Group #33239361 — subgroups

Subgroups — #33239361

Order 18688.

646 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 211 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	C₂²×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic
2	#22618798	C₂	1	2	yes	maximal
3	#22611801	C₂	1	4	yes	maximal, complemented
4	#22611789	C₂	1	4	yes	maximal, complemented
5	#22611650	C₂	1	2	yes	maximal
6	#22611527	C₂	1	1	yes	maximal, characteristic
7	C₇₃×#2158	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₉₂	C₂²	1	2	yes	—
9	#17989064	—	2	2	no	—
10	C₂×Dic₅₈₄	C₄	1	8	yes	complemented
11	C₂×Dic₅₈₄	C₂²	1	2	yes	—
12	C₂×Dic₅₈₄	C₂²	1	2	yes	—
13	C₂×#14556531	C₂²	1	1	yes	characteristic
14	#17988693	—	2	4	no	—
15	#17988681	C₂²	1	8	yes	—
16	#17988595	—	2	2	no	—
17	#17988542	C₂²	1	1	yes	characteristic
18	#17988390	C₂²	1	8	yes	—
19	#17988362	C₂²	1	2	yes	—
20	C₁₆×Dic₇₃	C₂²	1	2	yes	—
21	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
22	#427×C₇₃	C₂²	1	2	yes	—
23	#367×C₇₃	C₂²	1	2	yes	—
24	#366×C₇₃	C₂²	1	2	yes	—
25	C₂²×Dic₁₄₆	—	2	2	no	—
26	C₂×Dic₂₉₂	—	2	2	no	—
27	C₂×Dic₂₉₂	—	2	4	no	—
28	C₂×Dic₂₉₂	C₂×C₄	1	8	yes	—
29	C₂×Dic₂₉₂	C₂³	1	4	yes	—
30	C₂×#11924012	D₄	1	1	yes	characteristic
31	C₄×Dic₁₄₆	—	4	4	no	—
32	#14556556	—	2	8	no	—
33	#14556550	D₄	1	1	yes	characteristic
34	#14556547	—	4	4	no	—
35	Dic₅₈₄	C₂×C₄	1	8	yes	—
36	Dic₅₈₄	C₂×C₄	1	8	yes	—
37	#14556531	—	2	2	no	—
38	C₂×#11924003	C₂³	1	2	yes	—
39	#14556514	—	2	8	no	—
40	#14556501	D₄	1	2	yes	—
41	C₈×Dic₇₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
43	#117×C₇₃	—	2	2	no	—
44	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
45	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
47	#107×C₇₃	C₂³	1	4	yes	—
48	#106×C₇₃	—	2	2	no	—
49	C₂×Dic₁₄₆	—	2	8	no	—
50	C₂×Dic₁₄₆	—	2	4	no	—
51	C₂×Dic₁₄₆	—	4	2	no	—
52	C₂×Dic₁₄₆	—	4	4	no	—
53	C₂²×Dic₇₃	—	4	2	no	—
54	Dic₂₉₂	—	2	8	no	—
55	Dic₂₉₂	—	2	4	no	—
56	Dic₂₉₂	C₂²×C₄	1	8	yes	—
57	#11924012	—	2	2	no	—
58	C₂×#10596831	C₂×D₄	1	2	yes	—
59	C₄⋊Dic₇₃	D₈	1	2	yes	—
60	#11924007	—	4	8	no	—
61	C₂²⋊Dic₇₃	D₈	1	2	yes	—
62	C₄×Dic₇₃	—	4	8	no	—
63	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
64	#11924003	—	2	2	no	—
65	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₃	#41	1	2	yes	—
69	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₇₃	—	2	4	no	—
71	C₂²⋊C₄×C₇₃	—	4	2	no	—
72	C₄²×C₇₃	—	4	2	no	—
73	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
74	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	—	4	4	no	—
77	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#135	1	2	yes	—
80	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₇₃	—	4	4	no	—
82	C₂×Dic₇₃	—	4	8	no	—
83	C₂×Dic₇₃	C₂×D₈	1	4	yes	—
84	Dic₁₄₆	—	2	8	no	—
85	Dic₁₄₆	—	4	4	no	—
86	Dic₁₄₆	—	4	8	no	—
87	C₈×C₇₃	C₄×D₄	1	2	yes	—
88	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
90	#10596831	—	2	2	no	—
91	C₂²×C₇₃	—	2	2	no	—
92	C₂²×C₇₃	C₄×D₈	1	2	yes	—
93	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₇₃	—	8	4	no	—
95	C₄×C₇₃	C₄×D₈	1	2	yes	—
96	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
97	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
98	Dic₇₃	—	2	4	no	—
99	Dic₇₃	—	4	8	no	—
100	#33458	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	—	2	2	no	—
102	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
103	C₂×C₇₃	#3417	1	2	yes	—
104	C₂²×Q₃₂	—	73	1	no	—
105	#2950	—	73	2	no	—
106	#2176	—	73	4	no	—
107	C₄×Q₃₂	—	73	4	no	—
108	#2158	D₇₃	1	1	yes	characteristic
109	#2141	—	73	2	no	—
110	#2108	—	73	1	no	—
111	C₇₃	#33458	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₁₆	—	73	2	no	—
113	#519	—	146	2	no	—
114	C₂×Q₃₂	—	73	8	no	—
115	C₂×Q₃₂	—	73	2	no	—
116	C₂×Q₃₂	—	73	2	no	—
117	C₂×M₃₂	—	73	1	no	—
118	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
119	#441	—	146	4	no	—
120	C₄×Q₁₆	—	73	8	no	—
121	#427	C₂×D₇₃	1	2	yes	—
122	#419	—	146	2	no	—
123	#405	—	73	1	no	—
124	#367	C₂×D₇₃	1	2	yes	—
125	#366	C₂×D₇₃	1	2	yes	—
126	#358	—	73	8	no	—
127	#346	—	73	2	no	—
128	C₄×C₁₆	—	73	2	no	—
129	C₂²×Q₈	—	146	2	no	—
130	C₂×Q₁₆	—	73	8	no	—
131	C₂×Q₁₆	—	73	4	no	—
132	C₂×Q₁₆	—	146	2	no	—
133	C₂×Q₁₆	—	146	4	no	—
134	C₂×M₁₆	—	73	1	no	—
135	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
136	C₄×Q₈	—	292	4	no	—
137	C₄×Q₈	—	146	8	no	—
138	#117	—	2	2	no	—
139	#117	—	292	4	no	—
140	#117	—	73	1	no	—
141	Q₃₂	—	73	8	no	—
142	Q₃₂	—	73	8	no	—
143	M₃₂	—	146	2	no	—
144	C₂×C₁₆	—	73	2	no	—
145	C₂×C₁₆	C₄×D₇₃	1	4	yes	—
146	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
148	#107	C₂²×D₇₃	1	4	yes	—
149	#106	—	2	2	no	—
150	#103	—	146	8	no	—
151	#97	—	73	2	no	—
152	C₄×C₈	—	73	2	no	—
153	C₂×Q₈	—	292	2	no	—
154	C₂×Q₈	—	292	4	no	—
155	C₂×Q₈	—	146	8	no	—
156	C₂×Q₈	—	146	4	no	—
157	C₂²×C₄	—	292	2	no	—
158	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
159	Q₁₆	—	73	8	no	—
160	Q₁₆	—	146	8	no	—
161	Q₁₆	—	146	4	no	—
162	M₁₆	—	146	2	no	—
163	C₂×C₈	—	73	2	no	—
164	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
165	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
166	C₂×C₈	#11924029	1	2	yes	—
167	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	292	8	no	—
170	C₄⋊C₄	—	73	2	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	73	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	—	292	8	no	—
175	C₄²	—	73	2	no	—
176	C₁₆	—	146	2	no	—
177	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
178	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
179	C₂³	D₈×D₇₃	1	1	yes	characteristic
180	Q₈	—	292	8	no	—
181	Q₈	—	292	4	no	—
182	Q₈	—	146	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	292	8	no	—
185	C₂×C₄	—	292	4	no	—
186	C₂×C₄	—	73	4	no	—
187	C₂×C₄	#14556553	1	2	yes	—
188	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
189	C₂×C₄	#14556614	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
191	C₈	—	146	2	no	—
192	C₈	#14556553	1	2	yes	—
193	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
194	C₈	C₂×#11924029	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	#17988669	1	2	yes	—
197	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
198	C₄	—	8	4	no	—
199	C₄	—	292	8	no	—
200	C₄	—	146	4	no	—
201	C₄	#17988669	1	2	yes	—
202	C₄	C₂×#14556553	1	1	yes	characteristic
203	C₄	C₂×#14556614	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×#17988669	1	1	yes	characteristic
206	C₂	#22623081	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₈₄

C₂

yes

maximal, characteristic

#22618798

C₂

yes

maximal

#22611801

C₂

yes

maximal, complemented

#22611789

C₂

yes

maximal, complemented

#22611650

C₂

yes

maximal

#22611527

C₂

yes

maximal, characteristic

C₇₃×#2158

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₂

C₂²

yes

—

#17989064

—

C₂×Dic₅₈₄

C₄

yes

complemented

C₂×Dic₅₈₄

C₂²

yes

—

C₂×Dic₅₈₄

C₂²

yes

—

C₂×#14556531

C₂²

yes

characteristic

#17988693

—

#17988681

C₂²

yes

—

#17988595

—

#17988542

C₂²

yes

characteristic

#17988390

C₂²

yes

—

#17988362

C₂²

yes

—

C₁₆×Dic₇₃

C₂²

yes

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

#427×C₇₃

C₂²

yes

—

#367×C₇₃

C₂²

yes

—

#366×C₇₃

C₂²

yes

—

C₂²×Dic₁₄₆

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

C₂×C₄

yes

—

C₂×Dic₂₉₂

C₂³

yes

—

C₂×#11924012

D₄

yes

characteristic

C₄×Dic₁₄₆

—

#14556556

—

#14556550

D₄

yes

characteristic

#14556547

—

Dic₅₈₄

C₂×C₄

yes

—

Dic₅₈₄

C₂×C₄

yes

—

#14556531

—

C₂×#11924003

C₂³

yes

—

#14556514

—

#14556501

D₄

yes

—

C₈×Dic₇₃

C₂³

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

#117×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

C₂³

yes

—

#106×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

C₂²×C₄

yes

—

#11924012

—

C₂×#10596831

C₂×D₄

yes

—

C₄⋊Dic₇₃

D₈

yes

—

#11924007

—

C₂²⋊Dic₇₃

D₈

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×D₄

yes

—

#11924003

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₁₆×C₇₃

C₂²×C₄

yes

—

C₁₆×C₇₃

C₂²×C₄

yes

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

C₂²×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₈

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

C₄×D₄

yes

—

C₈×C₇₃

C₂³×C₄

yes

characteristic

C₈×C₇₃

C₂×#41

yes

characteristic

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×D₈

yes

—

C₂²×C₇₃

C₂²×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₄×D₈

yes

—

C₄×C₇₃

C₂×C₄×D₄

yes

characteristic

C₄×C₇₃

C₂×#135

yes

characteristic

Dic₇₃

—

Dic₇₃

—

100

#33458

—

maximal, Sylow

101

C₂×C₇₃

—

102

C₂×C₇₃

C₂×C₄×D₈

yes

characteristic

103

C₂×C₇₃

#3417

yes

—

104

C₂²×Q₃₂

—

105

#2950

—

106

#2176

—

107

C₄×Q₃₂

—

108

#2158

D₇₃

yes

characteristic

109

#2141

—

110

#2108

—

111

C₇₃

#33458

yes

characteristic, complemented, Sylow

112

C₂²×Q₁₆

—

113

#519

—

146

—

114

C₂×Q₃₂

—

115

C₂×Q₃₂

—

116

C₂×Q₃₂

—

117

C₂×M₃₂

—

118

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

119

#441

—

146

—

120

C₄×Q₁₆

—

121

#427

C₂×D₇₃

yes

—

122

#419

—

146

—

123

#405

—

124

#367

C₂×D₇₃

yes

—

125

#366

C₂×D₇₃

yes

—

126

#358

—

127

#346

—

128

C₄×C₁₆

—

129

C₂²×Q₈

—

146

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

146

—

133

C₂×Q₁₆

—

146

—

134

C₂×M₁₆

—

135

C₂²×C₈

C₂²×D₇₃

yes

characteristic

136

C₄×Q₈

—

292

—

137

C₄×Q₈

—

146

—

138

#117

—

139

#117

—

292

—

140

#117

—

141

Q₃₂

—

142

Q₃₂

—

143

M₃₂

—

146

—

144

C₂×C₁₆

—

145

C₂×C₁₆

C₄×D₇₃

yes

—

146

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

147

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

148

#107

C₂²×D₇₃

yes

—

149

#106

—

150

#103

—

146

—

151

#97

—

152

C₄×C₈

—

153

C₂×Q₈

—

292

—

154

C₂×Q₈

—

292

—

155

C₂×Q₈

—

146

—

156

C₂×Q₈

—

146

—

157

C₂²×C₄

—

292

—

158

C₂²×C₄

D₄×D₇₃

yes

characteristic

159

Q₁₆

—

160

Q₁₆

—

146

—

161

Q₁₆

—

146

—

162

M₁₆

—

146

—

163

C₂×C₈

—

164

C₂×C₈

C₂×C₄×D₇₃

yes

—

165

C₂×C₈

D₄×D₇₃

yes

characteristic

166

C₂×C₈

#11924029

yes

—

167

C₂×C₈

C₂³×D₇₃

yes

characteristic

168

C₄⋊C₄

—

169

C₄⋊C₄

—

292

—

170

C₄⋊C₄

—

171

C₂²⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

—

174

C₄²

—

292

—

175

C₄²

—

176

C₁₆

—

146

—

177

C₁₆

C₂×C₄×D₇₃

yes

—

178

C₁₆

C₂×C₄×D₇₃

yes

—

179

C₂³

D₈×D₇₃

yes

characteristic

180

Q₈

—

292

—

181

Q₈

—

292

—

182

Q₈

—

146

—

183

C₂×C₄

—

184

C₂×C₄

—

292

—

185

C₂×C₄

—

292

—

186

C₂×C₄

—

187

C₂×C₄

#14556553

yes

—

188

C₂×C₄

D₈×D₇₃

yes

characteristic

189

C₂×C₄

#14556614

yes

—

190

C₂×C₄

C₂×D₄×D₇₃

yes

characteristic

191

C₈

—

146

—

192

C₈

#14556553

yes

—

193

C₈

C₂²×C₄×D₇₃

yes

characteristic

194

C₈

C₂×#11924029

yes

characteristic

195

C₂²

—

196

C₂²

#17988669

yes

—

197

C₂²

C₂×D₈×D₇₃

yes

characteristic

198

C₄

—

199

C₄

—

292

—

200

C₄

—

146

—

201

C₄

#17988669

yes

—

202

C₄

C₂×#14556553

yes

characteristic

203

C₄

C₂×#14556614

yes

characteristic

204

C₂

—

205

C₂

C₂×#17988669

yes

characteristic

206

C₂

#22623081

yes

—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2158	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂³×C₇₃	C₂×D₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2158	D₇₃
hypocenter	C₇₃	#33458
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂²

C₂×D₈×D₇₃

commutator

C₈×C₇₃

C₂³×C₄

Frattini

C₂×C₈

C₂³×D₇₃

Fitting

C₇₃×#2158

C₂

socle

C₂²×C₇₃

C₂²×D₈

prime radical

C₂³×C₇₃

C₂×D₈

cosocle

C₂×C₈×C₇₃

C₂⁴

D₇₃

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
3. #22611801	C₂	semidirect
4. #22611789	C₂	semidirect
10. C₂×Dic₅₈₄	C₄	semidirect
111. C₇₃	#33458	semidirect

H ≅ G/N

type

3. #22611801

C₂

semidirect

4. #22611789

C₂

semidirect

10. C₂×Dic₅₈₄

C₄

semidirect

111. C₇₃

#33458

semidirect

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

subgroup	index
C₂²×Dic₅₈₄	2
#22618798	2
#22611801	2
#22611789	2
#22611650	2
#22611527	2
C₇₃×#2158	2
#33458	73

subgroup

index

C₂²×Dic₅₈₄

C₇₃×#2158

Subgroups — #33239361

Conjugacy classes of subgroups

Complements

Maximal subgroups