Group #32990109 — subgroups

Subgroups — #32990109

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	#22610909	C₂	1	2	yes	maximal
2	#22610082	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	C₂×#14556520	—	2	2	no	—
9	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
10	#17988346	—	2	2	no	—
11	#17988346	—	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	#14556520	—	4	4	no	—
29	C₂×Q₁₆×C₇₃	—	2	2	no	—
30	C₂²×C₈×C₇₃	D₄	1	2	yes	—
31	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
32	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
33	#125×C₇₃	—	2	2	no	—
34	C₂²⋊Q₈×C₇₃	—	2	2	no	—
35	C₂²⋊Q₈×C₇₃	—	2	2	no	—
36	C₄×Q₈×C₇₃	—	2	2	no	—
37	C₄×Q₈×C₇₃	—	2	2	no	—
38	#117×C₇₃	—	2	2	no	—
39	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
40	#107×C₇₃	—	2	2	no	—
41	C₄⋊C₈×C₇₃	—	2	2	no	—
42	#103×C₇₃	—	2	4	no	—
43	C₂²×Dic₇₃	—	2	1	no	—
44	C₂²×Dic₇₃	—	2	1	no	—
45	#11924012	—	4	2	no	—
46	#11924012	—	4	2	no	—
47	C₂×#10596831	—	4	2	no	—
48	C₂×#10596831	—	4	1	no	—
49	C₂×#10596831	—	4	2	no	—
50	C₄×Dic₇₃	—	2	2	no	—
51	C₄×Dic₇₃	—	2	2	no	—
52	C₄×Dic₇₃	—	2	1	no	—
53	C₄×Dic₇₃	—	2	2	no	—
54	C₄×Dic₇₃	—	4	2	no	—
55	C₄×Dic₇₃	Q₁₆	1	2	yes	complemented
56	C₂×Q₈×C₇₃	—	2	2	no	—
57	C₂×Q₈×C₇₃	—	2	2	no	—
58	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
61	Q₁₆×C₇₃	—	8	2	no	—
62	C₂×C₈×C₇₃	—	2	2	no	—
63	C₂×C₈×C₇₃	—	4	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	—	2	2	no	—
68	C₄⋊C₄×C₇₃	—	4	2	no	—
69	C₄⋊C₄×C₇₃	—	4	2	no	—
70	C₂²⋊C₄×C₇₃	—	4	2	no	—
71	C₄²×C₇₃	—	4	2	no	—
72	C₄²×C₇₃	C₂×D₄	1	2	yes	—
73	C₄²×C₇₃	#41	1	2	yes	—
74	C₂³×C₇₃	#121	1	1	yes	characteristic
75	Q₈×C₇₃	—	4	2	no	—
76	Q₈×C₇₃	—	4	2	no	—
77	Q₈×C₇₃	—	8	2	no	—
78	C₂×C₄×C₇₃	—	2	1	no	—
79	C₂×C₄×C₇₃	—	2	1	no	—
80	C₂×C₄×C₇₃	—	4	1	no	—
81	C₂×C₄×C₇₃	—	4	2	no	—
82	C₂×C₄×C₇₃	—	4	2	no	—
83	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
84	C₂×C₄×C₇₃	#121	1	2	yes	—
85	C₂×C₄×C₇₃	#123	1	2	yes	—
86	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
87	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
89	C₂×C₄×C₇₃	#137	1	2	yes	—
90	C₂×Dic₇₃	—	2	2	no	—
91	C₂×Dic₇₃	—	2	2	no	—
92	C₂×Dic₇₃	—	2	2	no	—
93	C₂×Dic₇₃	—	2	2	no	—
94	C₂×Dic₇₃	—	4	1	no	—
95	C₂×Dic₇₃	—	4	1	no	—
96	C₈×C₇₃	—	4	2	no	—
97	#10596831	—	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	2	no	—
105	C₄×C₇₃	—	2	1	no	—
106	C₄×C₇₃	—	8	2	no	—
107	C₄×C₇₃	—	8	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	—	2	4	no	—
113	Dic₇₃	—	4	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	D₇₃	1	1	yes	characteristic
121	#1994	—	73	2	no	—
122	#1842	—	73	1	no	—
123	#1550	—	73	2	no	—
124	#1292	—	73	1	no	—
125	C₇₃	#10522	1	1	yes	characteristic, complemented, Sylow
126	#533	C₂×D₇₃	1	2	yes	—
127	#477	—	2	2	no	—
128	#467	—	2	2	no	—
129	#462	—	2	2	no	—
130	#423	—	73	2	no	—
131	C₂×C₄⋊C₈	—	73	1	no	—
132	C₂×C₄⋊C₈	—	146	2	no	—
133	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
134	C₄³	—	73	1	no	—
135	#339	—	146	2	no	—
136	#339	—	146	2	no	—
137	#333	—	146	2	no	—
138	#332	—	146	2	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×M₁₆	—	146	2	no	—
141	C₂²×C₈	—	146	1	no	—
142	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
143	C₄⋊Q₈	C₂²⋊D₇₃	1	2	yes	—
144	C₄⋊Q₈	C₂²⋊D₇₃	1	2	yes	—
145	#125	—	2	2	no	—
146	C₂²⋊Q₈	—	2	2	no	—
147	C₂²⋊Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	—	2	2	no	—
150	#117	—	2	2	no	—
151	C₂×C₄²	—	73	1	no	—
152	C₂×C₄²	—	73	1	no	—
153	C₂×C₄²	—	146	1	no	—
154	C₂×C₄²	—	146	1	no	—
155	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
156	#107	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	292	4	no	—
159	C₄⋊C₈	—	146	2	no	—
160	C₄⋊C₈	—	146	2	no	—
161	C₄⋊C₈	—	146	2	no	—
162	#103	—	2	4	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂²×C₄	—	146	1	no	—
166	C₂²×C₄	—	146	1	no	—
167	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
169	C₂²×C₄	#11924027	1	1	yes	characteristic
170	Q₁₆	—	8	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	—	292	2	no	—
176	C₂×C₈	—	292	1	no	—
177	C₂×C₈	—	292	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	—	292	2	no	—
187	C₄²	—	73	2	no	—
188	C₄²	—	146	2	no	—
189	C₄²	—	146	2	no	—
190	C₄²	—	146	2	no	—
191	C₄²	—	146	1	no	—
192	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
193	C₄²	#11924027	1	2	yes	—
194	C₂³	#14556562	1	1	yes	characteristic
195	Q₈	—	4	2	no	—
196	Q₈	—	4	2	no	—
197	Q₈	—	8	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	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	—	146	2	no	—
208	C₂×C₄	—	146	2	no	—
209	C₂×C₄	#14556560	1	1	yes	characteristic
210	C₂×C₄	#14556563	1	2	yes	—
211	C₂×C₄	#14556572	1	2	yes	—
212	C₂×C₄	#14556578	1	1	yes	characteristic
213	C₂×C₄	Q₁₆×D₇₃	1	1	yes	characteristic
214	C₂×C₄	#14556614	1	1	yes	characteristic
215	C₂×C₄	#14556623	1	2	yes	—
216	C₈	—	4	2	no	—
217	C₈	—	584	1	no	—
218	C₈	—	584	2	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#17988501	1	1	yes	characteristic
222	C₂²	#17988825	1	1	yes	characteristic
223	C₂²	#17988842	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	2	no	—
227	C₄	—	8	2	no	—
228	C₄	—	292	2	no	—
229	C₄	—	146	4	no	—
230	C₄	#17988755	1	1	yes	characteristic
231	C₄	#17988770	1	1	yes	characteristic
232	C₄	#17988795	1	2	yes	—
233	C₄	#17988811	1	2	yes	—
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	#22611152	1	1	yes	characteristic
237	C₂	#22611215	1	1	yes	characteristic
238	C₂	#22611251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610909

C₂

yes

maximal

#22610082

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

C₂×#14556520

—

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

—

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

D₇₃

yes

characteristic

121

#1994

—

122

#1842

—

123

#1550

—

124

#1292

—

125

C₇₃

#10522

yes

characteristic, complemented, Sylow

126

#533

C₂×D₇₃

yes

—

127

#477

—

128

#467

—

129

#462

—

130

#423

—

131

C₂×C₄⋊C₈

—

132

C₂×C₄⋊C₈

—

146

—

133

C₂×C₄⋊C₈

C₂×D₇₃

yes

characteristic

134

C₄³

—

135

#339

—

146

—

136

#339

—

146

—

137

#333

—

146

—

138

#332

—

146

—

139

C₂×Q₁₆

—

140

C₂×M₁₆

—

146

—

141

C₂²×C₈

—

146

—

142

C₂²×C₈

C₂²⋊D₇₃

yes

—

143

C₄⋊Q₈

C₂²⋊D₇₃

yes

—

144

C₄⋊Q₈

C₂²⋊D₇₃

yes

—

145

#125

—

146

C₂²⋊Q₈

—

147

C₂²⋊Q₈

—

148

C₄×Q₈

—

149

C₄×Q₈

—

150

#117

—

151

C₂×C₄²

—

152

C₂×C₄²

—

153

C₂×C₄²

—

146

—

154

C₂×C₄²

—

146

—

155

C₂×C₄²

C₂²×D₇₃

yes

characteristic

156

#107

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

292

—

159

C₄⋊C₈

—

146

—

160

C₄⋊C₈

—

146

—

161

C₄⋊C₈

—

146

—

162

#103

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂²×C₄

—

146

—

166

C₂²×C₄

—

146

—

167

C₂²×C₄

D₄×D₇₃

yes

characteristic

168

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

169

C₂²×C₄

#11924027

yes

characteristic

170

Q₁₆

—

171

M₁₆

—

292

—

172

M₁₆

—

292

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

292

—

176

C₂×C₈

—

292

—

177

C₂×C₈

—

292

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₂²⋊C₄

—

185

C₄²

—

186

C₄²

—

292

—

187

C₄²

—

188

C₄²

—

146

—

189

C₄²

—

146

—

190

C₄²

—

146

—

191

C₄²

—

146

—

192

C₄²

C₂×C₂²⋊D₇₃

yes

—

193

C₄²

#11924027

yes

—

194

C₂³

#14556562

yes

characteristic

195

Q₈

—

196

Q₈

—

197

Q₈

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

292

—

204

C₂×C₄

—

292

—

205

C₂×C₄

—

146

—

206

C₂×C₄

—

146

—

207

C₂×C₄

—

146

—

208

C₂×C₄

—

146

—

209

C₂×C₄

#14556560

yes

characteristic

210

C₂×C₄

#14556563

yes

—

211

C₂×C₄

#14556572

yes

—

212

C₂×C₄

#14556578

yes

characteristic

213

C₂×C₄

Q₁₆×D₇₃

yes

characteristic

214

C₂×C₄

#14556614

yes

characteristic

215

C₂×C₄

#14556623

yes

—

216

C₈

—

217

C₈

—

584

—

218

C₈

—

584

—

219

C₂²

—

220

C₂²

—

221

C₂²

#17988501

yes

characteristic

222

C₂²

#17988825

yes

characteristic

223

C₂²

#17988842

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

292

—

229

C₄

—

146

—

230

C₄

#17988755

yes

characteristic

231

C₄

#17988770

yes

characteristic

232

C₄

#17988795

yes

—

233

C₄

#17988811

yes

—

234

C₂

—

235

C₂

—

236

C₂

#22611152

yes

characteristic

237

C₂

#22611215

yes

characteristic

238

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	#10522
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
55. C₄×Dic₇₃	Q₁₆	semidirect
125. C₇₃	#10522	semidirect

H ≅ G/N

type

55. C₄×Dic₇₃

Q₁₆

semidirect

125. C₇₃

#10522

semidirect

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

subgroup	index
#22610909	2
#22610082	2
#22608621	2
#22607806	2
C₇₃×#3249	2
#10522	73

subgroup

index

C₇₃×#3249

Subgroups — #32990109

Conjugacy classes of subgroups

Complements

Maximal subgroups