Group #11786324 — subgroups

Subgroups — #11786324

Order 1152.

318 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 134 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×#108	C₂	1	1	yes	maximal, characteristic
2	#10588802	C₂	1	2	yes	maximal
3	C₂×#63381	C₂	1	2	yes	maximal
4	C₂×#63375	C₂	1	1	yes	maximal, characteristic
5	C₂×#63372	C₂	1	1	yes	maximal, characteristic
6	#70694	—	3	1	no	maximal
7	C₂×C₉×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
10	C₂×#3667	C₂²	1	2	yes	—
11	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₉×#108	—	2	2	no	—
13	#63381	—	2	4	no	—
14	#63375	C₄	1	4	yes	—
15	Dic₉⋊C₈	—	2	4	no	—
16	#63372	—	2	2	no	—
17	C₂×C₃×#108	D₃	1	1	yes	characteristic
18	C₂×#822	—	3	2	no	—
19	#5118	—	3	2	no	—
20	C₂×#819	—	3	1	no	—
21	C₂×#815	—	3	1	no	—
22	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₉	Q₈	1	2	yes	—
24	C₉×M₁₆	—	2	2	no	—
25	C₉×M₁₆	—	2	2	no	—
26	C₂×C₈×C₉	—	2	2	no	—
27	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₉	D₄	1	2	yes	—
29	C₂×C₈×C₉	Q₈	1	2	yes	—
30	C₄×Dic₉	C₂×C₄	1	2	yes	—
31	C₄×Dic₉	D₄	1	2	yes	—
32	#3667	—	2	2	no	—
33	#3667	—	2	2	no	—
34	C₂×C₉⋊C₈	—	2	2	no	—
35	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₉⋊C₈	D₄	1	2	yes	—
37	C₂×C₉⋊C₈	Q₈	1	2	yes	—
38	#1849	—	9	1	no	Sylow
39	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
40	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂×C₄×Dic₃	—	3	1	no	—
42	C₂×#208	—	3	2	no	—
43	C₂²×C₃⋊C₈	—	3	1	no	—
44	C₃×#108	—	2	2	no	—
45	#822	—	6	4	no	—
46	#819	—	6	2	no	—
47	#815	—	3	4	no	—
48	Dic₃⋊C₈	—	6	4	no	—
49	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×Dic₉	—	4	2	no	—
57	C₂×Dic₉	C₄⋊C₄	1	4	yes	—
58	C₈×C₉	—	4	2	no	—
59	C₈×C₉	C₄⋊C₄	1	4	yes	—
60	C₉⋊C₈	—	4	2	no	—
61	C₉⋊C₈	C₄⋊C₄	1	4	yes	—
62	C₂×#108	—	9	2	no	—
63	C₂×#108	—	9	1	no	—
64	C₂×#108	D₉	1	1	yes	characteristic
65	#423	—	9	2	no	—
66	C₂×#97	—	9	1	no	—
67	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
68	C₂²×Dic₃	—	3	2	no	—
69	C₃×M₁₆	—	2	2	no	—
70	C₃×M₁₆	—	2	2	no	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	Dic₆	1	2	yes	—
73	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
74	C₂×C₃×C₈	#67	1	2	yes	—
75	C₄×Dic₃	—	3	2	no	—
76	C₄×Dic₃	—	3	2	no	—
77	#208	—	6	2	no	—
78	#208	—	6	2	no	—
79	C₂×C₃⋊C₈	—	3	2	no	—
80	C₂×C₃⋊C₈	—	3	2	no	—
81	C₂×C₃⋊C₈	—	3	2	no	—
82	C₂×C₃⋊C₈	—	6	2	no	—
83	C₂²×C₉	—	2	2	no	—
84	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
87	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₉	#127	1	1	yes	characteristic
90	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
91	Dic₉	—	2	4	no	—
92	C₂×M₁₆	—	9	2	no	—
93	C₂×M₁₆	C₂×D₉	1	2	yes	—
94	C₂²×C₈	—	9	1	no	—
95	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
96	C₂×C₄²	—	9	1	no	—
97	#108	—	2	2	no	—
98	#108	—	18	4	no	—
99	#108	—	18	2	no	—
100	C₄⋊C₈	—	18	4	no	—
101	#97	—	9	4	no	—
102	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
103	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
104	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
105	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
106	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
107	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
109	C₂×Dic₃	—	3	4	no	—
110	C₂×Dic₃	—	12	2	no	—
111	C₃×C₈	—	4	2	no	—
112	C₃×C₈	Dic₃⋊C₄	1	4	yes	—
113	C₃⋊C₈	—	3	4	no	—
114	C₃⋊C₈	—	12	2	no	—
115	C₂×C₉	—	2	1	no	—
116	C₂×C₉	—	2	1	no	—
117	C₂×C₉	#384	1	1	yes	characteristic
118	C₂×C₉	#430	1	2	yes	—
119	C₂²×C₄	—	9	2	no	—
120	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
121	M₁₆	—	2	2	no	—
122	M₁₆	—	2	2	no	—
123	M₁₆	—	18	2	no	—
124	M₁₆	—	18	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	9	2	no	—
127	C₂×C₈	—	9	2	no	—
128	C₂×C₈	—	9	2	no	—
129	C₂×C₈	—	18	2	no	—
130	C₂×C₈	Dic₁₈	1	2	yes	—
131	C₂×C₈	C₄×D₉	1	2	yes	—
132	C₂×C₈	#611	1	2	yes	—
133	C₄²	—	9	2	no	—
134	C₄²	—	9	2	no	—
135	C₂²×C₃	—	2	2	no	—
136	C₂²×C₃	#888	1	1	yes	characteristic
137	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₃	#944	1	1	yes	characteristic
139	C₃×C₄	#888	1	1	yes	characteristic
140	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
141	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
142	C₃×C₄	#940	1	1	yes	characteristic
143	Dic₃	—	6	4	no	—
144	C₉	#1849	1	1	yes	characteristic, complemented, Sylow
145	C₂³	Q₈×D₉	1	1	yes	characteristic
146	C₂×C₄	—	9	4	no	—
147	C₂×C₄	—	36	2	no	—
148	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
150	C₂×C₄	D₄×D₉	1	1	yes	characteristic
151	C₂×C₄	D₄×D₉	1	1	yes	characteristic
152	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
153	C₂×C₄	C₂×#611	1	1	yes	characteristic
154	C₈	—	4	2	no	—
155	C₈	—	9	4	no	—
156	C₈	—	36	2	no	—
157	C₈	Dic₉⋊C₄	1	4	yes	—
158	C₂×C₃	—	2	1	no	—
159	C₂×C₃	—	2	1	no	—
160	C₂×C₃	#4893	1	2	yes	—
161	C₂×C₃	#4975	1	1	yes	characteristic
162	C₂²	—	2	2	no	—
163	C₂²	#63450	1	1	yes	characteristic
164	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
165	C₂²	#63506	1	1	yes	characteristic
166	C₄	—	18	4	no	—
167	C₄	#63450	1	1	yes	characteristic
168	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
169	C₄	C₂×Dic₉⋊C₄	1	1	yes	characteristic
170	C₄	#63502	1	1	yes	characteristic
171	C₃	#70694	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#10588573	1	2	yes	—
175	C₂	#10588655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×#108

C₂

yes

maximal, characteristic

#10588802

C₂

yes

maximal

C₂×#63381

C₂

yes

maximal

C₂×#63375

C₂

yes

maximal, characteristic

C₂×#63372

C₂

yes

maximal, characteristic

#70694

—

maximal

C₂×C₉×M₁₆

C₂²

yes

—

C₂²×C₈×C₉

C₂²

yes

characteristic

C₂×C₄×Dic₉

C₂²

yes

characteristic

C₂×#3667

C₂²

yes

—

C₂²×C₉⋊C₈

C₂²

yes

—

—

C₄

yes

—

Dic₉⋊C₈

—

#63372

—

C₂×C₃×#108

D₃

yes

—

—

—

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂²×Dic₉

Q₈

yes

—

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂×C₄

yes

—

C₂×C₈×C₉

D₄

yes

—

C₂×C₈×C₉

Q₈

yes

—

C₄×Dic₉

C₂×C₄

yes

—

C₄×Dic₉

D₄

yes

—

#3667

—

#3667

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

C₂×C₄

yes

—

C₂×C₉⋊C₈

D₄

yes

—

C₂×C₉⋊C₈

Q₈

yes

—

#1849

—

Sylow

C₂×C₃×M₁₆

C₂×D₃

yes

—

C₂²×C₃×C₈

C₂×D₃

yes

characteristic

C₂×C₄×Dic₃

—

C₂×#208

—

C₂²×C₃⋊C₈

—

—

—

—

—

Dic₃⋊C₈

—

C₂³×C₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₉

C₂×Q₈

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

C₄⋊C₄

yes

—

C₈×C₉

—

C₈×C₉

C₄⋊C₄

yes

—

C₉⋊C₈

—

C₉⋊C₈

C₄⋊C₄

yes

—

C₂×#108

—

C₂×#108

—

C₂×#108

D₉

yes

characteristic

#423

—

C₂×#97

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂²×Dic₃

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

Dic₆

yes

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₂×C₃×C₈

#67

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

—

C₂²×C₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉

C₄⋊Q₈

yes

characteristic

C₂²×C₉

C₄⋊Q₈

yes

characteristic

C₄×C₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₉

#127

yes

characteristic

C₄×C₉

C₄⋊Q₈

yes

characteristic

Dic₉

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉

yes

characteristic

C₂×C₄²

—

#108

—

#108

—

#108

—

100

C₄⋊C₈

—

101

#97

—

102

C₂³×C₃

D₃×Q₈

yes

characteristic

103

C₂×C₃×C₄

C₂×Dic₆

yes

characteristic

104

C₂×C₃×C₄

C₂×C₄×D₃

yes

characteristic

105

C₂×C₃×C₄

D₃×D₄

yes

characteristic

106

C₂×C₃×C₄

D₃×D₄

yes

characteristic

107

C₂×C₃×C₄

D₃×Q₈

yes

characteristic

108

C₂×C₃×C₄

C₂×#67

yes

characteristic

109

C₂×Dic₃

—

110

C₂×Dic₃

—

111

C₃×C₈

—

112

C₃×C₈

Dic₃⋊C₄

yes

—

113

C₃⋊C₈

—

114

C₃⋊C₈

—

115

C₂×C₉

—

116

C₂×C₉

—

117

C₂×C₉

#384

yes

characteristic

118

C₂×C₉

#430

yes

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂²×D₉

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

Dic₁₈

yes

—

131

C₂×C₈

C₄×D₉

yes

—

132

C₂×C₈

#611

yes

—

133

C₄²

—

134

C₄²

—

135

C₂²×C₃

—

136

C₂²×C₃

#888

yes

characteristic

137

C₂²×C₃

D₃×C₄⋊C₄

yes

characteristic

138

C₂²×C₃

#944

yes

characteristic

139

C₃×C₄

#888

yes

characteristic

140

C₃×C₄

D₃×C₄⋊C₄

yes

characteristic

141

C₃×C₄

C₂×Dic₃⋊C₄

yes

characteristic

142

C₃×C₄

#940

yes

characteristic

143

Dic₃

—

144

C₉

#1849

yes

characteristic, complemented, Sylow

145

C₂³

Q₈×D₉

yes

characteristic

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

C₂×Dic₁₈

yes

characteristic

149

C₂×C₄

C₂×C₄×D₉

yes

characteristic

150

C₂×C₄

D₄×D₉

yes

characteristic

151

C₂×C₄

D₄×D₉

yes

characteristic

152

C₂×C₄

Q₈×D₉

yes

characteristic

153

C₂×C₄

C₂×#611

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

Dic₉⋊C₄

yes

—

158

C₂×C₃

—

159

C₂×C₃

—

160

C₂×C₃

#4893

yes

—

161

C₂×C₃

#4975

yes

characteristic

162

C₂²

—

163

C₂²

#63450

yes

characteristic

164

C₂²

C₄⋊C₄×D₉

yes

characteristic

165

C₂²

#63506

yes

characteristic

166

C₄

—

167

C₄

#63450

yes

characteristic

168

C₄

C₄⋊C₄×D₉

yes

characteristic

169

C₄

C₂×Dic₉⋊C₄

yes

characteristic

170

C₄

#63502

yes

characteristic

171

C₃

#70694

yes

characteristic

172

C₂

—

173

C₂

—

174

C₂

#10588573

yes

—

175

C₂

#10588655

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#108	C₂
socle	C₂²×C₃	D₃×C₄⋊C₄
prime radical	C₂³×C₃	D₃×Q₈
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#108	D₉
hypocenter	C₉	#1849
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₉

commutator

C₂×C₄×C₉

C₂²×C₄

Frattini

C₂²×C₃×C₄

C₂²×D₃

Fitting

C₂×C₉×#108

C₂

socle

C₂²×C₃

D₃×C₄⋊C₄

prime radical

C₂³×C₃

D₃×Q₈

cosocle

C₂²×C₄×C₉

C₂³

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
144. C₉	#1849	semidirect

H ≅ G/N

type

144. C₉

#1849

semidirect

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

subgroup	index
C₂×C₉×#108	2
#10588802	2
C₂×#63381	2
C₂×#63375	2
C₂×#63372	2
#70694	3

subgroup

index

C₂×C₉×#108

Subgroups — #11786324

Conjugacy classes of subgroups

Complements

Maximal subgroups