Group #13143223 — subgroups

Subgroups — #10588800⋊C₃ #13143223

Order 1728.

340 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 162 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×M₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×#3667⋊C₃	C₂	1	2	yes	maximal
3	#63372⋊C₃	C₂	1	4	yes	maximal, complemented
4	C₃×#5125	—	3	1	no	maximal
5	#10588800	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	M₁₆×3₋¹⁺²	C₄	1	4	yes	—
8	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
9	#3667⋊C₃	—	2	2	no	—
10	#3667⋊C₃	C₂²	1	4	yes	—
11	C₂×3₋¹⁺²⋊C₈	—	2	2	no	—
12	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic
13	C₂×C₃×#208	—	3	2	no	—
14	C₃×#819	—	3	4	no	—
15	C₂×C₉×M₁₆	—	2	1	no	—
16	C₂×C₉×M₁₆	C₂×C₃	1	1	yes	characteristic
17	C₂×#3667	C₂×C₃	1	2	yes	—
18	#63372	C₂×C₃	1	4	yes	complemented
19	C₂³×3₋¹⁺²	Q₈	1	1	yes	characteristic
20	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
21	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
22	C₂×C₄×3₋¹⁺²	Q₈	1	1	yes	characteristic
23	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
24	C₈×3₋¹⁺²	C₂×C₄	1	4	yes	—
25	3₋¹⁺²⋊C₈	—	2	4	no	—
26	C₃×#430	—	9	1	no	—
27	#5125	—	3	1	no	—
28	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
29	C₃²×M₁₆	Dic₃	1	4	yes	—
30	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
31	C₃×#208	—	3	4	no	—
32	C₃×#208	—	6	2	no	—
33	C₂×C₃×C₃⋊C₈	—	6	2	no	—
34	C₂²×C₄×C₉	—	2	1	no	—
35	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
36	C₉×M₁₆	—	2	4	no	—
37	C₉×M₁₆	C₃×C₄	1	4	yes	—
38	C₂×C₈×C₉	—	2	2	no	—
39	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
40	#3667	—	2	2	no	—
41	#3667	C₂²×C₃	1	4	yes	—
42	C₂×C₉⋊C₈	—	2	2	no	—
43	C₂²×3₋¹⁺²	—	4	1	no	—
44	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
45	C₂²×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
46	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
47	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
48	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×M₁₆	—	3	1	no	—
50	C₂×C₃×M₁₆	—	9	2	no	—
51	C₂×C₃×M₁₆	C₃×D₃	1	1	yes	characteristic
52	C₂×#208	—	3	2	no	—
53	C₃×#108	—	9	4	no	—
54	#819	—	3	4	no	—
55	C₂³×C₃²	Dic₆	1	1	yes	characteristic
56	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
57	C₂×C₃²×C₄	D₁₂	1	2	yes	—
58	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
59	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
60	C₂³×C₉	—	2	1	no	—
61	C₂³×C₉	C₃×Q₈	1	1	yes	characteristic
62	C₃²×C₈	C₂×Dic₃	1	4	yes	—
63	C₃×C₃⋊C₈	—	6	4	no	—
64	C₂×C₄×C₉	—	2	1	no	—
65	C₂×C₄×C₉	—	2	2	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	2	1	no	—
68	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
69	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
70	C₂×C₄×C₉	C₃×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
72	C₈×C₉	—	2	4	no	—
73	C₈×C₉	C₂×C₃×C₄	1	4	yes	—
74	C₉⋊C₈	—	2	4	no	—
75	#430	—	9	1	no	Sylow
76	C₂×3₋¹⁺²	—	2	1	no	—
77	C₂×3₋¹⁺²	—	2	2	no	—
78	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₃×C₄	—	3	1	no	—
80	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
81	C₃×M₁₆	—	3	4	no	—
82	C₃×M₁₆	—	9	4	no	—
83	C₃×M₁₆	—	18	2	no	—
84	C₃×M₁₆	C₃×Dic₃	1	4	yes	—
85	C₂×C₃×C₈	—	3	2	no	—
86	C₂×C₃×C₈	—	18	2	no	—
87	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
88	#208	—	3	4	no	—
89	#208	—	6	2	no	—
90	C₂×C₃⋊C₈	—	6	2	no	—
91	C₂²×C₃²	—	4	1	no	—
92	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
93	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
94	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
95	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
96	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
97	C₂²×C₉	—	2	2	no	—
98	C₂²×C₉	—	2	1	no	—
99	C₂²×C₉	—	4	1	no	—
100	C₂²×C₉	—	4	2	no	—
101	C₂²×C₉	C₃×C₄⋊C₄	1	2	yes	—
102	C₂²×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
103	C₄×C₉	—	2	2	no	—
104	C₄×C₉	—	2	1	no	—
105	C₄×C₉	—	2	1	no	—
106	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
107	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
108	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
109	C₂×M₁₆	—	9	2	no	—
110	C₂×M₁₆	AGL(1,ℤ/9)	1	1	yes	characteristic
111	#108	—	9	4	no	—
112	3₋¹⁺²	#430	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₃	—	3	1	no	—
114	C₂³×C₃	C₃×Dic₆	1	1	yes	characteristic
115	C₂×C₃×C₄	—	3	1	no	—
116	C₂×C₃×C₄	—	3	2	no	—
117	C₂×C₃×C₄	—	3	2	no	—
118	C₂×C₃×C₄	—	3	1	no	—
119	C₂×C₃×C₄	C₃×Dic₆	1	1	yes	characteristic
120	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
121	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
122	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
123	C₃×C₈	—	3	4	no	—
124	C₃×C₈	—	18	4	no	—
125	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
126	C₃⋊C₈	—	6	4	no	—
127	C₂×C₃²	—	2	2	no	—
128	C₂×C₃²	—	2	1	no	—
129	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
130	C₂×C₉	—	2	2	no	—
131	C₂×C₉	—	2	2	no	—
132	C₂×C₉	—	2	1	no	—
133	C₂×C₉	—	2	1	no	—
134	C₂×C₉	—	4	2	no	—
135	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
137	M₁₆	—	9	4	no	—
138	M₁₆	—	18	2	no	—
139	M₁₆	3₋¹⁺²⋊C₄	1	4	yes	—
140	C₂×C₈	—	18	2	no	—
141	C₂×C₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
142	C₂²×C₃	—	3	1	no	—
143	C₂²×C₃	—	3	2	no	—
144	C₂²×C₃	—	4	1	no	—
145	C₂²×C₃	—	12	1	no	—
146	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
147	C₂²×C₃	C₂×C₃×Dic₆	1	1	yes	characteristic
148	C₃×C₄	—	3	1	no	—
149	C₃×C₄	—	3	1	no	—
150	C₃×C₄	—	3	2	no	—
151	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
152	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
153	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
154	C₃²	#5125	1	1	yes	characteristic
155	C₉	—	2	1	no	—
156	C₉	C₃×#430	1	1	yes	characteristic, complemented
157	C₂³	3₋¹⁺²⋊Q₈	1	1	yes	characteristic
158	C₂×C₄	3₋¹⁺²⋊Q₈	1	1	yes	characteristic
159	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	2	yes	—
160	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
161	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
162	C₈	—	18	4	no	—
163	C₈	C₂×3₋¹⁺²⋊C₄	1	4	yes	—
164	C₂×C₃	—	2	2	no	—
165	C₂×C₃	—	2	1	no	—
166	C₂×C₃	—	3	1	no	—
167	C₂×C₃	—	6	1	no	—
168	C₂×C₃	—	6	2	no	—
169	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
170	C₂²	—	4	1	no	—
171	C₂²	C₄⋊(3₋¹⁺²⋊C₄)	1	2	yes	—
172	C₂²	C₂×3₋¹⁺²⋊Q₈	1	1	yes	characteristic
173	C₄	C₄⋊(3₋¹⁺²⋊C₄)	1	2	yes	—
174	C₄	C₂×C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
175	C₄	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
176	C₃	—	3	1	no	—
177	C₃	C₃×#5125	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	2	no	—
180	C₂	C₂×C₄⋊(3₋¹⁺²⋊C₄)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×3₋¹⁺²

C₂

yes

maximal, characteristic

C₂×#3667⋊C₃

C₂

yes

maximal

#63372⋊C₃

C₂

yes

maximal, complemented

C₃×#5125

—

maximal

#10588800

C₃

yes

maximal, characteristic, complemented

C₂²×C₄×3₋¹⁺²

C₂²

yes

characteristic

M₁₆×3₋¹⁺²

C₄

yes

—

C₂×C₈×3₋¹⁺²

C₂²

yes

—

#3667⋊C₃

—

#3667⋊C₃

C₂²

yes

—

C₂×3₋¹⁺²⋊C₈

—

C₂×C₃²×M₁₆

D₃

yes

characteristic

C₂×C₃×#208

—

C₃×#819

—

C₂×C₉×M₁₆

—

C₂×C₉×M₁₆

C₂×C₃

yes

characteristic

C₂×#3667

C₂×C₃

yes

—

#63372

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

Q₈

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₂×C₄×3₋¹⁺²

Q₈

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂³

yes

characteristic

C₈×3₋¹⁺²

C₂×C₄

yes

—

3₋¹⁺²⋊C₈

—

C₃×#430

—

#5125

—

C₂²×C₃²×C₄

C₂×D₃

yes

characteristic

C₃²×M₁₆

Dic₃

yes

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃×#208

—

C₃×#208

—

C₂×C₃×C₃⋊C₈

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

characteristic

C₉×M₁₆

—

C₉×M₁₆

C₃×C₄

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂²×C₃

yes

—

#3667

—

#3667

C₂²×C₃

yes

—

C₂×C₉⋊C₈

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₄⋊C₄

yes

—

C₂²×3₋¹⁺²

C₂×Q₈

yes

characteristic

C₄×3₋¹⁺²

C₄⋊C₄

yes

—

C₄×3₋¹⁺²

C₂²×C₄

yes

characteristic

C₄×3₋¹⁺²

C₂×D₄

yes

characteristic

C₂×C₃×M₁₆

—

C₂×C₃×M₁₆

—

C₂×C₃×M₁₆

C₃×D₃

yes

—

—

—

C₂³×C₃²

Dic₆

yes

characteristic

C₂×C₃²×C₄

Dic₆

yes

characteristic

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂³×C₉

—

C₂³×C₉

C₃×Q₈

yes

characteristic

C₃²×C₈

C₂×Dic₃

yes

—

C₃×C₃⋊C₈

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₂×C₄×C₉

C₃×Q₈

yes

characteristic

C₂×C₄×C₉

C₂³×C₃

yes

characteristic

C₈×C₉

—

C₈×C₉

C₂×C₃×C₄

yes

—

C₉⋊C₈

—

#430

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

C₃×Dic₃

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃×D₃

yes

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₄⋊Dic₃

yes

—

C₂²×C₃²

C₂×Dic₆

yes

characteristic

C₃²×C₄

C₄⋊Dic₃

yes

—

C₃²×C₄

C₂×D₁₂

yes

characteristic

C₃²×C₄

C₂²×Dic₃

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

100

C₂²×C₉

—

101

C₂²×C₉

C₃×C₄⋊C₄

yes

—

102

C₂²×C₉

C₂×C₃×Q₈

yes

characteristic

103

C₄×C₉

—

104

C₄×C₉

—

105

C₄×C₉

—

106

C₄×C₉

C₃×C₄⋊C₄

yes

—

107

C₄×C₉

C₂²×C₃×C₄

yes

characteristic

108

C₄×C₉

C₂×C₃×D₄

yes

characteristic

109

C₂×M₁₆

—

110

C₂×M₁₆

AGL(1,ℤ/9)

yes

characteristic

111

#108

—

112

3₋¹⁺²

#430

yes

characteristic, complemented, Sylow

113

C₂³×C₃

—

114

C₂³×C₃

C₃×Dic₆

yes

characteristic

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

—

118

C₂×C₃×C₄

—

119

C₂×C₃×C₄

C₃×Dic₆

yes

characteristic

120

C₂×C₃×C₄

C₃×D₁₂

yes

—

121

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

122

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

123

C₃×C₈

—

124

C₃×C₈

—

125

C₃×C₈

C₂×C₃×Dic₃

yes

—

126

C₃⋊C₈

—

127

C₂×C₃²

—

128

C₂×C₃²

—

129

C₂×C₃²

C₂×C₄⋊Dic₃

yes

characteristic

130

C₂×C₉

—

131

C₂×C₉

—

132

C₂×C₉

—

133

C₂×C₉

—

134

C₂×C₉

—

135

C₂×C₉

C₂×C₃×C₄⋊C₄

yes

characteristic

136

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

characteristic

137

M₁₆

—

138

M₁₆

—

139

M₁₆

3₋¹⁺²⋊C₄

yes

—

140

C₂×C₈

—

141

C₂×C₈

C₂×AGL(1,ℤ/9)

yes

—

142

C₂²×C₃

—

143

C₂²×C₃

—

144

C₂²×C₃

—

145

C₂²×C₃

—

146

C₂²×C₃

C₃×C₄⋊Dic₃

yes

—

147

C₂²×C₃

C₂×C₃×Dic₆

yes

characteristic

148

C₃×C₄

—

149

C₃×C₄

—

150

C₃×C₄

—

151

C₃×C₄

C₃×C₄⋊Dic₃

yes

—

152

C₃×C₄

C₂×C₃×D₁₂

yes

characteristic

153

C₃×C₄

C₂²×C₃×Dic₃

yes

characteristic

154

C₃²

#5125

yes

characteristic

155

C₉

—

156

C₉

C₃×#430

yes

characteristic, complemented

157

C₂³

3₋¹⁺²⋊Q₈

yes

characteristic

158

C₂×C₄

3₋¹⁺²⋊Q₈

yes

characteristic

159

C₂×C₄

C₄⋊AGL(1,ℤ/9)

yes

—

160

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

—

161

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

characteristic

162

C₈

—

163

C₈

C₂×3₋¹⁺²⋊C₄

yes

—

164

C₂×C₃

—

165

C₂×C₃

—

166

C₂×C₃

—

167

C₂×C₃

—

168

C₂×C₃

—

169

C₂×C₃

C₂×C₃×C₄⋊Dic₃

yes

characteristic

170

C₂²

—

171

C₂²

C₄⋊(3₋¹⁺²⋊C₄)

yes

—

172

C₂²

C₂×3₋¹⁺²⋊Q₈

yes

characteristic

173

C₄

C₄⋊(3₋¹⁺²⋊C₄)

yes

—

174

C₄

C₂×C₄⋊AGL(1,ℤ/9)

yes

characteristic

175

C₄

C₂²×3₋¹⁺²⋊C₄

yes

characteristic

176

C₃

—

177

C₃

C₃×#5125

yes

characteristic

178

C₂

—

179

C₂

—

180

C₂

C₂×C₄⋊(3₋¹⁺²⋊C₄)

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₄⋊AGL(1,ℤ/9)
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×M₁₆×3₋¹⁺²	C₂
socle	C₂×C₃	C₂×C₃×C₄⋊Dic₃
prime radical	C₂³×C₃²	Dic₆
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂×M₁₆	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#430
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

C₂×C₄⋊AGL(1,ℤ/9)

commutator

C₄×C₉

C₂²×C₃×C₄

Frattini

C₂×C₃×C₄

C₂²×C₃×D₃

Fitting

C₂×M₁₆×3₋¹⁺²

C₂

socle

C₂×C₃

C₂×C₃×C₄⋊Dic₃

prime radical

C₂³×C₃²

Dic₆

cosocle

C₂×C₄×C₉

C₂³×C₃

C₂×M₁₆

C₉

G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
3. #63372⋊C₃	C₂	semidirect
5. #10588800	C₃	semidirect
18. #63372	C₂×C₃	semidirect
112. 3₋¹⁺²	#430	semidirect
156. C₉	C₃×#430	semidirect

H ≅ G/N

type

3. #63372⋊C₃

C₂

semidirect

5. #10588800

C₃

semidirect

18. #63372

C₂×C₃

semidirect

112. 3₋¹⁺²

#430

semidirect

156. C₉

C₃×#430

semidirect

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

subgroup	index
C₂×M₁₆×3₋¹⁺²	2
C₂×#3667⋊C₃	2
#63372⋊C₃	2
C₃×#5125	3
#10588800	3

subgroup

index

C₂×M₁₆×3₋¹⁺²

C₂×#3667⋊C₃

#63372⋊C₃

C₃×#5125

#10588800

Subgroups — #10588800⋊C₃ #13143223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588800⋊C3 #13143223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588800⋊C₃ #13143223