Group #33004749 — subgroups

Subgroups — #33004749

Order 18688.

516 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 99 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, complemented
2	C₂×#17988863	C₂	1	2	yes	maximal
3	C₂×#17988402	C₂	1	2	yes	maximal, complemented
4	#22611544	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556624	—	2	2	no	—
7	C₂²×Dic₂₉₂	C₂²	1	2	yes	—
8	C₂×#14556597	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₈₄	—	2	4	no	—
10	C₂×Dic₅₈₄	—	2	2	no	—
11	C₂×#14556531	C₂²	1	1	yes	characteristic
12	#17988863	—	2	4	no	—
13	#17988863	—	2	4	no	—
14	C₂×#14556509	—	2	2	no	—
15	#17988402	—	2	4	no	—
16	#17988366	—	2	2	no	—
17	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×#108×C₇₃	C₂²	1	2	yes	—
19	C₂×#11924028	D₄	1	1	yes	characteristic
20	C₂²×Dic₁₄₆	—	2	2	no	—
21	#14556624	—	2	8	no	—
22	#14556624	—	4	4	no	—
23	C₂×Dic₂₉₂	—	2	2	no	—
24	C₂×Dic₂₉₂	—	2	2	no	—
25	C₂×Dic₂₉₂	—	2	2	no	—
26	C₂×Dic₂₉₂	—	2	2	no	—
27	C₂×Dic₂₉₂	—	2	2	no	—
28	C₂×Dic₂₉₂	—	4	2	no	—
29	C₂×Dic₂₉₂	D₄	1	4	yes	—
30	C₂×#11924017	—	4	2	no	—
31	#14556597	—	2	2	no	—
32	#14556597	—	2	2	no	—
33	C₂×#11924012	D₄	1	1	yes	characteristic
34	C₂×#11924011	—	2	1	no	—
35	C₂×C₈×D₇₃	—	2	1	no	—
36	Dic₅₈₄	—	2	8	no	—
37	#14556531	—	2	2	no	—
38	C₂×#11924003	—	2	1	no	—
39	#14556509	—	2	8	no	—
40	C₂×M₁₆×C₇₃	—	2	2	no	—
41	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
42	C₂×C₁₆×C₇₃	—	2	2	no	—
43	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
44	#108×C₇₃	—	2	2	no	—
45	#108×C₇₃	D₄	1	4	yes	—
46	#11924028	—	2	2	no	—
47	#11924028	—	2	2	no	—
48	C₂×Dic₁₄₆	—	2	4	no	—
49	C₂×Dic₁₄₆	—	4	2	no	—
50	C₂×Dic₁₄₆	—	4	2	no	—
51	C₂×Dic₁₄₆	—	4	2	no	—
52	C₂×Dic₁₄₆	—	4	2	no	—
53	C₂×Dic₁₄₆	—	4	2	no	—
54	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
55	C₂×D₂₉₂	D₈	1	1	yes	characteristic
56	C₂×C₂²⋊D₇₃	—	2	1	no	—
57	C₂²×Dic₇₃	—	4	2	no	—
58	C₂×C₄×D₇₃	—	2	1	no	—
59	Dic₂₉₂	—	2	4	no	—
60	Dic₂₉₂	—	2	4	no	—
61	Dic₂₉₂	—	4	8	no	—
62	Dic₂₉₂	—	4	2	no	—
63	Dic₂₉₂	—	4	2	no	—
64	#11924017	—	4	8	no	—
65	#11924012	—	2	2	no	—
66	#11924011	—	4	2	no	—
67	C₂×#10596831	—	2	1	no	—
68	C₈×D₇₃	—	4	1	no	—
69	C₈×D₇₃	—	4	1	no	—
70	#11924003	—	4	1	no	—
71	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
72	M₁₆×C₇₃	—	2	4	no	—
73	M₁₆×C₇₃	—	4	2	no	—
74	C₂×C₈×C₇₃	—	4	2	no	—
75	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
79	C₁₆×C₇₃	—	2	2	no	—
80	C₂³×C₇₃	#135	1	1	yes	characteristic
81	C₂²×D₇₃	—	2	1	no	—
82	C₂×C₄×C₇₃	#120	1	2	yes	—
83	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
86	C₂²⋊D₇₃	—	4	2	no	—
87	C₂×Dic₇₃	—	2	1	no	—
88	C₂×Dic₇₃	—	4	4	no	—
89	C₂×Dic₇₃	—	8	2	no	—
90	C₂×Dic₇₃	—	8	2	no	—
91	D₂₉₂	—	2	2	no	—
92	C₄×D₇₃	—	4	1	no	—
93	C₄×D₇₃	—	4	1	no	—
94	Dic₁₄₆	—	2	2	no	—
95	Dic₁₄₆	—	4	4	no	—
96	Dic₁₄₆	—	4	4	no	—
97	Dic₁₄₆	—	8	2	no	—
98	Dic₁₄₆	—	8	2	no	—
99	C₈×C₇₃	—	4	4	no	—
100	C₈×C₇₃	#120	1	1	yes	characteristic
101	C₈×C₇₃	#121	1	2	yes	—
102	C₈×C₇₃	#127	1	1	yes	characteristic
103	#10596831	—	4	1	no	—
104	C₂²×C₇₃	—	2	2	no	—
105	C₂²×C₇₃	#449	1	2	yes	—
106	C₂²×C₇₃	#466	1	1	yes	characteristic
107	C₂×D₇₃	—	4	2	no	—
108	C₂×D₇₃	—	4	1	no	—
109	C₄×C₇₃	#392	1	1	yes	characteristic
110	C₄×C₇₃	#447	1	2	yes	—
111	C₄×C₇₃	#469	1	1	yes	characteristic
112	Dic₇₃	—	4	1	no	—
113	Dic₇₃	—	8	4	no	—
114	#13315	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	—	2	2	no	—
116	C₂×C₇₃	#1999	1	1	yes	characteristic
117	C₂×C₇₃	#2222	1	2	yes	—
118	D₇₃	—	8	1	no	—
119	C₂²×Q₃₂	—	73	1	no	—
120	C₂×#473	—	73	2	no	—
121	C₂×#362	—	73	2	no	—
122	#2113	—	73	1	no	—
123	#1381	D₇₃	1	1	yes	characteristic, complemented
124	C₇₃	#13315	1	1	yes	characteristic, complemented, Sylow
125	C₂×#137	—	146	2	no	—
126	C₂²×Q₁₆	—	73	2	no	—
127	C₂×#131	—	73	1	no	—
128	C₂×Q₃₂	—	146	4	no	—
129	C₂×Q₃₂	—	146	2	no	—
130	C₂×M₃₂	—	73	1	no	—
131	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
132	#473	—	146	4	no	—
133	#473	—	146	4	no	—
134	C₂×#108	C₂×D₇₃	1	2	yes	—
135	C₂×#101	—	146	2	no	—
136	#362	—	146	4	no	—
137	C₂²⋊C₁₆	—	146	2	no	—
138	C₂×#41	—	73	1	no	—
139	C₂²×Q₈	—	146	2	no	—
140	#137	—	292	4	no	—
141	#137	—	146	8	no	—
142	C₂×Q₁₆	—	292	2	no	—
143	C₂×Q₁₆	—	73	4	no	—
144	C₂×Q₁₆	—	146	2	no	—
145	C₂×Q₁₆	—	146	2	no	—
146	C₂×Q₁₆	—	146	2	no	—
147	C₂×Q₁₆	—	146	2	no	—
148	C₂×Q₁₆	—	146	2	no	—
149	C₂×SD₁₆	—	292	2	no	—
150	#131	—	146	2	no	—
151	#131	—	146	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	73	1	no	—
154	C₂×M₁₆	—	146	1	no	—
155	C₂²×C₈	—	146	1	no	—
156	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
157	Q₃₂	—	146	8	no	—
158	M₃₂	—	146	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	146	1	no	—
161	C₂×C₁₆	D₂₉₂	1	2	yes	—
162	#108	—	2	2	no	—
163	#108	D₂₉₂	1	4	yes	—
164	#101	—	146	8	no	—
165	#41	—	146	2	no	—
166	#41	—	146	2	no	—
167	C₂×Q₈	—	292	2	no	—
168	C₂×Q₈	—	292	2	no	—
169	C₂×Q₈	—	292	2	no	—
170	C₂×Q₈	—	292	2	no	—
171	C₂×Q₈	—	292	2	no	—
172	C₂×Q₈	—	73	1	no	—
173	C₂×Q₈	—	146	4	no	—
174	C₂×D₄	—	73	1	no	—
175	C₂×D₄	—	146	1	no	—
176	C₂²×C₄	—	292	2	no	—
177	C₂²×C₄	—	146	1	no	—
178	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
179	Q₁₆	—	292	8	no	—
180	Q₁₆	—	292	2	no	—
181	Q₁₆	—	292	2	no	—
182	Q₁₆	—	146	4	no	—
183	Q₁₆	—	146	4	no	—
184	SD₁₆	—	292	8	no	—
185	M₁₆	—	2	4	no	—
186	M₁₆	—	4	2	no	—
187	M₁₆	—	292	2	no	—
188	M₁₆	—	146	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	292	1	no	—
191	C₂×C₈	—	292	1	no	—
192	C₂×C₈	—	146	1	no	—
193	C₂×C₈	D₄×D₇₃	1	2	yes	—
194	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
195	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
196	C₂×C₈	#11924029	1	1	yes	characteristic
197	C₁₆	—	2	2	no	—
198	C₁₆	—	292	1	no	—
199	C₂³	—	146	1	no	—
200	C₂³	#14556614	1	1	yes	characteristic
201	Q₈	—	292	4	no	—
202	Q₈	—	292	4	no	—
203	Q₈	—	584	2	no	—
204	Q₈	—	584	2	no	—
205	Q₈	—	146	2	no	—
206	D₄	—	292	2	no	—
207	D₄	—	146	2	no	—
208	C₂×C₄	—	292	1	no	—
209	C₂×C₄	—	292	1	no	—
210	C₂×C₄	—	292	4	no	—
211	C₂×C₄	—	584	2	no	—
212	C₂×C₄	—	584	2	no	—
213	C₂×C₄	—	146	1	no	—
214	C₂×C₄	#14556561	1	2	yes	—
215	C₂×C₄	#14556566	1	1	yes	characteristic
216	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
217	C₂×C₄	#14556618	1	2	yes	—
218	C₈	—	4	4	no	—
219	C₈	—	292	1	no	—
220	C₈	#14556561	1	1	yes	characteristic
221	C₈	#14556566	1	2	yes	—
222	C₈	#14556588	1	1	yes	characteristic
223	C₂²	—	2	2	no	—
224	C₂²	—	292	2	no	—
225	C₂²	—	292	1	no	—
226	C₂²	#17988735	1	2	yes	—
227	C₂²	#17988824	1	1	yes	characteristic
228	C₄	—	292	1	no	—
229	C₄	—	584	4	no	—
230	C₄	#17988497	1	1	yes	characteristic
231	C₄	#17988722	1	2	yes	—
232	C₄	#17988841	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	—	584	1	no	—
235	C₂	#22610930	1	1	yes	characteristic
236	C₂	#22612057	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₈₄

C₂

yes

maximal, characteristic, complemented

C₂×#17988863

C₂

yes

maximal

C₂×#17988402

C₂

yes

maximal, complemented

#22611544

C₂

yes

maximal, characteristic

C₇₃×#1381

C₂

yes

maximal, characteristic, complemented

C₂×#14556624

—

C₂²×Dic₂₉₂

C₂²

yes

—

C₂×#14556597

C₂²

yes

characteristic

C₂×Dic₅₈₄

—

C₂×Dic₅₈₄

—

C₂×#14556531

C₂²

yes

—

—

—

—

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#108×C₇₃

C₂²

yes

—

C₂×#11924028

D₄

yes

characteristic

C₂²×Dic₁₄₆

—

#14556624

—

#14556624

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

D₄

yes

—

—

—

—

D₄

yes

characteristic

C₂×#11924011

—

C₂×C₈×D₇₃

—

Dic₅₈₄

—

#14556531

—

C₂×#11924003

—

#14556509

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

#108×C₇₃

—

#108×C₇₃

D₄

yes

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₈

yes

characteristic

C₂×D₂₉₂

D₈

yes

characteristic

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

—

—

—

—

C₈×D₇₃

—

C₈×D₇₃

—

#11924003

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

#41

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#135

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

#120

yes

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

100

C₈×C₇₃

#120

yes

characteristic

101

C₈×C₇₃

#121

yes

—

102

C₈×C₇₃

#127

yes

characteristic

103

#10596831

—

104

C₂²×C₇₃

—

105

C₂²×C₇₃

#449

yes

—

106

C₂²×C₇₃

#466

yes

characteristic

107

C₂×D₇₃

—

108

C₂×D₇₃

—

109

C₄×C₇₃

#392

yes

characteristic

110

C₄×C₇₃

#447

yes

—

111

C₄×C₇₃

#469

yes

characteristic

112

Dic₇₃

—

113

Dic₇₃

—

114

#13315

—

maximal, Sylow

115

C₂×C₇₃

—

116

C₂×C₇₃

#1999

yes

characteristic

117

C₂×C₇₃

#2222

yes

—

118

D₇₃

—

119

C₂²×Q₃₂

—

120

C₂×#473

—

121

C₂×#362

—

122

#2113

—

123

#1381

D₇₃

yes

characteristic, complemented

124

C₇₃

#13315

yes

characteristic, complemented, Sylow

125

C₂×#137

—

146

—

126

C₂²×Q₁₆

—

127

C₂×#131

—

128

C₂×Q₃₂

—

146

—

129

C₂×Q₃₂

—

146

—

130

C₂×M₃₂

—

131

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

132

#473

—

146

—

133

#473

—

146

—

134

C₂×#108

C₂×D₇₃

yes

—

135

C₂×#101

—

146

—

136

#362

—

146

—

137

C₂²⋊C₁₆

—

146

—

138

C₂×#41

—

139

C₂²×Q₈

—

146

—

140

#137

—

292

—

141

#137

—

146

—

142

C₂×Q₁₆

—

292

—

143

C₂×Q₁₆

—

144

C₂×Q₁₆

—

146

—

145

C₂×Q₁₆

—

146

—

146

C₂×Q₁₆

—

146

—

147

C₂×Q₁₆

—

146

—

148

C₂×Q₁₆

—

146

—

149

C₂×SD₁₆

—

292

—

150

#131

—

146

—

151

#131

—

146

—

152

C₂×M₁₆

—

153

C₂×M₁₆

—

154

C₂×M₁₆

—

146

—

155

C₂²×C₈

—

146

—

156

C₂²×C₈

C₂²×D₇₃

yes

characteristic

157

Q₃₂

—

146

—

158

M₃₂

—

146

—

159

C₂×C₁₆

—

160

C₂×C₁₆

—

146

—

161

C₂×C₁₆

D₂₉₂

yes

—

162

#108

—

163

#108

D₂₉₂

yes

—

164

#101

—

146

—

165

#41

—

146

—

166

#41

—

146

—

167

C₂×Q₈

—

292

—

168

C₂×Q₈

—

292

—

169

C₂×Q₈

—

292

—

170

C₂×Q₈

—

292

—

171

C₂×Q₈

—

292

—

172

C₂×Q₈

—

173

C₂×Q₈

—

146

—

174

C₂×D₄

—

175

C₂×D₄

—

146

—

176

C₂²×C₄

—

292

—

177

C₂²×C₄

—

146

—

178

C₂²×C₄

D₄×D₇₃

yes

characteristic

179

Q₁₆

—

292

—

180

Q₁₆

—

292

—

181

Q₁₆

—

292

—

182

Q₁₆

—

146

—

183

Q₁₆

—

146

—

184

SD₁₆

—

292

—

185

M₁₆

—

186

M₁₆

—

187

M₁₆

—

292

—

188

M₁₆

—

146

—

189

C₂×C₈

—

190

C₂×C₈

—

292

—

191

C₂×C₈

—

292

—

192

C₂×C₈

—

146

—

193

C₂×C₈

D₄×D₇₃

yes

—

194

C₂×C₈

C₂×D₂₉₂

yes

characteristic

195

C₂×C₈

C₂×D₂₉₂

yes

—

196

C₂×C₈

#11924029

yes

characteristic

197

C₁₆

—

198

C₁₆

—

292

—

199

C₂³

—

146

—

200

C₂³

#14556614

yes

characteristic

201

Q₈

—

292

—

202

Q₈

—

292

—

203

Q₈

—

584

—

204

Q₈

—

584

—

205

Q₈

—

146

—

206

D₄

—

292

—

207

D₄

—

146

—

208

C₂×C₄

—

292

—

209

C₂×C₄

—

292

—

210

C₂×C₄

—

292

—

211

C₂×C₄

—

584

—

212

C₂×C₄

—

584

—

213

C₂×C₄

—

146

—

214

C₂×C₄

#14556561

yes

—

215

C₂×C₄

#14556566

yes

characteristic

216

C₂×C₄

D₈×D₇₃

yes

characteristic

217

C₂×C₄

#14556618

yes

—

218

C₈

—

219

C₈

—

292

—

220

C₈

#14556561

yes

characteristic

221

C₈

#14556566

yes

—

222

C₈

#14556588

yes

characteristic

223

C₂²

—

224

C₂²

—

292

—

225

C₂²

—

292

—

226

C₂²

#17988735

yes

—

227

C₂²

#17988824

yes

characteristic

228

C₄

—

292

—

229

C₄

—

584

—

230

C₄

#17988497

yes

characteristic

231

C₄

#17988722

yes

—

232

C₄

#17988841

yes

characteristic

233

C₂

—

234

C₂

—

584

—

235

C₂

#22610930

yes

characteristic

236

C₂

#22612057

yes

—

	subgroup	G/H
center	C₂²	#17988824
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1381	C₂
socle	C₂²×C₇₃	#466
prime radical	C₂×#11924028	D₄
hypercenter	#1381	D₇₃
hypocenter	C₇₃	#13315
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#17988824

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1381

C₂

C₂²×C₇₃

D₄

D₇₃

C₇₃

G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
1. C₂²×Dic₅₈₄	C₂	semidirect
3. C₂×#17988402	C₂	semidirect
5. C₇₃×#1381	C₂	semidirect
123. #1381	D₇₃	semidirect
124. C₇₃	#13315	semidirect

H ≅ G/N

type

1. C₂²×Dic₅₈₄

C₂

semidirect

3. C₂×#17988402

C₂

semidirect

5. C₇₃×#1381

C₂

semidirect

123. #1381

D₇₃

semidirect

124. C₇₃

#13315

semidirect

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

subgroup	index
C₂²×Dic₅₈₄	2
C₂×#17988863	2
C₂×#17988402	2
#22611544	2
C₇₃×#1381	2
#13315	73

subgroup

index

C₂²×Dic₅₈₄

C₇₃×#1381

Subgroups — #33004749

Conjugacy classes of subgroups

Complements

Maximal subgroups