Group #11766750 — subgroups

Subgroups — C₉×#2029 #11766750

Order 1152.

699 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 699 conjugacy classes of subgroups — too many to draw. Only its 126 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, complemented
2	C₉×#379	C₂	1	3	yes	maximal
3	C₉×#342	C₂	1	3	yes	maximal, complemented
4	C₃×#2029	C₃	1	1	yes	maximal, characteristic
5	C₂⁵×C₉	C₂²	1	1	yes	characteristic
6	C₉×C₂²⋊Q₈	—	2	6	no	—
7	C₂×C₉×C₂²⋊C₄	—	2	6	no	—
8	C₂×C₉×C₂²⋊C₄	C₂²	1	3	yes	—
9	C₂×C₉×C₂²⋊C₄	C₂²	1	3	yes	—
10	C₃×#543	C₂×C₃	1	1	yes	characteristic
11	C₃×#379	C₂×C₃	1	3	yes	—
12	C₃×#342	C₂×C₃	1	3	yes	—
13	C₂⁴×C₉	—	2	3	no	—
14	C₂⁴×C₉	—	2	3	no	—
15	C₂⁴×C₉	—	4	4	no	—
16	C₂⁴×C₉	Q₈	1	2	yes	complemented
17	C₂⁴×C₉	C₂³	1	1	yes	characteristic
18	C₂×Q₈×C₉	—	2	2	no	—
19	C₂²×C₄×C₉	—	2	3	no	—
20	C₂²×C₄×C₉	D₄	1	6	yes	—
21	C₉×C₄⋊C₄	—	2	6	no	—
22	C₉×C₂²⋊C₄	—	2	6	no	—
23	C₉×C₂²⋊C₄	—	2	6	no	—
24	C₉×C₂²⋊C₄	—	4	12	no	—
25	#2029	C₉	1	1	yes	characteristic, direct product factor, Sylow
26	C₂⁵×C₃	C₂²×C₃	1	1	yes	characteristic
27	C₃×C₂²⋊Q₈	—	2	6	no	—
28	C₂×C₃×C₂²⋊C₄	—	2	6	no	—
29	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
30	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
31	C₂³×C₉	—	2	12	no	—
32	C₂³×C₉	—	2	6	no	—
33	C₂³×C₉	—	4	4	no	—
34	C₂³×C₉	—	4	2	no	—
35	C₂³×C₉	—	4	6	no	—
36	C₂³×C₉	—	4	12	no	—
37	C₂³×C₉	—	4	1	no	—
38	C₂³×C₉	—	4	3	no	—
39	C₂³×C₉	C₂×D₄	1	3	yes	—
40	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₉	#41	1	3	yes	—
42	Q₈×C₉	—	8	2	no	—
43	C₂×C₄×C₉	—	2	6	no	—
44	C₂×C₄×C₉	—	4	6	no	—
45	C₂×C₄×C₉	—	4	6	no	—
46	C₂×C₄×C₉	—	4	3	no	—
47	#543	C₂×C₉	1	1	yes	characteristic, complemented
48	#379	C₂×C₉	1	3	yes	—
49	#342	C₂×C₉	1	3	yes	complemented
50	C₂⁴×C₃	—	2	3	no	—
51	C₂⁴×C₃	—	2	3	no	—
52	C₂⁴×C₃	—	4	4	no	—
53	C₂⁴×C₃	C₃×Q₈	1	2	yes	—
54	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
55	C₂×C₃×Q₈	—	2	2	no	—
56	C₂²×C₃×C₄	—	2	3	no	—
57	C₂²×C₃×C₄	C₃×D₄	1	6	yes	—
58	C₃×C₄⋊C₄	—	2	6	no	—
59	C₃×C₂²⋊C₄	—	2	6	no	—
60	C₃×C₂²⋊C₄	—	2	6	no	—
61	C₃×C₂²⋊C₄	—	4	12	no	—
62	C₂²×C₉	—	2	6	no	—
63	C₂²×C₉	—	2	12	no	—
64	C₂²×C₉	—	4	1	no	—
65	C₂²×C₉	—	4	3	no	—
66	C₂²×C₉	—	4	12	no	—
67	C₂²×C₉	—	4	6	no	—
68	C₂²×C₉	—	4	2	no	—
69	C₂²×C₉	—	4	4	no	—
70	C₂²×C₉	#120	1	1	yes	characteristic
71	C₂²×C₉	C₂²⋊Q₈	1	3	yes	—
72	C₂²×C₉	#124	1	3	yes	—
73	C₄×C₉	—	4	6	no	—
74	C₄×C₉	—	8	3	no	—
75	C₂⁵	C₂²×C₉	1	1	yes	characteristic
76	C₂²⋊Q₈	—	2	6	no	—
77	C₂×C₂²⋊C₄	—	2	6	no	—
78	C₂×C₂²⋊C₄	C₂²×C₉	1	3	yes	—
79	C₂×C₂²⋊C₄	C₂²×C₉	1	3	yes	—
80	C₂³×C₃	—	2	12	no	—
81	C₂³×C₃	—	2	6	no	—
82	C₂³×C₃	—	4	4	no	—
83	C₂³×C₃	—	4	2	no	—
84	C₂³×C₃	—	4	6	no	—
85	C₂³×C₃	—	4	12	no	—
86	C₂³×C₃	—	4	1	no	—
87	C₂³×C₃	—	4	3	no	—
88	C₂³×C₃	C₂×C₃×D₄	1	3	yes	—
89	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
90	C₂³×C₃	C₃×#41	1	3	yes	—
91	C₃×Q₈	—	8	2	no	—
92	C₂×C₃×C₄	—	2	6	no	—
93	C₂×C₃×C₄	—	4	6	no	—
94	C₂×C₃×C₄	—	4	6	no	—
95	C₂×C₃×C₄	—	4	3	no	—
96	C₂×C₉	—	2	3	no	—
97	C₂×C₉	—	2	3	no	—
98	C₂×C₉	—	4	4	no	—
99	C₂×C₉	#393	1	1	yes	characteristic
100	C₂×C₉	B(4,𝔽₂)	1	2	yes	—
101	C₂⁴	—	2	3	no	—
102	C₂⁴	—	2	3	no	—
103	C₂⁴	—	4	4	no	—
104	C₂⁴	Q₈×C₉	1	2	yes	complemented
105	C₂⁴	C₂³×C₉	1	1	yes	characteristic
106	C₂×Q₈	—	2	2	no	—
107	C₂²×C₄	—	2	3	no	—
108	C₂²×C₄	D₄×C₉	1	6	yes	—
109	C₄⋊C₄	—	2	6	no	—
110	C₂²⋊C₄	—	2	6	no	—
111	C₂²⋊C₄	—	2	6	no	—
112	C₂²⋊C₄	—	4	12	no	—
113	C₂²×C₃	—	2	6	no	—
114	C₂²×C₃	—	2	12	no	—
115	C₂²×C₃	—	4	1	no	—
116	C₂²×C₃	—	4	3	no	—
117	C₂²×C₃	—	4	12	no	—
118	C₂²×C₃	—	4	6	no	—
119	C₂²×C₃	—	4	2	no	—
120	C₂²×C₃	—	4	4	no	—
121	C₂²×C₃	C₃×#120	1	1	yes	characteristic
122	C₂²×C₃	C₃×C₂²⋊Q₈	1	3	yes	—
123	C₂²×C₃	C₃×#124	1	3	yes	—
124	C₃×C₄	—	4	6	no	—
125	C₃×C₄	—	8	3	no	—
126	C₉	#2029	1	1	yes	characteristic, direct product factor, Sylow
127	C₂³	—	2	12	no	—
128	C₂³	—	2	6	no	—
129	C₂³	—	4	4	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	4	6	no	—
132	C₂³	—	4	12	no	—
133	C₂³	—	4	1	no	—
134	C₂³	—	4	3	no	—
135	C₂³	C₂×D₄×C₉	1	3	yes	—
136	C₂³	C₂×Q₈×C₉	1	1	yes	characteristic
137	C₂³	C₉×#41	1	3	yes	—
138	Q₈	—	8	2	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	4	6	no	—
141	C₂×C₄	—	4	6	no	—
142	C₂×C₄	—	4	3	no	—
143	C₂×C₃	—	2	3	no	—
144	C₂×C₃	—	2	3	no	—
145	C₂×C₃	—	4	4	no	—
146	C₂×C₃	C₃×#393	1	1	yes	characteristic
147	C₂×C₃	C₃×B(4,𝔽₂)	1	2	yes	—
148	C₂²	—	2	6	no	—
149	C₂²	—	2	12	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	3	no	—
152	C₂²	—	4	12	no	—
153	C₂²	—	4	6	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	4	no	—
156	C₂²	C₉×#120	1	1	yes	characteristic
157	C₂²	C₉×C₂²⋊Q₈	1	3	yes	—
158	C₂²	C₉×#124	1	3	yes	—
159	C₄	—	4	6	no	—
160	C₄	—	8	3	no	—
161	C₃	C₃×#2029	1	1	yes	characteristic
162	C₂	—	2	3	no	—
163	C₂	—	2	3	no	—
164	C₂	—	4	4	no	—
165	C₂	C₉×#393	1	1	yes	characteristic
166	C₂	C₉×B(4,𝔽₂)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#543

C₂

yes

maximal, characteristic, complemented

C₉×#379

C₂

yes

maximal

C₉×#342

C₂

yes

maximal, complemented

C₃×#2029

C₃

yes

maximal, characteristic

C₂⁵×C₉

C₂²

yes

characteristic

C₉×C₂²⋊Q₈

—

C₂×C₉×C₂²⋊C₄

—

C₂×C₉×C₂²⋊C₄

C₂²

yes

—

C₂×C₉×C₂²⋊C₄

C₂²

yes

—

C₃×#543

C₂×C₃

yes

characteristic

C₃×#379

C₂×C₃

yes

—

C₃×#342

C₂×C₃

yes

—

C₂⁴×C₉

—

C₂⁴×C₉

—

C₂⁴×C₉

—

C₂⁴×C₉

Q₈

yes

complemented

C₂⁴×C₉

C₂³

yes

characteristic

C₂×Q₈×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

D₄

yes

—

C₉×C₄⋊C₄

—

C₉×C₂²⋊C₄

—

C₉×C₂²⋊C₄

—

C₉×C₂²⋊C₄

—

#2029

C₉

yes

characteristic, direct product factor, Sylow

C₂⁵×C₃

C₂²×C₃

yes

characteristic

C₃×C₂²⋊Q₈

—

C₂×C₃×C₂²⋊C₄

—

C₂×C₃×C₂²⋊C₄

C₂²×C₃

yes

—

C₂×C₃×C₂²⋊C₄

C₂²×C₃

yes

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

C₂×D₄

yes

—

C₂³×C₉

C₂×Q₈

yes

characteristic

C₂³×C₉

#41

yes

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

#543

C₂×C₉

yes

characteristic, complemented

#379

C₂×C₉

yes

—

#342

C₂×C₉

yes

complemented

C₂⁴×C₃

—

C₂⁴×C₃

—

C₂⁴×C₃

—

C₂⁴×C₃

C₃×Q₈

yes

—

C₂⁴×C₃

C₂³×C₃

yes

characteristic

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₃×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

#120

yes

characteristic

C₂²×C₉

C₂²⋊Q₈

yes

—

C₂²×C₉

#124

yes

—

C₄×C₉

—

C₄×C₉

—

C₂⁵

C₂²×C₉

yes

characteristic

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂²×C₉

yes

—

C₂×C₂²⋊C₄

C₂²×C₉

yes

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂×C₃×D₄

yes

—

C₂³×C₃

C₂×C₃×Q₈

yes

characteristic

C₂³×C₃

C₃×#41

yes

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

#393

yes

characteristic

100

C₂×C₉

B(4,𝔽₂)

yes

—

101

C₂⁴

—

102

C₂⁴

—

103

C₂⁴

—

104

C₂⁴

Q₈×C₉

yes

complemented

105

C₂⁴

C₂³×C₉

yes

characteristic

106

C₂×Q₈

—

107

C₂²×C₄

—

108

C₂²×C₄

D₄×C₉

yes

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²×C₃

—

114

C₂²×C₃

—

115

C₂²×C₃

—

116

C₂²×C₃

—

117

C₂²×C₃

—

118

C₂²×C₃

—

119

C₂²×C₃

—

120

C₂²×C₃

—

121

C₂²×C₃

C₃×#120

yes

characteristic

122

C₂²×C₃

C₃×C₂²⋊Q₈

yes

—

123

C₂²×C₃

C₃×#124

yes

—

124

C₃×C₄

—

125

C₃×C₄

—

126

C₉

#2029

yes

characteristic, direct product factor, Sylow

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

C₂×D₄×C₉

yes

—

136

C₂³

C₂×Q₈×C₉

yes

characteristic

137

C₂³

C₉×#41

yes

—

138

Q₈

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₃

—

144

C₂×C₃

—

145

C₂×C₃

—

146

C₂×C₃

C₃×#393

yes

characteristic

147

C₂×C₃

C₃×B(4,𝔽₂)

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

C₉×#120

yes

characteristic

157

C₂²

C₉×C₂²⋊Q₈

yes

—

158

C₂²

C₉×#124

yes

—

159

C₄

—

160

C₄

—

161

C₃

C₃×#2029

yes

characteristic

162

C₂

—

163

C₂

—

164

C₂

—

165

C₂

C₉×#393

yes

characteristic

166

C₂

C₉×B(4,𝔽₂)

yes

—

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₂⁴	C₂³×C₉
Frattini, cosocle	C₂⁴×C₃	C₂³×C₃
socle	C₂²×C₃	C₃×#120
prime radical	C₂⁵×C₃	C₂²×C₃
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂⁴×C₉ ◁ G

subgroup

G/H

center

C₂²×C₉

#120

commutator

C₂⁴

C₂³×C₉

Frattini, cosocle

C₂⁴×C₃

C₂³×C₃

socle

C₂²×C₃

C₃×#120

prime radical

C₂⁵×C₃

C₂²×C₃

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₉ ◁ C₂⁴×C₉ ◁ G

N	H ≅ G/N	type
1. C₉×#543	C₂	semidirect
3. C₉×#342	C₂	semidirect
16. C₂⁴×C₉	Q₈	semidirect
25. #2029	C₉	direct (N × H)
47. #543	C₂×C₉	semidirect
49. #342	C₂×C₉	semidirect
104. C₂⁴	Q₈×C₉	semidirect
126. C₉	#2029	direct (N × H)

H ≅ G/N

type

1. C₉×#543

C₂

semidirect

3. C₉×#342

C₂

semidirect

16. C₂⁴×C₉

Q₈

semidirect

25. #2029

C₉

direct (N × H)

47. #543

C₂×C₉

semidirect

49. #342

C₂×C₉

semidirect

104. C₂⁴

Q₈×C₉

semidirect

126. C₉

#2029

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×#379	2
C₉×#342	2
C₃×#2029	3

subgroup

index

Subgroups — C₉×#2029 #11766750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#2029 #11766750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2029 #11766750