Group #13144527 — subgroups

Subgroups — #13144527

Order 1728.

590 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 191 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	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic
2	#11721706	C₂	1	1	yes	maximal, characteristic, complemented
3	#11720576	C₂	1	4	yes	maximal, complemented
4	C₄×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#10592769	—	3	1	no	maximal
6	#10588750	—	3	3	no	maximal
7	C₂×C₃×D₄×C₉	—	2	1	no	—
8	C₂×C₃×D₄×C₉	—	2	1	no	—
9	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
10	C₂×#6226	—	2	1	no	—
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
13	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
15	C₃²×#127	D₃	1	1	yes	characteristic
16	#64078	—	3	1	no	—
17	#63659	—	3	4	no	—
18	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
19	C₉×#127	D₃	1	3	yes	—
20	#63431	—	3	3	no	—
21	#63392	—	3	12	no	—
22	C₄×C₉⋊C₈	—	3	3	no	—
23	C₂³×C₃×C₉	—	2	2	no	—
24	C₃×D₄×C₉	—	2	4	no	—
25	C₃×D₄×C₉	—	4	2	no	—
26	C₃×D₄×C₉	—	4	2	no	—
27	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₃⋊Dic₉	—	2	2	no	—
31	#6226	—	4	2	no	—
32	C₃⋊(C₉⋊C₈)	—	2	2	no	—
33	#5070	—	9	3	no	—
34	#5070	—	9	1	no	—
35	C₂×C₃²×D₄	—	2	1	no	—
36	C₂×C₃²×D₄	—	2	1	no	—
37	C₂×C₃²×D₄	C₂×D₃	1	2	yes	—
38	C₂×#634	—	6	1	no	—
39	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
40	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
41	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
43	C₂×D₄×C₉	—	2	3	no	—
44	C₂×D₄×C₉	—	2	3	no	—
45	C₂×D₄×C₉	C₂×D₃	1	6	yes	—
46	C₂×Dic₁₈	—	6	3	no	—
47	C₄²×C₉	C₂×D₃	1	3	yes	—
48	C₄⋊Dic₉	—	3	6	no	—
49	C₄⋊Dic₉	—	6	3	no	—
50	C₂×C₉⋊C₈	—	3	6	no	—
51	C₂²×C₃×C₉	—	4	2	no	—
52	C₂²×C₃×C₉	—	4	2	no	—
53	C₂²×C₃×C₉	—	4	2	no	—
54	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
55	C₃×C₄×C₉	D₈	1	2	yes	—
56	C₃×C₄×C₉	SD₁₆	1	2	yes	—
57	C₃×C₄×C₉	#41	1	2	yes	—
58	C₃⋊Dic₉	—	4	2	no	—
59	C₃×#127	D₉	1	3	yes	—
60	C₃×#127	C₃⋊D₃	1	1	yes	characteristic
61	#869	—	9	3	no	—
62	#869	—	9	1	no	—
63	#832	—	9	12	no	—
64	#832	—	9	4	no	—
65	C₄×C₃⋊C₈	—	9	3	no	—
66	C₄×C₃⋊C₈	—	9	1	no	—
67	C₂³×C₃²	—	2	2	no	—
68	C₃²×D₄	—	2	4	no	—
69	C₃²×D₄	—	4	2	no	—
70	C₃²×D₄	—	4	2	no	—
71	C₂×C₃²×C₄	#67	1	1	yes	characteristic
72	C₂×C₃²×C₄	#67	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂×C₃⋊Dic₃	—	6	2	no	—
75	#634	—	12	2	no	—
76	C₂³×C₉	—	2	6	no	—
77	C₃⋊(C₃⋊C₈)	—	6	2	no	—
78	D₄×C₉	—	2	12	no	—
79	D₄×C₉	—	4	6	no	—
80	D₄×C₉	—	4	6	no	—
81	C₂×C₄×C₉	#67	1	3	yes	—
82	C₂×C₄×C₉	#67	1	3	yes	—
83	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
84	C₂×Dic₉	—	6	6	no	—
85	Dic₁₈	—	12	6	no	—
86	C₉⋊C₈	—	6	6	no	—
87	#486	—	27	1	no	Sylow
88	C₂×C₃×C₉	—	8	2	no	—
89	C₂×C₃×C₉	#124	1	1	yes	characteristic
90	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
91	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×C₃×D₄	—	2	3	no	—
93	C₂×C₃×D₄	—	2	3	no	—
94	C₂×C₃×D₄	—	2	1	no	—
95	C₂×C₃×D₄	—	2	1	no	—
96	C₂×C₃×D₄	C₂×D₉	1	6	yes	—
97	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
98	C₂×Dic₆	—	18	3	no	—
99	C₂×Dic₆	—	18	1	no	—
100	C₃×C₄²	C₂×D₉	1	3	yes	—
101	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
102	C₄⋊Dic₃	—	9	6	no	—
103	C₄⋊Dic₃	—	9	2	no	—
104	C₄⋊Dic₃	—	18	3	no	—
105	C₄⋊Dic₃	—	18	1	no	—
106	C₂×C₃⋊C₈	—	9	6	no	—
107	C₂×C₃⋊C₈	—	9	2	no	—
108	C₂²×C₃²	—	4	2	no	—
109	C₂²×C₃²	—	4	2	no	—
110	C₂²×C₃²	—	4	2	no	—
111	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
112	C₃²×C₄	#213	1	2	yes	—
113	C₃²×C₄	#214	1	2	yes	—
114	C₃²×C₄	#237	1	2	yes	—
115	C₃⋊Dic₃	—	12	2	no	—
116	C₂²×C₉	—	4	6	no	—
117	C₂²×C₉	—	4	6	no	—
118	C₂²×C₉	—	4	6	no	—
119	C₂²×C₉	C₂×#67	1	3	yes	—
120	C₄×C₉	#213	1	6	yes	—
121	C₄×C₉	#214	1	6	yes	—
122	C₄×C₉	#237	1	6	yes	—
123	Dic₉	—	12	6	no	—
124	C₄⋊Q₈	—	27	1	no	—
125	#127	C₃⋊D₉	1	1	yes	characteristic
126	#102	—	27	4	no	—
127	C₄×C₈	—	27	1	no	—
128	C₃×C₉	#486	1	1	yes	characteristic, complemented, Sylow
129	C₂³×C₃	—	2	6	no	—
130	C₂³×C₃	—	2	2	no	—
131	C₃×D₄	—	2	12	no	—
132	C₃×D₄	—	2	4	no	—
133	C₃×D₄	—	4	6	no	—
134	C₃×D₄	—	4	6	no	—
135	C₃×D₄	—	4	2	no	—
136	C₃×D₄	—	4	2	no	—
137	C₂×C₃×C₄	#611	1	3	yes	—
138	C₂×C₃×C₄	#611	1	3	yes	—
139	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
140	C₂×C₃×C₄	#638	1	1	yes	characteristic
141	C₂×C₃×C₄	#638	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
143	C₂×Dic₃	—	18	6	no	—
144	C₂×Dic₃	—	18	2	no	—
145	Dic₆	—	36	6	no	—
146	Dic₆	—	36	2	no	—
147	C₃⋊C₈	—	18	6	no	—
148	C₃⋊C₈	—	18	2	no	—
149	C₂×C₃²	—	8	2	no	—
150	C₂×C₃²	C₂×#213	1	1	yes	characteristic
151	C₂×C₃²	C₂×#214	1	1	yes	characteristic
152	C₂×C₃²	#936	1	1	yes	characteristic
153	C₂×C₉	—	8	6	no	—
154	C₂×C₉	C₂×#213	1	3	yes	—
155	C₂×C₉	C₂×#214	1	3	yes	—
156	C₂×C₉	#936	1	3	yes	—
157	C₂×Q₈	—	54	1	no	—
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	C₂×C₃⋊D₉	1	2	yes	—
161	C₂×C₈	—	27	2	no	—
162	C₄⋊C₄	—	27	2	no	—
163	C₄⋊C₄	—	54	1	no	—
164	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
165	C₂²×C₃	—	4	6	no	—
166	C₂²×C₃	—	4	6	no	—
167	C₂²×C₃	—	4	6	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	—	4	2	no	—
170	C₂²×C₃	—	4	2	no	—
171	C₂²×C₃	C₂×#611	1	3	yes	—
172	C₂²×C₃	C₂×#638	1	1	yes	characteristic
173	C₃×C₄	#3672	1	6	yes	—
174	C₃×C₄	#3673	1	6	yes	—
175	C₃×C₄	#3699	1	6	yes	—
176	C₃×C₄	#3753	1	2	yes	—
177	C₃×C₄	#3754	1	2	yes	—
178	C₃×C₄	#3830	1	2	yes	—
179	Dic₃	—	36	6	no	—
180	Dic₃	—	36	2	no	—
181	C₃²	#5070	1	1	yes	characteristic
182	C₉	#5070	1	3	yes	complemented
183	C₂³	—	2	2	no	—
184	Q₈	—	108	2	no	—
185	D₄	—	2	4	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	C₂×C₄	—	54	2	no	—
189	C₂×C₄	#6233	1	1	yes	characteristic
190	C₂×C₄	#6233	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
192	C₈	—	54	2	no	—
193	C₂×C₃	—	8	2	no	—
194	C₂×C₃	—	8	6	no	—
195	C₂×C₃	C₂×#3672	1	3	yes	—
196	C₂×C₃	C₂×#3673	1	3	yes	—
197	C₂×C₃	#63498	1	3	yes	—
198	C₂×C₃	C₂×#3753	1	1	yes	characteristic
199	C₂×C₃	C₂×#3754	1	1	yes	characteristic
200	C₂×C₃	#64145	1	1	yes	characteristic
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	C₂×#6233	1	1	yes	characteristic
205	C₄	—	108	2	no	—
206	C₄	#88543	1	2	yes	—
207	C₄	#88544	1	2	yes	—
208	C₄	#88742	1	2	yes	—
209	C₃	#10588750	1	3	yes	complemented
210	C₃	#10592769	1	1	yes	characteristic
211	C₂	—	8	2	no	—
212	C₂	C₂×#88543	1	1	yes	characteristic
213	C₂	C₂×#88544	1	1	yes	characteristic
214	C₂	#11721836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×#127

C₂

yes

maximal, characteristic

#11721706

C₂

yes

maximal, characteristic, complemented

#11720576

C₂

yes

maximal, complemented

C₄×C₃⋊(C₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

—

—

C₂×C₃×D₄×C₉

—

C₂×C₃×D₄×C₉

—

C₂×C₃×D₄×C₉

C₂²

yes

—

C₂×#6226

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₄⋊(C₃⋊Dic₉)

—

C₄⋊(C₃⋊Dic₉)

C₂²

yes

—

C₂×C₃⋊(C₉⋊C₈)

C₂²

yes

—

C₃²×#127

D₃

yes

characteristic

#64078

—

#63659

—

C₄×C₃⋊(C₃⋊C₈)

—

C₉×#127

D₃

yes

—

#63431

—

#63392

—

C₄×C₉⋊C₈

—

C₂³×C₃×C₉

—

C₃×D₄×C₉

—

C₃×D₄×C₉

—

C₃×D₄×C₉

—

C₂×C₃×C₄×C₉

D₄

yes

characteristic

C₂×C₃×C₄×C₉

D₄

yes

characteristic

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₃⋊Dic₉

—

#6226

—

C₃⋊(C₉⋊C₈)

—

#5070

—

#5070

—

C₂×C₃²×D₄

—

C₂×C₃²×D₄

—

C₂×C₃²×D₄

C₂×D₃

yes

—

C₂×#634

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

C₂×D₃

yes

—

C₂×Dic₁₈

—

C₄²×C₉

C₂×D₃

yes

—

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₂×C₉⋊C₈

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₃×C₄×C₉

D₈

yes

—

C₃×C₄×C₉

SD₁₆

yes

—

C₃×C₄×C₉

#41

yes

—

C₃⋊Dic₉

—

C₃×#127

D₉

yes

—

C₃×#127

C₃⋊D₃

yes

—

—

—

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

yes

C₂×C₃²×C₄

yes

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₂³×C₉

—

C₃⋊(C₃⋊C₈)

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

#67

yes

—

C₂×C₄×C₉

#67

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

—

C₂×Dic₉

—

Dic₁₈

—

C₉⋊C₈

—

#486

—

Sylow

C₂×C₃×C₉

—

C₂×C₃×C₉

#124

yes

characteristic

C₂×C₃×C₉

C₂×D₈

yes

characteristic

C₂×C₃×C₉

C₂×SD₁₆

yes

characteristic

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₂×D₉

yes

—

C₂×C₃×D₄

C₂×C₃⋊D₃

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

100

C₃×C₄²

C₂×D₉

yes

—

101

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

102

C₄⋊Dic₃

—

103

C₄⋊Dic₃

—

104

C₄⋊Dic₃

—

105

C₄⋊Dic₃

—

106

C₂×C₃⋊C₈

—

107

C₂×C₃⋊C₈

—

108

C₂²×C₃²

—

109

C₂²×C₃²

—

110

C₂²×C₃²

—

111

C₂²×C₃²

C₂×#67

yes

characteristic

112

C₃²×C₄

#213

yes

—

113

C₃²×C₄

#214

yes

—

114

C₃²×C₄

#237

yes

—

115

C₃⋊Dic₃

—

116

C₂²×C₉

—

117

C₂²×C₉

—

118

C₂²×C₉

—

119

C₂²×C₉

C₂×#67

yes

—

120

C₄×C₉

#213

yes

—

121

C₄×C₉

#214

yes

—

122

C₄×C₉

#237

yes

—

123

Dic₉

—

124

C₄⋊Q₈

—

125

#127

C₃⋊D₉

yes

characteristic

126

#102

—

127

C₄×C₈

—

128

C₃×C₉

#486

yes

characteristic, complemented, Sylow

129

C₂³×C₃

—

130

C₂³×C₃

—

131

C₃×D₄

—

132

C₃×D₄

—

133

C₃×D₄

—

134

C₃×D₄

—

135

C₃×D₄

—

136

C₃×D₄

—

137

C₂×C₃×C₄

#611

yes

—

138

C₂×C₃×C₄

#611

yes

—

139

C₂×C₃×C₄

C₂²×D₉

yes

—

140

C₂×C₃×C₄

#638

yes

characteristic

141

C₂×C₃×C₄

#638

yes

characteristic

142

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

143

C₂×Dic₃

—

144

C₂×Dic₃

—

145

Dic₆

—

146

Dic₆

—

147

C₃⋊C₈

—

148

C₃⋊C₈

—

149

C₂×C₃²

—

150

C₂×C₃²

C₂×#213

yes

characteristic

151

C₂×C₃²

C₂×#214

yes

characteristic

152

C₂×C₃²

#936

yes

characteristic

153

C₂×C₉

—

154

C₂×C₉

C₂×#213

yes

—

155

C₂×C₉

C₂×#214

yes

—

156

C₂×C₉

#936

yes

—

157

C₂×Q₈

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

C₂×C₃⋊D₉

yes

—

161

C₂×C₈

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄²

C₂×C₃⋊D₉

yes

characteristic

165

C₂²×C₃

—

166

C₂²×C₃

—

167

C₂²×C₃

—

168

C₂²×C₃

—

169

C₂²×C₃

—

170

C₂²×C₃

—

171

C₂²×C₃

C₂×#611

yes

—

172

C₂²×C₃

C₂×#638

yes

characteristic

173

C₃×C₄

#3672

yes

—

174

C₃×C₄

#3673

yes

—

175

C₃×C₄

#3699

yes

—

176

C₃×C₄

#3753

yes

—

177

C₃×C₄

#3754

yes

—

178

C₃×C₄

#3830

yes

—

179

Dic₃

—

180

Dic₃

—

181

C₃²

#5070

yes

characteristic

182

C₉

#5070

yes

complemented

183

C₂³

—

184

Q₈

—

108

—

185

D₄

—

186

D₄

—

187

D₄

—

188

C₂×C₄

—

189

C₂×C₄

#6233

yes

characteristic

190

C₂×C₄

#6233

yes

characteristic

191

C₂×C₄

C₂²×C₃⋊D₉

yes

characteristic

192

C₈

—

193

C₂×C₃

—

194

C₂×C₃

—

195

C₂×C₃

C₂×#3672

yes

—

196

C₂×C₃

C₂×#3673

yes

—

197

C₂×C₃

#63498

yes

—

198

C₂×C₃

C₂×#3753

yes

characteristic

199

C₂×C₃

C₂×#3754

yes

characteristic

200

C₂×C₃

#64145

yes

characteristic

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

C₂×#6233

yes

characteristic

205

C₄

—

108

—

206

C₄

#88543

yes

—

207

C₄

#88544

yes

—

208

C₄

#88742

yes

—

209

C₃

#10588750

yes

complemented

210

C₃

#10592769

yes

characteristic

211

C₂

—

212

C₂

C₂×#88543

yes

characteristic

213

C₂

C₂×#88544

yes

characteristic

214

C₂

#11721836

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6233
commutator, cosocle	C₂×C₃×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×#127	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₃²×#127	D₃
hypercenter	#127	C₃⋊D₉
hypocenter	C₃×C₉	#486
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×#6233

commutator, cosocle

C₂×C₃×C₄×C₉

C₂³

Frattini

C₂×C₃×C₄

C₂²×C₃⋊D₃

Fitting

C₃×C₉×#127

C₂

socle

C₂²×C₃²

C₂×#67

prime radical

C₃²×#127

D₃

C₃⋊D₉

C₃×C₉

G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
2. #11721706	C₂	semidirect
3. #11720576	C₂	semidirect
4. C₄×C₃⋊(C₉⋊C₈)	C₂	semidirect
128. C₃×C₉	#486	semidirect
182. C₉	#5070	semidirect
209. C₃	#10588750	semidirect

H ≅ G/N

type

2. #11721706

C₂

semidirect

3. #11720576

C₂

semidirect

4. C₄×C₃⋊(C₉⋊C₈)

C₂

semidirect

128. C₃×C₉

#486

semidirect

182. C₉

#5070

semidirect

209. C₃

#10588750

semidirect

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

subgroup	index
C₃×C₉×#127	2
#11721706	2
#11720576	2
C₄×C₃⋊(C₉⋊C₈)	2
#10592769	3
#10588750	3

subgroup

index

C₃×C₉×#127

C₄×C₃⋊(C₉⋊C₈)

Subgroups — #13144527

Conjugacy classes of subgroups

Complements

Maximal subgroups