Group #32990112 — subgroups

Subgroups — #32990112

Order 18688.

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

normal subgroups only

This group has 402 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	#22621276	C₂	1	1	yes	maximal, characteristic
2	#22610908	C₂	1	2	yes	maximal
3	#22608619	C₂	1	2	yes	maximal
4	#22607806	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1842	C₂	1	1	yes	maximal, characteristic
6	#17989144	C₂²	1	2	yes	—
7	#17988883	—	2	2	no	—
8	#17988828	—	2	2	no	—
9	#17988798	—	2	2	no	—
10	#17988611	C₂²	1	2	yes	—
11	#17988347	—	2	2	no	—
12	#17988347	—	2	2	no	—
13	#17988335	—	2	2	no	—
14	#17988334	—	2	2	no	—
15	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
16	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₄³×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₉₂	—	2	2	no	—
20	C₂×#11924012	—	2	2	no	—
21	#14556591	D₄	1	2	yes	—
22	#14556591	D₄	1	2	yes	—
23	#14556583	—	2	2	no	—
24	#14556570	—	2	2	no	—
25	#14556570	—	2	2	no	—
26	C₄×Dic₁₄₆	—	2	2	no	—
27	C₄×Dic₁₄₆	—	2	2	no	—
28	#14556550	—	2	2	no	—
29	#14556525	—	2	2	no	—
30	C₄⋊#10596831	—	2	2	no	—
31	C₄⋊#10596831	—	2	2	no	—
32	#14556516	—	2	4	no	—
33	C₂²×C₈×C₇₃	—	2	1	no	—
34	C₂²×C₈×C₇₃	D₄	1	2	yes	—
35	C₂×C₄²×C₇₃	—	2	1	no	—
36	C₂×C₄²×C₇₃	—	2	1	no	—
37	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₇₃	—	2	2	no	—
41	C₄⋊C₈×C₇₃	—	2	2	no	—
42	C₄⋊C₈×C₇₃	—	4	4	no	—
43	C₂×Dic₁₄₆	—	2	2	no	—
44	C₂×Dic₁₄₆	—	2	2	no	—
45	Dic₂₉₂	—	8	2	no	—
46	#11924012	—	4	2	no	—
47	#11924012	—	4	2	no	—
48	C₂×#10596831	—	4	2	no	—
49	C₄⋊Dic₇₃	—	2	2	no	—
50	C₄⋊Dic₇₃	—	2	2	no	—
51	C₄⋊Dic₇₃	—	2	2	no	—
52	C₄⋊Dic₇₃	—	2	2	no	—
53	#11924007	—	4	2	no	—
54	#11924007	—	4	2	no	—
55	C₂²⋊Dic₇₃	—	4	2	no	—
56	C₄×Dic₇₃	—	4	2	no	—
57	C₂²×C₄×C₇₃	—	2	1	no	—
58	C₂²×C₄×C₇₃	—	2	1	no	—
59	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
62	C₂×C₈×C₇₃	—	2	2	no	—
63	C₂×C₈×C₇₃	—	4	2	no	—
64	C₂×C₈×C₇₃	—	4	2	no	—
65	C₂×C₈×C₇₃	—	4	1	no	—
66	C₄²×C₇₃	—	2	2	no	—
67	C₄²×C₇₃	—	2	2	no	—
68	C₄²×C₇₃	—	2	1	no	—
69	C₄²×C₇₃	—	2	2	no	—
70	C₄²×C₇₃	—	4	2	no	—
71	C₄²×C₇₃	Q₁₆	1	2	yes	complemented
72	C₄²×C₇₃	C₂×D₄	1	2	yes	—
73	C₄²×C₇₃	#41	1	2	yes	—
74	C₂³×C₇₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	—	2	2	no	—
78	C₂×C₄×C₇₃	—	2	1	no	—
79	C₂×C₄×C₇₃	—	2	1	no	—
80	C₂×C₄×C₇₃	—	2	2	no	—
81	C₂×C₄×C₇₃	—	4	1	no	—
82	C₂×C₄×C₇₃	—	4	1	no	—
83	C₂×C₄×C₇₃	—	4	1	no	—
84	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
85	C₂×C₄×C₇₃	#121	1	2	yes	—
86	C₂×C₄×C₇₃	#123	1	2	yes	—
87	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
88	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
90	C₂×C₄×C₇₃	#137	1	2	yes	—
91	C₂×Dic₇₃	—	4	2	no	—
92	C₂×Dic₇₃	—	4	2	no	—
93	Dic₁₄₆	—	4	2	no	—
94	Dic₁₄₆	—	4	2	no	—
95	Dic₁₄₆	—	8	2	no	—
96	C₈×C₇₃	—	4	2	no	—
97	C₈×C₇₃	—	8	1	no	—
98	#10596831	—	8	2	no	—
99	C₂²×C₇₃	—	2	1	no	—
100	C₂²×C₇₃	—	2	1	no	—
101	C₂²×C₇₃	#394	1	1	yes	characteristic
102	C₂²×C₇₃	#467	1	1	yes	characteristic
103	C₂²×C₇₃	#470	1	1	yes	characteristic
104	C₄×C₇₃	—	2	4	no	—
105	C₄×C₇₃	—	2	2	no	—
106	C₄×C₇₃	—	2	1	no	—
107	C₄×C₇₃	—	4	2	no	—
108	C₄×C₇₃	#453	1	1	yes	characteristic
109	C₄×C₇₃	#456	1	1	yes	characteristic
110	C₄×C₇₃	#462	1	2	yes	—
111	C₄×C₇₃	#464	1	2	yes	—
112	Dic₇₃	—	8	2	no	—
113	Dic₇₃	—	8	2	no	—
114	#10522	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	—	2	1	no	—
116	C₂×C₇₃	—	2	1	no	—
117	C₂×C₇₃	#2039	1	1	yes	characteristic
118	C₂×C₇₃	#2050	1	1	yes	characteristic
119	C₂×C₇₃	#2056	1	1	yes	characteristic
120	#3249	—	73	1	no	—
121	#1994	—	73	2	no	—
122	#1842	D₇₃	1	1	yes	characteristic
123	#1550	—	73	2	no	—
124	#1292	—	73	1	no	—
125	C₇₃	#10522	1	1	yes	characteristic, complemented, Sylow
126	#533	—	73	2	no	—
127	#477	—	146	2	no	—
128	#467	—	146	2	no	—
129	#462	—	146	2	no	—
130	#423	—	73	2	no	—
131	C₂×C₄⋊C₈	—	2	2	no	—
132	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
133	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
134	C₄³	C₂×D₇₃	1	1	yes	characteristic
135	#339	—	146	2	no	—
136	#339	—	146	2	no	—
137	#333	—	146	2	no	—
138	#332	—	146	2	no	—
139	C₂×Q₁₆	—	146	2	no	—
140	C₂×M₁₆	—	146	2	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
143	C₄⋊Q₈	—	73	2	no	—
144	C₄⋊Q₈	—	73	2	no	—
145	#125	—	146	2	no	—
146	C₂²⋊Q₈	—	146	2	no	—
147	C₂²⋊Q₈	—	146	2	no	—
148	C₄×Q₈	—	146	2	no	—
149	C₄×Q₈	—	146	2	no	—
150	#117	—	146	2	no	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
154	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
155	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
156	#107	—	146	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	4	4	no	—
160	C₄⋊C₈	—	146	2	no	—
161	C₄⋊C₈	—	146	2	no	—
162	#103	—	146	4	no	—
163	C₂×Q₈	—	146	2	no	—
164	C₂×Q₈	—	146	2	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
169	C₂²×C₄	#11924028	1	1	yes	characteristic
170	Q₁₆	—	584	2	no	—
171	M₁₆	—	292	2	no	—
172	M₁₆	—	292	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	292	2	no	—
178	C₄⋊C₄	—	292	2	no	—
179	C₄⋊C₄	—	292	2	no	—
180	C₄⋊C₄	—	146	2	no	—
181	C₄⋊C₄	—	146	2	no	—
182	C₄⋊C₄	—	146	2	no	—
183	C₄⋊C₄	—	146	2	no	—
184	C₂²⋊C₄	—	292	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	4	2	no	—
190	C₄²	—	292	2	no	—
191	C₄²	Dic₂₉₂	1	2	yes	complemented
192	C₄²	D₄×D₇₃	1	2	yes	—
193	C₄²	#11924029	1	2	yes	—
194	C₂³	#14556565	1	1	yes	characteristic
195	Q₈	—	292	2	no	—
196	Q₈	—	292	2	no	—
197	Q₈	—	584	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	292	2	no	—
208	C₂×C₄	—	292	2	no	—
209	C₂×C₄	#14556559	1	1	yes	characteristic
210	C₂×C₄	#14556566	1	2	yes	—
211	C₂×C₄	#14556576	1	2	yes	—
212	C₂×C₄	#14556580	1	1	yes	characteristic
213	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
214	C₂×C₄	#14556613	1	1	yes	characteristic
215	C₂×C₄	#14556624	1	2	yes	—
216	C₈	—	4	2	no	—
217	C₈	—	8	1	no	—
218	C₈	—	584	2	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#17988504	1	1	yes	characteristic
222	C₂²	#17988828	1	1	yes	characteristic
223	C₂²	#17988845	1	1	yes	characteristic
224	C₄	—	2	4	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	584	2	no	—
229	C₄	—	584	2	no	—
230	C₄	#17988754	1	1	yes	characteristic
231	C₄	#17988769	1	1	yes	characteristic
232	C₄	#17988798	1	2	yes	—
233	C₄	#17988810	1	2	yes	—
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	#22611155	1	1	yes	characteristic
237	C₂	#22611218	1	1	yes	characteristic
238	C₂	#22611254	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22621276

C₂

yes

maximal, characteristic

#22610908

C₂

yes

maximal

#22608619

C₂

yes

maximal

#22607806

C₂

yes

maximal, characteristic

C₇₃×#1842

C₂

yes

maximal, characteristic

#17989144

C₂²

yes

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂×C₄⋊C₈×C₇₃

—

C₂×C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄³×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×#11924012

—

#14556591

D₄

yes

—

#14556591

D₄

yes

—

#14556583

—

#14556570

—

#14556570

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

—

—

—

—

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

Q₁₆

yes

complemented

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×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

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

100

C₂²×C₇₃

—

101

C₂²×C₇₃

#394

yes

characteristic

102

C₂²×C₇₃

#467

yes

characteristic

103

C₂²×C₇₃

#470

yes

characteristic

104

C₄×C₇₃

—

105

C₄×C₇₃

—

106

C₄×C₇₃

—

107

C₄×C₇₃

—

108

C₄×C₇₃

#453

yes

characteristic

109

C₄×C₇₃

#456

yes

characteristic

110

C₄×C₇₃

#462

yes

—

111

C₄×C₇₃

#464

yes

—

112

Dic₇₃

—

113

Dic₇₃

—

114

#10522

—

maximal, Sylow

115

C₂×C₇₃

—

116

C₂×C₇₃

—

117

C₂×C₇₃

#2039

yes

characteristic

118

C₂×C₇₃

#2050

yes

characteristic

119

C₂×C₇₃

#2056

yes

characteristic

120

#3249

—

121

#1994

—

122

#1842

D₇₃

yes

characteristic

123

#1550

—

124

#1292

—

125

C₇₃

#10522

yes

characteristic, complemented, Sylow

126

#533

—

127

#477

—

146

—

128

#467

—

146

—

129

#462

—

146

—

130

#423

—

131

C₂×C₄⋊C₈

—

132

C₂×C₄⋊C₈

C₂×D₇₃

yes

characteristic

133

C₂×C₄⋊C₈

C₂×D₇₃

yes

characteristic

134

C₄³

C₂×D₇₃

yes

characteristic

135

#339

—

146

—

136

#339

—

146

—

137

#333

—

146

—

138

#332

—

146

—

139

C₂×Q₁₆

—

146

—

140

C₂×M₁₆

—

146

—

141

C₂²×C₈

—

142

C₂²×C₈

C₂²⋊D₇₃

yes

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

#125

—

146

—

146

C₂²⋊Q₈

—

146

—

147

C₂²⋊Q₈

—

146

—

148

C₄×Q₈

—

146

—

149

C₄×Q₈

—

146

—

150

#117

—

146

—

151

C₂×C₄²

—

152

C₂×C₄²

—

153

C₂×C₄²

D₂₉₂

yes

characteristic

154

C₂×C₄²

D₂₉₂

yes

characteristic

155

C₂×C₄²

C₂²×D₇₃

yes

characteristic

156

#107

—

146

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

C₄⋊C₈

—

146

—

161

C₄⋊C₈

—

146

—

162

#103

—

146

—

163

C₂×Q₈

—

146

—

164

C₂×Q₈

—

146

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

168

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

169

C₂²×C₄

#11924028

yes

characteristic

170

Q₁₆

—

584

—

171

M₁₆

—

292

—

172

M₁₆

—

292

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

292

—

178

C₄⋊C₄

—

292

—

179

C₄⋊C₄

—

292

—

180

C₄⋊C₄

—

146

—

181

C₄⋊C₄

—

146

—

182

C₄⋊C₄

—

146

—

183

C₄⋊C₄

—

146

—

184

C₂²⋊C₄

—

292

—

185

C₄²

—

186

C₄²

—

187

C₄²

—

188

C₄²

—

189

C₄²

—

190

C₄²

—

292

—

191

C₄²

Dic₂₉₂

yes

complemented

192

C₄²

D₄×D₇₃

yes

—

193

C₄²

#11924029

yes

—

194

C₂³

#14556565

yes

characteristic

195

Q₈

—

292

—

196

Q₈

—

292

—

197

Q₈

—

584

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

292

—

208

C₂×C₄

—

292

—

209

C₂×C₄

#14556559

yes

characteristic

210

C₂×C₄

#14556566

yes

—

211

C₂×C₄

#14556576

yes

—

212

C₂×C₄

#14556580

yes

characteristic

213

C₂×C₄

C₂×Dic₂₉₂

yes

characteristic

214

C₂×C₄

#14556613

yes

characteristic

215

C₂×C₄

#14556624

yes

—

216

C₈

—

217

C₈

—

218

C₈

—

584

—

219

C₂²

—

220

C₂²

—

221

C₂²

#17988504

yes

characteristic

222

C₂²

#17988828

yes

characteristic

223

C₂²

#17988845

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

584

—

229

C₄

—

584

—

230

C₄

#17988754

yes

characteristic

231

C₄

#17988769

yes

characteristic

232

C₄

#17988798

yes

—

233

C₄

#17988810

yes

—

234

C₂

—

235

C₂

—

236

C₂

#22611155

yes

characteristic

237

C₂

#22611218

yes

characteristic

238

C₂

#22611254

yes

characteristic

	subgroup	G/H
center	C₂²	#17988504
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1842	C₂
socle	C₂²×C₇₃	#394
prime radical	C₂³×C₇₃	#121
hypercenter	#1842	D₇₃
hypocenter	C₇₃	#10522
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1842

subgroup

G/H

center

C₂²

#17988504

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1842

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1842

N	H ≅ G/N	type
71. C₄²×C₇₃	Q₁₆	semidirect
125. C₇₃	#10522	semidirect
191. C₄²	Dic₂₉₂	semidirect

H ≅ G/N

type

71. C₄²×C₇₃

Q₁₆

semidirect

125. C₇₃

#10522

semidirect

191. C₄²

Dic₂₉₂

semidirect

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

subgroup	index
#22621276	2
#22610908	2
#22608619	2
#22607806	2
C₇₃×#1842	2
#10522	73

subgroup

index

C₇₃×#1842

Subgroups — #32990112

Conjugacy classes of subgroups

Complements

Maximal subgroups