Group #72219975 — subgroups

Subgroups — #72219975

Order 2592.

315 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 81 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×D₄×C₉⋊C₉	C₂	1	1	yes	maximal, characteristic
2	#3667⋊C₉	C₂	1	2	yes	maximal, complemented
3	C₉×#833	—	3	1	no	maximal
4	C₃×#63391	C₃	1	1	yes	maximal, characteristic
5	C₂³×C₉⋊C₉	C₄	1	2	yes	—
6	D₄×C₉⋊C₉	—	2	2	no	—
7	C₂×C₄×C₉⋊C₉	C₂²	1	1	yes	characteristic
8	(C₉⋊C₈)⋊C₉	—	2	2	no	—
9	C₂×C₃×D₄×C₉	—	2	1	no	—
10	C₂×C₃×D₄×C₉	D₃	1	1	yes	characteristic
11	C₂×C₃×D₄×C₉	C₂×C₃	1	1	yes	characteristic
12	C₉×#208	—	3	2	no	—
13	C₃×#3667	C₂×C₃	1	2	yes	—
14	C₂²×C₉⋊C₉	—	2	2	no	—
15	C₂²×C₉⋊C₉	—	4	2	no	—
16	C₂²×C₉⋊C₉	C₂×C₄	1	1	yes	characteristic
17	C₄×C₉⋊C₉	D₄	1	2	yes	—
18	C₃×#833	—	3	1	no	—
19	C₉×#100	—	9	1	no	—
20	#63391	C₉	1	1	yes	characteristic, complemented
21	C₂³×C₃×C₉	—	2	2	no	—
22	C₂³×C₃×C₉	Dic₃	1	2	yes	—
23	C₂³×C₃×C₉	C₃×C₄	1	2	yes	—
24	C₃×D₄×C₉	—	2	4	no	—
25	C₃×D₄×C₉	—	2	2	no	—
26	C₃×D₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	—	2	1	no	—
28	C₂×C₃×C₄×C₉	C₂×D₃	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
30	C₉×C₃⋊C₈	—	6	2	no	—
31	C₃×C₉⋊C₈	—	2	2	no	—
32	C₂×C₉⋊C₉	—	2	1	no	—
33	C₂×C₉⋊C₉	—	4	2	no	—
34	C₂×C₉⋊C₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₃²×D₄	C₃×D₃	1	1	yes	characteristic
36	C₃×#208	—	3	2	no	—
37	C₂×D₄×C₉	—	2	1	no	—
38	C₂×D₄×C₉	—	3	1	no	—
39	C₂×D₄×C₉	—	6	1	no	—
40	C₂×D₄×C₉	C₂×C₉	1	1	yes	characteristic
41	C₉×M₁₆	—	9	2	no	—
42	#3667	C₂×C₉	1	2	yes	complemented
43	C₂²×C₃×C₉	—	2	1	no	—
44	C₂²×C₃×C₉	—	2	4	no	—
45	C₂²×C₃×C₉	—	2	2	no	—
46	C₂²×C₃×C₉	—	2	2	no	—
47	C₂²×C₃×C₉	—	4	4	no	—
48	C₂²×C₃×C₉	—	4	2	no	—
49	C₂²×C₃×C₉	—	4	2	no	—
50	C₂²×C₃×C₉	C₂×Dic₃	1	1	yes	characteristic
51	C₂²×C₃×C₉	C₂×C₃×C₄	1	1	yes	characteristic
52	C₃×C₄×C₉	—	2	2	no	—
53	C₃×C₄×C₉	#67	1	2	yes	—
54	C₃×C₄×C₉	C₃×D₄	1	2	yes	—
55	C₃×#100	—	9	1	no	—
56	#833	—	3	1	no	—
57	C₉⋊C₉	#100	1	1	yes	characteristic, complemented, Sylow
58	C₂³×C₃²	C₃×Dic₃	1	2	yes	—
59	C₃²×D₄	—	2	2	no	—
60	C₂×C₃²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
61	C₂³×C₉	—	2	2	no	—
62	C₂³×C₉	—	3	2	no	—
63	C₂³×C₉	—	6	2	no	—
64	C₂³×C₉	C₄×C₉	1	2	yes	—
65	C₃×C₃⋊C₈	—	6	2	no	—
66	D₄×C₉	—	2	4	no	—
67	D₄×C₉	—	2	2	no	—
68	D₄×C₉	—	6	2	no	—
69	D₄×C₉	—	6	4	no	—
70	C₂×C₄×C₉	—	2	1	no	—
71	C₂×C₄×C₉	—	3	1	no	—
72	C₂×C₄×C₉	—	6	1	no	—
73	C₂×C₄×C₉	C₂²×C₉	1	1	yes	characteristic
74	C₈×C₉	—	18	2	no	—
75	C₉⋊C₈	—	2	2	no	—
76	C₂×C₃×C₉	—	2	1	no	—
77	C₂×C₃×C₉	—	2	2	no	—
78	C₂×C₃×C₉	—	2	1	no	—
79	C₂×C₃×C₉	—	2	1	no	—
80	C₂×C₃×C₉	—	4	2	no	—
81	C₂×C₃×C₉	—	4	4	no	—
82	C₂×C₃×C₉	—	4	2	no	—
83	C₂×C₃×C₉	#217	1	1	yes	characteristic
84	C₂×C₃×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×D₄	—	2	1	no	—
86	C₂×C₃×D₄	D₃×C₉	1	1	yes	characteristic
87	C₂×C₃×D₄	AGL(1,ℤ/9)	1	1	yes	characteristic
88	C₃×M₁₆	—	9	2	no	—
89	#208	—	3	2	no	—
90	C₂²×C₃²	—	2	2	no	—
91	C₂²×C₃²	—	4	2	no	—
92	C₂²×C₃²	C₂×C₃×Dic₃	1	1	yes	characteristic
93	C₃²×C₄	C₃×#67	1	2	yes	—
94	C₂²×C₉	—	2	4	no	—
95	C₂²×C₉	—	2	2	no	—
96	C₂²×C₉	—	2	1	no	—
97	C₂²×C₉	—	3	1	no	—
98	C₂²×C₉	—	4	2	no	—
99	C₂²×C₉	—	4	4	no	—
100	C₂²×C₉	—	6	2	no	—
101	C₂²×C₉	—	6	1	no	—
102	C₂²×C₉	—	6	4	no	—
103	C₂²×C₉	—	12	2	no	—
104	C₂²×C₉	—	12	4	no	—
105	C₂²×C₉	C₂×C₄×C₉	1	1	yes	characteristic
106	C₄×C₉	—	2	2	no	—
107	C₄×C₉	—	3	2	no	—
108	C₄×C₉	—	6	2	no	—
109	C₄×C₉	D₄×C₉	1	2	yes	—
110	#100	—	9	1	no	Sylow
111	C₃×C₉	—	2	1	no	—
112	C₃×C₉	#833	1	1	yes	characteristic
113	C₃×C₉	C₃×#100	1	1	yes	characteristic
114	C₂³×C₃	—	2	2	no	—
115	C₂³×C₃	C₉×Dic₃	1	2	yes	—
116	C₂³×C₃	3₋¹⁺²⋊C₄	1	2	yes	—
117	C₃×D₄	—	2	4	no	—
118	C₃×D₄	—	2	2	no	—
119	C₃×D₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	C₂×D₃×C₉	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
123	C₃×C₈	—	18	2	no	—
124	C₃⋊C₈	—	6	2	no	—
125	C₂×C₃²	—	2	1	no	—
126	C₂×C₃²	—	4	2	no	—
127	C₂×C₃²	C₃×#217	1	1	yes	characteristic
128	C₂×C₉	—	2	2	no	—
129	C₂×C₉	—	2	1	no	—
130	C₂×C₉	—	2	1	no	—
131	C₂×C₉	—	3	1	no	—
132	C₂×C₉	—	4	4	no	—
133	C₂×C₉	—	4	2	no	—
134	C₂×C₉	—	6	1	no	—
135	C₂×C₉	—	6	1	no	—
136	C₂×C₉	—	6	2	no	—
137	C₂×C₉	—	12	2	no	—
138	C₂×C₉	—	12	4	no	—
139	C₂×C₉	C₉×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×D₄	D₉⋊C₉	1	1	yes	characteristic
141	M₁₆	—	9	2	no	—
142	C₂²×C₃	—	2	4	no	—
143	C₂²×C₃	—	2	2	no	—
144	C₂²×C₃	—	2	2	no	—
145	C₂²×C₃	—	2	1	no	—
146	C₂²×C₃	—	4	2	no	—
147	C₂²×C₃	—	4	2	no	—
148	C₂²×C₃	—	4	4	no	—
149	C₂²×C₃	C₂×C₉×Dic₃	1	1	yes	characteristic
150	C₂²×C₃	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
151	C₃×C₄	—	2	2	no	—
152	C₃×C₄	C₉×#67	1	2	yes	—
153	C₃×C₄	#611⋊C₃	1	2	yes	—
154	C₃²	C₃×#833	1	1	yes	characteristic
155	C₉	—	2	1	no	—
156	C₉	—	3	1	no	—
157	C₉	—	6	1	no	—
158	C₉	C₉×#100	1	1	yes	characteristic, complemented
159	C₂³	Dic₉⋊C₉	1	2	yes	—
160	D₄	—	2	2	no	—
161	C₂×C₄	C₂×D₉⋊C₉	1	1	yes	characteristic
162	C₈	—	18	2	no	—
163	C₂×C₃	—	2	2	no	—
164	C₂×C₃	—	2	1	no	—
165	C₂×C₃	—	2	1	no	—
166	C₂×C₃	—	2	1	no	—
167	C₂×C₃	—	4	4	no	—
168	C₂×C₃	—	4	2	no	—
169	C₂×C₃	—	4	2	no	—
170	C₂×C₃	C₉×#217	1	1	yes	characteristic
171	C₂×C₃	#3676⋊C₃	1	1	yes	characteristic
172	C₂²	—	2	2	no	—
173	C₂²	—	4	2	no	—
174	C₂²	C₂×Dic₉⋊C₉	1	1	yes	characteristic
175	C₄	#611⋊C₉	1	2	yes	—
176	C₃	—	2	1	no	—
177	C₃	C₉×#833	1	1	yes	characteristic
178	C₃	#63391⋊C₃	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	4	2	no	—
181	C₂	#3676⋊C₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×C₉⋊C₉

C₂

yes

maximal, characteristic

#3667⋊C₉

C₂

yes

maximal, complemented

C₉×#833

—

maximal

C₃×#63391

C₃

yes

maximal, characteristic

C₂³×C₉⋊C₉

C₄

yes

—

D₄×C₉⋊C₉

—

C₂×C₄×C₉⋊C₉

C₂²

yes

characteristic

(C₉⋊C₈)⋊C₉

—

C₂×C₃×D₄×C₉

—

C₂×C₃×D₄×C₉

D₃

yes

characteristic

C₂×C₃×D₄×C₉

C₂×C₃

yes

characteristic

C₉×#208

—

C₃×#3667

C₂×C₃

yes

—

C₂²×C₉⋊C₉

—

C₂²×C₉⋊C₉

—

C₂²×C₉⋊C₉

C₂×C₄

yes

characteristic

C₄×C₉⋊C₉

D₄

yes

—

C₃×#833

—

C₉×#100

—

#63391

C₉

yes

characteristic, complemented

C₂³×C₃×C₉

—

C₂³×C₃×C₉

Dic₃

yes

—

C₂³×C₃×C₉

C₃×C₄

yes

—

C₃×D₄×C₉

—

C₃×D₄×C₉

—

C₃×D₄×C₉

—

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

C₂×D₃

yes

characteristic

C₂×C₃×C₄×C₉

C₂²×C₃

yes

characteristic

C₉×C₃⋊C₈

—

C₃×C₉⋊C₈

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

C₂²⋊C₄

yes

characteristic

C₂×C₃²×D₄

C₃×D₃

yes

characteristic

C₃×#208

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

C₂×C₉

yes

characteristic

C₉×M₁₆

—

#3667

C₂×C₉

yes

complemented

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×Dic₃

yes

characteristic

C₂²×C₃×C₉

C₂×C₃×C₄

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

#67

yes

—

C₃×C₄×C₉

C₃×D₄

yes

—

C₃×#100

—

#833

—

C₉⋊C₉

#100

yes

characteristic, complemented, Sylow

C₂³×C₃²

C₃×Dic₃

yes

—

C₃²×D₄

—

C₂×C₃²×C₄

C₂×C₃×D₃

yes

characteristic

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

C₄×C₉

yes

—

C₃×C₃⋊C₈

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²×C₉

yes

characteristic

C₈×C₉

—

C₉⋊C₈

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

#217

yes

characteristic

C₂×C₃×C₉

C₃×C₂²⋊C₄

yes

characteristic

C₂×C₃×D₄

—

C₂×C₃×D₄

D₃×C₉

yes

characteristic

C₂×C₃×D₄

AGL(1,ℤ/9)

yes

characteristic

C₃×M₁₆

—

#208

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₃×Dic₃

yes

characteristic

C₃²×C₄

C₃×#67

yes

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

100

C₂²×C₉

—

101

C₂²×C₉

—

102

C₂²×C₉

—

103

C₂²×C₉

—

104

C₂²×C₉

—

105

C₂²×C₉

C₂×C₄×C₉

yes

characteristic

106

C₄×C₉

—

107

C₄×C₉

—

108

C₄×C₉

—

109

C₄×C₉

D₄×C₉

yes

—

110

#100

—

Sylow

111

C₃×C₉

—

112

C₃×C₉

#833

yes

characteristic

113

C₃×C₉

C₃×#100

yes

characteristic

114

C₂³×C₃

—

115

C₂³×C₃

C₉×Dic₃

yes

—

116

C₂³×C₃

3₋¹⁺²⋊C₄

yes

—

117

C₃×D₄

—

118

C₃×D₄

—

119

C₃×D₄

—

120

C₂×C₃×C₄

—

121

C₂×C₃×C₄

C₂×D₃×C₉

yes

characteristic

122

C₂×C₃×C₄

C₂×AGL(1,ℤ/9)

yes

characteristic

123

C₃×C₈

—

124

C₃⋊C₈

—

125

C₂×C₃²

—

126

C₂×C₃²

—

127

C₂×C₃²

C₃×#217

yes

characteristic

128

C₂×C₉

—

129

C₂×C₉

—

130

C₂×C₉

—

131

C₂×C₉

—

132

C₂×C₉

—

133

C₂×C₉

—

134

C₂×C₉

—

135

C₂×C₉

—

136

C₂×C₉

—

137

C₂×C₉

—

138

C₂×C₉

—

139

C₂×C₉

C₉×C₂²⋊C₄

yes

characteristic

140

C₂×D₄

D₉⋊C₉

yes

characteristic

141

M₁₆

—

142

C₂²×C₃

—

143

C₂²×C₃

—

144

C₂²×C₃

—

145

C₂²×C₃

—

146

C₂²×C₃

—

147

C₂²×C₃

—

148

C₂²×C₃

—

149

C₂²×C₃

C₂×C₉×Dic₃

yes

characteristic

150

C₂²×C₃

C₂×3₋¹⁺²⋊C₄

yes

characteristic

151

C₃×C₄

—

152

C₃×C₄

C₉×#67

yes

—

153

C₃×C₄

#611⋊C₃

yes

—

154

C₃²

C₃×#833

yes

characteristic

155

C₉

—

156

C₉

—

157

C₉

—

158

C₉

C₉×#100

yes

characteristic, complemented

159

C₂³

Dic₉⋊C₉

yes

—

160

D₄

—

161

C₂×C₄

C₂×D₉⋊C₉

yes

characteristic

162

C₈

—

163

C₂×C₃

—

164

C₂×C₃

—

165

C₂×C₃

—

166

C₂×C₃

—

167

C₂×C₃

—

168

C₂×C₃

—

169

C₂×C₃

—

170

C₂×C₃

C₉×#217

yes

characteristic

171

C₂×C₃

#3676⋊C₃

yes

characteristic

172

C₂²

—

173

C₂²

—

174

C₂²

C₂×Dic₉⋊C₉

yes

characteristic

175

C₄

#611⋊C₉

yes

—

176

C₃

—

177

C₃

C₉×#833

yes

characteristic

178

C₃

#63391⋊C₃

yes

characteristic

179

C₂

—

180

C₂

—

181

C₂

#3676⋊C₉

yes

characteristic

	subgroup	G/H
center	C₂×C₃	#3676⋊C₃
commutator	C₂²×C₉	C₂×C₄×C₉
Frattini	C₂×C₃²×C₄	C₂×C₃×D₃
Fitting	C₂×D₄×C₉⋊C₉	C₂
socle	C₂×C₃²	C₃×#217
prime radical	C₂×C₃²×D₄	C₃×D₃
cosocle	C₂×C₃×C₄×C₉	C₂²×C₃
hypercenter	C₂×C₃×D₄	AGL(1,ℤ/9)
hypocenter	C₉	C₉×#100
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×D₄

subgroup

G/H

center

C₂×C₃

#3676⋊C₃

commutator

C₂²×C₉

C₂×C₄×C₉

Frattini

C₂×C₃²×C₄

C₂×C₃×D₃

Fitting

C₂×D₄×C₉⋊C₉

C₂

socle

C₂×C₃²

C₃×#217

prime radical

C₂×C₃²×D₄

C₃×D₃

cosocle

C₂×C₃×C₄×C₉

C₂²×C₃

C₂×C₃×D₄

C₉

G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×D₄

N	H ≅ G/N	type
2. #3667⋊C₉	C₂	semidirect
20. #63391	C₉	semidirect
42. #3667	C₂×C₉	semidirect
57. C₉⋊C₉	#100	semidirect
158. C₉	C₉×#100	semidirect

H ≅ G/N

type

2. #3667⋊C₉

C₂

semidirect

20. #63391

C₉

semidirect

42. #3667

C₂×C₉

semidirect

57. C₉⋊C₉

#100

semidirect

158. C₉

C₉×#100

semidirect

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

subgroup	index
C₂×D₄×C₉⋊C₉	2
#3667⋊C₉	2
C₉×#833	3
C₃×#63391	3

subgroup

index

C₂×D₄×C₉⋊C₉

#3667⋊C₉

C₉×#833

C₃×#63391

Subgroups — #72219975

Conjugacy classes of subgroups

Complements

Maximal subgroups