C₃×C₉	C₃×Dic₃	1	6	yes	—
74	C₂×C₃×C₉	C₃²×C₄	1	2	yes	—
75	C₂×C₃×C₉	C₂×C₃×D₃	1	3	yes	—
76	C₂×C₃×C₉	C₂²×C₃²	1	1	yes	characteristic
77	C₂²×C₃²	—	2	1	no	—
78	C₂²×C₃²	—	3	1	no	—
79	C₂²×C₃²	—	6	1	no	—
80	C₂²×C₃²	C₃×D₉	1	1	yes	characteristic
81	C₂²×C₃²	C₃²×D₃	1	1	yes	characteristic
82	C₃²×C₄	—	9	2	no	—
83	C₃×Dic₃	—	3	2	no	—
84	C₃×Dic₃	—	9	2	no	—
85	C₂²×C₉	—	2	3	no	—
86	C₂²×C₉	—	2	1	no	—
87	C₂²×C₉	—	6	3	no	—
88	C₂²×C₉	—	6	1	no	—
89	C₂²×C₉	C₃×D₉	1	3	yes	—
90	C₂²×C₉	C₂×3₋¹⁺²	1	1	yes	characteristic
91	C₄×C₉	—	9	6	no	—
92	Dic₉	C₂×3₋¹⁺²	1	2	yes	direct product factor
93	C₃³	C₂×C₃×Dic₃	1	1	yes	characteristic
94	3₋¹⁺²	—	2	1	no	—
95	3₋¹⁺²	—	2	3	no	—
96	3₋¹⁺²	C₂×Dic₉	1	1	yes	characteristic, direct product factor
97	C₃×C₉	—	2	3	no	—
98	C₃×C₉	—	2	1	no	—
99	C₃×C₉	—	3	1	no	—
100	C₃×C₉	—	6	1	no	—
101	C₃×C₉	C₂×C₃×Dic₃	1	3	yes	—
102	C₃×C₉	C₂×C₃²×C₄	1	1	yes	characteristic
103	C₂×C₃×C₄	—	9	1	no	—
104	C₂×C₃×C₄	—	27	1	no	—
105	C₂×Dic₃	—	3	1	no	—
106	C₂×C₃²	—	2	2	no	—
107	C₂×C₃²	—	2	1	no	—
108	C₂×C₃²	—	3	2	no	—
109	C₂×C₃²	—	3	1	no	—
110	C₂×C₃²	—	6	2	no	—
111	C₂×C₃²	—	6	1	no	—
112	C₂×C₃²	C₃×Dic₉	1	2	yes	—
113	C₂×C₃²	C₂×C₃×D₉	1	1	yes	characteristic
114	C₂×C₃²	C₃²×Dic₃	1	2	yes	—
115	C₂×C₃²	C₂×C₃²×D₃	1	1	yes	characteristic
116	C₂×C₉	—	2	6	no	—
117	C₂×C₉	—	2	2	no	—
118	C₂×C₉	—	2	3	no	—
119	C₂×C₉	—	2	1	no	—
120	C₂×C₉	—	6	3	no	—
121	C₂×C₉	—	6	6	no	—
122	C₂×C₉	—	6	2	no	—
123	C₂×C₉	—	6	1	no	—
124	C₂×C₉	C₃×Dic₉	1	6	yes	complemented
125	C₂×C₉	C₄×3₋¹⁺²	1	2	yes	complemented
126	C₂×C₉	C₂×C₃×D₉	1	3	yes	—
127	C₂×C₉	C₂²×3₋¹⁺²	1	1	yes	characteristic
128	C₂²×C₃	—	2	1	no	—
129	C₂²×C₃	—	3	1	no	—
130	C₂²×C₃	—	6	1	no	—
131	C₂²×C₃	C₃²×D₉	1	1	yes	characteristic
132	C₂²×C₃	D₃×3₋¹⁺²	1	1	yes	characteristic
133	C₃×C₄	—	9	2	no	—
134	C₃×C₄	—	27	2	no	—
135	Dic₃	—	3	2	no	—
136	C₃²	—	2	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	6	1	no	—
139	C₃²	C₂×C₃×Dic₉	1	1	yes	characteristic
140	C₃²	C₂×C₃²×Dic₃	1	1	yes	characteristic
141	C₉	—	2	3	no	—
142	C₉	—	2	1	no	—
143	C₉	—	6	3	no	—
144	C₉	—	6	1	no	—
145	C₉	C₂×C₃×Dic₉	1	3	yes	complemented
146	C₉	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic, complemented
147	C₂×C₄	—	9	1	no	Sylow
148	C₂×C₃	—	2	1	no	—
149	C₂×C₃	—	2	2	no	—
150	C₂×C₃	—	3	1	no	—
151	C₂×C₃	—	3	2	no	—
152	C₂×C₃	—	6	1	no	—
153	C₂×C₃	—	6	2	no	—
154	C₂×C₃	C₃²×Dic₉	1	2	yes	—
155	C₂×C₃	Dic₃×3₋¹⁺²	1	2	yes	—
156	C₂×C₃	C₂×C₃²×D₉	1	1	yes	characteristic
157	C₂×C₃	C₂×D₃×3₋¹⁺²	1	1	yes	characteristic
158	C₂²	D₉×3₋¹⁺²	1	1	yes	characteristic
159	C₄	—	9	2	no	—
160	C₃	—	2	1	no	—
161	C₃	—	3	1	no	—
162	C₃	—	6	1	no	—
163	C₃	C₂×C₃²×Dic₉	1	1	yes	characteristic
164	C₃	C₂×Dic₃×3₋¹⁺²	1	1	yes	characteristic
165	C₂	3₋¹⁺²×Dic₉	1	2	yes	direct product factor
166	C₂	C₂×D₉×3₋¹⁺²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₉×3₋¹⁺²

C₂

yes

maximal, characteristic

3₋¹⁺²×Dic₉

C₂

yes

maximal, direct product factor

C₂×Dic₃×3₋¹⁺²

—

maximal

C₂×C₃²×Dic₉

C₃

yes

maximal, characteristic

C₂×C₉×Dic₉

C₃

yes

maximal, complemented

C₂×C₉×3₋¹⁺²

C₄

yes

complemented

C₂×C₉×3₋¹⁺²

C₂²

yes

characteristic

C₂²×C₃×3₋¹⁺²

D₃

yes

characteristic

C₂²×C₃²×C₉

C₂×C₃

yes

characteristic

Dic₃×3₋¹⁺²

—

C₃²×Dic₉

C₂×C₃

yes

—

C₂²×C₉⋊C₉

—

C₂²×C₃²⋊C₉

—

C₂²×C₉²

C₂×C₃

yes

—

C₉×Dic₉

C₂×C₃

yes

complemented

C₉×3₋¹⁺²

C₂×C₄

yes

characteristic, complemented, Sylow

C₂×C₃²×Dic₃

—

C₂×C₄×3₋¹⁺²

—

C₂×C₉×Dic₃

—

C₂×C₃×Dic₉

—

C₂×C₃×Dic₉

C₃²

yes

characteristic

C₂×C₃×3₋¹⁺²

Dic₃

yes

—

C₂×C₃×3₋¹⁺²

C₂×D₃

yes

characteristic

C₂×C₃²×C₉

C₃×C₄

yes

—

C₂×C₃²×C₉

C₂²×C₃

yes

characteristic

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₉²

C₃×C₄

yes

complemented

C₂×C₉²

C₂²×C₃

yes

—

C₂²×C₃³

C₃×D₃

yes

characteristic

C₃²×Dic₃

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

D₉

yes

characteristic

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₃×D₃

yes

—

C₂²×C₃×C₉

C₂×C₃²

yes

characteristic

C₄×3₋¹⁺²

—

C₉×Dic₃

—

C₃×Dic₉

—

C₃×Dic₉

C₂×C₃²

yes

—

C₃×3₋¹⁺²

C₂×Dic₃

yes

characteristic

C₃²×C₉

C₂×C₃×C₄

yes

characteristic

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

C₂×C₃×C₄

yes

complemented

C₂×C₃²×C₄

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂×C₄×C₉

—

C₂×Dic₉

3₋¹⁺²

yes

characteristic, direct product factor

C₂×C₃³

C₃×Dic₃

yes

—

C₂×C₃³

C₂×C₃×D₃

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

Dic₉

yes

direct product factor

C₂×3₋¹⁺²

C₂×D₉

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×Dic₃

yes

—

C₂×C₃×C₉

C₃²×C₄

yes

—

C₂×C₃×C₉

C₂×C₃×D₃

yes

—

C₂×C₃×C₉

C₂²×C₃²

yes

characteristic

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃×D₉

yes

characteristic

C₂²×C₃²

C₃²×D₃

yes

characteristic

C₃²×C₄

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×D₉

yes

—

C₂²×C₉

C₂×3₋¹⁺²

yes

characteristic

C₄×C₉

—

Dic₉

C₂×3₋¹⁺²

yes

direct product factor

C₃³

C₂×C₃×Dic₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₂×Dic₉

yes

characteristic, direct product factor

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

C₂×C₃×Dic₃

yes

—

102

C₃×C₉

C₂×C₃²×C₄

yes

characteristic

103

C₂×C₃×C₄

—

104

C₂×C₃×C₄

—

105

C₂×Dic₃

—

106

C₂×C₃²

—

107

C₂×C₃²

—

108

C₂×C₃²

—

109

C₂×C₃²

—

110

C₂×C₃²

—

111

C₂×C₃²

—

112

C₂×C₃²

C₃×Dic₉

yes

—

113

C₂×C₃²

C₂×C₃×D₉

yes

characteristic

114

C₂×C₃²

C₃²×Dic₃

yes

—

115

C₂×C₃²

C₂×C₃²×D₃

yes

characteristic

116

C₂×C₉

—

117

C₂×C₉

—

118

C₂×C₉

—

119

C₂×C₉

—

120

C₂×C₉

—

121

C₂×C₉

—

122

C₂×C₉

—

123

C₂×C₉

—

124

C₂×C₉

C₃×Dic₉

yes

complemented

125

C₂×C₉

C₄×3₋¹⁺²

yes

complemented

126

C₂×C₉

C₂×C₃×D₉

yes

—

127

C₂×C₉

C₂²×3₋¹⁺²

yes

characteristic

128

C₂²×C₃

—

129

C₂²×C₃

—

130

C₂²×C₃

—

131

C₂²×C₃

C₃²×D₉

yes

characteristic

132

C₂²×C₃

D₃×3₋¹⁺²

yes

characteristic

133

C₃×C₄

—

134

C₃×C₄

—

135

Dic₃

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

C₂×C₃×Dic₉

yes

characteristic

140

C₃²

C₂×C₃²×Dic₃

yes

characteristic

141

C₉

—

142

C₉

—

143

C₉

—

144

C₉

—

145

C₉

C₂×C₃×Dic₉

yes

complemented

146

C₉

C₂×C₄×3₋¹⁺²

yes

characteristic, complemented

147

C₂×C₄

—

Sylow

148

C₂×C₃

—

149

C₂×C₃

—

150

C₂×C₃

—

151

C₂×C₃

—

152

C₂×C₃

—

153

C₂×C₃

—

154

C₂×C₃

C₃²×Dic₉

yes

—

155

C₂×C₃

Dic₃×3₋¹⁺²

yes

—

156

C₂×C₃

C₂×C₃²×D₉

yes

characteristic

157

C₂×C₃

C₂×D₃×3₋¹⁺²

yes

characteristic

158

C₂²

D₉×3₋¹⁺²

yes

characteristic

159

C₄

—

160

C₃

—

161

C₃

—

162

C₃

—

163

C₃

C₂×C₃²×Dic₉

yes

characteristic

164

C₃

C₂×Dic₃×3₋¹⁺²

yes

characteristic

165

C₂

3₋¹⁺²×Dic₉

yes

direct product factor

166

C₂

C₂×D₉×3₋¹⁺²

yes

characteristic

	subgroup	G/H
center	C₂²×C₃	C₃²×D₉
commutator	C₃×C₉	C₂×C₃²×C₄
Frattini	C₂×C₃²	C₂×C₃²×D₃
Fitting	C₂²×C₉×3₋¹⁺²	C₂
socle	C₂²×C₃²	C₃²×D₃
prime radical	C₂²×C₃³	C₃×D₃
cosocle	C₂×C₃×C₉	C₂²×C₃²
hypercenter	C₂²×3₋¹⁺²	D₉
hypocenter	C₉	C₂×C₄×3₋¹⁺²
LCS	G ▷ C₃×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×3₋¹⁺²

subgroup

G/H

center

C₂²×C₃

C₃²×D₉

commutator

C₃×C₉

C₂×C₃²×C₄

Frattini

C₂×C₃²

C₂×C₃²×D₃

Fitting

C₂²×C₉×3₋¹⁺²

C₂

socle

C₂²×C₃²

C₃²×D₃

prime radical

C₂²×C₃³

C₃×D₃

cosocle

C₂×C₃×C₉

C₂²×C₃²

hypercenter

C₂²×3₋¹⁺²

D₉

hypocenter

C₉

C₂×C₄×3₋¹⁺²

LCS

G ▷ C₃×C₉ ▷ C₉

UCS

C₁ ◁ C₂²×C₃ ◁ C₂²×3₋¹⁺²

N	H ≅ G/N	type
2. 3₋¹⁺²×Dic₉	C₂	direct (N × H)
5. C₂×C₉×Dic₉	C₃	semidirect
6. C₂×C₉×3₋¹⁺²	C₄	semidirect
15. C₉×Dic₉	C₂×C₃	semidirect
16. C₉×3₋¹⁺²	C₂×C₄	semidirect
30. C₂×C₉²	C₃×C₄	semidirect
51. C₉²	C₂×C₃×C₄	semidirect
56. C₂×Dic₉	3₋¹⁺²	direct (N × H)
63. C₂×3₋¹⁺²	Dic₉	direct (N × H)
92. Dic₉	C₂×3₋¹⁺²	direct (N × H)
96. 3₋¹⁺²	C₂×Dic₉	direct (N × H)
124. C₂×C₉	C₃×Dic₉	semidirect
125. C₂×C₉	C₄×3₋¹⁺²	semidirect
145. C₉	C₂×C₃×Dic₉	semidirect
146. C₉	C₂×C₄×3₋¹⁺²	semidirect
165. C₂	3₋¹⁺²×Dic₉	direct (N × H)

H ≅ G/N

type

2. 3₋¹⁺²×Dic₉

C₂

direct (N × H)

5. C₂×C₉×Dic₉

C₃

semidirect

6. C₂×C₉×3₋¹⁺²

C₄

semidirect

15. C₉×Dic₉

C₂×C₃

semidirect

16. C₉×3₋¹⁺²

C₂×C₄

semidirect

30. C₂×C₉²

C₃×C₄

semidirect

51. C₉²

C₂×C₃×C₄

semidirect

56. C₂×Dic₉

3₋¹⁺²

direct (N × H)

63. C₂×3₋¹⁺²

Dic₉

direct (N × H)

92. Dic₉

C₂×3₋¹⁺²

direct (N × H)

96. 3₋¹⁺²

C₂×Dic₉

direct (N × H)

124. C₂×C₉

C₃×Dic₉

semidirect

125. C₂×C₉

C₄×3₋¹⁺²

semidirect

145. C₉

C₂×C₃×Dic₉

semidirect

146. C₉

C₂×C₄×3₋¹⁺²

semidirect

165. C₂

3₋¹⁺²×Dic₉

direct (N × H)

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

subgroup	index
C₂²×C₉×3₋¹⁺²	2
3₋¹⁺²×Dic₉	2
C₂×Dic₃×3₋¹⁺²	3
C₂×C₃²×Dic₉	3
C₂×C₉×Dic₉	3

subgroup

index

C₂²×C₉×3₋¹⁺²

3₋¹⁺²×Dic₉

C₂×Dic₃×3₋¹⁺²

C₂×C₃²×Dic₉

C₂×C₉×Dic₉

Subgroups — C₂×3₋¹⁺²×Dic₉ #14517419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×3−1+2×Dic9 #14517419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×3₋¹⁺²×Dic₉ #14517419