Group #72243884 — subgroups

Subgroups — C₂×#14550736 #72243884

Order 4374.

516 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 154 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	#14550736	C₂	1	1	yes	maximal, characteristic, direct product factor, Sylow
2	C₂×#10624887	C₃	1	1	yes	maximal, characteristic
3	C₂×#10624709	C₃	1	3	yes	maximal, complemented
4	C₂×#10624704	C₃	1	3	yes	maximal
5	C₂×#10624472	C₃	1	3	yes	maximal, complemented
6	C₂×#10624465	C₃	1	2	yes	maximal, complemented
7	C₂×#10624465	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624887	C₂×C₃	1	1	yes	characteristic
9	#10624709	C₂×C₃	1	3	yes	complemented
10	#10624704	C₂×C₃	1	3	yes	—
11	#10624472	C₂×C₃	1	3	yes	complemented
12	#10624465	C₂×C₃	1	2	yes	complemented
13	#10624465	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂×C₃×#737	C₃²	1	1	yes	characteristic
15	C₂×C₃×C₂₇⋊C₃	C₃²	1	6	yes	—
16	C₂×C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
17	C₂×#6900	—	3	1	no	—
18	C₂×#6899	—	3	2	no	—
19	C₂×#6896	—	3	3	no	—
20	C₂×#6894	—	3	2	no	—
21	C₂×#6894	—	3	3	no	—
22	C₂×C₉⋊H(3,𝔽₃)	—	3	1	no	—
23	C₂×#6887	C₃²	1	1	yes	characteristic
24	C₂×#6887	C₃²	1	1	yes	characteristic
25	C₂×C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
26	C₂×#6875	—	3	9	no	—
27	C₂×#6873	—	3	9	no	—
28	C₃×#737	C₂×C₃²	1	1	yes	characteristic
29	C₃×C₂₇⋊C₃	C₂×C₃²	1	6	yes	—
30	C₃×C₂₇⋊C₃	C₂×C₃²	1	3	yes	—
31	#6900	—	3	1	no	—
32	#6899	—	3	2	no	—
33	#6896	—	3	3	no	—
34	#6894	—	3	2	no	—
35	#6894	—	3	3	no	—
36	C₉⋊H(3,𝔽₃)	—	3	1	no	—
37	#6887	C₂×C₃²	1	1	yes	characteristic
38	#6887	C₂×C₃²	1	1	yes	characteristic
39	C₃×C₉⋊C₉	C₂×C₃²	1	1	yes	characteristic
40	#6875	—	3	9	no	—
41	#6873	—	3	9	no	—
42	C₂×#737	—	3	3	no	—
43	C₂×#737	—	3	6	no	—
44	C₂×C₃×3₋¹⁺²	—	3	3	no	—
45	C₂×C₃×3₋¹⁺²	—	3	1	no	—
46	C₂×C₃×3₋¹⁺²	—	3	1	no	—
47	C₂×C₃×3₋¹⁺²	3₋¹⁺²	1	2	yes	—
48	C₂×C₃×H(3,𝔽₃)	3₋¹⁺²	1	1	yes	characteristic
49	C₂×C₃²×C₉	—	3	1	no	—
50	C₂×C₃²×C₉	—	3	1	no	—
51	C₂×C₃²×C₉	C₃³	1	1	yes	characteristic
52	C₂×C₂₇⋊C₃	—	3	3	no	—
53	C₂×C₂₇⋊C₃	—	3	6	no	—
54	C₂×C₂₇⋊C₃	—	3	6	no	—
55	C₂×C₂₇⋊C₃	—	3	6	no	—
56	C₂×C₂₇⋊C₃	—	3	6	no	—
57	C₂×C₃×C₂₇	—	3	6	no	—
58	C₂×C₃×C₂₇	3₋¹⁺²	1	9	yes	complemented
59	C₂×C₉⋊C₉	—	3	12	no	—
60	C₂×C₉⋊C₉	—	3	6	no	—
61	C₂×C₉⋊C₉	—	3	1	no	—
62	C₂×C₉⋊C₉	—	3	1	no	—
63	C₂×C₉⋊C₉	—	3	1	no	—
64	C₂×C₉⋊C₉	—	3	1	no	—
65	C₂×C₉⋊C₉	—	3	1	no	—
66	C₂×C₉⋊C₉	H(3,𝔽₃)	1	3	yes	—
67	C₂×C₉⋊C₉	3₋¹⁺²	1	2	yes	—
68	C₂×C₉⋊C₉	3₋¹⁺²	1	1	yes	characteristic
69	C₂×C₃²⋊C₉	—	3	6	no	—
70	C₂×C₃²⋊C₉	—	3	1	no	—
71	C₂×C₃²⋊C₉	—	3	1	no	—
72	C₂×C₉²	—	3	1	no	—
73	C₂×C₉²	3₋¹⁺²	1	3	yes	—
74	#737	—	3	3	no	—
75	#737	—	3	6	no	—
76	C₃×3₋¹⁺²	—	3	3	no	—
77	C₃×3₋¹⁺²	—	3	1	no	—
78	C₃×3₋¹⁺²	—	3	1	no	—
79	C₃×3₋¹⁺²	C₂×3₋¹⁺²	1	2	yes	—
80	C₃×H(3,𝔽₃)	C₂×3₋¹⁺²	1	1	yes	characteristic
81	C₃²×C₉	—	3	1	no	—
82	C₃²×C₉	—	3	1	no	—
83	C₃²×C₉	C₂×C₃³	1	1	yes	characteristic
84	C₂₇⋊C₃	—	3	3	no	—
85	C₂₇⋊C₃	—	3	6	no	—
86	C₂₇⋊C₃	—	3	6	no	—
87	C₂₇⋊C₃	—	3	6	no	—
88	C₂₇⋊C₃	—	3	6	no	—
89	C₃×C₂₇	—	3	6	no	—
90	C₃×C₂₇	C₂×3₋¹⁺²	1	9	yes	complemented
91	C₉⋊C₉	—	3	12	no	—
92	C₉⋊C₉	—	3	6	no	—
93	C₉⋊C₉	—	3	1	no	—
94	C₉⋊C₉	—	3	1	no	—
95	C₉⋊C₉	—	3	1	no	—
96	C₉⋊C₉	—	3	1	no	—
97	C₉⋊C₉	—	3	1	no	—
98	C₉⋊C₉	C₂×H(3,𝔽₃)	1	3	yes	—
99	C₉⋊C₉	C₂×3₋¹⁺²	1	2	yes	—
100	C₉⋊C₉	C₂×3₋¹⁺²	1	1	yes	characteristic
101	C₃²⋊C₉	—	3	6	no	—
102	C₃²⋊C₉	—	3	1	no	—
103	C₃²⋊C₉	—	3	1	no	—
104	C₉²	—	3	1	no	—
105	C₉²	C₂×3₋¹⁺²	1	3	yes	—
106	C₂×C₃³	—	3	1	no	—
107	C₂×C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
108	C₂×3₋¹⁺²	—	3	6	no	—
109	C₂×3₋¹⁺²	—	9	3	no	—
110	C₂×3₋¹⁺²	—	9	2	no	—
111	C₂×3₋¹⁺²	—	9	1	no	—
112	C₂×3₋¹⁺²	—	27	3	no	—
113	C₂×H(3,𝔽₃)	—	3	3	no	—
114	C₂×C₃×C₉	—	3	3	no	—
115	C₂×C₃×C₉	—	3	3	no	—
116	C₂×C₃×C₉	—	3	2	no	—
117	C₂×C₃×C₉	—	3	2	no	—
118	C₂×C₃×C₉	—	3	1	no	—
119	C₂×C₃×C₉	—	9	3	no	—
120	C₂×C₃×C₉	—	27	1	no	—
121	C₂×C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
122	C₂×C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
123	C₂×C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
124	C₂×C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
125	C₂×C₃×C₉	#737	1	6	yes	—
126	C₂×C₂₇	—	3	9	no	—
127	C₂×C₂₇	—	9	6	no	—
128	C₃³	—	3	1	no	—
129	C₃³	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic
130	3₋¹⁺²	—	3	6	no	—
131	3₋¹⁺²	—	9	3	no	—
132	3₋¹⁺²	—	9	2	no	—
133	3₋¹⁺²	—	9	1	no	—
134	3₋¹⁺²	—	27	3	no	—
135	H(3,𝔽₃)	—	3	3	no	—
136	C₃×C₉	—	3	3	no	—
137	C₃×C₉	—	3	3	no	—
138	C₃×C₉	—	3	2	no	—
139	C₃×C₉	—	3	2	no	—
140	C₃×C₉	—	3	1	no	—
141	C₃×C₉	—	9	3	no	—
142	C₃×C₉	—	27	1	no	—
143	C₃×C₉	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
144	C₃×C₉	C₂×C₃×3₋¹⁺²	1	3	yes	—
145	C₃×C₉	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic
146	C₃×C₉	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic
147	C₃×C₉	C₂×#737	1	6	yes	—
148	C₂₇	—	3	9	no	—
149	C₂₇	—	9	6	no	—
150	C₂×C₃²	—	9	1	no	—
151	C₂×C₃²	—	9	3	no	—
152	C₂×C₃²	—	9	1	no	—
153	C₂×C₃²	#6891	1	1	yes	characteristic
154	C₂×C₃²	#6894	1	3	yes	—
155	C₂×C₉	—	9	2	no	—
156	C₂×C₉	—	9	1	no	—
157	C₂×C₉	—	9	3	no	—
158	C₂×C₉	—	9	3	no	—
159	C₂×C₉	—	9	2	no	—
160	C₂×C₉	C₉⋊H(3,𝔽₃)	1	3	yes	—
161	C₂×C₉	#6896	1	3	yes	—
162	C₂×C₉	#6899	1	2	yes	—
163	C₂×C₉	#6900	1	1	yes	characteristic
164	C₃²	—	9	1	no	—
165	C₃²	—	9	3	no	—
166	C₃²	—	9	1	no	—
167	C₃²	C₂×#6891	1	1	yes	characteristic
168	C₃²	C₂×#6894	1	3	yes	—
169	C₉	—	9	2	no	—
170	C₉	—	9	1	no	—
171	C₉	—	9	3	no	—
172	C₉	—	9	3	no	—
173	C₉	—	9	2	no	—
174	C₉	C₂×C₉⋊H(3,𝔽₃)	1	3	yes	—
175	C₉	C₂×#6896	1	3	yes	—
176	C₉	C₂×#6899	1	2	yes	—
177	C₉	C₂×#6900	1	1	yes	characteristic
178	C₂×C₃	—	3	1	no	—
179	C₂×C₃	—	3	3	no	—
180	C₂×C₃	—	27	1	no	—
181	C₂×C₃	#10624565	1	1	yes	characteristic
182	C₃	—	3	1	no	—
183	C₃	—	3	3	no	—
184	C₃	—	27	1	no	—
185	C₃	C₂×#10624565	1	1	yes	characteristic
186	C₂	#14550736	1	1	yes	characteristic, direct product factor, Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#14550736

C₂

yes

maximal, characteristic, direct product factor, Sylow

C₂×#10624887

C₃

yes

maximal, characteristic

C₂×#10624709

C₃

yes

maximal, complemented

C₂×#10624704

C₃

yes

maximal

C₂×#10624472

C₃

yes

maximal, complemented

C₂×#10624465

C₃

yes

maximal, complemented

C₂×#10624465

C₃

yes

maximal, characteristic, complemented

#10624887

C₂×C₃

yes

characteristic

#10624709

C₂×C₃

yes

complemented

#10624704

C₂×C₃

yes

—

#10624472

C₂×C₃

yes

complemented

#10624465

C₂×C₃

yes

complemented

#10624465

C₂×C₃

yes

characteristic, complemented

C₂×C₃×#737

C₃²

yes

characteristic

C₂×C₃×C₂₇⋊C₃

C₃²

yes

—

C₂×C₃×C₂₇⋊C₃

C₃²

yes

—

—

—

—

—

—

C₂×C₉⋊H(3,𝔽₃)

—

C₂×#6887

C₃²

yes

characteristic

C₂×#6887

C₃²

yes

characteristic

C₂×C₃×C₉⋊C₉

C₃²

yes

—

—

C₂×C₃²

yes

characteristic

C₃×C₂₇⋊C₃

C₂×C₃²

yes

—

C₃×C₂₇⋊C₃

C₂×C₃²

yes

—

—

—

—

—

—

C₉⋊H(3,𝔽₃)

—

#6887

C₂×C₃²

yes

characteristic

#6887

C₂×C₃²

yes

characteristic

C₃×C₉⋊C₉

C₂×C₃²

yes

—

—

—

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

3₋¹⁺²

yes

—

C₂×C₃×H(3,𝔽₃)

3₋¹⁺²

yes

characteristic

C₂×C₃²×C₉

—

C₂×C₃²×C₉

—

C₂×C₃²×C₉

C₃³

yes

characteristic

C₂×C₂₇⋊C₃

—

C₂×C₂₇⋊C₃

—

C₂×C₂₇⋊C₃

—

C₂×C₂₇⋊C₃

—

C₂×C₂₇⋊C₃

—

C₂×C₃×C₂₇

—

C₂×C₃×C₂₇

3₋¹⁺²

yes

complemented

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

H(3,𝔽₃)

yes

—

C₂×C₉⋊C₉

3₋¹⁺²

yes

—

C₂×C₉⋊C₉

3₋¹⁺²

yes

characteristic

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₉²

—

C₂×C₉²

3₋¹⁺²

yes

—

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₂×3₋¹⁺²

yes

—

C₃×H(3,𝔽₃)

C₂×3₋¹⁺²

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₂×C₃³

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₃×C₂₇

—

C₃×C₂₇

C₂×3₋¹⁺²

yes

complemented

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₂×H(3,𝔽₃)

yes

—

C₉⋊C₉

C₂×3₋¹⁺²

yes

—

100

C₉⋊C₉

C₂×3₋¹⁺²

yes

characteristic

101

C₃²⋊C₉

—

102

C₃²⋊C₉

—

103

C₃²⋊C₉

—

104

C₉²

—

105

C₉²

C₂×3₋¹⁺²

yes

—

106

C₂×C₃³

—

107

C₂×C₃³

C₃×3₋¹⁺²

yes

characteristic

108

C₂×3₋¹⁺²

—

109

C₂×3₋¹⁺²

—

110

C₂×3₋¹⁺²

—

111

C₂×3₋¹⁺²

—

112

C₂×3₋¹⁺²

—

113

C₂×H(3,𝔽₃)

—

114

C₂×C₃×C₉

—

115

C₂×C₃×C₉

—

116

C₂×C₃×C₉

—

117

C₂×C₃×C₉

—

118

C₂×C₃×C₉

—

119

C₂×C₃×C₉

—

120

C₂×C₃×C₉

—

121

C₂×C₃×C₉

C₃×H(3,𝔽₃)

yes

characteristic

122

C₂×C₃×C₉

C₃×3₋¹⁺²

yes

—

123

C₂×C₃×C₉

C₃×3₋¹⁺²

yes

characteristic

124

C₂×C₃×C₉

C₃×3₋¹⁺²

yes

characteristic

125

C₂×C₃×C₉

#737

yes

—

126

C₂×C₂₇

—

127

C₂×C₂₇

—

128

C₃³

—

129

C₃³

C₂×C₃×3₋¹⁺²

yes

characteristic

130

3₋¹⁺²

—

131

3₋¹⁺²

—

132

3₋¹⁺²

—

133

3₋¹⁺²

—

134

3₋¹⁺²

—

135

H(3,𝔽₃)

—

136

C₃×C₉

—

137

C₃×C₉

—

138

C₃×C₉

—

139

C₃×C₉

—

140

C₃×C₉

—

141

C₃×C₉

—

142

C₃×C₉

—

143

C₃×C₉

C₂×C₃×H(3,𝔽₃)

yes

characteristic

144

C₃×C₉

C₂×C₃×3₋¹⁺²

yes

—

145

C₃×C₉

C₂×C₃×3₋¹⁺²

yes

characteristic

146

C₃×C₉

C₂×C₃×3₋¹⁺²

yes

characteristic

147

C₃×C₉

C₂×#737

yes

—

148

C₂₇

—

149

C₂₇

—

150

C₂×C₃²

—

151

C₂×C₃²

—

152

C₂×C₃²

—

153

C₂×C₃²

#6891

yes

characteristic

154

C₂×C₃²

#6894

yes

—

155

C₂×C₉

—

156

C₂×C₉

—

157

C₂×C₉

—

158

C₂×C₉

—

159

C₂×C₉

—

160

C₂×C₉

C₉⋊H(3,𝔽₃)

yes

—

161

C₂×C₉

#6896

yes

—

162

C₂×C₉

#6899

yes

—

163

C₂×C₉

#6900

yes

characteristic

164

C₃²

—

165

C₃²

—

166

C₃²

—

167

C₃²

C₂×#6891

yes

characteristic

168

C₃²

C₂×#6894

yes

—

169

C₉

—

170

C₉

—

171

C₉

—

172

C₉

—

173

C₉

—

174

C₉

C₂×C₉⋊H(3,𝔽₃)

yes

—

175

C₉

C₂×#6896

yes

—

176

C₉

C₂×#6899

yes

—

177

C₉

C₂×#6900

yes

characteristic

178

C₂×C₃

—

179

C₂×C₃

—

180

C₂×C₃

—

181

C₂×C₃

#10624565

yes

characteristic

182

C₃

—

183

C₃

—

184

C₃

—

185

C₃

C₂×#10624565

yes

characteristic

186

C₂

#14550736

yes

characteristic, direct product factor, Sylow

	subgroup	G/H
center, socle	C₂×C₃	#10624565
commutator, Frattini, cosocle	C₃²×C₉	C₂×C₃³
prime radical	C₂×C₃×H(3,𝔽₃)	3₋¹⁺²
LCS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃²×C₉ ◁ G

subgroup

G/H

center, socle

C₂×C₃

#10624565

commutator, Frattini, cosocle

C₃²×C₉

C₂×C₃³

prime radical

C₂×C₃×H(3,𝔽₃)

3₋¹⁺²

LCS

G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₂×C₃ ◁ C₂×C₃²×C₉ ◁ G

N	H ≅ G/N	type
1. #14550736	C₂	direct (N × H)
3. C₂×#10624709	C₃	semidirect
5. C₂×#10624472	C₃	semidirect
6. C₂×#10624465	C₃	semidirect
7. C₂×#10624465	C₃	semidirect
9. #10624709	C₂×C₃	semidirect
11. #10624472	C₂×C₃	semidirect
12. #10624465	C₂×C₃	semidirect
13. #10624465	C₂×C₃	semidirect
58. C₂×C₃×C₂₇	3₋¹⁺²	semidirect
90. C₃×C₂₇	C₂×3₋¹⁺²	semidirect
186. C₂	#14550736	direct (N × H)

H ≅ G/N

type

1. #14550736

C₂

direct (N × H)

3. C₂×#10624709

C₃

semidirect

5. C₂×#10624472

C₃

semidirect

6. C₂×#10624465

C₃

semidirect

7. C₂×#10624465

C₃

semidirect

9. #10624709

C₂×C₃

semidirect

11. #10624472

C₂×C₃

semidirect

12. #10624465

C₂×C₃

semidirect

13. #10624465

C₂×C₃

semidirect

58. C₂×C₃×C₂₇

3₋¹⁺²

semidirect

90. C₃×C₂₇

C₂×3₋¹⁺²

semidirect

186. C₂

#14550736

direct (N × H)

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

subgroup	index
#14550736	2
C₂×#10624887	3
C₂×#10624709	3
C₂×#10624704	3
C₂×#10624472	3
C₂×#10624465	3

subgroup

index

Subgroups — C₂×#14550736 #72243884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14550736 #72243884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14550736 #72243884