Group #11766783 — subgroups

Subgroups — C₉×#2062 #11766783

Order 1152.

471 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 138 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×#543	C₂	1	1	yes	maximal, characteristic
2	C₉×#385	C₂	1	3	yes	maximal
3	C₉×#342	C₂	1	3	yes	maximal
4	C₃×#2062	C₃	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
6	C₉×C₂²⋊Q₈	—	2	3	no	—
7	C₉×C₂²⋊Q₈	—	2	3	no	—
8	C₂×C₉×C₄⋊C₄	—	2	3	no	—
9	C₂×C₉×C₂²⋊C₄	C₂²	1	3	yes	—
10	C₂×C₉×C₂²⋊C₄	C₂²	1	3	yes	—
11	C₉×#95	—	2	3	no	—
12	C₃×#543	C₂×C₃	1	1	yes	characteristic
13	C₃×#385	C₂×C₃	1	3	yes	—
14	C₃×#342	C₂×C₃	1	3	yes	—
15	C₂⁴×C₉	C₂³	1	1	yes	characteristic
16	C₂×Q₈×C₉	—	2	1	no	—
17	C₂×Q₈×C₉	—	2	1	no	—
18	C₂²×C₄×C₉	—	2	3	no	—
19	C₂²×C₄×C₉	—	2	3	no	—
20	C₂²×C₄×C₉	—	4	2	no	—
21	C₂²×C₄×C₉	D₄	1	3	yes	—
22	C₂²×C₄×C₉	D₄	1	3	yes	—
23	C₉×C₄⋊C₄	—	2	3	no	—
24	C₉×C₄⋊C₄	—	2	3	no	—
25	C₉×C₄⋊C₄	—	4	6	no	—
26	C₉×C₂²⋊C₄	—	2	6	no	—
27	C₉×C₂²⋊C₄	—	2	3	no	—
28	C₉×C₂²⋊C₄	Q₈	1	6	yes	complemented
29	#2062	C₉	1	1	yes	characteristic, direct product factor, Sylow
30	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
31	C₃×C₂²⋊Q₈	—	2	3	no	—
32	C₃×C₂²⋊Q₈	—	2	3	no	—
33	C₂×C₃×C₄⋊C₄	—	2	3	no	—
34	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
35	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
36	C₃×#95	—	2	3	no	—
37	C₂³×C₉	—	4	1	no	—
38	C₂³×C₉	—	4	1	no	—
39	C₂³×C₉	C₂×D₄	1	3	yes	—
40	C₂³×C₉	C₂×Q₈	1	3	yes	—
41	C₂³×C₉	#41	1	1	yes	characteristic
42	Q₈×C₉	—	8	1	no	—
43	Q₈×C₉	—	8	1	no	—
44	C₂×C₄×C₉	—	2	6	no	—
45	C₂×C₄×C₉	—	2	3	no	—
46	C₂×C₄×C₉	—	2	3	no	—
47	C₂×C₄×C₉	—	2	6	no	—
48	C₂×C₄×C₉	—	4	3	no	—
49	C₂×C₄×C₉	—	4	3	no	—
50	C₂×C₄×C₉	—	4	3	no	—
51	C₂×C₄×C₉	—	4	1	no	—
52	C₂×C₄×C₉	—	4	3	no	—
53	C₂×C₄×C₉	—	4	3	no	—
54	#543	C₂×C₉	1	1	yes	characteristic
55	#385	C₂×C₉	1	3	yes	—
56	#342	C₂×C₉	1	3	yes	—
57	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
58	C₂×C₃×Q₈	—	2	1	no	—
59	C₂×C₃×Q₈	—	2	1	no	—
60	C₂²×C₃×C₄	—	2	3	no	—
61	C₂²×C₃×C₄	—	2	3	no	—
62	C₂²×C₃×C₄	—	4	2	no	—
63	C₂²×C₃×C₄	C₃×D₄	1	3	yes	—
64	C₂²×C₃×C₄	C₃×D₄	1	3	yes	—
65	C₃×C₄⋊C₄	—	2	3	no	—
66	C₃×C₄⋊C₄	—	2	3	no	—
67	C₃×C₄⋊C₄	—	4	6	no	—
68	C₃×C₂²⋊C₄	—	2	6	no	—
69	C₃×C₂²⋊C₄	—	2	3	no	—
70	C₃×C₂²⋊C₄	C₃×Q₈	1	6	yes	—
71	C₂²×C₉	—	2	3	no	—
72	C₂²×C₉	—	2	3	no	—
73	C₂²×C₉	—	4	3	no	—
74	C₂²×C₉	—	4	1	no	—
75	C₂²×C₉	#120	1	1	yes	characteristic
76	C₂²×C₉	C₂²⋊Q₈	1	3	yes	—
77	C₂²×C₉	C₄⋊Q₈	1	3	yes	—
78	C₄×C₉	—	4	6	no	—
79	C₄×C₉	—	4	2	no	—
80	C₄×C₉	—	4	3	no	—
81	C₄×C₉	—	4	3	no	—
82	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
83	C₂²⋊Q₈	—	2	3	no	—
84	C₂²⋊Q₈	—	2	3	no	—
85	C₂×C₄⋊C₄	—	2	3	no	—
86	C₂×C₂²⋊C₄	C₂²×C₉	1	3	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₉	1	3	yes	—
88	#95	—	2	3	no	—
89	C₂³×C₃	—	4	1	no	—
90	C₂³×C₃	—	4	1	no	—
91	C₂³×C₃	C₂×C₃×D₄	1	3	yes	—
92	C₂³×C₃	C₂×C₃×Q₈	1	3	yes	—
93	C₂³×C₃	C₃×#41	1	1	yes	characteristic
94	C₃×Q₈	—	8	1	no	—
95	C₃×Q₈	—	8	1	no	—
96	C₂×C₃×C₄	—	2	6	no	—
97	C₂×C₃×C₄	—	2	3	no	—
98	C₂×C₃×C₄	—	2	3	no	—
99	C₂×C₃×C₄	—	2	6	no	—
100	C₂×C₃×C₄	—	4	3	no	—
101	C₂×C₃×C₄	—	4	3	no	—
102	C₂×C₃×C₄	—	4	3	no	—
103	C₂×C₃×C₄	—	4	1	no	—
104	C₂×C₃×C₄	—	4	3	no	—
105	C₂×C₃×C₄	—	4	3	no	—
106	C₂×C₉	—	2	3	no	—
107	C₂×C₉	—	2	3	no	—
108	C₂×C₉	#395	1	1	yes	characteristic
109	C₂×C₉	B(4,𝔽₂)	1	1	yes	characteristic
110	C₂×C₉	#458	1	1	yes	characteristic
111	C₂⁴	C₂³×C₉	1	1	yes	characteristic
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	2	1	no	—
114	C₂²×C₄	—	2	3	no	—
115	C₂²×C₄	—	2	3	no	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	D₄×C₉	1	3	yes	—
118	C₂²×C₄	D₄×C₉	1	3	yes	—
119	C₄⋊C₄	—	2	3	no	—
120	C₄⋊C₄	—	2	3	no	—
121	C₄⋊C₄	—	4	6	no	—
122	C₂²⋊C₄	—	2	6	no	—
123	C₂²⋊C₄	—	2	3	no	—
124	C₂²⋊C₄	Q₈×C₉	1	6	yes	complemented
125	C₂²×C₃	—	2	3	no	—
126	C₂²×C₃	—	2	3	no	—
127	C₂²×C₃	—	4	3	no	—
128	C₂²×C₃	—	4	1	no	—
129	C₂²×C₃	C₃×#120	1	1	yes	characteristic
130	C₂²×C₃	C₃×C₂²⋊Q₈	1	3	yes	—
131	C₂²×C₃	C₃×C₄⋊Q₈	1	3	yes	—
132	C₃×C₄	—	4	6	no	—
133	C₃×C₄	—	4	2	no	—
134	C₃×C₄	—	4	3	no	—
135	C₃×C₄	—	4	3	no	—
136	C₉	#2062	1	1	yes	characteristic, direct product factor, Sylow
137	C₂³	—	4	1	no	—
138	C₂³	—	4	1	no	—
139	C₂³	C₂×D₄×C₉	1	3	yes	—
140	C₂³	C₂×Q₈×C₉	1	3	yes	—
141	C₂³	C₉×#41	1	1	yes	characteristic
142	Q₈	—	8	1	no	—
143	Q₈	—	8	1	no	—
144	C₂×C₄	—	2	6	no	—
145	C₂×C₄	—	2	3	no	—
146	C₂×C₄	—	2	3	no	—
147	C₂×C₄	—	2	6	no	—
148	C₂×C₄	—	4	3	no	—
149	C₂×C₄	—	4	3	no	—
150	C₂×C₄	—	4	3	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	3	no	—
153	C₂×C₄	—	4	3	no	—
154	C₂×C₃	—	2	3	no	—
155	C₂×C₃	—	2	3	no	—
156	C₂×C₃	C₃×#395	1	1	yes	characteristic
157	C₂×C₃	C₃×B(4,𝔽₂)	1	1	yes	characteristic
158	C₂×C₃	C₃×#458	1	1	yes	characteristic
159	C₂²	—	2	3	no	—
160	C₂²	—	2	3	no	—
161	C₂²	—	4	3	no	—
162	C₂²	—	4	1	no	—
163	C₂²	C₉×#120	1	1	yes	characteristic
164	C₂²	C₉×C₂²⋊Q₈	1	3	yes	—
165	C₂²	C₉×C₄⋊Q₈	1	3	yes	—
166	C₄	—	4	6	no	—
167	C₄	—	4	2	no	—
168	C₄	—	4	3	no	—
169	C₄	—	4	3	no	—
170	C₃	C₃×#2062	1	1	yes	characteristic
171	C₂	—	2	3	no	—
172	C₂	—	2	3	no	—
173	C₂	C₉×#395	1	1	yes	characteristic
174	C₂	C₉×B(4,𝔽₂)	1	1	yes	characteristic
175	C₂	C₉×#458	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#543

C₂

yes

maximal, characteristic

C₉×#385

C₂

yes

maximal

C₉×#342

C₂

yes

maximal

C₃×#2062

C₃

yes

maximal, characteristic

C₂³×C₄×C₉

C₂²

yes

characteristic

C₉×C₂²⋊Q₈

—

C₉×C₂²⋊Q₈

—

C₂×C₉×C₄⋊C₄

—

C₂×C₉×C₂²⋊C₄

C₂²

yes

—

C₂×C₉×C₂²⋊C₄

C₂²

yes

—

C₉×#95

—

C₃×#543

C₂×C₃

yes

characteristic

C₃×#385

C₂×C₃

yes

—

C₃×#342

C₂×C₃

yes

—

C₂⁴×C₉

C₂³

yes

characteristic

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

D₄

yes

—

C₂²×C₄×C₉

D₄

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₂²⋊C₄

—

C₉×C₂²⋊C₄

—

C₉×C₂²⋊C₄

Q₈

yes

complemented

#2062

C₉

yes

characteristic, direct product factor, Sylow

C₂³×C₃×C₄

C₂²×C₃

yes

characteristic

C₃×C₂²⋊Q₈

—

C₃×C₂²⋊Q₈

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₂²⋊C₄

C₂²×C₃

yes

—

C₂×C₃×C₂²⋊C₄

C₂²×C₃

yes

—

C₃×#95

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

C₂×D₄

yes

—

C₂³×C₉

C₂×Q₈

yes

—

C₂³×C₉

#41

yes

characteristic

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

#543

C₂×C₉

yes

characteristic

#385

C₂×C₉

yes

—

#342

C₂×C₉

yes

—

C₂⁴×C₃

C₂³×C₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₃×D₄

yes

—

C₂²×C₃×C₄

C₃×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

C₃×Q₈

yes

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

#120

yes

characteristic

C₂²×C₉

C₂²⋊Q₈

yes

—

C₂²×C₉

C₄⋊Q₈

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₂³×C₄

C₂²×C₉

yes

characteristic

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

C₂²×C₉

yes

—

C₂×C₂²⋊C₄

C₂²×C₉

yes

—

#95

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂×C₃×D₄

yes

—

C₂³×C₃

C₂×C₃×Q₈

yes

—

C₂³×C₃

C₃×#41

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

—

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

—

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

—

106

C₂×C₉

—

107

C₂×C₉

—

108

C₂×C₉

#395

yes

characteristic

109

C₂×C₉

B(4,𝔽₂)

yes

characteristic

110

C₂×C₉

#458

yes

characteristic

111

C₂⁴

C₂³×C₉

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

D₄×C₉

yes

—

118

C₂²×C₄

D₄×C₉

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

Q₈×C₉

yes

complemented

125

C₂²×C₃

—

126

C₂²×C₃

—

127

C₂²×C₃

—

128

C₂²×C₃

—

129

C₂²×C₃

C₃×#120

yes

characteristic

130

C₂²×C₃

C₃×C₂²⋊Q₈

yes

—

131

C₂²×C₃

C₃×C₄⋊Q₈

yes

—

132

C₃×C₄

—

133

C₃×C₄

—

134

C₃×C₄

—

135

C₃×C₄

—

136

C₉

#2062

yes

characteristic, direct product factor, Sylow

137

C₂³

—

138

C₂³

—

139

C₂³

C₂×D₄×C₉

yes

—

140

C₂³

C₂×Q₈×C₉

yes

—

141

C₂³

C₉×#41

yes

characteristic

142

Q₈

—

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₃

—

155

C₂×C₃

—

156

C₂×C₃

C₃×#395

yes

characteristic

157

C₂×C₃

C₃×B(4,𝔽₂)

yes

characteristic

158

C₂×C₃

C₃×#458

yes

characteristic

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

C₉×#120

yes

characteristic

164

C₂²

C₉×C₂²⋊Q₈

yes

—

165

C₂²

C₉×C₄⋊Q₈

yes

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₃

C₃×#2062

yes

characteristic

171

C₂

—

172

C₂

—

173

C₂

C₉×#395

yes

characteristic

174

C₂

C₉×B(4,𝔽₂)

yes

characteristic

175

C₂

C₉×#458

yes

characteristic

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₂⁴	C₂³×C₉
Frattini, prime radical, cosocle	C₂⁴×C₃	C₂³×C₃
socle	C₂²×C₃	C₃×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂⁴×C₉ ◁ G

subgroup

G/H

center

C₂²×C₉

#120

commutator

C₂⁴

C₂³×C₉

Frattini, prime radical, cosocle

C₂⁴×C₃

C₂³×C₃

C₂²×C₃

G ▷ C₂⁴ ▷ C₂ ▷ C₁

C₁ ◁ C₂²×C₉ ◁ C₂⁴×C₉ ◁ G

N	H ≅ G/N	type
28. C₉×C₂²⋊C₄	Q₈	semidirect
29. #2062	C₉	direct (N × H)
124. C₂²⋊C₄	Q₈×C₉	semidirect
136. C₉	#2062	direct (N × H)

H ≅ G/N

type

28. C₉×C₂²⋊C₄

Q₈

semidirect

29. #2062

C₉

direct (N × H)

124. C₂²⋊C₄

Q₈×C₉

semidirect

136. C₉

#2062

direct (N × H)

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

subgroup	index
C₉×#543	2
C₉×#385	2
C₉×#342	2
C₃×#2062	3

subgroup

index

Subgroups — C₉×#2062 #11766783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#2062 #11766783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2062 #11766783