Group #11786272 — subgroups

Subgroups — #11786272

Order 1152.

384 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 158 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	C₉×#384	C₂	1	1	yes	maximal, characteristic
2	C₂×#63367	C₂	1	2	yes	maximal
3	C₂×#63366	C₂	1	2	yes	maximal
4	#10588616	C₂	1	1	yes	maximal, characteristic
5	C₂×#63362	C₂	1	1	yes	maximal, characteristic
6	#69578	—	3	1	no	maximal
7	C₂×C₉×C₄⋊C₄	—	2	2	no	—
8	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
12	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
13	C₉⋊#95	—	2	2	no	—
14	#63367	—	2	4	no	—
15	#63366	—	2	4	no	—
16	#63362	C₄	1	4	yes	complemented
17	C₃×#384	D₃	1	1	yes	characteristic
18	C₂×#808	—	3	2	no	—
19	C₂×#807	—	3	2	no	—
20	#4936	—	3	1	no	—
21	C₂×#803	—	3	1	no	—
22	C₂²×C₄×C₉	—	2	2	no	—
23	C₂²×C₄×C₉	D₄	1	2	yes	—
24	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₉	—	2	1	no	—
26	C₂²×Dic₉	—	2	1	no	—
27	C₉×C₄⋊C₄	—	2	2	no	—
28	C₉×C₄⋊C₄	—	2	2	no	—
29	C₉×C₄⋊C₄	—	4	4	no	—
30	C₄²×C₉	C₂×C₄	1	2	yes	—
31	C₄²×C₉	Q₈	1	2	yes	—
32	C₄⋊Dic₉	—	2	2	no	—
33	C₄⋊Dic₉	—	2	2	no	—
34	C₂×C₉⋊C₈	C₂×C₄	1	4	yes	—
35	C₂×C₉⋊C₈	D₄	1	4	yes	—
36	C₂×C₉⋊C₈	Q₈	1	4	yes	—
37	#1847	—	9	1	no	Sylow
38	C₂×C₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
40	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂×C₄⋊Dic₃	—	3	1	no	—
42	C₂×C₄⋊Dic₃	—	3	1	no	—
43	C₂²×C₃⋊C₈	—	3	2	no	—
44	C₃⋊#95	—	6	2	no	—
45	#808	—	6	4	no	—
46	#807	—	6	4	no	—
47	#803	—	3	4	no	—
48	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₉	—	2	2	no	—
50	C₂×C₄×C₉	—	2	2	no	—
51	C₂×C₄×C₉	—	4	2	no	—
52	C₂×C₄×C₉	—	4	4	no	—
53	C₂×C₄×C₉	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
58	C₂×Dic₉	—	4	1	no	—
59	C₂×Dic₉	—	4	1	no	—
60	C₂×Dic₉	—	4	2	no	—
61	C₂×Dic₉	—	4	2	no	—
62	C₉⋊C₈	C₄⋊C₄	1	8	yes	complemented
63	C₂×#107	—	9	2	no	—
64	C₂×#106	—	9	2	no	—
65	C₂×#97	—	9	1	no	—
66	#384	D₉	1	1	yes	characteristic
67	#382	—	9	1	no	—
68	C₂²×C₃×C₄	—	2	2	no	—
69	C₂²×C₃×C₄	#67	1	2	yes	—
70	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
71	C₂²×Dic₃	—	6	1	no	—
72	C₂²×Dic₃	—	6	1	no	—
73	C₃×C₄⋊C₄	—	2	2	no	—
74	C₃×C₄⋊C₄	—	2	2	no	—
75	C₃×C₄⋊C₄	—	4	4	no	—
76	C₃×C₄²	Dic₆	1	2	yes	—
77	C₃×C₄²	C₄×D₃	1	2	yes	—
78	C₄⋊Dic₃	—	6	2	no	—
79	C₄⋊Dic₃	—	6	2	no	—
80	C₂×C₃⋊C₈	—	3	4	no	—
81	C₂×C₃⋊C₈	—	3	4	no	—
82	C₂×C₃⋊C₈	—	3	4	no	—
83	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₉	#127	1	1	yes	characteristic
85	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
87	C₂²×C₉	#137	1	2	yes	—
88	C₄×C₉	—	2	4	no	—
89	C₄×C₉	—	8	2	no	—
90	C₄×C₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₄×C₉	C₄⋊Q₈	1	2	yes	—
92	Dic₉	—	8	1	no	—
93	Dic₉	—	8	1	no	—
94	C₂²×C₈	—	9	2	no	—
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	9	1	no	—
97	C₂×C₄⋊C₄	—	9	1	no	—
98	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
99	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
100	#107	—	18	4	no	—
101	#106	—	18	4	no	—
102	#97	—	9	4	no	—
103	#95	—	18	2	no	—
104	C₂³×C₃	C₂×#67	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	4	4	no	—
108	C₂×C₃×C₄	—	4	2	no	—
109	C₂×C₃×C₄	Dic₃⋊C₄	1	4	yes	—
110	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
112	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
113	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
114	C₂×Dic₃	—	12	2	no	—
115	C₂×Dic₃	—	12	1	no	—
116	C₂×Dic₃	—	12	1	no	—
117	C₂×Dic₃	—	12	2	no	—
118	C₃⋊C₈	—	3	8	no	—
119	C₂×C₉	#384	1	1	yes	characteristic
120	C₂×C₉	#428	1	2	yes	—
121	C₂×C₉	#496	1	1	yes	characteristic
122	C₂×C₉	#497	1	1	yes	characteristic
123	C₂×C₉	#501	1	2	yes	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	18	1	no	—
126	C₂²×C₄	—	18	1	no	—
127	C₂²×C₄	#611	1	2	yes	—
128	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
129	C₂×C₈	—	9	4	no	—
130	C₂×C₈	—	9	4	no	—
131	C₂×C₈	—	9	4	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	18	2	no	—
136	C₄⋊C₄	—	18	2	no	—
137	C₄²	Dic₁₈	1	2	yes	—
138	C₄²	C₄×D₉	1	2	yes	—
139	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
140	C₂²×C₃	#932	1	2	yes	—
141	C₂²×C₃	#940	1	1	yes	characteristic
142	C₂²×C₃	#942	1	2	yes	—
143	C₂²×C₃	#944	1	1	yes	characteristic
144	C₃×C₄	—	2	4	no	—
145	C₃×C₄	—	8	2	no	—
146	C₃×C₄	#888	1	2	yes	—
147	C₃×C₄	D₃×C₄⋊C₄	1	2	yes	—
148	Dic₃	—	24	1	no	—
149	Dic₃	—	24	1	no	—
150	C₉	#1847	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₂×#611	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	36	2	no	—
157	C₂×C₄	—	36	1	no	—
158	C₂×C₄	—	36	1	no	—
159	C₂×C₄	—	36	2	no	—
160	C₂×C₄	Dic₉⋊C₄	1	4	yes	—
161	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
162	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
163	C₂×C₄	D₄×D₉	1	2	yes	—
164	C₂×C₄	Q₈×D₉	1	2	yes	—
165	C₈	—	9	8	no	—
166	C₂×C₃	#4964	1	2	yes	—
167	C₂×C₃	#4975	1	1	yes	characteristic
168	C₂×C₃	#5074	1	1	yes	characteristic
169	C₂×C₃	#5081	1	2	yes	—
170	C₂×C₃	#5086	1	1	yes	characteristic
171	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
172	C₂²	#63495	1	2	yes	—
173	C₂²	#63502	1	1	yes	characteristic
174	C₂²	#63505	1	2	yes	—
175	C₂²	#63506	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	8	2	no	—
178	C₄	—	72	1	no	—
179	C₄	—	72	1	no	—
180	C₄	#63450	1	2	yes	—
181	C₄	C₄⋊C₄×D₉	1	2	yes	—
182	C₃	#69578	1	1	yes	characteristic
183	C₂	#10588644	1	2	yes	—
184	C₂	#10588655	1	1	yes	characteristic
185	C₂	#10588756	1	1	yes	characteristic
186	C₂	#10588761	1	2	yes	—
187	C₂	#10588771	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#384

C₂

yes

maximal, characteristic

C₂×#63367

C₂

yes

maximal

C₂×#63366

C₂

yes

maximal

#10588616

C₂

yes

maximal, characteristic

C₂×#63362

C₂

yes

maximal, characteristic

#69578

—

maximal

C₂×C₉×C₄⋊C₄

—

C₂×C₉×C₄⋊C₄

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

—

—

—

—

C₄

yes

complemented

C₃×#384

D₃

yes

—

—

—

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

D₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂²×Dic₉

—

C₂²×Dic₉

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

C₂×C₄

yes

—

C₄²×C₉

Q₈

yes

—

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₂×C₉⋊C₈

C₂×C₄

yes

—

C₂×C₉⋊C₈

D₄

yes

—

C₂×C₉⋊C₈

Q₈

yes

—

#1847

—

Sylow

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₃×C₄²

C₂×D₃

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂²×C₃⋊C₈

—

—

—

—

—

C₂³×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₄⋊C₄

yes

—

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₉

C₂×Q₈

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₉⋊C₈

C₄⋊C₄

yes

complemented

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

D₉

yes

characteristic

#382

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

#67

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

Dic₆

yes

—

C₃×C₄²

C₄×D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉

#127

yes

characteristic

C₂²×C₉

C₄⋊Q₈

yes

characteristic

C₂²×C₉

AGL(1,ℤ/8)

yes

—

C₂²×C₉

#137

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×C₄⋊C₄

yes

—

C₄×C₉

C₄⋊Q₈

yes

—

Dic₉

—

Dic₉

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉

yes

—

C₂×C₄²

C₂×D₉

yes

characteristic

100

#107

—

101

#106

—

102

#97

—

103

#95

—

104

C₂³×C₃

C₂×#67

yes

characteristic

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

110

C₂×C₃×C₄

C₂×Dic₆

yes

characteristic

111

C₂×C₃×C₄

C₂×C₄×D₃

yes

characteristic

112

C₂×C₃×C₄

D₃×D₄

yes

—

113

C₂×C₃×C₄

D₃×Q₈

yes

—

114

C₂×Dic₃

—

115

C₂×Dic₃

—

116

C₂×Dic₃

—

117

C₂×Dic₃

—

118

C₃⋊C₈

—

119

C₂×C₉

#384

yes

characteristic

120

C₂×C₉

#428

yes

—

121

C₂×C₉

#496

yes

characteristic

122

C₂×C₉

#497

yes

characteristic

123

C₂×C₉

#501

yes

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

#611

yes

—

128

C₂²×C₄

C₂²×D₉

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

Dic₁₈

yes

—

138

C₄²

C₄×D₉

yes

—

139

C₂²×C₃

C₂×Dic₃⋊C₄

yes

characteristic

140

C₂²×C₃

#932

yes

—

141

C₂²×C₃

#940

yes

characteristic

142

C₂²×C₃

#942

yes

—

143

C₂²×C₃

#944

yes

characteristic

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

#888

yes

—

147

C₃×C₄

D₃×C₄⋊C₄

yes

—

148

Dic₃

—

149

Dic₃

—

150

C₉

#1847

yes

characteristic, complemented, Sylow

151

C₂³

C₂×#611

yes

characteristic

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

Dic₉⋊C₄

yes

—

161

C₂×C₄

C₂×Dic₁₈

yes

characteristic

162

C₂×C₄

C₂×C₄×D₉

yes

characteristic

163

C₂×C₄

D₄×D₉

yes

—

164

C₂×C₄

Q₈×D₉

yes

—

165

C₈

—

166

C₂×C₃

#4964

yes

—

167

C₂×C₃

#4975

yes

characteristic

168

C₂×C₃

#5074

yes

characteristic

169

C₂×C₃

#5081

yes

—

170

C₂×C₃

#5086

yes

characteristic

171

C₂²

C₂×Dic₉⋊C₄

yes

characteristic

172

C₂²

#63495

yes

—

173

C₂²

#63502

yes

characteristic

174

C₂²

#63505

yes

—

175

C₂²

#63506

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#63450

yes

—

181

C₄

C₄⋊C₄×D₉

yes

—

182

C₃

#69578

yes

characteristic

183

C₂

#10588644

yes

—

184

C₂

#10588655

yes

characteristic

185

C₂

#10588756

yes

characteristic

186

C₂

#10588761

yes

—

187

C₂

#10588771

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#384	C₂
socle, prime radical	C₂³×C₃	C₂×#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#384	D₉
hypocenter	C₉	#1847
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×#611

commutator

C₂×C₄×C₉

C₂²×C₄

Frattini

C₂²×C₃×C₄

C₂²×D₃

Fitting

C₉×#384

C₂

socle, prime radical

C₂³×C₃

C₂×#67

cosocle

C₂²×C₄×C₉

C₂³

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
16. #63362	C₄	semidirect
62. C₉⋊C₈	C₄⋊C₄	semidirect
150. C₉	#1847	semidirect

H ≅ G/N

type

16. #63362

C₄

semidirect

62. C₉⋊C₈

C₄⋊C₄

semidirect

150. C₉

#1847

semidirect

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

subgroup	index
C₉×#384	2
C₂×#63367	2
C₂×#63366	2
#10588616	2
C₂×#63362	2
#69578	3

subgroup

index

Subgroups — #11786272

Conjugacy classes of subgroups

Complements

Maximal subgroups