Group #33005365 — subgroups

Subgroups — #33005365

Order 18688.

540 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 97 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×#17988964	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988401	C₂	1	2	yes	maximal, complemented
3	#22611648	C₂	1	2	yes	maximal, complemented
4	#22610679	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₈₄	—	2	1	no	—
7	C₂²×D₅₈₄	C₂²	1	2	yes	—
8	C₂×#14556588	—	2	1	no	—
9	#17988964	—	2	4	no	—
10	C₂×#14556527	C₂²	1	1	yes	characteristic
11	#17988625	C₂²	1	1	yes	characteristic
12	#17988613	—	2	1	no	—
13	#17988401	—	2	4	no	—
14	#17988389	—	2	4	no	—
15	#17988347	—	2	1	no	—
16	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×D₂₉₂	—	2	2	no	—
19	C₂²×D₂₉₂	—	4	2	no	—
20	C₂×D₅₈₄	—	2	4	no	—
21	C₂×D₅₈₄	—	2	2	no	—
22	C₂×D₅₈₄	—	2	2	no	—
23	C₂×D₅₈₄	—	2	2	no	—
24	C₂×D₅₈₄	—	2	2	no	—
25	C₂×D₅₈₄	—	2	2	no	—
26	C₂×D₅₈₄	—	4	2	no	—
27	C₂×D₅₈₄	—	4	2	no	—
28	C₂×D₅₈₄	D₄	1	4	yes	—
29	C₂×#11924012	—	2	1	no	—
30	#14556588	—	2	4	no	—
31	#14556588	—	4	2	no	—
32	#14556550	—	2	1	no	—
33	#14556527	—	2	2	no	—
34	#14556525	D₄	1	2	yes	complemented
35	#14556523	—	2	1	no	—
36	#14556520	—	4	2	no	—
37	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
40	M₃₂×C₇₃	—	2	2	no	—
41	M₃₂×C₇₃	—	2	2	no	—
42	C₂×C₁₆×C₇₃	—	2	2	no	—
43	C₄×C₈×C₇₃	—	2	2	no	—
44	C₂³×D₇₃	—	4	2	no	—
45	C₂×D₂₉₂	—	2	4	no	—
46	C₂×D₂₉₂	—	4	2	no	—
47	C₂×D₂₉₂	—	4	2	no	—
48	C₂×D₂₉₂	—	4	2	no	—
49	C₂×D₂₉₂	—	4	2	no	—
50	C₂×D₂₉₂	—	4	8	no	—
51	C₂×D₂₉₂	—	4	2	no	—
52	C₂×D₂₉₂	—	8	8	no	—
53	D₅₈₄	—	2	4	no	—
54	D₅₈₄	—	2	4	no	—
55	D₅₈₄	—	4	8	no	—
56	D₅₈₄	—	4	2	no	—
57	D₅₈₄	—	4	2	no	—
58	#11924012	—	4	1	no	—
59	#11924012	—	4	1	no	—
60	C₂×#10596831	—	4	1	no	—
61	C₄⋊Dic₇₃	—	2	2	no	—
62	C₂²⋊Dic₇₃	—	4	1	no	—
63	C₄×Dic₇₃	—	4	1	no	—
64	C₂²×C₄×C₇₃	—	2	1	no	—
65	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	—	2	1	no	—
67	C₂×C₈×C₇₃	—	2	1	no	—
68	C₂×C₈×C₇₃	D₈	1	2	yes	—
69	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₇₃	#41	1	2	yes	—
72	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
73	C₄²×C₇₃	—	2	1	no	—
74	C₄²×C₇₃	SD₁₆	1	2	yes	—
75	C₁₆×C₇₃	—	4	2	no	—
76	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂²×D₇₃	—	4	4	no	—
78	C₂²×D₇₃	—	8	8	no	—
79	C₂²×D₇₃	—	8	2	no	—
80	C₂²×D₇₃	—	8	2	no	—
81	C₂×C₄×C₇₃	—	2	4	no	—
82	C₂×C₄×C₇₃	—	2	2	no	—
83	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
84	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
85	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×Dic₇₃	—	4	2	no	—
90	D₂₉₂	—	4	4	no	—
91	D₂₉₂	—	4	4	no	—
92	D₂₉₂	—	8	16	no	—
93	D₂₉₂	—	8	2	no	—
94	D₂₉₂	—	8	2	no	—
95	C₈×C₇₃	—	2	2	no	—
96	C₈×C₇₃	#121	1	2	yes	—
97	C₈×C₇₃	#123	1	2	yes	—
98	#10596831	—	8	1	no	—
99	C₂²×C₇₃	—	2	2	no	—
100	C₂²×C₇₃	#450	1	1	yes	characteristic
101	C₂²×C₇₃	#459	1	1	yes	characteristic
102	C₂²×C₇₃	#490	1	1	yes	characteristic
103	C₂×D₇₃	—	8	8	no	—
104	C₂×D₇₃	—	8	4	no	—
105	C₂×D₇₃	—	16	4	no	—
106	C₂×D₇₃	—	16	4	no	—
107	C₄×C₇₃	—	2	4	no	—
108	C₄×C₇₃	#394	1	1	yes	characteristic
109	C₄×C₇₃	#447	1	1	yes	characteristic
110	C₄×C₇₃	#460	1	1	yes	characteristic
111	C₄×C₇₃	#486	1	1	yes	characteristic
112	Dic₇₃	—	8	2	no	—
113	#13413	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	—	2	1	no	—
115	C₂×C₇₃	—	2	1	no	—
116	C₂×C₇₃	#2041	1	1	yes	characteristic
117	C₂×C₇₃	#2213	1	2	yes	—
118	D₇₃	—	16	4	no	—
119	C₂×#493	—	73	1	no	—
120	C₂×#361	—	73	2	no	—
121	#2140	—	73	2	no	—
122	#1945	—	73	1	no	—
123	#1376	D₇₃	1	1	yes	characteristic, complemented
124	C₇₃	#13413	1	1	yes	characteristic, complemented, Sylow
125	C₂²×D₈	—	73	2	no	—
126	C₂²×D₈	—	146	1	no	—
127	C₂×#127	—	146	1	no	—
128	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
129	#493	—	146	4	no	—
130	C₂×#108	—	73	1	no	—
131	#427	—	73	1	no	—
132	#424	—	146	1	no	—
133	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
134	#361	—	146	4	no	—
135	#357	—	146	4	no	—
136	#339	—	146	1	no	—
137	C₂²×D₄	—	292	2	no	—
138	C₂²×D₄	—	146	2	no	—
139	C₂×D₈	—	292	2	no	—
140	C₂×D₈	—	292	2	no	—
141	C₂×D₈	—	73	4	no	—
142	C₂×D₈	—	146	4	no	—
143	C₂×D₈	—	146	2	no	—
144	C₂×D₈	—	146	2	no	—
145	C₂×D₈	—	146	2	no	—
146	C₂×D₈	—	146	2	no	—
147	C₂×D₈	—	146	2	no	—
148	C₂×M₁₆	—	146	1	no	—
149	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
151	#127	—	292	2	no	—
152	#127	—	146	4	no	—
153	#117	—	146	1	no	—
154	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	#108	—	146	2	no	—
159	#107	—	73	2	no	—
160	#106	—	146	1	no	—
161	C₄⋊C₈	—	292	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂⁴	—	292	2	no	—
164	C₂×D₄	—	292	8	no	—
165	C₂×D₄	—	292	2	no	—
166	C₂×D₄	—	292	2	no	—
167	C₂×D₄	—	292	2	no	—
168	C₂×D₄	—	292	2	no	—
169	C₂×D₄	—	292	2	no	—
170	C₂×D₄	—	584	8	no	—
171	C₂×D₄	—	146	4	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
174	D₈	—	292	8	no	—
175	D₈	—	292	2	no	—
176	D₈	—	292	2	no	—
177	D₈	—	146	4	no	—
178	D₈	—	146	4	no	—
179	M₁₆	—	292	1	no	—
180	M₁₆	—	292	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	292	1	no	—
184	C₂×C₈	D₅₈₄	1	2	yes	—
185	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
186	C₂×C₈	D₄×D₇₃	1	2	yes	—
187	C₂×C₈	#11924028	1	2	yes	—
188	C₂×C₈	#11924029	1	1	yes	characteristic
189	C₄⋊C₄	—	146	2	no	—
190	C₂²⋊C₄	—	292	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	292	1	no	—
193	C₄²	#11924017	1	2	yes	—
194	C₁₆	—	4	2	no	—
195	C₂³	—	292	4	no	—
196	C₂³	—	584	8	no	—
197	C₂³	—	584	2	no	—
198	C₂³	—	584	2	no	—
199	C₂³	#14556619	1	1	yes	characteristic
200	D₄	—	292	4	no	—
201	D₄	—	292	4	no	—
202	D₄	—	584	16	no	—
203	D₄	—	584	2	no	—
204	D₄	—	584	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	292	2	no	—
208	C₂×C₄	#14556561	1	1	yes	characteristic
209	C₂×C₄	#14556566	1	1	yes	characteristic
210	C₂×C₄	#14556579	1	1	yes	characteristic
211	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
212	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
213	C₂×C₄	#14556619	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	584	1	no	—
216	C₈	#14556564	1	2	yes	—
217	C₈	#14556576	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	584	8	no	—
220	C₂²	—	584	4	no	—
221	C₂²	—	1168	4	no	—
222	C₂²	—	1168	4	no	—
223	C₂²	#17988740	1	1	yes	characteristic
224	C₂²	#17988783	1	1	yes	characteristic
225	C₂²	#17988952	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	584	2	no	—
228	C₄	#17988505	1	1	yes	characteristic
229	C₄	#17988723	1	1	yes	characteristic
230	C₄	#17988788	1	1	yes	characteristic
231	C₄	#17988934	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	1168	4	no	—
235	C₂	#22611166	1	1	yes	characteristic
236	C₂	#22612008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988964

C₂

yes

maximal, characteristic

C₂×#17988401

C₂

yes

maximal, complemented

#22611648

C₂

yes

maximal, complemented

#22610679

C₂

yes

maximal, characteristic, complemented

C₇₃×#1376

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₈₄

—

C₂²×D₅₈₄

C₂²

yes

—

C₂×#14556588

—

#17988964

—

C₂×#14556527

C₂²

yes

characteristic

#17988625

C₂²

yes

—

—

—

—

C₂×M₃₂×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂²×D₂₉₂

—

C₂²×D₂₉₂

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

D₄

yes

—

—

—

—

—

—

D₄

yes

complemented

#14556523

—

#14556520

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

M₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

#121

yes

—

C₈×C₇₃

#123

yes

—

#10596831

—

C₂²×C₇₃

—

100

C₂²×C₇₃

#450

yes

characteristic

101

C₂²×C₇₃

#459

yes

characteristic

102

C₂²×C₇₃

#490

yes

characteristic

103

C₂×D₇₃

—

104

C₂×D₇₃

—

105

C₂×D₇₃

—

106

C₂×D₇₃

—

107

C₄×C₇₃

—

108

C₄×C₇₃

#394

yes

characteristic

109

C₄×C₇₃

#447

yes

characteristic

110

C₄×C₇₃

#460

yes

characteristic

111

C₄×C₇₃

#486

yes

characteristic

112

Dic₇₃

—

113

#13413

—

maximal, Sylow

114

C₂×C₇₃

—

115

C₂×C₇₃

—

116

C₂×C₇₃

#2041

yes

characteristic

117

C₂×C₇₃

#2213

yes

—

118

D₇₃

—

119

C₂×#493

—

120

C₂×#361

—

121

#2140

—

122

#1945

—

123

#1376

D₇₃

yes

characteristic, complemented

124

C₇₃

#13413

yes

characteristic, complemented, Sylow

125

C₂²×D₈

—

126

C₂²×D₈

—

146

—

127

C₂×#127

—

146

—

128

C₂×M₃₂

C₂×D₇₃

yes

—

129

#493

—

146

—

130

C₂×#108

—

131

#427

—

132

#424

—

146

—

133

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

134

#361

—

146

—

135

#357

—

146

—

136

#339

—

146

—

137

C₂²×D₄

—

292

—

138

C₂²×D₄

—

146

—

139

C₂×D₈

—

292

—

140

C₂×D₈

—

292

—

141

C₂×D₈

—

142

C₂×D₈

—

146

—

143

C₂×D₈

—

146

—

144

C₂×D₈

—

146

—

145

C₂×D₈

—

146

—

146

C₂×D₈

—

146

—

147

C₂×D₈

—

146

—

148

C₂×M₁₆

—

146

—

149

C₂²×C₈

D₂₉₂

yes

characteristic

150

C₂²×C₈

C₂²×D₇₃

yes

characteristic

151

#127

—

292

—

152

#127

—

146

—

153

#117

—

146

—

154

C₂×C₄²

D₂₉₂

yes

characteristic

155

M₃₂

—

156

M₃₂

—

157

C₂×C₁₆

—

158

#108

—

146

—

159

#107

—

160

#106

—

146

—

161

C₄⋊C₈

—

292

—

162

C₄×C₈

—

163

C₂⁴

—

292

—

164

C₂×D₄

—

292

—

165

C₂×D₄

—

292

—

166

C₂×D₄

—

292

—

167

C₂×D₄

—

292

—

168

C₂×D₄

—

292

—

169

C₂×D₄

—

292

—

170

C₂×D₄

—

584

—

171

C₂×D₄

—

146

—

172

C₂²×C₄

—

173

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

174

D₈

—

292

—

175

D₈

—

292

—

176

D₈

—

292

—

177

D₈

—

146

—

178

D₈

—

146

—

179

M₁₆

—

292

—

180

M₁₆

—

292

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

292

—

184

C₂×C₈

D₅₈₄

yes

—

185

C₂×C₈

D₄×D₇₃

yes

characteristic

186

C₂×C₈

D₄×D₇₃

yes

—

187

C₂×C₈

#11924028

yes

—

188

C₂×C₈

#11924029

yes

characteristic

189

C₄⋊C₄

—

146

—

190

C₂²⋊C₄

—

292

—

191

C₄²

—

192

C₄²

—

292

—

193

C₄²

#11924017

yes

—

194

C₁₆

—

195

C₂³

—

292

—

196

C₂³

—

584

—

197

C₂³

—

584

—

198

C₂³

—

584

—

199

C₂³

#14556619

yes

characteristic

200

D₄

—

292

—

201

D₄

—

292

—

202

D₄

—

584

—

203

D₄

—

584

—

204

D₄

—

584

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

292

—

208

C₂×C₄

#14556561

yes

characteristic

209

C₂×C₄

#14556566

yes

characteristic

210

C₂×C₄

#14556579

yes

characteristic

211

C₂×C₄

C₂×D₅₈₄

yes

characteristic

212

C₂×C₄

C₂×#11924017

yes

characteristic

213

C₂×C₄

#14556619

yes

characteristic

214

C₈

—

215

C₈

—

584

—

216

C₈

#14556564

yes

—

217

C₈

#14556576

yes

—

218

C₂²

—

219

C₂²

—

584

—

220

C₂²

—

584

—

221

C₂²

—

1168

—

222

C₂²

—

1168

—

223

C₂²

#17988740

yes

characteristic

224

C₂²

#17988783

yes

characteristic

225

C₂²

#17988952

yes

characteristic

226

C₄

—

227

C₄

—

584

—

228

C₄

#17988505

yes

characteristic

229

C₄

#17988723

yes

characteristic

230

C₄

#17988788

yes

characteristic

231

C₄

#17988934

yes

characteristic

232

C₂

—

233

C₂

—

234

C₂

—

1168

—

235

C₂

#22611166

yes

characteristic

236

C₂

#22612008

yes

—

	subgroup	G/H
center	C₂²	#17988783
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1376	C₂
socle	C₂²×C₇₃	#459
prime radical	C₂×#17988964	C₂
hypercenter	#1376	D₇₃
hypocenter	C₇₃	#13413
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#17988783

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1376

C₂

C₂²×C₇₃

C₂

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
2. C₂×#17988401	C₂	semidirect
3. #22611648	C₂	semidirect
4. #22610679	C₂	semidirect
5. C₇₃×#1376	C₂	semidirect
34. #14556525	D₄	semidirect
123. #1376	D₇₃	semidirect
124. C₇₃	#13413	semidirect

H ≅ G/N

type

2. C₂×#17988401

C₂

semidirect

3. #22611648

C₂

semidirect

4. #22610679

C₂

semidirect

5. C₇₃×#1376

C₂

semidirect

34. #14556525

D₄

semidirect

123. #1376

D₇₃

semidirect

124. C₇₃

#13413

semidirect

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

subgroup	index
C₂×#17988964	2
C₂×#17988401	2
#22611648	2
#22610679	2
C₇₃×#1376	2
#13413	73

subgroup

index

C₇₃×#1376

Subgroups — #33005365

Conjugacy classes of subgroups

Complements

Maximal subgroups