Group #13142438 — subgroups

Subgroups — #10588612⋊C₃ #13142438

Order 1728.

588 conjugacy classes of subgroups; only the 244 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 244 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	C₂×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×3₋¹⁺²⋊C₄	C₂	1	2	yes	maximal
3	3₋¹⁺²⋊#95	C₂	1	4	yes	maximal
4	C₃×#4932	—	3	1	no	maximal
5	#10588612	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×3₋¹⁺²	C₂²	1	2	yes	—
7	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂²×3₋¹⁺²⋊C₄	C₂²	1	4	yes	—
9	C₄²×3₋¹⁺²	C₄	1	4	yes	complemented
10	C₄×3₋¹⁺²⋊C₄	C₄	1	8	yes	complemented
11	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₃	—	3	2	no	—
13	C₃×C₃⋊#95	—	3	4	no	—
14	C₂×C₄²×C₉	—	2	1	no	—
15	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
16	C₂×C₄×Dic₉	C₂×C₃	1	2	yes	—
17	C₉⋊#95	C₂×C₃	1	4	yes	—
18	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
19	C₂×C₄×3₋¹⁺²	—	2	4	no	—
20	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	—
21	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	—
22	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
23	C₂×3₋¹⁺²⋊C₄	—	2	8	no	—
24	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	8	yes	—
25	C₃×#376	—	9	1	no	—
26	#4932	—	3	1	no	—
27	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
28	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	—	3	4	no	—
30	C₃²×C₄²	Dic₃	1	4	yes	—
31	C₃×C₄×Dic₃	—	3	8	no	—
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	—	2	1	no	—
34	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
35	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
36	C₂²×Dic₉	C₂²×C₃	1	4	yes	—
37	C₄²×C₉	—	2	4	no	—
38	C₄²×C₉	C₃×C₄	1	4	yes	complemented
39	C₄×Dic₉	C₃×C₄	1	8	yes	complemented
40	C₂²×3₋¹⁺²	C₂²×C₄	1	2	yes	—
41	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
42	C₂²×3₋¹⁺²	#41	1	4	yes	—
43	C₄×3₋¹⁺²	—	2	4	no	—
44	C₄×3₋¹⁺²	C₄²	1	4	yes	—
45	3₋¹⁺²⋊C₄	—	2	8	no	—
46	C₂×C₃×C₄²	—	3	1	no	—
47	C₂×C₃×C₄²	—	9	2	no	—
48	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
49	C₂×C₄×Dic₃	—	3	2	no	—
50	C₃×#95	—	9	4	no	—
51	C₃⋊#95	—	3	4	no	—
52	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
53	C₂×C₃²×C₄	—	2	4	no	—
54	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
55	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
56	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
57	C₂×C₃×Dic₃	—	3	8	no	—
58	C₂×C₃×Dic₃	—	6	8	no	—
59	C₂³×C₉	—	2	1	no	—
60	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
61	C₂×C₄×C₉	—	2	8	no	—
62	C₂×C₄×C₉	—	2	4	no	—
63	C₂×C₄×C₉	—	2	4	no	—
64	C₂×C₄×C₉	—	2	4	no	—
65	C₂×C₄×C₉	—	2	2	no	—
66	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
67	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
68	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
69	C₂×Dic₉	—	2	8	no	—
70	C₂×Dic₉	C₂×C₃×C₄	1	8	yes	—
71	#376	—	9	1	no	Sylow
72	C₂×3₋¹⁺²	C₂×C₄²	1	1	yes	characteristic
73	C₂×3₋¹⁺²	#117	1	4	yes	—
74	C₂×3₋¹⁺²	#117	1	2	yes	—
75	C₂²×C₃×C₄	—	3	2	no	—
76	C₂²×C₃×C₄	—	3	1	no	—
77	C₂²×C₃×C₄	—	9	4	no	—
78	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
79	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
80	C₂²×Dic₃	—	3	4	no	—
81	C₃×C₄²	—	3	4	no	—
82	C₃×C₄²	—	9	8	no	—
83	C₃×C₄²	C₃×Dic₃	1	4	yes	—
84	C₄×Dic₃	—	3	8	no	—
85	C₂²×C₃²	C₂×C₄×D₃	1	2	yes	—
86	C₂²×C₃²	#235	1	4	yes	—
87	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
88	C₃²×C₄	—	2	4	no	—
89	C₃²×C₄	C₄×Dic₃	1	4	yes	—
90	C₃×Dic₃	—	6	8	no	—
91	C₂²×C₉	—	2	2	no	—
92	C₂²×C₉	—	2	4	no	—
93	C₂²×C₉	—	2	1	no	—
94	C₂²×C₉	C₂²×C₃×C₄	1	2	yes	—
95	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
96	C₂²×C₉	C₃×#41	1	4	yes	—
97	C₄×C₉	—	2	8	no	—
98	C₄×C₉	—	2	4	no	—
99	C₄×C₉	—	2	4	no	—
100	C₄×C₉	C₃×C₄²	1	4	yes	—
101	Dic₉	—	2	8	no	—
102	C₂×C₄²	—	9	2	no	—
103	C₂×C₄²	AGL(1,ℤ/9)	1	1	yes	characteristic
104	#95	—	9	4	no	—
105	3₋¹⁺²	#376	1	1	yes	characteristic, complemented, Sylow
106	C₂³×C₃	—	3	1	no	—
107	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	4	no	—
109	C₂×C₃×C₄	—	3	4	no	—
110	C₂×C₃×C₄	—	3	2	no	—
111	C₂×C₃×C₄	—	3	4	no	—
112	C₂×C₃×C₄	—	6	4	no	—
113	C₂×C₃×C₄	—	9	8	no	—
114	C₂×C₃×C₄	—	18	8	no	—
115	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
116	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
117	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
118	C₂×Dic₃	—	3	8	no	—
119	C₂×Dic₃	—	6	8	no	—
120	C₂×C₃²	#872	1	4	yes	—
121	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
122	C₂×C₃²	#926	1	2	yes	—
123	C₂×C₉	—	2	1	no	—
124	C₂×C₉	—	2	2	no	—
125	C₂×C₉	—	2	4	no	—
126	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
127	C₂×C₉	C₃×#117	1	2	yes	—
128	C₂×C₉	C₃×#117	1	4	yes	—
129	C₂²×C₄	—	9	4	no	—
130	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	2	yes	—
131	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
132	C₄²	—	9	8	no	—
133	C₄²	3₋¹⁺²⋊C₄	1	4	yes	complemented
134	C₂²×C₃	—	3	2	no	—
135	C₂²×C₃	—	3	4	no	—
136	C₂²×C₃	—	3	1	no	—
137	C₂²×C₃	C₂×C₃×C₄×D₃	1	2	yes	—
138	C₂²×C₃	C₃×#235	1	4	yes	—
139	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
140	C₃×C₄	—	2	4	no	—
141	C₃×C₄	—	3	4	no	—
142	C₃×C₄	—	6	4	no	—
143	C₃×C₄	—	18	8	no	—
144	C₃×C₄	C₃×C₄×Dic₃	1	4	yes	—
145	Dic₃	—	6	8	no	—
146	C₃²	#4932	1	1	yes	characteristic
147	C₉	—	2	1	no	—
148	C₉	C₃×#376	1	1	yes	characteristic, complemented
149	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	9	8	no	—
152	C₂×C₄	—	18	8	no	—
153	C₂×C₄	C₄×AGL(1,ℤ/9)	1	4	yes	—
154	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	4	yes	—
155	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
156	C₂×C₃	—	3	1	no	—
157	C₂×C₃	—	3	2	no	—
158	C₂×C₃	—	3	4	no	—
159	C₂×C₃	C₃×#872	1	4	yes	—
160	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
161	C₂×C₃	C₃×#926	1	2	yes	—
162	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	2	yes	—
163	C₂²	#3697⋊C₃	1	4	yes	—
164	C₂²	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
165	C₄	—	2	4	no	—
166	C₄	—	18	8	no	—
167	C₄	C₄×3₋¹⁺²⋊C₄	1	4	yes	—
168	C₃	—	3	1	no	—
169	C₃	C₃×#4932	1	1	yes	characteristic
170	C₂	#63434⋊C₃	1	4	yes	—
171	C₂	C₂×C₄×3₋¹⁺²⋊C₄	1	1	yes	characteristic
172	C₂	#63488⋊C₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×3₋¹⁺²

C₂

yes

maximal, characteristic

C₂×C₄×3₋¹⁺²⋊C₄

C₂

yes

maximal

3₋¹⁺²⋊#95

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₂²×C₄×3₋¹⁺²

C₂²

yes

characteristic

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

—

C₄²×3₋¹⁺²

C₄

yes

complemented

C₄×3₋¹⁺²⋊C₄

C₄

yes

complemented

C₂×C₃²×C₄²

D₃

yes

characteristic

C₂×C₃×C₄×Dic₃

—

C₃×C₃⋊#95

—

C₂×C₄²×C₉

—

C₂×C₄²×C₉

C₂×C₃

yes

characteristic

C₂×C₄×Dic₉

C₂×C₃

yes

—

C₉⋊#95

C₂×C₃

yes

—

C₂³×3₋¹⁺²

C₂³

yes

characteristic

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

C₂×C₄

yes

—

C₃×#376

—

#4932

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃²×C₄

C₂×D₃

yes

characteristic

C₂²×C₃×Dic₃

—

C₃²×C₄²

Dic₃

yes

—

C₃×C₄×Dic₃

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₂²×C₄×C₉

C₂²×C₃

yes

characteristic

C₂²×Dic₉

C₂²×C₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₃×C₄

yes

complemented

C₄×Dic₉

C₃×C₄

yes

complemented

C₂²×3₋¹⁺²

C₂²×C₄

yes

—

C₂²×3₋¹⁺²

C₂²×C₄

yes

characteristic

C₂²×3₋¹⁺²

#41

yes

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₄²

yes

—

3₋¹⁺²⋊C₄

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₃×D₃

yes

characteristic

C₂×C₄×Dic₃

—

C₃×#95

—

C₃⋊#95

—

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

C₂³×C₃

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×Dic₉

—

C₂×Dic₉

C₂×C₃×C₄

yes

—

#376

—

Sylow

C₂×3₋¹⁺²

C₂×C₄²

yes

characteristic

C₂×3₋¹⁺²

#117

yes

—

C₂×3₋¹⁺²

#117

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

characteristic

C₂²×Dic₃

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₃×Dic₃

yes

—

C₄×Dic₃

—

C₂²×C₃²

C₂×C₄×D₃

yes

—

C₂²×C₃²

#235

yes

—

C₂²×C₃²

C₂²×Dic₃

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₄×Dic₃

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂²×C₃×C₄

yes

—

C₂²×C₉

C₂²×C₃×C₄

yes

characteristic

C₂²×C₉

C₃×#41

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

100

C₄×C₉

C₃×C₄²

yes

—

101

Dic₉

—

102

C₂×C₄²

—

103

C₂×C₄²

AGL(1,ℤ/9)

yes

characteristic

104

#95

—

105

3₋¹⁺²

#376

yes

characteristic, complemented, Sylow

106

C₂³×C₃

—

107

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

C₃×C₄×D₃

yes

—

116

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

117

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

118

C₂×Dic₃

—

119

C₂×Dic₃

—

120

C₂×C₃²

#872

yes

—

121

C₂×C₃²

C₂×C₄×Dic₃

yes

characteristic

122

C₂×C₃²

#926

yes

—

123

C₂×C₉

—

124

C₂×C₉

—

125

C₂×C₉

—

126

C₂×C₉

C₂×C₃×C₄²

yes

characteristic

127

C₂×C₉

C₃×#117

yes

—

128

C₂×C₉

C₃×#117

yes

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

—

131

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

characteristic

132

C₄²

—

133

C₄²

3₋¹⁺²⋊C₄

yes

complemented

134

C₂²×C₃

—

135

C₂²×C₃

—

136

C₂²×C₃

—

137

C₂²×C₃

C₂×C₃×C₄×D₃

yes

—

138

C₂²×C₃

C₃×#235

yes

—

139

C₂²×C₃

C₂²×C₃×Dic₃

yes

characteristic

140

C₃×C₄

—

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

C₃×C₄×Dic₃

yes

—

145

Dic₃

—

146

C₃²

#4932

yes

characteristic

147

C₉

—

148

C₉

C₃×#376

yes

characteristic, complemented

149

C₂³

C₂²×AGL(1,ℤ/9)

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₄×AGL(1,ℤ/9)

yes

—

154

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

—

155

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

—

156

C₂×C₃

—

157

C₂×C₃

—

158

C₂×C₃

—

159

C₂×C₃

C₃×#872

yes

—

160

C₂×C₃

C₂×C₃×C₄×Dic₃

yes

characteristic

161

C₂×C₃

C₃×#926

yes

—

162

C₂²

C₂×C₄×AGL(1,ℤ/9)

yes

—

163

C₂²

#3697⋊C₃

yes

—

164

C₂²

C₂²×3₋¹⁺²⋊C₄

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

C₄×3₋¹⁺²⋊C₄

yes

—

168

C₃

—

169

C₃

C₃×#4932

yes

characteristic

170

C₂

#63434⋊C₃

yes

—

171

C₂

C₂×C₄×3₋¹⁺²⋊C₄

yes

characteristic

172

C₂

#63488⋊C₃

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂×C₃×C₄²
Frattini, socle	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₄²×3₋¹⁺²	C₂
prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₂×C₄²	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#376
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂²×C₄

C₂×AGL(1,ℤ/9)

commutator

C₂×C₉

C₂×C₃×C₄²

Frattini, socle

C₂³×C₃

C₂²×C₃×D₃

Fitting

C₂×C₄²×3₋¹⁺²

C₂

prime radical

C₂³×C₃²

C₂²×D₃

cosocle

C₂³×C₉

C₂³×C₃

C₂×C₄²

C₉

G ▷ C₂×C₉ ▷ C₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
5. #10588612	C₃	semidirect
9. C₄²×3₋¹⁺²	C₄	semidirect
10. C₄×3₋¹⁺²⋊C₄	C₄	semidirect
38. C₄²×C₉	C₃×C₄	semidirect
39. C₄×Dic₉	C₃×C₄	semidirect
105. 3₋¹⁺²	#376	semidirect
133. C₄²	3₋¹⁺²⋊C₄	semidirect
148. C₉	C₃×#376	semidirect

H ≅ G/N

type

5. #10588612

C₃

semidirect

9. C₄²×3₋¹⁺²

C₄

semidirect

10. C₄×3₋¹⁺²⋊C₄

C₄

semidirect

38. C₄²×C₉

C₃×C₄

semidirect

39. C₄×Dic₉

C₃×C₄

semidirect

105. 3₋¹⁺²

#376

semidirect

133. C₄²

3₋¹⁺²⋊C₄

semidirect

148. C₉

C₃×#376

semidirect

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

subgroup	index
C₂×C₄²×3₋¹⁺²	2
C₂×C₄×3₋¹⁺²⋊C₄	2
3₋¹⁺²⋊#95	2
C₃×#4932	3
#10588612	3

subgroup

index

C₂×C₄²×3₋¹⁺²

C₂×C₄×3₋¹⁺²⋊C₄

3₋¹⁺²⋊#95

C₃×#4932

#10588612

Subgroups — #10588612⋊C₃ #13142438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588612⋊C3 #13142438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588612⋊C₃ #13142438