Group #14518428 — subgroups

Subgroups — C₂×3₋¹⁺²×C₂²⋊C₉ #14518428

Order 1944.

448 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 178 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	3₋¹⁺²×C₂²⋊C₉	C₂	1	1	yes	maximal, characteristic, direct product factor
2	C₂³×C₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₂×C₃²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal
5	C₂×C₉⋊(C₂²⋊C₉)	C₃	1	6	yes	maximal, complemented
6	C₂×C₉×C₂²⋊C₉	C₃	1	3	yes	maximal, complemented
7	C₂×C₉×3₋¹⁺²	—	4	1	no	maximal
8	C₂²×C₃×3₋¹⁺²	—	3	1	no	—
9	C₂²×C₃×3₋¹⁺²	—	3	1	no	—
10	C₂²×C₃×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic
11	C₃²×C₂²⋊C₉	C₂×C₃	1	1	yes	characteristic
12	C₃²⋊(C₂²⋊C₉)	C₂×C₃	1	2	yes	—
13	C₉⋊(C₂²⋊C₉)	C₂×C₃	1	6	yes	complemented
14	C₉×C₂²⋊C₉	C₂×C₃	1	3	yes	complemented
15	C₉×3₋¹⁺²	—	4	1	no	Sylow
16	C₂³×C₃³	C₃²	1	1	yes	characteristic
17	C₂³×3₋¹⁺²	C₉	1	9	yes	complemented
18	C₂³×C₃×C₉	C₃²	1	3	yes	—
19	C₂×C₃×C₂²⋊C₉	—	3	3	no	—
20	C₂×C₃×C₂²⋊C₉	C₃²	1	6	yes	—
21	C₂×C₃×C₂²⋊C₉	C₃²	1	2	yes	—
22	C₂×C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
23	C₂×C₃×3₋¹⁺²	—	3	1	no	—
24	C₂×C₃×3₋¹⁺²	—	3	1	no	—
25	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
26	C₂×C₃²×C₉	—	4	1	no	—
27	C₂×C₉⋊C₉	—	4	6	no	—
28	C₂×C₃²⋊C₉	—	4	2	no	—
29	C₂×C₉²	—	4	3	no	—
30	C₂²×C₃³	—	3	1	no	—
31	C₂²×C₃³	—	3	1	no	—
32	C₂²×C₃³	C₂×C₃²	1	1	yes	characteristic
33	C₂²×3₋¹⁺²	—	3	9	no	—
34	C₂²×3₋¹⁺²	—	3	9	no	—
35	C₂²×3₋¹⁺²	C₂×C₉	1	9	yes	complemented
36	C₂²×C₃×C₉	—	3	3	no	—
37	C₂²×C₃×C₉	—	3	3	no	—
38	C₂²×C₃×C₉	C₂×C₃²	1	3	yes	—
39	C₃×C₂²⋊C₉	—	3	3	no	—
40	C₃×C₂²⋊C₉	C₂×C₃²	1	6	yes	—
41	C₃×C₂²⋊C₉	C₂×C₃²	1	2	yes	—
42	C₃×C₂²⋊C₉	C₂×C₃²	1	1	yes	characteristic
43	C₃×3₋¹⁺²	C₂×A₄	1	1	yes	characteristic
44	C₃²×C₉	—	4	1	no	—
45	C₉⋊C₉	—	4	6	no	—
46	C₃²⋊C₉	—	4	2	no	—
47	C₉²	—	4	3	no	—
48	C₂³×C₃²	—	3	1	no	—
49	C₂³×C₃²	—	3	2	no	—
50	C₂³×C₃²	C₃×C₉	1	3	yes	—
51	C₂³×C₃²	C₃³	1	1	yes	characteristic
52	C₂³×C₉	C₃×C₉	1	9	yes	complemented
53	C₂×C₂²⋊C₉	—	3	2	no	—
54	C₂×C₂²⋊C₉	—	3	6	no	—
55	C₂×C₂²⋊C₉	3₋¹⁺²	1	3	yes	direct product factor
56	C₂×C₃³	—	3	1	no	—
57	C₂×C₃³	—	3	1	no	—
58	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
59	C₂×3₋¹⁺²	—	3	9	no	—
60	C₂×3₋¹⁺²	—	3	9	no	—
61	C₂×3₋¹⁺²	C₂²⋊C₉	1	9	yes	direct product factor
62	C₂×C₃×C₉	—	3	3	no	—
63	C₂×C₃×C₉	—	3	3	no	—
64	C₂×C₃×C₉	—	4	1	no	—
65	C₂×C₃×C₉	—	4	6	no	—
66	C₂×C₃×C₉	—	4	2	no	—
67	C₂×C₃×C₉	—	12	3	no	—
68	C₂×C₃×C₉	C₃×A₄	1	3	yes	—
69	C₂²×C₃²	—	3	1	no	—
70	C₂²×C₃²	—	3	1	no	—
71	C₂²×C₃²	—	3	1	no	—
72	C₂²×C₃²	—	3	2	no	—
73	C₂²×C₃²	—	3	3	no	—
74	C₂²×C₃²	—	3	3	no	—
75	C₂²×C₃²	—	9	1	no	—
76	C₂²×C₃²	—	9	2	no	—
77	C₂²×C₃²	—	9	1	no	—
78	C₂²×C₃²	—	9	2	no	—
79	C₂²×C₃²	C₂×C₃×C₉	1	3	yes	—
80	C₂²×C₃²	C₂×C₃³	1	1	yes	characteristic
81	C₂²×C₉	—	3	9	no	—
82	C₂²×C₉	—	3	9	no	—
83	C₂²×C₉	C₂×C₃×C₉	1	9	yes	complemented
84	C₂²⋊C₉	—	3	2	no	—
85	C₂²⋊C₉	—	3	6	no	—
86	C₂²⋊C₉	C₂×3₋¹⁺²	1	3	yes	direct product factor
87	C₃³	C₂×C₃×A₄	1	1	yes	characteristic
88	3₋¹⁺²	C₂×C₂²⋊C₉	1	9	yes	direct product factor
89	C₃×C₉	—	4	1	no	—
90	C₃×C₉	—	4	6	no	—
91	C₃×C₉	—	4	2	no	—
92	C₃×C₉	—	12	3	no	—
93	C₃×C₉	C₂×C₃×A₄	1	3	yes	—
94	C₂³×C₃	—	3	3	no	—
95	C₂³×C₃	C₃²×C₉	1	1	yes	characteristic
96	C₂³×C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
97	C₂³×C₃	#737	1	2	yes	—
98	C₂×C₃²	—	3	1	no	—
99	C₂×C₃²	—	3	1	no	—
100	C₂×C₃²	—	3	1	no	—
101	C₂×C₃²	—	3	2	no	—
102	C₂×C₃²	—	3	3	no	—
103	C₂×C₃²	—	3	3	no	—
104	C₂×C₃²	—	9	1	no	—
105	C₂×C₃²	—	9	2	no	—
106	C₂×C₃²	—	9	1	no	—
107	C₂×C₃²	—	9	2	no	—
108	C₂×C₃²	C₃×C₂²⋊C₉	1	3	yes	—
109	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
110	C₂×C₉	—	3	9	no	—
111	C₂×C₉	—	3	9	no	—
112	C₂×C₉	—	4	3	no	—
113	C₂×C₉	—	12	2	no	—
114	C₂×C₉	—	12	6	no	—
115	C₂×C₉	C₃×C₂²⋊C₉	1	9	yes	complemented
116	C₂²×C₃	—	3	1	no	—
117	C₂²×C₃	—	3	1	no	—
118	C₂²×C₃	—	3	2	no	—
119	C₂²×C₃	—	3	1	no	—
120	C₂²×C₃	—	3	2	no	—
121	C₂²×C₃	—	3	1	no	—
122	C₂²×C₃	—	3	3	no	—
123	C₂²×C₃	—	9	3	no	—
124	C₂²×C₃	—	9	3	no	—
125	C₂²×C₃	C₂×C₃²×C₉	1	1	yes	characteristic
126	C₂²×C₃	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic
127	C₂²×C₃	C₂×#737	1	2	yes	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	2	no	—
130	C₃²	C₂×C₃×C₂²⋊C₉	1	3	yes	—
131	C₃²	C₂×C₃²×A₄	1	1	yes	characteristic
132	C₉	—	4	3	no	—
133	C₉	—	12	2	no	—
134	C₉	—	12	6	no	—
135	C₉	C₂×C₃×C₂²⋊C₉	1	9	yes	complemented
136	C₂³	C₉×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
137	C₂×C₃	—	3	1	no	—
138	C₂×C₃	—	3	1	no	—
139	C₂×C₃	—	3	2	no	—
140	C₂×C₃	—	3	2	no	—
141	C₂×C₃	—	3	1	no	—
142	C₂×C₃	—	3	1	no	—
143	C₂×C₃	—	3	3	no	—
144	C₂×C₃	—	9	3	no	—
145	C₂×C₃	—	9	3	no	—
146	C₂×C₃	A₄×3₋¹⁺²	1	1	yes	characteristic
147	C₂×C₃	#66440	1	2	yes	—
148	C₂×C₃	C₃²×C₂²⋊C₉	1	1	yes	characteristic
149	C₂²	—	3	1	no	—
150	C₂²	—	3	1	no	—
151	C₂²	C₂×C₉×3₋¹⁺²	1	1	yes	characteristic, complemented
152	C₃	—	3	3	no	—
153	C₃	C₂×A₄×3₋¹⁺²	1	1	yes	characteristic
154	C₃	C₂×#66440	1	2	yes	—
155	C₃	C₂×C₃²×C₂²⋊C₉	1	1	yes	characteristic
156	C₂	—	3	1	no	—
157	C₂	—	3	1	no	—
158	C₂	3₋¹⁺²×C₂²⋊C₉	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×C₂²⋊C₉

C₂

yes

maximal, characteristic, direct product factor

C₂³×C₃×3₋¹⁺²

C₃

yes

maximal, characteristic

C₂×C₃²×C₂²⋊C₉

C₃

yes

maximal, characteristic

C₂×C₃²⋊(C₂²⋊C₉)

C₃

yes

maximal

C₂×C₉⋊(C₂²⋊C₉)

C₃

yes

maximal, complemented

C₂×C₉×C₂²⋊C₉

C₃

yes

maximal, complemented

C₂×C₉×3₋¹⁺²

—

maximal

C₂²×C₃×3₋¹⁺²

—

C₂²×C₃×3₋¹⁺²

—

C₂²×C₃×3₋¹⁺²

C₂×C₃

yes

characteristic

C₃²×C₂²⋊C₉

C₂×C₃

yes

characteristic

C₃²⋊(C₂²⋊C₉)

C₂×C₃

yes

—

C₉⋊(C₂²⋊C₉)

C₂×C₃

yes

complemented

C₉×C₂²⋊C₉

C₂×C₃

yes

complemented

C₉×3₋¹⁺²

—

Sylow

C₂³×C₃³

C₃²

yes

characteristic

C₂³×3₋¹⁺²

C₉

yes

complemented

C₂³×C₃×C₉

C₃²

yes

—

C₂×C₃×C₂²⋊C₉

—

C₂×C₃×C₂²⋊C₉

C₃²

yes

—

C₂×C₃×C₂²⋊C₉

C₃²

yes

—

C₂×C₃×C₂²⋊C₉

C₃²

yes

characteristic

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

A₄

yes

characteristic

C₂×C₃²×C₉

—

C₂×C₉⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₉²

—

C₂²×C₃³

—

C₂²×C₃³

—

C₂²×C₃³

C₂×C₃²

yes

characteristic

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₂×C₉

yes

complemented

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×C₃²

yes

—

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

C₂×C₃²

yes

—

C₃×C₂²⋊C₉

C₂×C₃²

yes

—

C₃×C₂²⋊C₉

C₂×C₃²

yes

characteristic

C₃×3₋¹⁺²

C₂×A₄

yes

characteristic

C₃²×C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₂³×C₃²

—

C₂³×C₃²

—

C₂³×C₃²

C₃×C₉

yes

—

C₂³×C₃²

C₃³

yes

characteristic

C₂³×C₉

C₃×C₉

yes

complemented

C₂×C₂²⋊C₉

—

C₂×C₂²⋊C₉

—

C₂×C₂²⋊C₉

3₋¹⁺²

yes

direct product factor

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₃×A₄

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂²⋊C₉

yes

direct product factor

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×A₄

yes

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₃×C₉

yes

—

C₂²×C₃²

C₂×C₃³

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×C₉

yes

complemented

C₂²⋊C₉

—

C₂²⋊C₉

—

C₂²⋊C₉

C₂×3₋¹⁺²

yes

direct product factor

C₃³

C₂×C₃×A₄

yes

characteristic

3₋¹⁺²

C₂×C₂²⋊C₉

yes

direct product factor

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₂×C₃×A₄

yes

—

C₂³×C₃

—

C₂³×C₃

C₃²×C₉

yes

characteristic

C₂³×C₃

C₃×3₋¹⁺²

yes

characteristic

C₂³×C₃

#737

yes

—

C₂×C₃²

—

C₂×C₃²

—

100

C₂×C₃²

—

101

C₂×C₃²

—

102

C₂×C₃²

—

103

C₂×C₃²

—

104

C₂×C₃²

—

105

C₂×C₃²

—

106

C₂×C₃²

—

107

C₂×C₃²

—

108

C₂×C₃²

C₃×C₂²⋊C₉

yes

—

109

C₂×C₃²

C₃²×A₄

yes

characteristic

110

C₂×C₉

—

111

C₂×C₉

—

112

C₂×C₉

—

113

C₂×C₉

—

114

C₂×C₉

—

115

C₂×C₉

C₃×C₂²⋊C₉

yes

complemented

116

C₂²×C₃

—

117

C₂²×C₃

—

118

C₂²×C₃

—

119

C₂²×C₃

—

120

C₂²×C₃

—

121

C₂²×C₃

—

122

C₂²×C₃

—

123

C₂²×C₃

—

124

C₂²×C₃

—

125

C₂²×C₃

C₂×C₃²×C₉

yes

characteristic

126

C₂²×C₃

C₂×C₃×3₋¹⁺²

yes

characteristic

127

C₂²×C₃

C₂×#737

yes

—

128

C₃²

—

129

C₃²

—

130

C₃²

C₂×C₃×C₂²⋊C₉

yes

—

131

C₃²

C₂×C₃²×A₄

yes

characteristic

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

C₂×C₃×C₂²⋊C₉

yes

complemented

136

C₂³

C₉×3₋¹⁺²

yes

characteristic, complemented, Sylow

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

—

142

C₂×C₃

—

143

C₂×C₃

—

144

C₂×C₃

—

145

C₂×C₃

—

146

C₂×C₃

A₄×3₋¹⁺²

yes

characteristic

147

C₂×C₃

#66440

yes

—

148

C₂×C₃

C₃²×C₂²⋊C₉

yes

characteristic

149

C₂²

—

150

C₂²

—

151

C₂²

C₂×C₉×3₋¹⁺²

yes

characteristic, complemented

152

C₃

—

153

C₃

C₂×A₄×3₋¹⁺²

yes

characteristic

154

C₃

C₂×#66440

yes

—

155

C₃

C₂×C₃²×C₂²⋊C₉

yes

characteristic

156

C₂

—

157

C₂

—

158

C₂

3₋¹⁺²×C₂²⋊C₉

yes

characteristic, direct product factor

	subgroup	G/H
center	C₂×C₃²	C₃²×A₄
commutator	C₂²×C₃	C₂×C₃²×C₉
Frattini	C₃²	C₂×C₃²×A₄
Fitting	C₂³×C₃×3₋¹⁺²	C₃
socle	C₂³×C₃²	C₃³
prime radical	C₂³×C₃³	C₃²
cosocle	C₂²×C₃²	C₂×C₃³
hypercenter	C₂×C₃×3₋¹⁺²	A₄
hypocenter	C₂²	C₂×C₉×3₋¹⁺²
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃×3₋¹⁺²

subgroup

G/H

center

C₂×C₃²

C₃²×A₄

commutator

C₂²×C₃

C₂×C₃²×C₉

Frattini

C₃²

C₂×C₃²×A₄

Fitting

C₂³×C₃×3₋¹⁺²

C₃

socle

C₂³×C₃²

C₃³

prime radical

C₂³×C₃³

C₃²

cosocle

C₂²×C₃²

C₂×C₃³

hypercenter

C₂×C₃×3₋¹⁺²

A₄

hypocenter

C₂²

C₂×C₉×3₋¹⁺²

LCS

G ▷ C₂²×C₃ ▷ C₂²

UCS

C₁ ◁ C₂×C₃² ◁ C₂×C₃×3₋¹⁺²

N	H ≅ G/N	type
1. 3₋¹⁺²×C₂²⋊C₉	C₂	direct (N × H)
5. C₂×C₉⋊(C₂²⋊C₉)	C₃	semidirect
6. C₂×C₉×C₂²⋊C₉	C₃	semidirect
13. C₉⋊(C₂²⋊C₉)	C₂×C₃	semidirect
14. C₉×C₂²⋊C₉	C₂×C₃	semidirect
17. C₂³×3₋¹⁺²	C₉	semidirect
35. C₂²×3₋¹⁺²	C₂×C₉	semidirect
52. C₂³×C₉	C₃×C₉	semidirect
55. C₂×C₂²⋊C₉	3₋¹⁺²	direct (N × H)
61. C₂×3₋¹⁺²	C₂²⋊C₉	direct (N × H)
83. C₂²×C₉	C₂×C₃×C₉	semidirect
86. C₂²⋊C₉	C₂×3₋¹⁺²	direct (N × H)
88. 3₋¹⁺²	C₂×C₂²⋊C₉	direct (N × H)
115. C₂×C₉	C₃×C₂²⋊C₉	semidirect
135. C₉	C₂×C₃×C₂²⋊C₉	semidirect
136. C₂³	C₉×3₋¹⁺²	semidirect
151. C₂²	C₂×C₉×3₋¹⁺²	semidirect
158. C₂	3₋¹⁺²×C₂²⋊C₉	direct (N × H)

H ≅ G/N

type

1. 3₋¹⁺²×C₂²⋊C₉

C₂

direct (N × H)

5. C₂×C₉⋊(C₂²⋊C₉)

C₃

semidirect

6. C₂×C₉×C₂²⋊C₉

C₃

semidirect

13. C₉⋊(C₂²⋊C₉)

C₂×C₃

semidirect

14. C₉×C₂²⋊C₉

C₂×C₃

semidirect

17. C₂³×3₋¹⁺²

C₉

semidirect

35. C₂²×3₋¹⁺²

C₂×C₉

semidirect

52. C₂³×C₉

C₃×C₉

semidirect

55. C₂×C₂²⋊C₉

3₋¹⁺²

direct (N × H)

61. C₂×3₋¹⁺²

C₂²⋊C₉

direct (N × H)

83. C₂²×C₉

C₂×C₃×C₉

semidirect

86. C₂²⋊C₉

C₂×3₋¹⁺²

direct (N × H)

88. 3₋¹⁺²

C₂×C₂²⋊C₉

direct (N × H)

115. C₂×C₉

C₃×C₂²⋊C₉

semidirect

135. C₉

C₂×C₃×C₂²⋊C₉

semidirect

136. C₂³

C₉×3₋¹⁺²

semidirect

151. C₂²

C₂×C₉×3₋¹⁺²

semidirect

158. C₂

3₋¹⁺²×C₂²⋊C₉

direct (N × H)

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

subgroup	index
3₋¹⁺²×C₂²⋊C₉	2
C₂³×C₃×3₋¹⁺²	3
C₂×C₃²×C₂²⋊C₉	3
C₂×C₃²⋊(C₂²⋊C₉)	3
C₂×C₉⋊(C₂²⋊C₉)	3
C₂×C₉×C₂²⋊C₉	3
C₂×C₉×3₋¹⁺²	4

subgroup

index

3₋¹⁺²×C₂²⋊C₉

C₂³×C₃×3₋¹⁺²

C₂×C₃²×C₂²⋊C₉

C₂×C₃²⋊(C₂²⋊C₉)

C₂×C₉⋊(C₂²⋊C₉)

C₂×C₉×C₂²⋊C₉

C₂×C₉×3₋¹⁺²

Subgroups — C₂×3₋¹⁺²×C₂²⋊C₉ #14518428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×3−1+2×C22⋊C9 #14518428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×3₋¹⁺²×C₂²⋊C₉ #14518428