Group #13142949 — subgroups

Subgroups — C₉×#5069 #13142949

Order 1728.

326 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₃×C₉×#125	C₂	1	1	yes	maximal, characteristic
2	C₉×#869	C₂	1	1	yes	maximal, characteristic
3	C₉×#810	C₂	1	4	yes	maximal
4	C₉×#803	C₂	1	1	yes	maximal, characteristic
5	C₃×#5069	C₃	1	1	yes	maximal, characteristic
6	C₉×#491	—	3	1	no	maximal
7	C₂×C₉×Dic₆	C₂²	1	2	yes	—
8	C₃×C₉×C₄⋊C₄	—	2	2	no	—
9	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₉×C₄⋊Dic₃	—	2	2	no	—
12	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃²×#125	C₂×C₃	1	1	yes	characteristic
14	C₃×#869	C₂×C₃	1	1	yes	characteristic
15	C₃×#810	C₂×C₃	1	4	yes	—
16	C₃×#803	C₂×C₃	1	1	yes	characteristic
17	C₉×C₄⋊Q₈	—	3	1	no	—
18	C₉×#125	—	2	1	no	—
19	C₉×#125	D₃	1	1	yes	characteristic
20	C₉×#103	—	3	4	no	—
21	C₉×#97	—	3	1	no	—
22	C₂×C₃×C₄×C₉	—	2	2	no	—
23	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
24	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₂×C₉×Dic₃	—	2	2	no	—
26	C₉×Dic₆	—	2	4	no	—
27	C₉×C₃⋊C₈	—	2	2	no	—
28	C₃×#491	—	3	1	no	—
29	#5069	C₉	1	1	yes	characteristic, direct product factor
30	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
31	C₃²×C₄⋊C₄	—	2	2	no	—
32	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
33	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
34	C₃×C₄⋊Dic₃	—	2	2	no	—
35	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
36	C₂×Q₈×C₉	—	3	2	no	—
37	C₂×C₈×C₉	—	3	2	no	—
38	C₉×C₄⋊C₄	—	2	4	no	—
39	C₉×C₄⋊C₄	—	2	2	no	—
40	C₉×C₄⋊C₄	—	2	2	no	—
41	C₉×C₄⋊C₄	—	6	2	no	—
42	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
43	C₄²×C₉	—	2	1	no	—
44	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
46	C₃×C₄×C₉	—	2	2	no	—
47	C₃×C₄×C₉	—	4	2	no	—
48	C₃×C₄×C₉	#41	1	2	yes	—
49	C₉×Dic₃	—	4	2	no	—
50	C₃×C₄⋊Q₈	—	3	1	no	—
51	C₃×#125	—	2	1	no	—
52	C₃×#125	C₂×C₉	1	1	yes	characteristic
53	C₃×#125	C₃×D₃	1	1	yes	characteristic
54	#869	C₂×C₉	1	1	yes	characteristic
55	C₃×#103	—	3	4	no	—
56	C₃×#97	—	3	1	no	—
57	#810	C₂×C₉	1	4	yes	—
58	#803	C₂×C₉	1	1	yes	characteristic
59	C₂×C₃²×C₄	—	2	2	no	—
60	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
61	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
62	C₂×C₃×Dic₃	—	2	2	no	—
63	C₃×Dic₆	—	2	4	no	—
64	C₃×C₃⋊C₈	—	2	2	no	—
65	Q₈×C₉	—	6	4	no	—
66	C₂×C₄×C₉	—	2	4	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	6	2	no	—
71	C₂×C₄×C₉	#67	1	2	yes	—
72	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
73	C₈×C₉	—	6	2	no	—
74	#491	—	3	1	no	Sylow
75	C₂×C₃×C₉	#124	1	1	yes	characteristic
76	C₂×C₃×C₉	#137	1	2	yes	—
77	C₂×C₃×Q₈	—	3	2	no	—
78	C₂×Dic₆	C₂²×C₉	1	2	yes	—
79	C₂×C₃×C₈	—	3	2	no	—
80	C₃×C₄⋊C₄	—	2	4	no	—
81	C₃×C₄⋊C₄	—	2	2	no	—
82	C₃×C₄⋊C₄	—	2	2	no	—
83	C₃×C₄⋊C₄	—	2	2	no	—
84	C₃×C₄⋊C₄	—	6	2	no	—
85	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
86	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
89	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
90	C₄⋊Dic₃	—	2	2	no	—
91	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
92	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
93	C₃²×C₄	—	2	2	no	—
94	C₃²×C₄	—	4	2	no	—
95	C₃²×C₄	C₃×#41	1	2	yes	—
96	C₃×Dic₃	—	4	2	no	—
97	C₂²×C₉	—	2	1	no	—
98	C₂²×C₉	C₂×#67	1	1	yes	characteristic
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	—	2	2	no	—
101	C₄×C₉	—	4	2	no	—
102	C₄×C₉	—	4	4	no	—
103	C₄×C₉	—	4	2	no	—
104	C₄×C₉	—	12	2	no	—
105	C₄×C₉	#239	1	2	yes	—
106	C₄⋊Q₈	—	3	1	no	—
107	#125	D₃×C₉	1	1	yes	characteristic
108	#103	—	3	4	no	—
109	#97	—	3	1	no	—
110	C₃×C₉	#491	1	1	yes	characteristic, complemented, Sylow
111	C₃×Q₈	—	6	4	no	—
112	C₂×C₃×C₄	—	2	4	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	6	2	no	—
118	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
119	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
120	C₂×C₃×C₄	C₃×#67	1	2	yes	—
121	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
122	C₂×Dic₃	—	2	2	no	—
123	Dic₆	—	2	4	no	—
124	C₃×C₈	—	6	2	no	—
125	C₃⋊C₈	—	2	2	no	—
126	C₂×C₃²	C₃×#124	1	1	yes	characteristic
127	C₂×C₃²	C₃×#137	1	2	yes	—
128	C₂×C₉	—	2	1	no	—
129	C₂×C₉	—	2	2	no	—
130	C₂×C₉	#947	1	1	yes	characteristic
131	C₂×C₉	#951	1	2	yes	—
132	C₂×Q₈	—	3	2	no	—
133	C₂×C₈	—	3	2	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	6	2	no	—
136	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
137	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
138	C₂²×C₃	—	2	1	no	—
139	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
140	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
141	C₃×C₄	—	2	2	no	—
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	2	2	no	—
144	C₃×C₄	—	4	2	no	—
145	C₃×C₄	—	4	4	no	—
146	C₃×C₄	—	4	2	no	—
147	C₃×C₄	—	4	2	no	—
148	C₃×C₄	—	12	2	no	—
149	C₃×C₄	C₉×#41	1	2	yes	—
150	C₃×C₄	C₃×#239	1	2	yes	—
151	Dic₃	—	4	2	no	—
152	C₃²	C₃×#491	1	1	yes	characteristic
153	C₉	—	2	1	no	—
154	C₉	#5069	1	1	yes	characteristic, direct product factor
155	Q₈	—	6	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	6	2	no	—
158	C₂×C₄	C₉×#67	1	2	yes	—
159	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
160	C₈	—	6	2	no	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	2	no	—
163	C₂×C₃	C₉×#124	1	1	yes	characteristic
164	C₂×C₃	C₉×#137	1	2	yes	—
165	C₂×C₃	C₃×#947	1	1	yes	characteristic
166	C₂×C₃	C₃×#951	1	2	yes	—
167	C₂²	C₂×C₉×#67	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	12	2	no	—
171	C₄	C₉×#239	1	2	yes	—
172	C₃	—	2	1	no	—
173	C₃	C₉×#491	1	1	yes	characteristic, complemented
174	C₃	C₃×#5069	1	1	yes	characteristic
175	C₂	C₉×#947	1	1	yes	characteristic
176	C₂	C₉×#951	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×#125

C₂

yes

maximal, characteristic

C₉×#869

C₂

yes

maximal, characteristic

C₉×#810

C₂

yes

maximal

C₉×#803

C₂

yes

maximal, characteristic

C₃×#5069

C₃

yes

maximal, characteristic

C₉×#491

—

maximal

C₂×C₉×Dic₆

C₂²

yes

—

C₃×C₉×C₄⋊C₄

—

C₃×C₉×C₄⋊C₄

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₉×C₄⋊Dic₃

—

C₂×C₉×C₃⋊C₈

C₂²

yes

—

C₃²×#125

C₂×C₃

yes

characteristic

C₃×#869

C₂×C₃

yes

characteristic

C₃×#810

C₂×C₃

yes

—

C₃×#803

C₂×C₃

yes

characteristic

C₉×C₄⋊Q₈

—

C₉×#125

—

C₉×#125

D₃

yes

characteristic

C₉×#103

—

C₉×#97

—

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₉×Dic₃

—

C₉×Dic₆

—

C₉×C₃⋊C₈

—

C₃×#491

—

#5069

C₉

yes

characteristic, direct product factor

C₂×C₃×Dic₆

C₂²×C₃

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

C₂²×C₃

yes

—

C₃²×C₄²

C₂²×C₃

yes

characteristic

C₃×C₄⋊Dic₃

—

C₂×C₃×C₃⋊C₈

C₂²×C₃

yes

—

C₂×Q₈×C₉

—

C₂×C₈×C₉

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₂×D₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₂×D₃

yes

characteristic

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

#41

yes

—

C₉×Dic₃

—

C₃×C₄⋊Q₈

—

C₃×#125

—

C₃×#125

C₂×C₉

yes

characteristic

C₃×#125

C₃×D₃

yes

characteristic

#869

C₂×C₉

yes

characteristic

C₃×#103

—

C₃×#97

—

#810

C₂×C₉

yes

—

#803

C₂×C₉

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₃×D₄

yes

—

C₂×C₃²×C₄

C₂³×C₃

yes

characteristic

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

#67

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₈×C₉

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

—

C₂×C₃×Q₈

—

C₂×Dic₆

C₂²×C₉

yes

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂²×C₉

yes

—

C₃×C₄⋊C₄

C₂×C₃×D₃

yes

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₉

yes

characteristic

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₄⋊Dic₃

—

C₂×C₃⋊C₈

C₂²×C₉

yes

—

C₂²×C₃²

C₂×C₃×D₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₃×#41

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂×#67

yes

characteristic

C₄×C₉

—

100

C₄×C₉

—

101

C₄×C₉

—

102

C₄×C₉

—

103

C₄×C₉

—

104

C₄×C₉

—

105

C₄×C₉

#239

yes

—

106

C₄⋊Q₈

—

107

#125

D₃×C₉

yes

characteristic

108

#103

—

109

#97

—

110

C₃×C₉

#491

yes

characteristic, complemented, Sylow

111

C₃×Q₈

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

—

118

C₂×C₃×C₄

D₄×C₉

yes

—

119

C₂×C₃×C₄

C₂³×C₉

yes

characteristic

120

C₂×C₃×C₄

C₃×#67

yes

—

121

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

122

C₂×Dic₃

—

123

Dic₆

—

124

C₃×C₈

—

125

C₃⋊C₈

—

126

C₂×C₃²

C₃×#124

yes

characteristic

127

C₂×C₃²

C₃×#137

yes

—

128

C₂×C₉

—

129

C₂×C₉

—

130

C₂×C₉

#947

yes

characteristic

131

C₂×C₉

#951

yes

—

132

C₂×Q₈

—

133

C₂×C₈

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

C₂×D₃×C₉

yes

—

137

C₄²

C₂×D₃×C₉

yes

characteristic

138

C₂²×C₃

—

139

C₂²×C₃

C₂×D₄×C₉

yes

characteristic

140

C₂²×C₃

C₂×C₃×#67

yes

characteristic

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

—

147

C₃×C₄

—

148

C₃×C₄

—

149

C₃×C₄

C₉×#41

yes

—

150

C₃×C₄

C₃×#239

yes

—

151

Dic₃

—

152

C₃²

C₃×#491

yes

characteristic

153

C₉

—

154

C₉

#5069

yes

characteristic, direct product factor

155

Q₈

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

C₉×#67

yes

—

159

C₂×C₄

C₂²×D₃×C₉

yes

characteristic

160

C₈

—

161

C₂×C₃

—

162

C₂×C₃

—

163

C₂×C₃

C₉×#124

yes

characteristic

164

C₂×C₃

C₉×#137

yes

—

165

C₂×C₃

C₃×#947

yes

characteristic

166

C₂×C₃

C₃×#951

yes

—

167

C₂²

C₂×C₉×#67

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

C₉×#239

yes

—

172

C₃

—

173

C₃

C₉×#491

yes

characteristic, complemented

174

C₃

C₃×#5069

yes

characteristic

175

C₂

C₉×#947

yes

characteristic

176

C₂

C₉×#951

yes

—

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#125	C₂
socle, prime radical	C₂²×C₃²	C₂×C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#125	D₃
hypocenter	C₃	C₉×#491
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×#125

subgroup

G/H

center

C₂²×C₉

C₂×#67

commutator

C₂×C₃×C₄

C₂³×C₉

Frattini

C₂×C₃×C₄

C₂²×C₃×D₃

Fitting

C₃×C₉×#125

C₂

socle, prime radical

C₂²×C₃²

C₂×C₃×D₄

cosocle

C₂×C₃²×C₄

C₂³×C₃

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×#125

N	H ≅ G/N	type
29. #5069	C₉	direct (N × H)
110. C₃×C₉	#491	semidirect
154. C₉	#5069	direct (N × H)
173. C₃	C₉×#491	semidirect

H ≅ G/N

type

29. #5069

C₉

direct (N × H)

110. C₃×C₉

#491

semidirect

154. C₉

#5069

direct (N × H)

173. C₃

C₉×#491

semidirect

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

subgroup	index
C₃×C₉×#125	2
C₉×#869	2
C₉×#810	2
C₉×#803	2
C₃×#5069	3
C₉×#491	3

subgroup

index

C₃×C₉×#125

Subgroups — C₉×#5069 #13142949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#5069 #13142949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5069 #13142949