Group #11785513 — subgroups

Subgroups — #11785513

Order 1152.

378 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 152 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×C₉×#131	C₂	1	1	yes	maximal, characteristic
2	#10588915	C₂	1	2	yes	maximal
3	C₂×#63372	C₂	1	2	yes	maximal, complemented
4	C₉⋊#338	C₂	1	2	yes	maximal, complemented
5	#72014	—	3	1	no	maximal
6	C₂×C₉×#41	C₂²	1	1	yes	characteristic
7	C₉×#131	—	2	4	no	—
8	C₂×C₉×M₁₆	C₄	1	2	yes	—
9	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₄	1	2	yes	—
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×#3667	C₂²	1	4	yes	—
13	#63394	—	2	2	no	—
14	#63391	—	2	2	no	—
15	#63372	—	2	4	no	—
16	C₂×C₃×#131	D₃	1	1	yes	characteristic
17	#5237	—	3	2	no	—
18	C₂×#819	—	3	2	no	—
19	C₃⋊#338	—	3	2	no	—
20	C₉×#41	—	2	2	no	—
21	C₉×#41	—	2	2	no	—
22	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
23	C₂×D₄×C₉	D₄	1	1	yes	characteristic
24	C₂×D₄×C₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₉×M₁₆	—	2	2	no	—
28	C₉×M₁₆	—	2	4	no	—
29	C₂×C₈×C₉	—	2	4	no	—
30	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₉	D₄	1	4	yes	—
34	#3667	—	2	4	no	—
35	#3667	—	2	4	no	—
36	C₂×C₉⋊C₈	—	2	4	no	—
37	#1813	—	9	1	no	Sylow
38	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
39	C₃×#131	—	2	4	no	—
40	C₂×C₃×M₁₆	Dic₃	1	2	yes	—
41	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃×C₈	Dic₃	1	2	yes	—
43	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂×#208	—	3	4	no	—
45	#836	—	6	2	no	—
46	#833	—	6	2	no	—
47	#819	—	6	4	no	—
48	C₂³×C₉	C₄⋊C₄	1	2	yes	—
49	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
50	Q₈×C₉	—	2	2	no	—
51	D₄×C₉	—	2	4	no	—
52	D₄×C₉	—	2	2	no	—
53	C₂×C₄×C₉	—	2	2	no	—
54	C₂×C₄×C₉	—	2	2	no	—
55	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₉	#41	1	2	yes	—
60	C₈×C₉	—	2	2	no	—
61	C₈×C₉	C₂²⋊C₄	1	2	yes	—
62	C₈×C₉	C₂²⋊C₄	1	2	yes	—
63	C₉⋊C₈	—	4	4	no	—
64	C₂×#131	D₉	1	1	yes	characteristic
65	C₂×#108	—	9	2	no	—
66	#413	—	9	2	no	—
67	#338	—	9	2	no	—
68	C₃×#41	—	2	2	no	—
69	C₃×#41	—	2	2	no	—
70	C₂×C₃×Q₈	D₁₂	1	1	yes	characteristic
71	C₂×C₃×D₄	Dic₆	1	2	yes	—
72	C₂×C₃×D₄	D₁₂	1	1	yes	characteristic
73	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
74	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₃×M₁₆	—	2	2	no	—
76	C₃×M₁₆	—	2	4	no	—
77	C₂×C₃×C₈	—	2	4	no	—
78	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
79	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
81	C₂×C₃×C₈	#67	1	4	yes	—
82	#208	—	6	4	no	—
83	#208	—	6	4	no	—
84	C₂×C₃⋊C₈	—	6	4	no	—
85	C₂²×C₉	—	2	2	no	—
86	C₂²×C₉	—	4	1	no	—
87	C₂²×C₉	—	4	2	no	—
88	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
90	C₄×C₉	—	2	2	no	—
91	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₉	#117	1	1	yes	characteristic
93	C₄×C₉	#121	1	2	yes	—
94	C₂×#41	C₂×D₉	1	1	yes	characteristic
95	#131	—	2	4	no	—
96	C₂×M₁₆	—	9	4	no	—
97	C₂×M₁₆	Dic₉	1	2	yes	—
98	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
99	C₂²×C₈	Dic₉	1	2	yes	—
100	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
101	#108	—	18	4	no	—
102	#101	—	18	2	no	—
103	#100	—	18	2	no	—
104	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
105	C₂³×C₃	C₂×Dic₆	1	1	yes	characteristic
106	C₃×Q₈	—	2	2	no	—
107	C₃×D₄	—	2	4	no	—
108	C₃×D₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
112	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
113	C₂×C₃×C₄	#235	1	2	yes	—
114	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×#67	1	2	yes	—
116	C₃×C₈	—	2	2	no	—
117	C₃×C₈	#217	1	2	yes	—
118	C₃×C₈	#217	1	2	yes	—
119	C₃⋊C₈	—	12	4	no	—
120	C₂×C₉	—	2	1	no	—
121	C₂×C₉	—	2	2	no	—
122	C₂×C₉	—	4	2	no	—
123	C₂×C₉	#380	1	1	yes	characteristic
124	#41	—	2	2	no	—
125	#41	—	2	2	no	—
126	C₂×Q₈	D₃₆	1	1	yes	characteristic
127	C₂×D₄	Dic₁₈	1	2	yes	—
128	C₂×D₄	D₃₆	1	1	yes	characteristic
129	C₂²×C₄	C₂×Dic₉	1	2	yes	—
130	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
131	M₁₆	—	2	2	no	—
132	M₁₆	—	2	4	no	—
133	M₁₆	—	18	4	no	—
134	M₁₆	—	18	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	18	4	no	—
137	C₂×C₈	C₂×Dic₉	1	2	yes	—
138	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
140	C₂×C₈	#611	1	4	yes	—
141	C₂²×C₃	—	2	2	no	—
142	C₂²×C₃	—	4	1	no	—
143	C₂²×C₃	—	4	2	no	—
144	C₂²×C₃	#924	1	2	yes	—
145	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
146	C₃×C₄	—	2	2	no	—
147	C₃×C₄	#926	1	1	yes	characteristic
148	C₃×C₄	#930	1	2	yes	—
149	C₃×C₄	C₂×#217	1	1	yes	characteristic
150	C₉	#1813	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₄⋊Dic₉	1	2	yes	—
152	C₂³	C₂×Dic₁₈	1	1	yes	characteristic
153	Q₈	—	2	2	no	—
154	D₄	—	2	4	no	—
155	D₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₄⋊Dic₉	1	2	yes	—
159	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
160	C₂×C₄	#3697	1	2	yes	—
161	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
162	C₂×C₄	C₂×#611	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	—	36	4	no	—
165	C₈	#3676	1	2	yes	—
166	C₈	#3676	1	2	yes	—
167	C₂×C₃	—	2	1	no	—
168	C₂×C₃	—	2	2	no	—
169	C₂×C₃	—	4	2	no	—
170	C₂×C₃	#5212	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	—	4	1	no	—
173	C₂²	—	4	2	no	—
174	C₂²	#63486	1	2	yes	—
175	C₂²	C₂×C₄⋊Dic₉	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	#63488	1	1	yes	characteristic
178	C₄	#63492	1	2	yes	—
179	C₄	C₂×#3676	1	1	yes	characteristic
180	C₃	#72014	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	2	no	—
183	C₂	—	4	2	no	—
184	C₂	#10588892	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×#131

C₂

yes

maximal, characteristic

#10588915

C₂

yes

maximal

C₂×#63372

C₂

yes

maximal, complemented

C₉⋊#338

C₂

yes

maximal, complemented

#72014

—

maximal

C₂×C₉×#41

C₂²

yes

characteristic

C₉×#131

—

C₂×C₉×M₁₆

C₄

yes

—

C₂×C₉×M₁₆

C₂²

yes

characteristic

C₂²×C₈×C₉

C₄

yes

—

C₂²×C₈×C₉

C₂²

yes

characteristic

C₂×#3667

C₂²

yes

—

#63394

—

#63391

—

#63372

—

C₂×C₃×#131

D₃

yes

—

—

—

C₉×#41

—

C₉×#41

—

C₂×Q₈×C₉

D₄

yes

characteristic

C₂×D₄×C₉

D₄

yes

characteristic

C₂×D₄×C₉

Q₈

yes

—

C₂²×C₄×C₉

C₂×C₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂×C₄

yes

—

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

D₄

yes

—

#3667

—

#3667

—

C₂×C₉⋊C₈

—

#1813

—

Sylow

C₂×C₃×#41

C₂×D₃

yes

characteristic

C₃×#131

—

C₂×C₃×M₁₆

Dic₃

yes

—

C₂×C₃×M₁₆

C₂×D₃

yes

characteristic

C₂²×C₃×C₈

Dic₃

yes

—

C₂²×C₃×C₈

C₂×D₃

yes

—

—

—

—

C₂³×C₉

C₄⋊C₄

yes

—

C₂³×C₉

C₂×Q₈

yes

characteristic

Q₈×C₉

—

D₄×C₉

—

D₄×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

characteristic

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

#41

yes

—

C₈×C₉

—

C₈×C₉

C₂²⋊C₄

yes

—

C₈×C₉

C₂²⋊C₄

yes

—

C₉⋊C₈

—

C₂×#131

D₉

yes

—

—

—

C₃×#41

—

C₃×#41

—

C₂×C₃×Q₈

D₁₂

yes

characteristic

C₂×C₃×D₄

Dic₆

yes

—

C₂×C₃×D₄

D₁₂

yes

characteristic

C₂²×C₃×C₄

C₂×Dic₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×Dic₃

yes

—

C₂×C₃×C₈

C₂×Dic₃

yes

characteristic

C₂×C₃×C₈

C₂×Dic₃

yes

characteristic

C₂×C₃×C₈

#67

yes

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉

C₂²⋊Q₈

yes

—

C₄×C₉

—

C₄×C₉

C₂×C₂²⋊C₄

yes

C₄×C₉

yes

C₄×C₉

yes

—

C₂×#41

C₂×D₉

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

Dic₉

yes

—

C₂×M₁₆

C₂×D₉

yes

characteristic

C₂²×C₈

Dic₉

yes

—

100

C₂²×C₈

C₂×D₉

yes

characteristic

101

#108

—

102

#101

—

103

#100

—

104

C₂³×C₃

C₄⋊Dic₃

yes

—

105

C₂³×C₃

C₂×Dic₆

yes

characteristic

106

C₃×Q₈

—

107

C₃×D₄

—

108

C₃×D₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

C₄⋊Dic₃

yes

—

112

C₂×C₃×C₄

C₂×D₁₂

yes

characteristic

113

C₂×C₃×C₄

#235

yes

—

114

C₂×C₃×C₄

C₂²×Dic₃

yes

characteristic

115

C₂×C₃×C₄

C₂×#67

yes

—

116

C₃×C₈

—

117

C₃×C₈

#217

yes

—

118

C₃×C₈

#217

yes

—

119

C₃⋊C₈

—

120

C₂×C₉

—

121

C₂×C₉

—

122

C₂×C₉

—

123

C₂×C₉

#380

yes

characteristic

124

#41

—

125

#41

—

126

C₂×Q₈

D₃₆

yes

characteristic

127

C₂×D₄

Dic₁₈

yes

—

128

C₂×D₄

D₃₆

yes

characteristic

129

C₂²×C₄

C₂×Dic₉

yes

—

130

C₂²×C₄

C₂²×D₉

yes

characteristic

131

M₁₆

—

132

M₁₆

—

133

M₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

C₂×Dic₉

yes

—

138

C₂×C₈

C₂×Dic₉

yes

characteristic

139

C₂×C₈

C₂×Dic₉

yes

characteristic

140

C₂×C₈

#611

yes

—

141

C₂²×C₃

—

142

C₂²×C₃

—

143

C₂²×C₃

—

144

C₂²×C₃

#924

yes

—

145

C₂²×C₃

C₂×C₄⋊Dic₃

yes

characteristic

146

C₃×C₄

—

147

C₃×C₄

#926

yes

characteristic

148

C₃×C₄

#930

yes

—

149

C₃×C₄

C₂×#217

yes

characteristic

150

C₉

#1813

yes

characteristic, complemented, Sylow

151

C₂³

C₄⋊Dic₉

yes

—

152

C₂³

C₂×Dic₁₈

yes

characteristic

153

Q₈

—

154

D₄

—

155

D₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

C₄⋊Dic₉

yes

—

159

C₂×C₄

C₂×D₃₆

yes

characteristic

160

C₂×C₄

#3697

yes

—

161

C₂×C₄

C₂²×Dic₉

yes

characteristic

162

C₂×C₄

C₂×#611

yes

—

163

C₈

—

164

C₈

—

165

C₈

#3676

yes

—

166

C₈

#3676

yes

—

167

C₂×C₃

—

168

C₂×C₃

—

169

C₂×C₃

—

170

C₂×C₃

#5212

yes

characteristic

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#63486

yes

—

175

C₂²

C₂×C₄⋊Dic₉

yes

characteristic

176

C₄

—

177

C₄

#63488

yes

characteristic

178

C₄

#63492

yes

—

179

C₄

C₂×#3676

yes

characteristic

180

C₃

#72014

yes

characteristic

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

#10588892

yes

characteristic

	subgroup	G/H
center	C₄	#63488
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#131	C₂
socle	C₂×C₃	#5212
prime radical	C₂×C₃×#41	C₂×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#131	D₉
hypocenter	C₉	#1813
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#63488

commutator

C₂×C₄×C₉

C₂²×C₄

Frattini

C₂²×C₃×C₄

C₂²×D₃

Fitting

C₂×C₉×#131

C₂

socle

C₂×C₃

#5212

prime radical

C₂×C₃×#41

C₂×D₃

cosocle

C₂²×C₄×C₉

C₂³

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
3. C₂×#63372	C₂	semidirect
4. C₉⋊#338	C₂	semidirect
150. C₉	#1813	semidirect

H ≅ G/N

type

3. C₂×#63372

C₂

semidirect

4. C₉⋊#338

C₂

semidirect

150. C₉

#1813

semidirect

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

subgroup	index
C₂×C₉×#131	2
#10588915	2
C₂×#63372	2
C₉⋊#338	2
#72014	3

subgroup

index

C₂×C₉×#131

Subgroups — #11785513

Conjugacy classes of subgroups

Complements

Maximal subgroups