Group #32989014 — subgroups

Subgroups — #32989014

Order 18688.

416 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 123 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	#22615173	C₂	1	1	yes	maximal, characteristic
2	#22610880	C₂	1	2	yes	maximal
3	#22607814	C₂	1	1	yes	maximal, characteristic
4	#22607771	C₂	1	2	yes	maximal
5	C₇₃×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
7	#17988526	—	2	2	no	—
8	#17988486	—	2	2	no	—
9	#17988452	—	2	2	no	—
10	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
11	#17988349	—	2	2	no	—
12	#17988349	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#103×C₇₃	—	2	2	no	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#384×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×#10596831	—	2	2	no	—
18	C₂×C₄⋊Dic₇₃	—	2	2	no	—
19	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
21	C₂×C₄×Dic₇₃	—	2	2	no	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₄⋊#10596831	—	2	2	no	—
24	#14556498	—	2	2	no	—
25	#14556498	—	2	4	no	—
26	#14556498	—	2	2	no	—
27	C₂²×Q₈×C₇₃	—	2	1	no	—
28	C₂²×C₈×C₇₃	—	2	1	no	—
29	C₄⋊Q₈×C₇₃	—	2	2	no	—
30	C₄⋊Q₈×C₇₃	—	2	1	no	—
31	C₄⋊Q₈×C₇₃	Q₈	1	2	yes	—
32	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
35	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
36	#103×C₇₃	—	4	4	no	—
37	C₄×C₈×C₇₃	—	2	2	no	—
38	C₂²×Dic₇₃	—	2	2	no	—
39	C₂²×Dic₇₃	—	2	2	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂×#10596831	—	4	2	no	—
43	C₄⋊Dic₇₃	—	2	2	no	—
44	C₄⋊Dic₇₃	—	2	2	no	—
45	C₄⋊Dic₇₃	—	2	2	no	—
46	C₄⋊Dic₇₃	—	2	2	no	—
47	C₄⋊Dic₇₃	—	8	2	no	—
48	C₄×Dic₇₃	—	4	2	no	—
49	C₄×Dic₇₃	—	4	2	no	—
50	C₂×Q₈×C₇₃	—	2	2	no	—
51	C₂×Q₈×C₇₃	—	4	2	no	—
52	C₂×Q₈×C₇₃	—	4	2	no	—
53	C₂×Q₈×C₇₃	—	4	1	no	—
54	C₂²×C₄×C₇₃	—	2	2	no	—
55	C₂²×C₄×C₇₃	—	4	1	no	—
56	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₂×C₈×C₇₃	—	4	1	no	—
60	C₂×C₈×C₇₃	—	4	1	no	—
61	C₄⋊C₄×C₇₃	—	2	4	no	—
62	C₄⋊C₄×C₇₃	—	4	2	no	—
63	C₄⋊C₄×C₇₃	—	4	2	no	—
64	C₄⋊C₄×C₇₃	—	4	2	no	—
65	C₄⋊C₄×C₇₃	—	4	2	no	—
66	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₇₃	#41	1	2	yes	—
68	C₄²×C₇₃	#41	1	1	yes	characteristic
69	C₂³×C₇₃	#120	1	1	yes	characteristic
70	Q₈×C₇₃	—	4	4	no	—
71	Q₈×C₇₃	—	8	2	no	—
72	C₂×C₄×C₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	—	2	1	no	—
74	C₂×C₄×C₇₃	—	4	2	no	—
75	C₂×C₄×C₇₃	—	4	4	no	—
76	C₂×C₄×C₇₃	—	4	2	no	—
77	C₂×C₄×C₇₃	—	8	1	no	—
78	C₂×C₄×C₇₃	—	8	1	no	—
79	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#124	1	2	yes	—
82	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₇₃	#135	1	2	yes	—
84	C₂×C₄×C₇₃	#135	1	2	yes	—
85	C₂×C₄×C₇₃	#137	1	2	yes	—
86	C₂×C₄×C₇₃	#137	1	2	yes	—
87	C₂×Dic₇₃	—	4	2	no	—
88	C₂×Dic₇₃	—	4	2	no	—
89	C₂×Dic₇₃	—	4	2	no	—
90	C₂×Dic₇₃	—	4	2	no	—
91	C₂×Dic₇₃	—	4	2	no	—
92	C₂×Dic₇₃	—	4	2	no	—
93	C₈×C₇₃	—	4	2	no	—
94	#10596831	—	8	2	no	—
95	C₂²×C₇₃	#393	1	1	yes	characteristic
96	C₂²×C₇₃	#452	1	2	yes	—
97	C₂²×C₇₃	#452	1	2	yes	—
98	C₂²×C₇₃	#453	1	1	yes	characteristic
99	C₂²×C₇₃	#454	1	1	yes	characteristic
100	C₄×C₇₃	—	2	1	no	—
101	C₄×C₇₃	—	2	1	no	—
102	C₄×C₇₃	—	8	2	no	—
103	C₄×C₇₃	—	8	2	no	—
104	C₄×C₇₃	#479	1	2	yes	—
105	C₄×C₇₃	#479	1	2	yes	—
106	C₄×C₇₃	#488	1	2	yes	—
107	C₄×C₇₃	#491	1	2	yes	—
108	Dic₇₃	—	8	2	no	—
109	Dic₇₃	—	8	2	no	—
110	#10352	—	73	1	no	maximal, Sylow
111	C₂×C₇₃	#1630	1	1	yes	characteristic
112	C₂×C₇₃	#1634	1	1	yes	characteristic
113	C₂×C₇₃	#1638	1	2	yes	—
114	C₂×C₇₃	#2026	1	2	yes	—
115	C₂×C₇₃	#2027	1	1	yes	characteristic
116	#2571	—	73	1	no	—
117	#1987	—	73	2	no	—
118	#1956	D₇₃	1	1	yes	characteristic
119	#1296	—	73	1	no	—
120	#1279	—	73	2	no	—
121	C₇₃	#10352	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
123	C₂×C₄⋊C₈	—	73	2	no	—
124	C₂×#103	—	2	2	no	—
125	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
126	#400	—	146	2	no	—
127	#389	—	146	2	no	—
128	#384	C₂×D₇₃	1	1	yes	characteristic
129	#382	—	146	2	no	—
130	C₄×C₄⋊C₄	—	73	2	no	—
131	#340	—	146	2	no	—
132	#340	—	146	2	no	—
133	C₂²×Q₈	—	2	1	no	—
134	C₂²×C₈	—	2	1	no	—
135	C₂²×C₈	—	146	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	—	2	1	no	—
138	C₄⋊Q₈	Dic₁₄₆	1	2	yes	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	73	2	no	—
142	C₂×C₄⋊C₄	—	73	2	no	—
143	C₂×C₄⋊C₄	—	146	2	no	—
144	C₂×C₄⋊C₄	C₂²⋊D₇₃	1	2	yes	—
145	C₂×C₄²	—	146	2	no	—
146	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
147	C₄⋊C₈	—	146	2	no	—
148	C₄⋊C₈	—	146	2	no	—
149	#103	—	4	4	no	—
150	C₄×C₈	—	2	2	no	—
151	#95	—	146	2	no	—
152	#95	—	146	4	no	—
153	#95	—	146	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	146	2	no	—
161	C₂²×C₄	—	146	2	no	—
162	C₂²×C₄	D₄×D₇₃	1	2	yes	—
163	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	292	2	no	—
168	C₂×C₈	—	292	2	no	—
169	C₂×C₈	—	292	2	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	584	2	no	—
176	C₄⋊C₄	—	146	2	no	—
177	C₄⋊C₄	—	146	2	no	—
178	C₄⋊C₄	—	146	2	no	—
179	C₄⋊C₄	—	146	2	no	—
180	C₄²	—	292	2	no	—
181	C₄²	—	292	2	no	—
182	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
183	C₄²	#11924027	1	2	yes	—
184	C₄²	#11924028	1	1	yes	characteristic
185	C₂³	#14556559	1	1	yes	characteristic
186	Q₈	—	4	4	no	—
187	Q₈	—	8	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	—	292	2	no	—
196	C₂×C₄	—	292	2	no	—
197	C₂×C₄	—	292	2	no	—
198	C₂×C₄	—	292	2	no	—
199	C₂×C₄	—	292	2	no	—
200	C₂×C₄	—	292	2	no	—
201	C₂×C₄	#14556568	1	2	yes	—
202	C₂×C₄	#14556570	1	1	yes	characteristic
203	C₂×C₄	#14556577	1	2	yes	—
204	C₂×C₄	#14556578	1	1	yes	characteristic
205	C₂×C₄	#14556612	1	2	yes	—
206	C₂×C₄	#14556614	1	2	yes	—
207	C₂×C₄	#14556620	1	2	yes	—
208	C₂×C₄	#14556623	1	2	yes	—
209	C₈	—	4	2	no	—
210	C₈	—	584	2	no	—
211	C₂²	#17988499	1	1	yes	characteristic
212	C₂²	#17988746	1	2	yes	—
213	C₂²	#17988748	1	2	yes	—
214	C₂²	#17988754	1	1	yes	characteristic
215	C₂²	#17988759	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	2	no	—
220	C₄	—	584	2	no	—
221	C₄	—	584	2	no	—
222	C₄	#17988892	1	2	yes	—
223	C₄	#17988896	1	2	yes	—
224	C₄	#17988943	1	2	yes	—
225	C₄	#17988954	1	2	yes	—
226	C₂	#22609050	1	1	yes	characteristic
227	C₂	#22609074	1	1	yes	characteristic
228	C₂	#22609100	1	2	yes	—
229	C₂	#22611080	1	2	yes	—
230	C₂	#22611087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22615173

C₂

yes

maximal, characteristic

#22610880

C₂

yes

maximal

#22607814

C₂

yes

maximal, characteristic

#22607771

C₂

yes

maximal

C₇₃×#1956

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596831

C₂²

yes

—

#17988526

—

#17988486

—

#17988452

—

C₄×C₄⋊Dic₇₃

C₂²

yes

—

#17988349

—

#17988349

—

C₂×C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₂×#103×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#384×C₇₃

C₂²

yes

characteristic

C₂²×#10596831

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

D₄

yes

—

C₂×C₄⋊Dic₇₃

D₄

yes

—

C₂×C₄×Dic₇₃

—

—

—

—

—

—

C₂²×Q₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

Q₈

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#103×C₇₃

—

C₄×C₈×C₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₄²×C₇₃

yes

C₂³×C₇₃

yes

Q₈×C₇₃

—

Q₈×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

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

#124

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#393

yes

characteristic

C₂²×C₇₃

#452

yes

—

C₂²×C₇₃

#452

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

100

C₄×C₇₃

—

101

C₄×C₇₃

—

102

C₄×C₇₃

—

103

C₄×C₇₃

—

104

C₄×C₇₃

#479

yes

—

105

C₄×C₇₃

#479

yes

—

106

C₄×C₇₃

#488

yes

—

107

C₄×C₇₃

#491

yes

—

108

Dic₇₃

—

109

Dic₇₃

—

110

#10352

—

maximal, Sylow

111

C₂×C₇₃

#1630

yes

characteristic

112

C₂×C₇₃

#1634

yes

characteristic

113

C₂×C₇₃

#1638

yes

—

114

C₂×C₇₃

#2026

yes

—

115

C₂×C₇₃

#2027

yes

characteristic

116

#2571

—

117

#1987

—

118

#1956

D₇₃

yes

characteristic

119

#1296

—

120

#1279

—

121

C₇₃

#10352

yes

characteristic, complemented, Sylow

122

C₂×C₄⋊Q₈

C₂×D₇₃

yes

characteristic

123

C₂×C₄⋊C₈

—

124

C₂×#103

—

125

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

126

#400

—

146

—

127

#389

—

146

—

128

#384

C₂×D₇₃

yes

characteristic

129

#382

—

146

—

130

C₄×C₄⋊C₄

—

131

#340

—

146

—

132

#340

—

146

—

133

C₂²×Q₈

—

134

C₂²×C₈

—

135

C₂²×C₈

—

146

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

Dic₁₄₆

yes

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄⋊C₄

—

143

C₂×C₄⋊C₄

—

146

—

144

C₂×C₄⋊C₄

C₂²⋊D₇₃

yes

—

145

C₂×C₄²

—

146

—

146

C₂×C₄²

C₂²×D₇₃

yes

characteristic

147

C₄⋊C₈

—

146

—

148

C₄⋊C₈

—

146

—

149

#103

—

150

C₄×C₈

—

151

#95

—

146

—

152

#95

—

146

—

153

#95

—

146

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

146

—

161

C₂²×C₄

—

146

—

162

C₂²×C₄

D₄×D₇₃

yes

—

163

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

292

—

168

C₂×C₈

—

292

—

169

C₂×C₈

—

292

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

584

—

176

C₄⋊C₄

—

146

—

177

C₄⋊C₄

—

146

—

178

C₄⋊C₄

—

146

—

179

C₄⋊C₄

—

146

—

180

C₄²

—

292

—

181

C₄²

—

292

—

182

C₄²

C₂×Dic₁₄₆

yes

characteristic

183

C₄²

#11924027

yes

—

184

C₄²

#11924028

yes

characteristic

185

C₂³

#14556559

yes

characteristic

186

Q₈

—

187

Q₈

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

292

—

196

C₂×C₄

—

292

—

197

C₂×C₄

—

292

—

198

C₂×C₄

—

292

—

199

C₂×C₄

—

292

—

200

C₂×C₄

—

292

—

201

C₂×C₄

#14556568

yes

—

202

C₂×C₄

#14556570

yes

characteristic

203

C₂×C₄

#14556577

yes

—

204

C₂×C₄

#14556578

yes

characteristic

205

C₂×C₄

#14556612

yes

—

206

C₂×C₄

#14556614

yes

—

207

C₂×C₄

#14556620

yes

—

208

C₂×C₄

#14556623

yes

—

209

C₈

—

210

C₈

—

584

—

211

C₂²

#17988499

yes

characteristic

212

C₂²

#17988746

yes

—

213

C₂²

#17988748

yes

—

214

C₂²

#17988754

yes

characteristic

215

C₂²

#17988759

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

584

—

221

C₄

—

584

—

222

C₄

#17988892

yes

—

223

C₄

#17988896

yes

—

224

C₄

#17988943

yes

—

225

C₄

#17988954

yes

—

226

C₂

#22609050

yes

characteristic

227

C₂

#22609074

yes

characteristic

228

C₂

#22609100

yes

—

229

C₂

#22611080

yes

—

230

C₂

#22611087

yes

characteristic

	subgroup	G/H
center	C₂³	#14556559
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1956	C₂
socle, prime radical	C₂³×C₇₃	#120
hypercenter	#1956	D₇₃
hypocenter	C₇₃	#10352
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#14556559

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1956

C₂

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
121. C₇₃	#10352	semidirect

H ≅ G/N

type

121. C₇₃

#10352

semidirect

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

subgroup	index
#22615173	2
#22610880	2
#22607814	2
#22607771	2
C₇₃×#1956	2
#10352	73

subgroup

index

C₇₃×#1956

Subgroups — #32989014

Conjugacy classes of subgroups

Complements

Maximal subgroups