Group #11785971 — subgroups

Subgroups — #11785971

Order 1152.

324 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 126 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×Dic₉⋊C₈	C₂	1	2	yes	maximal
2	#10588612	C₂	1	1	yes	maximal, characteristic
3	C₉×#336	C₂	1	1	yes	maximal, characteristic
4	#10588228	C₂	1	2	yes	maximal
5	#10588200	C₂	1	1	yes	maximal, characteristic
6	#68606	—	3	1	no	maximal
7	C₂²×C₈×C₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
11	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
12	C₉⋊#95	C₄	1	2	yes	—
13	C₉⋊#95	C₄	1	2	yes	—
14	Dic₉⋊C₈	—	2	4	no	—
15	C₂×Dic₃⋊C₈	—	3	2	no	—
16	#4932	—	3	1	no	—
17	C₃×#336	D₃	1	1	yes	characteristic
18	#4550	—	3	2	no	—
19	#4524	—	3	1	no	—
20	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₉	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₉	—	2	2	no	—
26	C₂×C₈×C₉	—	2	2	no	—
27	C₂×C₈×C₉	—	2	2	no	—
28	C₄²×C₉	—	2	2	no	—
29	C₄×Dic₉	—	2	2	no	—
30	C₄×Dic₉	D₄	1	2	yes	—
31	C₄×Dic₉	Q₈	1	2	yes	—
32	C₂×C₉⋊C₈	—	2	2	no	—
33	C₂×C₉⋊C₈	—	2	2	no	—
34	C₂×C₉⋊C₈	—	2	2	no	—
35	#1832	—	9	1	no	Sylow
36	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
37	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
38	C₂×C₄×Dic₃	—	3	1	no	—
39	C₂×C₄×Dic₃	—	3	1	no	—
40	C₂²×C₃⋊C₈	—	3	2	no	—
41	C₃⋊#95	—	3	2	no	—
42	C₃⋊#95	—	3	2	no	—
43	Dic₃⋊C₈	—	6	4	no	—
44	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₉	—	2	2	no	—
46	C₂×C₄×C₉	—	2	2	no	—
47	C₂×C₄×C₉	—	2	2	no	—
48	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₉	#41	1	2	yes	—
51	C₂×C₄×C₉	#41	1	2	yes	—
52	C₂×Dic₉	—	2	2	no	—
53	C₂×Dic₉	—	2	2	no	—
54	C₂×Dic₉	—	2	1	no	—
55	C₂×Dic₉	—	2	1	no	—
56	C₂×Dic₉	—	2	2	no	—
57	C₂×Dic₉	—	2	2	no	—
58	C₂×Dic₉	C₄⋊C₄	1	4	yes	—
59	C₈×C₉	—	4	2	no	—
60	C₉⋊C₈	—	4	2	no	—
61	C₂×C₄⋊C₈	—	9	2	no	—
62	#376	—	9	1	no	—
63	#336	—	9	2	no	—
64	#336	—	9	1	no	—
65	#336	D₉	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
67	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
68	C₂²×Dic₃	—	3	2	no	—
69	C₂²×Dic₃	—	3	1	no	—
70	C₂²×Dic₃	—	3	1	no	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	—	2	2	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₃×C₄²	—	2	2	no	—
75	C₄×Dic₃	—	3	2	no	—
76	C₄×Dic₃	—	3	2	no	—
77	C₄×Dic₃	—	6	2	no	—
78	C₂×C₃⋊C₈	—	6	2	no	—
79	C₂×C₃⋊C₈	—	6	2	no	—
80	C₂×C₃⋊C₈	—	6	2	no	—
81	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₉	#117	1	1	yes	characteristic
83	C₂²×C₉	#117	1	1	yes	characteristic
84	C₂²×C₉	#131	1	2	yes	—
85	C₂²×C₉	#131	1	2	yes	—
86	C₄×C₉	—	4	2	no	—
87	C₄×C₉	#124	1	2	yes	—
88	C₄×C₉	#125	1	2	yes	—
89	Dic₉	—	2	4	no	—
90	Dic₉	—	4	1	no	—
91	Dic₉	—	4	1	no	—
92	C₂²×C₈	—	9	2	no	—
93	C₂²×C₈	C₂×D₉	1	2	yes	—
94	C₂×C₄²	—	9	1	no	—
95	C₂×C₄²	—	9	1	no	—
96	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
97	C₄⋊C₈	—	18	4	no	—
98	#95	—	9	2	no	—
99	#95	—	9	2	no	—
100	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	#235	1	2	yes	—
105	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
106	C₂×C₃×C₄	#237	1	2	yes	—
107	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
108	C₂×Dic₃	—	3	4	no	—
109	C₂×Dic₃	—	6	2	no	—
110	C₂×Dic₃	—	6	2	no	—
111	C₂×Dic₃	—	6	1	no	—
112	C₂×Dic₃	—	6	1	no	—
113	C₂×Dic₃	—	6	2	no	—
114	C₂×Dic₃	—	6	2	no	—
115	C₃×C₈	—	4	2	no	—
116	C₃⋊C₈	—	12	2	no	—
117	C₂×C₉	#382	1	1	yes	characteristic
118	C₂×C₉	#424	1	2	yes	—
119	C₂×C₉	#433	1	1	yes	characteristic
120	C₂×C₉	#433	1	1	yes	characteristic
121	C₂×C₉	#433	1	2	yes	—
122	C₂²×C₄	—	9	2	no	—
123	C₂²×C₄	—	9	1	no	—
124	C₂²×C₄	—	9	1	no	—
125	C₂²×C₄	C₄×D₉	1	2	yes	—
126	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	18	2	no	—
131	C₂×C₈	—	18	2	no	—
132	C₂×C₈	—	18	2	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	9	2	no	—
135	C₄²	—	9	2	no	—
136	C₄²	—	18	2	no	—
137	C₂²×C₃	#872	1	1	yes	characteristic
138	C₂²×C₃	#877	1	1	yes	characteristic
139	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₃	#901	1	2	yes	—
141	C₂²×C₃	#907	1	2	yes	—
142	C₃×C₄	—	4	2	no	—
143	C₃×C₄	#885	1	2	yes	—
144	C₃×C₄	#889	1	2	yes	—
145	Dic₃	—	6	4	no	—
146	Dic₃	—	12	1	no	—
147	Dic₃	—	12	1	no	—
148	C₉	#1832	1	1	yes	characteristic, complemented, Sylow
149	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	9	4	no	—
154	C₂×C₄	—	18	2	no	—
155	C₂×C₄	—	18	2	no	—
156	C₂×C₄	—	18	1	no	—
157	C₂×C₄	—	18	1	no	—
158	C₂×C₄	—	18	2	no	—
159	C₂×C₄	—	18	2	no	—
160	C₂×C₄	#3697	1	2	yes	—
161	C₂×C₄	D₄×D₉	1	1	yes	characteristic
162	C₂×C₄	#3699	1	2	yes	—
163	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₈	—	36	2	no	—
166	C₂×C₃	#4650	1	1	yes	characteristic
167	C₂×C₃	#4690	1	1	yes	characteristic
168	C₂×C₃	#4709	1	1	yes	characteristic
169	C₂×C₃	#4720	1	2	yes	—
170	C₂×C₃	#4839	1	2	yes	—
171	C₂²	#63434	1	1	yes	characteristic
172	C₂²	#63439	1	1	yes	characteristic
173	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
174	C₂²	#63464	1	2	yes	—
175	C₂²	#63469	1	2	yes	—
176	C₄	—	4	2	no	—
177	C₄	—	18	4	no	—
178	C₄	—	36	1	no	—
179	C₄	—	36	1	no	—
180	C₄	#63447	1	2	yes	—
181	C₄	#63451	1	2	yes	—
182	C₃	#68606	1	1	yes	characteristic
183	C₂	#10588330	1	1	yes	characteristic
184	C₂	#10588372	1	1	yes	characteristic
185	C₂	#10588390	1	1	yes	characteristic
186	C₂	#10588400	1	2	yes	—
187	C₂	#10588523	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₉⋊C₈

C₂

yes

maximal

#10588612

C₂

yes

maximal, characteristic

C₉×#336

C₂

yes

maximal, characteristic

#10588228

C₂

yes

maximal

#10588200

C₂

yes

maximal, characteristic

#68606

—

maximal

C₂²×C₈×C₉

C₂²

yes

—

C₂×C₄²×C₉

C₂²

yes

characteristic

C₂×C₄×Dic₉

C₂²

yes

characteristic

C₂×C₄×Dic₉

C₂²

yes

characteristic

C₂²×C₉⋊C₈

C₂²

yes

—

C₉⋊#95

C₄

yes

—

C₉⋊#95

C₄

yes

—

Dic₉⋊C₈

—

C₂×Dic₃⋊C₈

—

#4932

—

C₃×#336

D₃

yes

characteristic

#4550

—

#4524

—

C₂²×C₄×C₉

C₂×C₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂²×Dic₉

C₂×C₄

yes

characteristic

C₂²×Dic₉

C₂×C₄

yes

characteristic

C₂²×Dic₉

C₂×C₄

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₄²×C₉

—

C₄×Dic₉

—

C₄×Dic₉

D₄

yes

—

C₄×Dic₉

Q₈

yes

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

#1832

—

Sylow

C₂²×C₃×C₈

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

characteristic

C₂×C₄×Dic₃

—

C₂×C₄×Dic₃

—

C₂²×C₃⋊C₈

—

C₃⋊#95

—

C₃⋊#95

—

Dic₃⋊C₈

—

C₂³×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₉

#41

yes

—

C₂×C₄×C₉

#41

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

C₄⋊C₄

yes

—

C₈×C₉

—

C₉⋊C₈

—

C₂×C₄⋊C₈

—

—

—

—

D₉

yes

characteristic

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

C₂×C₄⋊C₄

yes

C₂²×C₉

yes

C₂²×C₉

yes

C₂²×C₉

yes

—

C₂²×C₉

#131

yes

—

C₄×C₉

—

C₄×C₉

#124

yes

—

C₄×C₉

#125

yes

—

Dic₉

—

Dic₉

—

Dic₉

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉

yes

characteristic

C₄⋊C₈

—

#95

—

#95

—

100

C₂³×C₃

C₂×C₄×D₃

yes

characteristic

101

C₂×C₃×C₄

—

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

—

104

C₂×C₃×C₄

#235

yes

—

105

C₂×C₃×C₄

D₃×D₄

yes

characteristic

106

C₂×C₃×C₄

#237

yes

—

107

C₂×C₃×C₄

D₃×Q₈

yes

characteristic

108

C₂×Dic₃

—

109

C₂×Dic₃

—

110

C₂×Dic₃

—

111

C₂×Dic₃

—

112

C₂×Dic₃

—

113

C₂×Dic₃

—

114

C₂×Dic₃

—

115

C₃×C₈

—

116

C₃⋊C₈

—

117

C₂×C₉

#382

yes

characteristic

118

C₂×C₉

#424

yes

—

119

C₂×C₉

#433

yes

characteristic

120

C₂×C₉

#433

yes

characteristic

121

C₂×C₉

#433

yes

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₄×D₉

yes

—

126

C₂²×C₄

C₂²×D₉

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₂²×C₃

#872

yes

characteristic

138

C₂²×C₃

#877

yes

characteristic

139

C₂²×C₃

D₃×C₄⋊C₄

yes

characteristic

140

C₂²×C₃

#901

yes

—

141

C₂²×C₃

#907

yes

—

142

C₃×C₄

—

143

C₃×C₄

#885

yes

—

144

C₃×C₄

#889

yes

—

145

Dic₃

—

146

Dic₃

—

147

Dic₃

—

148

C₉

#1832

yes

characteristic, complemented, Sylow

149

C₂³

C₂×C₄×D₉

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

#3697

yes

—

161

C₂×C₄

D₄×D₉

yes

characteristic

162

C₂×C₄

#3699

yes

—

163

C₂×C₄

Q₈×D₉

yes

characteristic

164

C₈

—

165

C₈

—

166

C₂×C₃

#4650

yes

characteristic

167

C₂×C₃

#4690

yes

characteristic

168

C₂×C₃

#4709

yes

characteristic

169

C₂×C₃

#4720

yes

—

170

C₂×C₃

#4839

yes

—

171

C₂²

#63434

yes

characteristic

172

C₂²

#63439

yes

characteristic

173

C₂²

C₄⋊C₄×D₉

yes

characteristic

174

C₂²

#63464

yes

—

175

C₂²

#63469

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#63447

yes

—

181

C₄

#63451

yes

—

182

C₃

#68606

yes

characteristic

183

C₂

#10588330

yes

characteristic

184

C₂

#10588372

yes

characteristic

185

C₂

#10588390

yes

characteristic

186

C₂

#10588400

yes

—

187

C₂

#10588523

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂³×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#336	C₂
socle, prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#336	D₉
hypocenter	C₉	#1832
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center

C₂²×C₄

C₂²×D₉

commutator

C₂³×C₉

C₂²×C₄

Frattini

C₂²×C₃×C₄

C₂²×D₃

Fitting

C₉×#336

C₂

socle, prime radical

C₂³×C₃

C₂×C₄×D₃

cosocle

C₂²×C₄×C₉

C₂³

D₉

C₉

G ▷ C₂³×C₉ ▷ C₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
148. C₉	#1832	semidirect

H ≅ G/N

type

148. C₉

#1832

semidirect

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

subgroup	index
C₂×Dic₉⋊C₈	2
#10588612	2
C₉×#336	2
#10588228	2
#10588200	2
#68606	3

subgroup

index

C₂×Dic₉⋊C₈

Subgroups — #11785971

Conjugacy classes of subgroups

Complements

Maximal subgroups