Group #13135691 — subgroups

Subgroups — #10588237⋊C₃ #13135691

Order 1728.

316 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 114 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	#63488⋊C₃	C₂	1	2	yes	maximal
3	C₃×#4557	—	3	1	no	maximal
4	#10588237	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
6	M₁₆×3₋¹⁺²	—	2	2	no	—
7	C₂×C₈×3₋¹⁺²	C₄	1	2	yes	complemented
8	#3676⋊C₃	—	2	2	no	—
9	C₄⋊(3₋¹⁺²⋊C₄)	—	2	2	no	—
10	C₄×3₋¹⁺²⋊C₄	C₄	1	4	yes	complemented
11	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic
12	C₃×#926	—	3	2	no	—
13	C₂×C₉×M₁₆	—	2	1	no	—
14	C₂×C₉×M₁₆	C₂×C₃	1	1	yes	characteristic
15	#63488	C₂×C₃	1	2	yes	—
16	C₂³×3₋¹⁺²	D₄	1	1	yes	characteristic
17	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
19	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
20	C₂×C₄×3₋¹⁺²	Q₈	1	1	yes	characteristic
21	C₂×3₋¹⁺²⋊C₄	—	2	4	no	—
22	C₈×3₋¹⁺²	—	2	2	no	—
23	C₃×H(3,ℤ/4)	—	9	1	no	—
24	#4557	—	3	1	no	—
25	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
26	C₃²×M₁₆	—	2	2	no	—
27	C₂×C₃²×C₈	Dic₃	1	2	yes	—
28	C₃×#217	—	6	2	no	—
29	C₃×C₄⋊Dic₃	—	6	2	no	—
30	C₃×C₄×Dic₃	—	3	4	no	—
31	C₂²×C₄×C₉	—	2	1	no	—
32	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
33	C₉×M₁₆	—	2	4	no	—
34	C₉×M₁₆	—	2	2	no	—
35	C₂×C₈×C₉	—	2	2	no	—
36	C₂×C₈×C₉	C₃×C₄	1	2	yes	complemented
37	#3676	—	2	2	no	—
38	C₄⋊Dic₉	—	2	2	no	—
39	C₄×Dic₉	C₃×C₄	1	4	yes	complemented
40	C₂²×3₋¹⁺²	—	4	1	no	—
41	C₂²×3₋¹⁺²	C₂²⋊C₄	1	2	yes	—
42	C₂²×3₋¹⁺²	C₄⋊C₄	1	1	yes	characteristic
43	C₄×3₋¹⁺²	C₄²	1	1	yes	characteristic
44	C₄×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
46	3₋¹⁺²⋊C₄	—	4	4	no	—
47	C₂×C₃×M₁₆	—	3	1	no	—
48	C₂×C₃×M₁₆	C₃×D₃	1	1	yes	characteristic
49	C₃×#117	—	9	2	no	—
50	#926	—	3	2	no	—
51	C₂³×C₃²	D₁₂	1	1	yes	characteristic
52	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
54	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
55	C₂×C₃²×C₄	#67	1	2	yes	—
56	C₂×C₃×Dic₃	—	6	4	no	—
57	C₂³×C₉	—	2	1	no	—
58	C₂³×C₉	C₃×D₄	1	1	yes	characteristic
59	C₃²×C₈	—	2	2	no	—
60	C₂×C₄×C₉	—	2	1	no	—
61	C₂×C₄×C₉	—	2	2	no	—
62	C₂×C₄×C₉	—	2	1	no	—
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
65	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
66	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
67	C₂×C₄×C₉	C₃×Q₈	1	1	yes	characteristic
68	C₂×Dic₉	—	2	4	no	—
69	C₈×C₉	—	2	4	no	—
70	C₈×C₉	—	2	2	no	—
71	H(3,ℤ/4)	—	9	1	no	Sylow
72	C₂×3₋¹⁺²	—	2	1	no	—
73	C₂×3₋¹⁺²	—	2	2	no	—
74	C₂×3₋¹⁺²	#95	1	1	yes	characteristic
75	C₂²×C₃×C₄	—	3	1	no	—
76	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
77	C₃×M₁₆	—	2	2	no	—
78	C₃×M₁₆	—	6	2	no	—
79	C₂×C₃×C₈	—	3	2	no	—
80	C₂×C₃×C₈	C₃×Dic₃	1	2	yes	—
81	C₃×C₄⋊C₄	—	18	2	no	—
82	C₃×C₂²⋊C₄	—	18	2	no	—
83	C₃×C₄²	—	9	4	no	—
84	#217	—	6	2	no	—
85	C₄⋊Dic₃	—	6	2	no	—
86	C₄×Dic₃	—	3	4	no	—
87	C₂²×C₃²	—	4	1	no	—
88	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
89	C₂²×C₃²	(C₂×D₃)⋊C₄	1	2	yes	—
90	C₃²×C₄	C₄×Dic₃	1	1	yes	characteristic
91	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
92	C₃²×C₄	#217	1	1	yes	characteristic
93	C₃×Dic₃	—	12	4	no	—
94	C₂²×C₉	—	2	1	no	—
95	C₂²×C₉	—	2	2	no	—
96	C₂²×C₉	—	4	1	no	—
97	C₂²×C₉	—	4	2	no	—
98	C₂²×C₉	C₃×C₂²⋊C₄	1	2	yes	—
99	C₂²×C₉	C₃×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₉	—	2	1	no	—
101	C₄×C₉	—	2	2	no	—
102	C₄×C₉	—	2	1	no	—
103	C₄×C₉	C₃×C₄²	1	1	yes	characteristic
104	C₄×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
105	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
106	Dic₉	—	4	4	no	—
107	C₂×M₁₆	AGL(1,ℤ/9)	1	1	yes	characteristic
108	#117	—	9	2	no	—
109	3₋¹⁺²	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₃	—	3	1	no	—
111	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
112	C₂×C₃×C₄	—	3	1	no	—
113	C₂×C₃×C₄	—	3	2	no	—
114	C₂×C₃×C₄	—	3	1	no	—
115	C₂×C₃×C₄	—	3	2	no	—
116	C₂×C₃×C₄	—	18	4	no	—
117	C₂×C₃×C₄	C₃×Dic₆	1	1	yes	characteristic
118	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
119	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
120	C₂×C₃×C₄	C₃×#67	1	2	yes	—
121	C₂×Dic₃	—	6	4	no	—
122	C₃×C₈	—	2	2	no	—
123	C₃×C₈	—	6	2	no	—
124	C₂×C₃²	—	2	2	no	—
125	C₂×C₃²	—	2	1	no	—
126	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
127	C₂×C₉	—	2	2	no	—
128	C₂×C₉	—	2	2	no	—
129	C₂×C₉	—	2	1	no	—
130	C₂×C₉	—	2	1	no	—
131	C₂×C₉	—	4	2	no	—
132	C₂×C₉	C₃×#95	1	1	yes	characteristic
133	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	C₂×C₈	3₋¹⁺²⋊C₄	1	2	yes	complemented
136	C₄⋊C₄	—	18	2	no	—
137	C₂²⋊C₄	—	18	2	no	—
138	C₄²	—	9	4	no	—
139	C₂²×C₃	—	3	1	no	—
140	C₂²×C₃	—	3	2	no	—
141	C₂²×C₃	—	4	1	no	—
142	C₂²×C₃	—	12	1	no	—
143	C₂²×C₃	C₃×C₄⋊Dic₃	1	1	yes	characteristic
144	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
145	C₃×C₄	—	3	1	no	—
146	C₃×C₄	—	3	2	no	—
147	C₃×C₄	—	3	1	no	—
148	C₃×C₄	—	36	4	no	—
149	C₃×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
150	C₃×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
151	C₃×C₄	C₃×#217	1	1	yes	characteristic
152	Dic₃	—	12	4	no	—
153	C₃²	#4557	1	1	yes	characteristic
154	C₉	—	2	1	no	—
155	C₉	C₃×H(3,ℤ/4)	1	1	yes	characteristic, complemented
156	C₂³	C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
157	C₂×C₄	—	18	4	no	—
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	1	yes	characteristic
161	C₂×C₄	#611⋊C₃	1	2	yes	—
162	C₈	—	2	2	no	—
163	C₂×C₃	—	2	2	no	—
164	C₂×C₃	—	2	1	no	—
165	C₂×C₃	—	3	1	no	—
166	C₂×C₃	—	6	2	no	—
167	C₂×C₃	—	6	1	no	—
168	C₂×C₃	C₃×C₃⋊#95	1	1	yes	characteristic
169	C₂²	—	4	1	no	—
170	C₂²	C₄⋊(3₋¹⁺²⋊C₄)	1	1	yes	characteristic
171	C₂²	(C₂×AGL(1,ℤ/9))⋊C₄	1	2	yes	—
172	C₄	—	36	4	no	—
173	C₄	C₄×3₋¹⁺²⋊C₄	1	1	yes	characteristic
174	C₄	(3₋¹⁺²⋊C₄)⋊C₄	1	2	yes	—
175	C₄	#3676⋊C₃	1	1	yes	characteristic
176	C₃	—	3	1	no	—
177	C₃	C₃×#4557	1	1	yes	characteristic
178	C₂	—	2	2	no	—
179	C₂	—	2	1	no	—
180	C₂	3₋¹⁺²⋊#95	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×3₋¹⁺²

C₂

yes

maximal, characteristic

#63488⋊C₃

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂²×C₄×3₋¹⁺²

C₂²

yes

characteristic

M₁₆×3₋¹⁺²

—

C₂×C₈×3₋¹⁺²

C₄

yes

complemented

#3676⋊C₃

—

C₄⋊(3₋¹⁺²⋊C₄)

—

C₄×3₋¹⁺²⋊C₄

C₄

yes

complemented

C₂×C₃²×M₁₆

D₃

yes

characteristic

C₃×#926

—

C₂×C₉×M₁₆

—

C₂×C₉×M₁₆

C₂×C₃

yes

characteristic

#63488

C₂×C₃

yes

—

C₂³×3₋¹⁺²

D₄

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂×C₄

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

C₂×3₋¹⁺²⋊C₄

—

C₈×3₋¹⁺²

—

C₃×H(3,ℤ/4)

—

#4557

—

C₂²×C₃²×C₄

C₂×D₃

yes

characteristic

C₃²×M₁₆

—

C₂×C₃²×C₈

Dic₃

yes

—

C₃×#217

—

C₃×C₄⋊Dic₃

—

C₃×C₄×Dic₃

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

characteristic

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₃×C₄

yes

complemented

#3676

—

C₄⋊Dic₉

—

C₄×Dic₉

C₃×C₄

yes

complemented

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₂²⋊C₄

yes

—

C₂²×3₋¹⁺²

C₄⋊C₄

yes

characteristic

C₄×3₋¹⁺²

C₄²

yes

characteristic

C₄×3₋¹⁺²

C₂²⋊C₄

yes

characteristic

C₄×3₋¹⁺²

C₄⋊C₄

yes

—

3₋¹⁺²⋊C₄

—

C₂×C₃×M₁₆

—

C₂×C₃×M₁₆

C₃×D₃

yes

characteristic

C₃×#117

—

#926

—

C₂³×C₃²

D₁₂

yes

characteristic

C₂×C₃²×C₄

Dic₆

yes

characteristic

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

characteristic

C₂×C₃²×C₄

#67

yes

—

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

C₃×D₄

yes

characteristic

C₃²×C₈

—

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

characteristic

C₂×C₄×C₉

C₃×D₄

yes

—

C₂×C₄×C₉

C₃×Q₈

yes

characteristic

C₂×Dic₉

—

C₈×C₉

—

C₈×C₉

—

H(3,ℤ/4)

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

#95

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃×Dic₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

#217

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

—

C₂²×C₃²

C₄⋊Dic₃

yes

characteristic

C₂²×C₃²

(C₂×D₃)⋊C₄

yes

—

C₃²×C₄

C₄×Dic₃

yes

characteristic

C₃²×C₄

Dic₃⋊C₄

yes

—

C₃²×C₄

#217

yes

characteristic

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×C₂²⋊C₄

yes

—

C₂²×C₉

C₃×C₄⋊C₄

yes

characteristic

100

C₄×C₉

—

101

C₄×C₉

—

102

C₄×C₉

—

103

C₄×C₉

C₃×C₄²

yes

characteristic

104

C₄×C₉

C₃×C₂²⋊C₄

yes

characteristic

105

C₄×C₉

C₃×C₄⋊C₄

yes

—

106

Dic₉

—

107

C₂×M₁₆

AGL(1,ℤ/9)

yes

characteristic

108

#117

—

109

3₋¹⁺²

H(3,ℤ/4)

yes

characteristic, complemented, Sylow

110

C₂³×C₃

—

111

C₂³×C₃

C₃×D₁₂

yes

characteristic

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

C₃×Dic₆

yes

characteristic

118

C₂×C₃×C₄

C₃×C₄×D₃

yes

—

119

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

characteristic

120

C₂×C₃×C₄

C₃×#67

yes

—

121

C₂×Dic₃

—

122

C₃×C₈

—

123

C₃×C₈

—

124

C₂×C₃²

—

125

C₂×C₃²

—

126

C₂×C₃²

C₃⋊#95

yes

characteristic

127

C₂×C₉

—

128

C₂×C₉

—

129

C₂×C₉

—

130

C₂×C₉

—

131

C₂×C₉

—

132

C₂×C₉

C₃×#95

yes

characteristic

133

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

characteristic

134

M₁₆

—

135

C₂×C₈

3₋¹⁺²⋊C₄

yes

complemented

136

C₄⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₂²×C₃

—

140

C₂²×C₃

—

141

C₂²×C₃

—

142

C₂²×C₃

—

143

C₂²×C₃

C₃×C₄⋊Dic₃

yes

characteristic

144

C₂²×C₃

C₃×(C₂×D₃)⋊C₄

yes

—

145

C₃×C₄

—

146

C₃×C₄

—

147

C₃×C₄

—

148

C₃×C₄

—

149

C₃×C₄

C₃×C₄×Dic₃

yes

characteristic

150

C₃×C₄

C₃×Dic₃⋊C₄

yes

—

151

C₃×C₄

C₃×#217

yes

characteristic

152

Dic₃

—

153

C₃²

#4557

yes

characteristic

154

C₉

—

155

C₉

C₃×H(3,ℤ/4)

yes

characteristic, complemented

156

C₂³

C₄⋊AGL(1,ℤ/9)

yes

characteristic

157

C₂×C₄

—

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

characteristic

161

C₂×C₄

#611⋊C₃

yes

—

162

C₈

—

163

C₂×C₃

—

164

C₂×C₃

—

165

C₂×C₃

—

166

C₂×C₃

—

167

C₂×C₃

—

168

C₂×C₃

C₃×C₃⋊#95

yes

characteristic

169

C₂²

—

170

C₂²

C₄⋊(3₋¹⁺²⋊C₄)

yes

characteristic

171

C₂²

(C₂×AGL(1,ℤ/9))⋊C₄

yes

—

172

C₄

—

173

C₄

C₄×3₋¹⁺²⋊C₄

yes

characteristic

174

C₄

(3₋¹⁺²⋊C₄)⋊C₄

yes

—

175

C₄

#3676⋊C₃

yes

characteristic

176

C₃

—

177

C₃

C₃×#4557

yes

characteristic

178

C₂

—

179

C₂

—

180

C₂

3₋¹⁺²⋊#95

yes

characteristic

	subgroup	G/H
center	C₄	C₄×3₋¹⁺²⋊C₄
commutator	C₄×C₉	C₃×C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃×D₃
Fitting	C₂×M₁₆×3₋¹⁺²	C₂
socle	C₂×C₃	C₃×C₃⋊#95
prime radical	C₂³×C₃²	D₁₂
cosocle	C₂²×C₄×C₉	C₂²×C₃
hypercenter	C₂×M₁₆	AGL(1,ℤ/9)
hypocenter	C₉	C₃×H(3,ℤ/4)
LCS	G ▷ C₄×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

C₄×3₋¹⁺²⋊C₄

commutator

C₄×C₉

C₃×C₄²

Frattini

C₂²×C₃×C₄

C₂×C₃×D₃

Fitting

C₂×M₁₆×3₋¹⁺²

C₂

socle

C₂×C₃

C₃×C₃⋊#95

prime radical

C₂³×C₃²

D₁₂

cosocle

C₂²×C₄×C₉

C₂²×C₃

C₂×M₁₆

C₉

G ▷ C₄×C₉ ▷ C₉

C₁ ◁ C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
4. #10588237	C₃	semidirect
7. C₂×C₈×3₋¹⁺²	C₄	semidirect
10. C₄×3₋¹⁺²⋊C₄	C₄	semidirect
36. C₂×C₈×C₉	C₃×C₄	semidirect
39. C₄×Dic₉	C₃×C₄	semidirect
109. 3₋¹⁺²	H(3,ℤ/4)	semidirect
135. C₂×C₈	3₋¹⁺²⋊C₄	semidirect
155. C₉	C₃×H(3,ℤ/4)	semidirect

H ≅ G/N

type

4. #10588237

C₃

semidirect

7. C₂×C₈×3₋¹⁺²

C₄

semidirect

10. C₄×3₋¹⁺²⋊C₄

C₄

semidirect

36. C₂×C₈×C₉

C₃×C₄

semidirect

39. C₄×Dic₉

C₃×C₄

semidirect

109. 3₋¹⁺²

H(3,ℤ/4)

semidirect

135. C₂×C₈

3₋¹⁺²⋊C₄

semidirect

155. C₉

C₃×H(3,ℤ/4)

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
#63488⋊C₃	2
C₃×#4557	3
#10588237	3

subgroup

index

C₂×M₁₆×3₋¹⁺²

#63488⋊C₃

C₃×#4557

#10588237

Subgroups — #10588237⋊C₃ #13135691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588237⋊C3 #13135691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588237⋊C₃ #13135691