Group #11783484 — subgroups

Subgroups — #11783484

Order 1152.

408 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 166 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×#63375	C₂	1	2	yes	maximal
2	C₄²×Dic₉	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	#68597	—	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	C₄	1	2	yes	—
10	C₂×C₄×Dic₉	C₄	1	2	yes	—
11	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
13	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
14	#63375	C₄	1	8	yes	complemented
15	C₂×#815	—	3	2	no	—
16	C₄²×Dic₃	—	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	C₂×C₄	1	4	yes	—
28	C₄²×C₉	—	2	2	no	—
29	C₄×Dic₉	—	2	4	no	—
30	C₄×Dic₉	—	2	1	no	—
31	C₄×Dic₉	—	2	1	no	—
32	C₄×Dic₉	—	2	4	no	—
33	C₄×Dic₉	C₂×C₄	1	4	yes	—
34	C₂×C₉⋊C₈	—	2	2	no	—
35	C₂×C₉⋊C₈	—	2	2	no	—
36	C₂×C₉⋊C₈	C₂×C₄	1	4	yes	—
37	#1745	—	9	1	no	Sylow
38	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
39	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂×C₄×Dic₃	—	3	1	no	—
41	C₂×C₄×Dic₃	—	3	2	no	—
42	C₂×C₄×Dic₃	—	3	2	no	—
43	C₂×C₄×Dic₃	—	3	1	no	—
44	C₂²×C₃⋊C₈	—	3	2	no	—
45	#815	—	3	8	no	—
46	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₉	—	2	2	no	—
48	C₂×C₄×C₉	—	2	2	no	—
49	C₂×C₄×C₉	M₁₆	1	4	yes	—
50	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₉	#41	1	2	yes	—
52	C₂×C₄×C₉	#41	1	1	yes	characteristic
53	C₂×C₄×C₉	#41	1	1	yes	characteristic
54	C₂×Dic₉	—	2	2	no	—
55	C₂×Dic₉	—	2	2	no	—
56	C₂×Dic₉	—	2	2	no	—
57	C₂×Dic₉	—	2	2	no	—
58	C₂×Dic₉	C₄²	1	2	yes	—
59	C₂×Dic₉	C₄²	1	2	yes	—
60	C₂×Dic₉	M₁₆	1	4	yes	—
61	C₈×C₉	—	2	4	no	—
62	C₉⋊C₈	—	2	4	no	—
63	C₂×#97	—	9	2	no	—
64	C₄³	—	9	1	no	—
65	#336	—	9	2	no	—
66	#336	—	9	1	no	—
67	#336	D₉	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
69	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
70	C₂²×Dic₃	—	3	1	no	—
71	C₂²×Dic₃	—	3	1	no	—
72	C₂²×Dic₃	—	3	2	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	—	2	2	no	—
75	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
76	C₃×C₄²	—	2	2	no	—
77	C₄×Dic₃	—	3	4	no	—
78	C₄×Dic₃	—	6	4	no	—
79	C₄×Dic₃	—	6	4	no	—
80	C₄×Dic₃	—	6	1	no	—
81	C₄×Dic₃	—	6	1	no	—
82	C₂×C₃⋊C₈	—	3	4	no	—
83	C₂×C₃⋊C₈	—	6	2	no	—
84	C₂×C₃⋊C₈	—	6	2	no	—
85	C₂²×C₉	C₂×C₄²	1	1	yes	characteristic
86	C₂²×C₉	#117	1	1	yes	characteristic
87	C₂²×C₉	#117	1	1	yes	characteristic
88	C₂²×C₉	C₂×M₁₆	1	2	yes	—
89	C₂²×C₉	C₂×M₁₆	1	2	yes	—
90	C₄×C₉	—	2	4	no	—
91	C₄×C₉	#117	1	2	yes	—
92	C₄×C₉	#117	1	2	yes	—
93	Dic₉	—	2	2	no	—
94	Dic₉	—	2	2	no	—
95	Dic₉	—	2	4	no	—
96	C₂²×C₈	—	9	2	no	—
97	C₂²×C₈	C₂×D₉	1	2	yes	—
98	C₂×C₄²	—	9	1	no	—
99	C₂×C₄²	—	9	2	no	—
100	C₂×C₄²	—	9	2	no	—
101	C₂×C₄²	—	9	1	no	—
102	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
103	#97	—	9	8	no	—
104	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
108	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
109	C₂×C₃×C₄	#237	1	1	yes	characteristic
110	C₂×C₃×C₄	#237	1	2	yes	—
111	C₂×C₃×C₄	#239	1	1	yes	characteristic
112	C₂×Dic₃	—	3	2	no	—
113	C₂×Dic₃	—	3	2	no	—
114	C₂×Dic₃	—	3	4	no	—
115	C₂×Dic₃	—	6	2	no	—
116	C₂×Dic₃	—	6	2	no	—
117	C₂×Dic₃	—	6	2	no	—
118	C₂×Dic₃	—	6	2	no	—
119	C₃×C₈	—	2	4	no	—
120	C₃⋊C₈	—	6	4	no	—
121	C₂×C₉	#376	1	1	yes	characteristic
122	C₂×C₉	C₄×M₁₆	1	1	yes	characteristic
123	C₂×C₉	C₄×M₁₆	1	1	yes	characteristic
124	C₂×C₉	#432	1	2	yes	—
125	C₂×C₉	#432	1	2	yes	—
126	C₂²×C₄	—	9	1	no	—
127	C₂²×C₄	—	9	1	no	—
128	C₂²×C₄	—	9	2	no	—
129	C₂²×C₄	C₄×D₉	1	2	yes	—
130	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	9	4	no	—
134	C₂×C₈	—	18	2	no	—
135	C₂×C₈	—	18	2	no	—
136	C₂×C₈	C₄×D₉	1	4	yes	—
137	C₄²	—	2	2	no	—
138	C₄²	—	9	4	no	—
139	C₄²	—	18	4	no	—
140	C₄²	—	18	4	no	—
141	C₄²	—	18	1	no	—
142	C₄²	—	18	1	no	—
143	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
144	C₂²×C₃	#877	1	1	yes	characteristic
145	C₂²×C₃	#892	1	1	yes	characteristic
146	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
147	C₂²×C₃	D₃×M₁₆	1	2	yes	—
148	C₃×C₄	—	2	4	no	—
149	C₃×C₄	#877	1	2	yes	—
150	C₃×C₄	#892	1	2	yes	—
151	Dic₃	—	6	2	no	—
152	Dic₃	—	6	2	no	—
153	Dic₃	—	6	4	no	—
154	C₉	#1745	1	1	yes	characteristic, complemented, Sylow
155	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	9	2	no	—
159	C₂×C₄	—	9	2	no	—
160	C₂×C₄	—	9	4	no	—
161	C₂×C₄	—	18	2	no	—
162	C₂×C₄	—	18	2	no	—
163	C₂×C₄	—	18	2	no	—
164	C₂×C₄	—	18	2	no	—
165	C₂×C₄	#3663	1	4	yes	—
166	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
167	C₂×C₄	#3699	1	1	yes	characteristic
168	C₂×C₄	#3699	1	2	yes	—
169	C₂×C₄	#3701	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	—	18	4	no	—
172	C₂×C₃	#4648	1	1	yes	characteristic
173	C₂×C₃	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
174	C₂×C₃	#4707	1	1	yes	characteristic
175	C₂×C₃	#4719	1	2	yes	—
176	C₂×C₃	#4835	1	2	yes	—
177	C₂²	C₄²×D₉	1	1	yes	characteristic
178	C₂²	#63439	1	1	yes	characteristic
179	C₂²	#63454	1	1	yes	characteristic
180	C₂²	C₂×#3663	1	2	yes	—
181	C₂²	M₁₆×D₉	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	18	4	no	—
184	C₄	—	18	2	no	—
185	C₄	—	18	2	no	—
186	C₄	#63439	1	2	yes	—
187	C₄	#63454	1	2	yes	—
188	C₃	#68597	1	1	yes	characteristic
189	C₂	#10588328	1	1	yes	characteristic
190	C₂	C₄×#3663	1	1	yes	characteristic
191	C₂	#10588388	1	1	yes	characteristic
192	C₂	#10588399	1	2	yes	—
193	C₂	#10588519	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63375

C₂

yes

maximal

C₄²×Dic₉

C₂

yes

maximal, characteristic

C₉×#336

C₂

yes

maximal, characteristic

#10588228

C₂

yes

maximal

#10588200

C₂

yes

maximal, characteristic

#68597

—

maximal

C₂²×C₈×C₉

C₂²

yes

—

C₂×C₄²×C₉

C₂²

yes

characteristic

C₂×C₄×Dic₉

C₄

yes

—

C₂×C₄×Dic₉

C₄

yes

—

C₂×C₄×Dic₉

C₂²

yes

characteristic

C₂×C₄×Dic₉

C₂²

yes

characteristic

C₂²×C₉⋊C₈

C₂²

yes

—

#63375

C₄

yes

complemented

C₂×#815

—

C₄²×Dic₃

—

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

C₂×C₄

yes

—

C₄²×C₉

—

C₄×Dic₉

—

C₄×Dic₉

—

C₄×Dic₉

—

C₄×Dic₉

—

C₄×Dic₉

C₂×C₄

yes

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

C₂×C₄

yes

—

#1745

—

Sylow

C₂²×C₃×C₈

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

characteristic

C₂×C₄×Dic₃

—

C₂×C₄×Dic₃

—

C₂×C₄×Dic₃

—

C₂×C₄×Dic₃

—

C₂²×C₃⋊C₈

—

#815

—

C₂³×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

M₁₆

yes

—

C₂×C₄×C₉

C₂²×C₄

yes

—

C₂×C₄×C₉

#41

yes

—

C₂×C₄×C₉

yes

C₂×C₄×C₉

yes

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

C₄²

yes

—

C₂×Dic₉

C₄²

yes

—

C₂×Dic₉

M₁₆

yes

—

C₈×C₉

—

C₉⋊C₈

—

C₂×#97

—

C₄³

—

#336

—

#336

—

#336

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

C₄×D₃

yes

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

C₂×C₄²

yes

C₂²×C₉

yes

C₂²×C₉

yes

C₂²×C₉

C₂×M₁₆

yes

—

C₂²×C₉

C₂×M₁₆

yes

—

C₄×C₉

—

C₄×C₉

#117

yes

—

C₄×C₉

#117

yes

—

Dic₉

—

Dic₉

—

Dic₉

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂×D₉

yes

characteristic

103

#97

—

104

C₂³×C₃

C₂×C₄×D₃

yes

characteristic

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

ΓL(1,𝔽₂₅)

yes

—

108

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

109

C₂×C₃×C₄

#237

yes

characteristic

110

C₂×C₃×C₄

#237

yes

—

111

C₂×C₃×C₄

#239

yes

characteristic

112

C₂×Dic₃

—

113

C₂×Dic₃

—

114

C₂×Dic₃

—

115

C₂×Dic₃

—

116

C₂×Dic₃

—

117

C₂×Dic₃

—

118

C₂×Dic₃

—

119

C₃×C₈

—

120

C₃⋊C₈

—

121

C₂×C₉

#376

yes

characteristic

122

C₂×C₉

C₄×M₁₆

yes

characteristic

123

C₂×C₉

C₄×M₁₆

yes

characteristic

124

C₂×C₉

#432

yes

—

125

C₂×C₉

#432

yes

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₄×D₉

yes

—

130

C₂²×C₄

C₂²×D₉

yes

characteristic

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

C₄×D₉

yes

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₂²×C₃

C₄²×D₃

yes

characteristic

144

C₂²×C₃

#877

yes

characteristic

145

C₂²×C₃

#892

yes

characteristic

146

C₂²×C₃

C₂×ΓL(1,𝔽₂₅)

yes

—

147

C₂²×C₃

D₃×M₁₆

yes

—

148

C₃×C₄

—

149

C₃×C₄

#877

yes

—

150

C₃×C₄

#892

yes

—

151

Dic₃

—

152

Dic₃

—

153

Dic₃

—

154

C₉

#1745

yes

characteristic, complemented, Sylow

155

C₂³

C₂×C₄×D₉

yes

characteristic

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#3663

yes

—

166

C₂×C₄

C₂×C₄×D₉

yes

—

167

C₂×C₄

#3699

yes

characteristic

168

C₂×C₄

#3699

yes

—

169

C₂×C₄

#3701

yes

characteristic

170

C₈

—

171

C₈

—

172

C₂×C₃

#4648

yes

characteristic

173

C₂×C₃

C₄×ΓL(1,𝔽₂₅)

yes

characteristic

174

C₂×C₃

#4707

yes

characteristic

175

C₂×C₃

#4719

yes

—

176

C₂×C₃

#4835

yes

—

177

C₂²

C₄²×D₉

yes

characteristic

178

C₂²

#63439

yes

characteristic

179

C₂²

#63454

yes

characteristic

180

C₂²

C₂×#3663

yes

—

181

C₂²

M₁₆×D₉

yes

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

#63439

yes

—

187

C₄

#63454

yes

—

188

C₃

#68597

yes

characteristic

189

C₂

#10588328

yes

characteristic

190

C₂

C₄×#3663

yes

characteristic

191

C₂

#10588388

yes

characteristic

192

C₂

#10588399

yes

—

193

C₂

#10588519

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂×C₄²
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	#1745
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center

C₂²×C₄

C₂²×D₉

commutator

C₂²×C₉

C₂×C₄²

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 ▷ C₂²×C₉ ▷ C₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
14. #63375	C₄	semidirect
154. C₉	#1745	semidirect

H ≅ G/N

type

14. #63375

C₄

semidirect

154. C₉

#1745

semidirect

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

subgroup	index
C₂×#63375	2
C₄²×Dic₉	2
C₉×#336	2
#10588228	2
#10588200	2
#68597	3

subgroup

index

C₄²×Dic₉

Subgroups — #11783484

Conjugacy classes of subgroups

Complements

Maximal subgroups