Group #33002202 — subgroups

Subgroups — #33002202

Order 18688.

434 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 135 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×#17988542	C₂	1	1	yes	maximal, characteristic
2	#22611542	C₂	1	1	yes	maximal, characteristic
3	#22608014	C₂	1	4	yes	maximal, complemented
4	C₇₃×#2112	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	C₄	1	1	yes	characteristic
6	C₂×#14556550	C₄	1	1	yes	characteristic
7	C₂×#14556531	C₂²	1	2	yes	—
8	#17988542	—	2	2	no	—
9	#17988542	—	2	2	no	—
10	C₂×#14556501	C₂²	1	2	yes	—
11	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
12	C₂²⋊#11924003	—	2	2	no	—
13	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
15	C₂²⋊C₁₆×C₇₃	—	2	2	no	—
16	C₂×#11924012	—	2	4	no	—
17	C₂×#11924012	—	2	2	no	—
18	C₂²×#10596831	C₂×C₄	1	2	yes	—
19	#14556550	—	2	2	no	—
20	#14556550	—	2	2	no	—
21	C₂×C₄⋊Dic₇₃	C₈	1	2	yes	—
22	C₂×C₂²⋊Dic₇₃	C₈	1	2	yes	—
23	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
24	#14556531	—	2	4	no	—
25	C₂×#11924003	—	2	2	no	—
26	#14556501	—	2	2	no	—
27	#14556501	—	2	2	no	—
28	C₈×Dic₇₃	—	2	4	no	—
29	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	—	2	4	no	—
31	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₃	D₄	1	2	yes	—
33	C₂²×C₈×C₇₃	D₄	1	2	yes	—
34	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	M₃₂×C₇₃	—	2	4	no	—
36	C₂×C₁₆×C₇₃	—	2	2	no	—
37	C₂²×Dic₇₃	C₂×C₈	1	4	yes	—
38	#11924012	—	2	8	no	—
39	C₂×#10596831	—	2	2	no	—
40	C₂×#10596831	—	2	2	no	—
41	C₂×#10596831	M₁₆	1	4	yes	—
42	C₄⋊Dic₇₃	—	2	4	no	—
43	C₂²⋊Dic₇₃	—	2	4	no	—
44	C₄×Dic₇₃	—	2	2	no	—
45	C₄×Dic₇₃	—	2	2	no	—
46	#11924003	—	4	2	no	—
47	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	—	2	2	no	—
49	C₂²×C₄×C₇₃	—	2	2	no	—
50	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
53	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	—	2	2	no	—
55	C₂×C₈×C₇₃	—	2	4	no	—
56	C₂×C₈×C₇₃	—	2	4	no	—
57	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	#41	1	4	yes	—
61	C₁₆×C₇₃	—	4	2	no	—
62	C₂³×C₇₃	—	2	2	no	—
63	C₂³×C₇₃	—	4	2	no	—
64	C₂³×C₇₃	C₂²⋊C₈	1	2	yes	—
65	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	4	no	—
70	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#117	1	2	yes	—
73	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₇₃	—	2	4	no	—
76	C₂×Dic₇₃	—	4	4	no	—
77	C₈×C₇₃	—	2	2	no	—
78	C₈×C₇₃	#123	1	2	yes	—
79	C₈×C₇₃	#123	1	2	yes	—
80	#10596831	—	2	4	no	—
81	C₂²×C₇₃	—	2	2	no	—
82	C₂²×C₇₃	—	2	2	no	—
83	C₂²×C₇₃	—	2	4	no	—
84	C₂²×C₇₃	—	4	4	no	—
85	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
86	C₂²×C₇₃	#431	1	2	yes	—
87	C₄×C₇₃	—	2	2	no	—
88	C₄×C₇₃	#381	1	1	yes	characteristic
89	C₄×C₇₃	#407	1	1	yes	characteristic
90	C₄×C₇₃	#432	1	2	yes	—
91	Dic₇₃	—	4	4	no	—
92	#12847	—	73	1	no	maximal, Sylow
93	C₂×C₇₃	—	2	2	no	—
94	C₂×C₇₃	—	4	2	no	—
95	C₂×C₇₃	#1817	1	1	yes	characteristic
96	C₂×C₇₃	#2115	1	2	yes	—
97	C₂×#405	—	73	1	no	—
98	#2112	—	73	1	no	—
99	#2112	D₇₃	1	1	yes	characteristic
100	#1377	—	73	4	no	—
101	C₇₃	#12847	1	1	yes	characteristic, complemented, Sylow
102	C₂²×M₁₆	—	73	1	no	—
103	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
104	C₂×#117	—	73	1	no	—
105	C₂×M₃₂	—	73	2	no	—
106	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
107	#405	—	146	2	no	—
108	#405	—	146	2	no	—
109	C₂×#97	—	73	2	no	—
110	C₂×C₄×C₈	—	73	2	no	—
111	C₂²⋊C₁₆	—	2	2	no	—
112	C₂²⋊C₁₆	—	146	2	no	—
113	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
114	C₂×M₁₆	—	146	4	no	—
115	C₂×M₁₆	—	146	2	no	—
116	C₂²×C₈	—	2	4	no	—
117	C₂²×C₈	—	73	2	no	—
118	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
119	C₂²×C₈	D₂₉₂	1	2	yes	—
120	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
121	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
122	#117	—	146	2	no	—
123	#117	—	146	2	no	—
124	C₂×C₄⋊C₄	—	73	2	no	—
125	C₂×C₂²⋊C₄	—	73	2	no	—
126	C₂×C₄²	—	73	2	no	—
127	M₃₂	—	2	4	no	—
128	M₃₂	—	146	4	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	146	2	no	—
131	#97	—	146	2	no	—
132	#97	—	146	2	no	—
133	C₄×C₈	—	146	4	no	—
134	C₂⁴	C₈×D₇₃	1	1	yes	characteristic
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	73	4	no	—
138	C₂²×C₄	#11924005	1	2	yes	—
139	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
140	C₂²×C₄	#11924011	1	2	yes	—
141	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
142	M₁₆	—	146	8	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	73	4	no	—
147	C₂×C₈	—	146	2	no	—
148	C₂×C₈	—	146	2	no	—
149	C₂×C₈	#11924005	1	2	yes	—
150	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
151	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
152	C₂×C₈	#11924027	1	4	yes	—
153	C₄⋊C₄	—	146	4	no	—
154	C₂²⋊C₄	—	146	4	no	—
155	C₄²	—	146	2	no	—
156	C₄²	—	146	2	no	—
157	C₁₆	—	4	2	no	—
158	C₁₆	—	292	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	4	2	no	—
161	C₂³	#14556503	1	2	yes	—
162	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	292	4	no	—
168	C₂×C₄	—	146	4	no	—
169	C₂×C₄	#14556503	1	2	yes	—
170	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
171	C₂×C₄	#14556547	1	2	yes	—
172	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
173	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	146	4	no	—
176	C₈	#14556572	1	2	yes	—
177	C₈	#14556575	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	2	4	no	—
181	C₂²	—	4	4	no	—
182	C₂²	C₂×#14556503	1	1	yes	characteristic
183	C₂²	#17988637	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	292	4	no	—
186	C₄	#17988448	1	1	yes	characteristic
187	C₄	#17988550	1	1	yes	characteristic
188	C₄	#17988641	1	2	yes	—
189	C₂	—	2	2	no	—
190	C₂	—	4	2	no	—
191	C₂	#22609997	1	1	yes	characteristic
192	C₂	#22611551	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988542

C₂

yes

maximal, characteristic

#22611542

C₂

yes

maximal, characteristic

#22608014

C₂

yes

maximal, complemented

C₇₃×#2112

C₂

yes

maximal, characteristic

C₂²×#11924012

C₄

yes

characteristic

C₂×#14556550

C₄

yes

characteristic

C₂×#14556531

C₂²

yes

—

#17988542

—

#17988542

—

C₂×#14556501

C₂²

yes

—

C₂×C₈×Dic₇₃

C₄

yes

—

C₂²⋊#11924003

—

C₂³×C₈×C₇₃

C₂²

yes

characteristic

C₂×M₃₂×C₇₃

C₂²

yes

—

C₂²⋊C₁₆×C₇₃

—

C₂×#11924012

—

C₂×#11924012

—

C₂²×#10596831

C₂×C₄

yes

—

#14556550

—

#14556550

—

C₂×C₄⋊Dic₇₃

C₈

yes

—

C₂×C₂²⋊Dic₇₃

C₈

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₇₃

—

C₂³×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂²×Dic₇₃

C₂×C₈

yes

—

—

—

—

M₁₆

yes

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂⁴×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂²⋊C₈

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#123

yes

—

C₈×C₇₃

#123

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₃

#431

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12847

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1817

yes

characteristic

C₂×C₇₃

#2115

yes

—

C₂×#405

—

#2112

—

#2112

D₇₃

yes

characteristic

100

#1377

—

101

C₇₃

#12847

yes

characteristic, complemented, Sylow

102

C₂²×M₁₆

—

103

C₂³×C₈

C₂×D₇₃

yes

characteristic

104

C₂×#117

—

105

C₂×M₃₂

—

106

C₂×M₃₂

C₂×D₇₃

yes

—

107

#405

—

146

—

108

#405

—

146

—

109

C₂×#97

—

110

C₂×C₄×C₈

—

111

C₂²⋊C₁₆

—

112

C₂²⋊C₁₆

—

146

—

113

C₂³×C₄

C₄×D₇₃

yes

characteristic

114

C₂×M₁₆

—

146

—

115

C₂×M₁₆

—

146

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂²×C₈

C₄×D₇₃

yes

characteristic

119

C₂²×C₈

D₂₉₂

yes

—

120

C₂²×C₈

C₂²⋊D₇₃

yes

—

121

C₂²×C₈

C₂²×D₇₃

yes

characteristic

122

#117

—

146

—

123

#117

—

146

—

124

C₂×C₄⋊C₄

—

125

C₂×C₂²⋊C₄

—

126

C₂×C₄²

—

127

M₃₂

—

128

M₃₂

—

146

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

146

—

131

#97

—

146

—

132

#97

—

146

—

133

C₄×C₈

—

146

—

134

C₂⁴

C₈×D₇₃

yes

characteristic

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

#11924005

yes

—

139

C₂²×C₄

C₈×D₇₃

yes

characteristic

140

C₂²×C₄

#11924011

yes

—

141

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

142

M₁₆

—

146

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

146

—

148

C₂×C₈

—

146

—

149

C₂×C₈

#11924005

yes

—

150

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

151

C₂×C₈

C₂×D₂₉₂

yes

characteristic

152

C₂×C₈

#11924027

yes

—

153

C₄⋊C₄

—

146

—

154

C₂²⋊C₄

—

146

—

155

C₄²

—

146

—

156

C₄²

—

146

—

157

C₁₆

—

158

C₁₆

—

292

—

159

C₂³

—

160

C₂³

—

161

C₂³

#14556503

yes

—

162

C₂³

C₂×C₈×D₇₃

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

292

—

168

C₂×C₄

—

146

—

169

C₂×C₄

#14556503

yes

—

170

C₂×C₄

C₂×#11924005

yes

characteristic

171

C₂×C₄

#14556547

yes

—

172

C₂×C₄

M₁₆×D₇₃

yes

—

173

C₂×C₄

C₂×#11924011

yes

characteristic

174

C₈

—

175

C₈

—

146

—

176

C₈

#14556572

yes

—

177

C₈

#14556575

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

C₂×#14556503

yes

characteristic

183

C₂²

#17988637

yes

—

184

C₄

—

185

C₄

—

292

—

186

C₄

#17988448

yes

characteristic

187

C₄

#17988550

yes

characteristic

188

C₄

#17988641

yes

—

189

C₂

—

190

C₂

—

191

C₂

#22609997

yes

characteristic

192

C₂

#22611551

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#11924005
commutator	C₂³×C₇₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2112	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#2112	D₇₃
hypocenter	C₇₃	#12847
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ #2112

subgroup

G/H

center

C₂×C₄

C₂×#11924005

commutator

C₂³×C₇₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#2112

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₇₃

C₂×C₈

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ #2112

N	H ≅ G/N	type
3. #22608014	C₂	semidirect
101. C₇₃	#12847	semidirect

H ≅ G/N

type

3. #22608014

C₂

semidirect

101. C₇₃

#12847

semidirect

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

subgroup	index
C₂×#17988542	2
#22611542	2
#22608014	2
C₇₃×#2112	2
#12847	73

subgroup

index

C₇₃×#2112

Subgroups — #33002202

Conjugacy classes of subgroups

Complements

Maximal subgroups