Group #32990114 — subgroups

Subgroups — #32990114

Order 18688.

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

normal subgroups only

This group has 382 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	#22610909	C₂	1	2	yes	maximal
2	#22610047	C₂	1	1	yes	maximal, characteristic
3	#22608621	C₂	1	2	yes	maximal
4	#22607806	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#3249	C₂	1	1	yes	maximal, characteristic
6	#17988611	C₂²	1	2	yes	—
7	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
8	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
9	#17988346	—	2	2	no	—
10	#17988346	—	2	2	no	—
11	#17988341	—	2	2	no	—
12	#17988336	—	2	2	no	—
13	#17988333	—	2	2	no	—
14	#533×C₇₃	C₂²	1	2	yes	—
15	#477×C₇₃	—	2	2	no	—
16	#467×C₇₃	—	2	2	no	—
17	#462×C₇₃	—	2	2	no	—
18	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×#11924012	—	2	2	no	—
20	C₂²×#10596831	—	2	1	no	—
21	C₂×C₄×Dic₇₃	—	2	1	no	—
22	C₂×C₄×Dic₇₃	—	2	1	no	—
23	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
25	C₄⋊#10596831	—	2	2	no	—
26	C₄⋊#10596831	—	2	2	no	—
27	C₄⋊#10596831	—	2	2	no	—
28	C₂×Q₁₆×C₇₃	—	2	2	no	—
29	C₂²×C₈×C₇₃	D₄	1	2	yes	—
30	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
31	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
32	#125×C₇₃	—	2	2	no	—
33	C₂²⋊Q₈×C₇₃	—	2	2	no	—
34	C₂²⋊Q₈×C₇₃	—	2	2	no	—
35	C₄×Q₈×C₇₃	—	2	2	no	—
36	C₄×Q₈×C₇₃	—	2	2	no	—
37	#117×C₇₃	—	2	2	no	—
38	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
39	#107×C₇₃	—	2	2	no	—
40	C₄⋊C₈×C₇₃	—	2	2	no	—
41	#103×C₇₃	—	2	4	no	—
42	C₂²×Dic₇₃	—	2	1	no	—
43	C₂²×Dic₇₃	—	2	1	no	—
44	#11924012	—	4	2	no	—
45	#11924012	—	4	2	no	—
46	C₂×#10596831	—	4	2	no	—
47	C₂×#10596831	—	4	1	no	—
48	C₂×#10596831	—	4	2	no	—
49	C₄×Dic₇₃	—	2	1	no	—
50	C₄×Dic₇₃	—	2	2	no	—
51	C₄×Dic₇₃	—	4	2	no	—
52	C₄×Dic₇₃	—	4	1	no	—
53	C₄×Dic₇₃	—	4	1	no	—
54	C₄×Dic₇₃	Q₁₆	1	2	yes	complemented
55	C₂×Q₈×C₇₃	—	2	2	no	—
56	C₂×Q₈×C₇₃	—	2	2	no	—
57	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
60	Q₁₆×C₇₃	—	8	2	no	—
61	C₂×C₈×C₇₃	—	2	2	no	—
62	C₂×C₈×C₇₃	—	4	2	no	—
63	C₄⋊C₄×C₇₃	—	2	2	no	—
64	C₄⋊C₄×C₇₃	—	2	2	no	—
65	C₄⋊C₄×C₇₃	—	2	2	no	—
66	C₄⋊C₄×C₇₃	—	2	2	no	—
67	C₄⋊C₄×C₇₃	—	4	2	no	—
68	C₄⋊C₄×C₇₃	—	4	2	no	—
69	C₂²⋊C₄×C₇₃	—	4	2	no	—
70	C₄²×C₇₃	—	4	2	no	—
71	C₄²×C₇₃	C₂×D₄	1	2	yes	—
72	C₄²×C₇₃	#41	1	2	yes	—
73	C₂³×C₇₃	#121	1	1	yes	characteristic
74	Q₈×C₇₃	—	4	2	no	—
75	Q₈×C₇₃	—	4	2	no	—
76	Q₈×C₇₃	—	8	2	no	—
77	C₂×C₄×C₇₃	—	2	1	no	—
78	C₂×C₄×C₇₃	—	2	1	no	—
79	C₂×C₄×C₇₃	—	4	1	no	—
80	C₂×C₄×C₇₃	—	4	2	no	—
81	C₂×C₄×C₇₃	—	4	2	no	—
82	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
83	C₂×C₄×C₇₃	#121	1	2	yes	—
84	C₂×C₄×C₇₃	#123	1	2	yes	—
85	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
88	C₂×C₄×C₇₃	#137	1	2	yes	—
89	C₂×Dic₇₃	—	2	2	no	—
90	C₂×Dic₇₃	—	2	2	no	—
91	C₂×Dic₇₃	—	4	1	no	—
92	C₂×Dic₇₃	—	4	1	no	—
93	C₂×Dic₇₃	—	4	1	no	—
94	C₂×Dic₇₃	—	4	1	no	—
95	C₈×C₇₃	—	4	2	no	—
96	#10596831	—	8	1	no	—
97	#10596831	—	8	2	no	—
98	C₂²×C₇₃	—	2	1	no	—
99	C₂²×C₇₃	—	2	1	no	—
100	C₂²×C₇₃	#394	1	1	yes	characteristic
101	C₂²×C₇₃	#467	1	1	yes	characteristic
102	C₂²×C₇₃	#470	1	1	yes	characteristic
103	C₄×C₇₃	—	2	2	no	—
104	C₄×C₇₃	—	2	1	no	—
105	C₄×C₇₃	—	8	2	no	—
106	C₄×C₇₃	—	8	2	no	—
107	C₄×C₇₃	#454	1	1	yes	characteristic
108	C₄×C₇₃	#455	1	1	yes	characteristic
109	C₄×C₇₃	#462	1	2	yes	—
110	C₄×C₇₃	#464	1	2	yes	—
111	Dic₇₃	—	4	2	no	—
112	Dic₇₃	—	4	2	no	—
113	#10523	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	—	2	1	no	—
115	C₂×C₇₃	—	2	1	no	—
116	C₂×C₇₃	#2040	1	1	yes	characteristic
117	C₂×C₇₃	#2050	1	1	yes	characteristic
118	C₂×C₇₃	#2056	1	1	yes	characteristic
119	#3249	D₇₃	1	1	yes	characteristic
120	#1994	—	73	2	no	—
121	#1831	—	73	1	no	—
122	#1550	—	73	2	no	—
123	#1292	—	73	1	no	—
124	C₇₃	#10523	1	1	yes	characteristic, complemented, Sylow
125	#533	C₂×D₇₃	1	2	yes	—
126	#477	—	2	2	no	—
127	#467	—	2	2	no	—
128	#462	—	2	2	no	—
129	#423	—	73	2	no	—
130	C₂×C₄⋊C₈	—	73	1	no	—
131	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
132	C₄³	—	73	1	no	—
133	#339	—	146	2	no	—
134	#339	—	146	2	no	—
135	#336	—	146	2	no	—
136	#333	—	146	2	no	—
137	#332	—	146	2	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×M₁₆	—	146	2	no	—
140	C₂²×C₈	—	146	1	no	—
141	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
142	C₄⋊Q₈	C₂²⋊D₇₃	1	2	yes	—
143	C₄⋊Q₈	C₂²⋊D₇₃	1	2	yes	—
144	#125	—	2	2	no	—
145	C₂²⋊Q₈	—	2	2	no	—
146	C₂²⋊Q₈	—	2	2	no	—
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	#117	—	2	2	no	—
150	C₂×C₄²	—	73	1	no	—
151	C₂×C₄²	—	73	1	no	—
152	C₂×C₄²	—	146	1	no	—
153	C₂×C₄²	—	146	1	no	—
154	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
155	#107	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	146	2	no	—
158	C₄⋊C₈	—	146	2	no	—
159	C₄⋊C₈	—	146	2	no	—
160	#103	—	2	4	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂²×C₄	—	146	1	no	—
164	C₂²×C₄	—	146	1	no	—
165	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
167	C₂²×C₄	#11924027	1	1	yes	characteristic
168	Q₁₆	—	8	2	no	—
169	M₁₆	—	292	2	no	—
170	M₁₆	—	292	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	292	2	no	—
174	C₂×C₈	—	292	1	no	—
175	C₂×C₈	—	292	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	—	292	1	no	—
185	C₄²	—	292	1	no	—
186	C₄²	—	292	2	no	—
187	C₄²	—	73	2	no	—
188	C₄²	—	146	2	no	—
189	C₄²	—	146	1	no	—
190	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
191	C₄²	#11924027	1	2	yes	—
192	C₂³	#14556562	1	1	yes	characteristic
193	Q₈	—	4	2	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	8	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	292	1	no	—
202	C₂×C₄	—	292	1	no	—
203	C₂×C₄	—	292	1	no	—
204	C₂×C₄	—	292	1	no	—
205	C₂×C₄	—	146	2	no	—
206	C₂×C₄	—	146	2	no	—
207	C₂×C₄	#14556560	1	1	yes	characteristic
208	C₂×C₄	#14556563	1	2	yes	—
209	C₂×C₄	#14556572	1	2	yes	—
210	C₂×C₄	#14556578	1	1	yes	characteristic
211	C₂×C₄	Q₁₆×D₇₃	1	1	yes	characteristic
212	C₂×C₄	#14556614	1	1	yes	characteristic
213	C₂×C₄	#14556623	1	2	yes	—
214	C₈	—	4	2	no	—
215	C₈	—	584	1	no	—
216	C₈	—	584	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	#17988501	1	1	yes	characteristic
220	C₂²	#17988825	1	1	yes	characteristic
221	C₂²	#17988842	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	2	no	—
225	C₄	—	8	2	no	—
226	C₄	—	292	2	no	—
227	C₄	—	292	2	no	—
228	C₄	#17988760	1	1	yes	characteristic
229	C₄	#17988765	1	1	yes	characteristic
230	C₄	#17988795	1	2	yes	—
231	C₄	#17988811	1	2	yes	—
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	#22611157	1	1	yes	characteristic
235	C₂	#22611215	1	1	yes	characteristic
236	C₂	#22611251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610909

C₂

yes

maximal

#22610047

C₂

yes

maximal, characteristic

#22608621

C₂

yes

maximal

#22607806

C₂

yes

maximal, characteristic

C₇₃×#3249

C₂

yes

maximal, characteristic

#17988611

C₂²

yes

—

C₂×C₄⋊#10596831

C₂²

yes

characteristic

C₄²×Dic₇₃

C₂²

yes

—

—

—

—

—

#533×C₇₃

C₂²

yes

—

#477×C₇₃

—

#467×C₇₃

—

#462×C₇₃

—

C₂×C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×#11924012

—

C₂²×#10596831

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

D₄

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

—

—

—

C₂×Q₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

D₄

yes

—

#125×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

C₄⋊C₈×C₇₃

—

#103×C₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

—

—

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

Q₁₆

yes

complemented

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

characteristic

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

#121

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#120

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

100

C₂²×C₇₃

#394

yes

characteristic

101

C₂²×C₇₃

#467

yes

characteristic

102

C₂²×C₇₃

#470

yes

characteristic

103

C₄×C₇₃

—

104

C₄×C₇₃

—

105

C₄×C₇₃

—

106

C₄×C₇₃

—

107

C₄×C₇₃

#454

yes

characteristic

108

C₄×C₇₃

#455

yes

characteristic

109

C₄×C₇₃

#462

yes

—

110

C₄×C₇₃

#464

yes

—

111

Dic₇₃

—

112

Dic₇₃

—

113

#10523

—

maximal, Sylow

114

C₂×C₇₃

—

115

C₂×C₇₃

—

116

C₂×C₇₃

#2040

yes

characteristic

117

C₂×C₇₃

#2050

yes

characteristic

118

C₂×C₇₃

#2056

yes

characteristic

119

#3249

D₇₃

yes

characteristic

120

#1994

—

121

#1831

—

122

#1550

—

123

#1292

—

124

C₇₃

#10523

yes

characteristic, complemented, Sylow

125

#533

C₂×D₇₃

yes

—

126

#477

—

127

#467

—

128

#462

—

129

#423

—

130

C₂×C₄⋊C₈

—

131

C₂×C₄⋊C₈

C₂×D₇₃

yes

characteristic

132

C₄³

—

133

#339

—

146

—

134

#339

—

146

—

135

#336

—

146

—

136

#333

—

146

—

137

#332

—

146

—

138

C₂×Q₁₆

—

139

C₂×M₁₆

—

146

—

140

C₂²×C₈

—

146

—

141

C₂²×C₈

C₂²⋊D₇₃

yes

—

142

C₄⋊Q₈

C₂²⋊D₇₃

yes

—

143

C₄⋊Q₈

C₂²⋊D₇₃

yes

—

144

#125

—

145

C₂²⋊Q₈

—

146

C₂²⋊Q₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

#117

—

150

C₂×C₄²

—

151

C₂×C₄²

—

152

C₂×C₄²

—

146

—

153

C₂×C₄²

—

146

—

154

C₂×C₄²

C₂²×D₇₃

yes

characteristic

155

#107

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

146

—

158

C₄⋊C₈

—

146

—

159

C₄⋊C₈

—

146

—

160

#103

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂²×C₄

—

146

—

164

C₂²×C₄

—

146

—

165

C₂²×C₄

D₄×D₇₃

yes

characteristic

166

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

167

C₂²×C₄

#11924027

yes

characteristic

168

Q₁₆

—

169

M₁₆

—

292

—

170

M₁₆

—

292

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

292

—

174

C₂×C₈

—

292

—

175

C₂×C₈

—

292

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₂²⋊C₄

—

183

C₄²

—

184

C₄²

—

292

—

185

C₄²

—

292

—

186

C₄²

—

292

—

187

C₄²

—

188

C₄²

—

146

—

189

C₄²

—

146

—

190

C₄²

C₂×C₂²⋊D₇₃

yes

—

191

C₄²

#11924027

yes

—

192

C₂³

#14556562

yes

characteristic

193

Q₈

—

194

Q₈

—

195

Q₈

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

292

—

202

C₂×C₄

—

292

—

203

C₂×C₄

—

292

—

204

C₂×C₄

—

292

—

205

C₂×C₄

—

146

—

206

C₂×C₄

—

146

—

207

C₂×C₄

#14556560

yes

characteristic

208

C₂×C₄

#14556563

yes

—

209

C₂×C₄

#14556572

yes

—

210

C₂×C₄

#14556578

yes

characteristic

211

C₂×C₄

Q₁₆×D₇₃

yes

characteristic

212

C₂×C₄

#14556614

yes

characteristic

213

C₂×C₄

#14556623

yes

—

214

C₈

—

215

C₈

—

584

—

216

C₈

—

584

—

217

C₂²

—

218

C₂²

—

219

C₂²

#17988501

yes

characteristic

220

C₂²

#17988825

yes

characteristic

221

C₂²

#17988842

yes

characteristic

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

292

—

227

C₄

—

292

—

228

C₄

#17988760

yes

characteristic

229

C₄

#17988765

yes

characteristic

230

C₄

#17988795

yes

—

231

C₄

#17988811

yes

—

232

C₂

—

233

C₂

—

234

C₂

#22611157

yes

characteristic

235

C₂

#22611215

yes

characteristic

236

C₂

#22611251

yes

characteristic

	subgroup	G/H
center	C₂²	#17988501
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#3249	C₂
socle	C₂²×C₇₃	#394
prime radical	C₂³×C₇₃	#121
hypercenter	#3249	D₇₃
hypocenter	C₇₃	#10523
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3249

subgroup

G/H

center

C₂²

#17988501

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#3249

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3249

N	H ≅ G/N	type
54. C₄×Dic₇₃	Q₁₆	semidirect
124. C₇₃	#10523	semidirect

H ≅ G/N

type

54. C₄×Dic₇₃

Q₁₆

semidirect

124. C₇₃

#10523

semidirect

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

subgroup	index
#22610909	2
#22610047	2
#22608621	2
#22607806	2
C₇₃×#3249	2
#10523	73

subgroup

index

C₇₃×#3249

Subgroups — #32990114

Conjugacy classes of subgroups

Complements

Maximal subgroups