Group #33005063 — subgroups

Subgroups — #33005063

Order 18688.

382 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 115 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×#17988989	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988391	C₂	1	2	yes	maximal
3	#22608022	C₂	1	2	yes	maximal
4	#22608009	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1969	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556583	—	2	1	no	—
7	C₂²×#11924003	C₂²	1	2	yes	—
8	#17988989	—	2	4	no	—
9	C₂×#14556525	C₂²	1	2	yes	—
10	C₂×#14556523	—	2	1	no	—
11	#17988391	—	2	4	no	—
12	C₂²×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#103×C₇₃	—	2	1	no	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#391×C₇₃	—	2	1	no	—
17	#14556583	—	2	4	no	—
18	#14556583	—	4	2	no	—
19	C₂×C₄⋊Dic₇₃	—	2	2	no	—
20	C₂×#11924007	—	4	2	no	—
21	C₂×#11924003	—	2	2	no	—
22	C₂×#11924003	—	2	2	no	—
23	C₂×#11924003	—	2	2	no	—
24	#14556525	—	2	2	no	—
25	#14556525	D₄	1	4	yes	—
26	#14556523	—	2	4	no	—
27	C₂²×Q₈×C₇₃	—	2	1	no	—
28	C₂×Q₁₆×C₇₃	—	2	2	no	—
29	C₂×Q₁₆×C₇₃	—	2	2	no	—
30	C₂×Q₁₆×C₇₃	—	2	1	no	—
31	C₂×Q₁₆×C₇₃	Q₈	1	2	yes	—
32	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
35	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
36	#107×C₇₃	—	2	2	no	—
37	#103×C₇₃	—	4	2	no	—
38	C₄×C₈×C₇₃	—	2	2	no	—
39	#95×C₇₃	—	4	2	no	—
40	C₂²×Dic₇₃	—	4	2	no	—
41	C₄⋊Dic₇₃	—	2	4	no	—
42	C₄⋊Dic₇₃	—	4	2	no	—
43	#11924007	—	4	8	no	—
44	#11924003	—	4	2	no	—
45	C₂×Q₈×C₇₃	—	2	2	no	—
46	C₂×Q₈×C₇₃	—	4	1	no	—
47	C₂×Q₈×C₇₃	—	4	1	no	—
48	C₂×Q₈×C₇₃	—	4	1	no	—
49	C₂×Q₈×C₇₃	—	4	1	no	—
50	C₂×Q₈×C₇₃	—	4	1	no	—
51	C₂²×C₄×C₇₃	—	4	1	no	—
52	C₂²×C₄×C₇₃	—	4	1	no	—
53	C₂²×C₄×C₇₃	D₈	1	2	yes	—
54	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₇₃	—	2	4	no	—
56	Q₁₆×C₇₃	—	4	2	no	—
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	—	2	1	no	—
59	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
60	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₃	#41	1	2	yes	—
63	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
64	C₄⋊C₄×C₇₃	—	4	1	no	—
65	C₄⋊C₄×C₇₃	—	4	1	no	—
66	C₄²×C₇₃	—	2	1	no	—
67	C₄²×C₇₃	—	2	1	no	—
68	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
69	Q₈×C₇₃	—	4	2	no	—
70	Q₈×C₇₃	—	4	2	no	—
71	Q₈×C₇₃	—	8	1	no	—
72	Q₈×C₇₃	—	8	1	no	—
73	C₂×C₄×C₇₃	—	2	2	no	—
74	C₂×C₄×C₇₃	—	2	2	no	—
75	C₂×C₄×C₇₃	—	2	2	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	—	8	1	no	—
80	C₂×C₄×C₇₃	—	8	2	no	—
81	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
87	C₂×Dic₇₃	—	4	4	no	—
88	C₂×Dic₇₃	—	8	2	no	—
89	C₂×Dic₇₃	—	8	2	no	—
90	C₈×C₇₃	—	2	2	no	—
91	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
92	C₈×C₇₃	#124	1	2	yes	—
93	C₂²×C₇₃	#447	1	1	yes	characteristic
94	C₂²×C₇₃	#474	1	1	yes	characteristic
95	C₂²×C₇₃	#486	1	1	yes	characteristic
96	C₂²×C₇₃	#508	1	2	yes	—
97	C₂²×C₇₃	#510	1	2	yes	—
98	C₄×C₇₃	—	4	2	no	—
99	C₄×C₇₃	—	8	2	no	—
100	C₄×C₇₃	—	16	1	no	—
101	C₄×C₇₃	#393	1	1	yes	characteristic
102	C₄×C₇₃	#448	1	1	yes	characteristic
103	C₄×C₇₃	#476	1	1	yes	characteristic
104	C₄×C₇₃	#489	1	1	yes	characteristic
105	Dic₇₃	—	8	4	no	—
106	#13366	—	73	1	no	maximal, Sylow
107	C₂×C₇₃	#2023	1	1	yes	characteristic
108	C₂×C₇₃	#2189	1	2	yes	—
109	C₂×C₇₃	#2229	1	2	yes	—
110	C₂×C₇₃	#2242	1	1	yes	characteristic
111	C₂×C₇₃	#2245	1	1	yes	characteristic
112	C₂×#499	—	73	1	no	—
113	C₂×#358	—	73	2	no	—
114	#1969	D₇₃	1	1	yes	characteristic
115	#1380	—	73	2	no	—
116	#1374	—	73	1	no	—
117	C₇₃	#13366	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
119	C₂×#125	—	146	1	no	—
120	C₂²×C₁₆	—	73	2	no	—
121	#499	—	146	4	no	—
122	C₂×#107	—	73	2	no	—
123	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
124	C₂×#106	—	146	1	no	—
125	C₂×#103	—	2	1	no	—
126	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
127	#391	—	2	1	no	—
128	#358	—	146	4	no	—
129	C₂²×Q₈	—	2	1	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	1	no	—
133	C₂×Q₁₆	Dic₁₄₆	1	2	yes	—
134	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
136	#125	—	292	2	no	—
137	#125	—	146	4	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	292	2	no	—
140	C₂×C₄⋊C₄	—	146	2	no	—
141	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
142	C₂×C₁₆	—	146	2	no	—
143	C₂×C₁₆	—	146	2	no	—
144	C₂×C₁₆	—	146	2	no	—
145	#107	—	2	2	no	—
146	#107	—	73	4	no	—
147	#107	—	146	2	no	—
148	#106	—	146	4	no	—
149	#103	—	4	2	no	—
150	C₄×C₈	—	2	2	no	—
151	#95	—	4	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	1	no	—
154	C₂×Q₈	—	4	1	no	—
155	C₂×Q₈	—	4	1	no	—
156	C₂×Q₈	—	4	1	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	292	2	no	—
161	C₂²×C₄	#11924013	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
163	Q₁₆	—	2	4	no	—
164	Q₁₆	—	4	2	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	#11924016	1	2	yes	—
168	C₂×C₈	D₄×D₇₃	1	2	yes	—
169	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
170	C₂×C₈	#11924027	1	2	yes	—
171	C₂×C₈	#11924028	1	1	yes	characteristic
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	292	8	no	—
175	C₄⋊C₄	—	292	2	no	—
176	C₄⋊C₄	—	146	4	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	1	no	—
179	C₁₆	—	292	2	no	—
180	C₂³	C₂×#11924013	1	1	yes	characteristic
181	Q₈	—	4	2	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	8	1	no	—
184	Q₈	—	8	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	1	no	—
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	—	8	1	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	292	4	no	—
194	C₂×C₄	—	584	2	no	—
195	C₂×C₄	—	584	2	no	—
196	C₂×C₄	#14556559	1	1	yes	characteristic
197	C₂×C₄	#14556570	1	1	yes	characteristic
198	C₂×C₄	#14556578	1	1	yes	characteristic
199	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
200	C₂×C₄	#14556616	1	1	yes	characteristic
201	C₂×C₄	#14556621	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	#14556568	1	2	yes	—
204	C₈	#14556577	1	2	yes	—
205	C₂²	#17988720	1	1	yes	characteristic
206	C₂²	#17988867	1	1	yes	characteristic
207	C₂²	#17988933	1	1	yes	characteristic
208	C₂²	#17989017	1	2	yes	—
209	C₂²	#17989028	1	2	yes	—
210	C₄	—	4	2	no	—
211	C₄	—	8	2	no	—
212	C₄	—	16	1	no	—
213	C₄	—	584	4	no	—
214	C₄	#17988499	1	1	yes	characteristic
215	C₄	#17988725	1	1	yes	characteristic
216	C₄	#17988877	1	1	yes	characteristic
217	C₄	#17988946	1	1	yes	characteristic
218	C₂	#22611067	1	1	yes	characteristic
219	C₂	#22611869	1	2	yes	—
220	C₂	#22612095	1	2	yes	—
221	C₂	#22612162	1	1	yes	characteristic
222	C₂	#22612176	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988989

C₂

yes

maximal, characteristic

C₂×#17988391

C₂

yes

maximal

#22608022

C₂

yes

maximal

#22608009

C₂

yes

maximal, characteristic

C₇₃×#1969

C₂

yes

maximal, characteristic

C₂×#14556583

—

C₂²×#11924003

C₂²

yes

—

#17988989

—

C₂×#14556525

C₂²

yes

—

C₂×#14556523

—

#17988391

—

C₂²×Q₁₆×C₇₃

C₂²

yes

characteristic

C₂×#107×C₇₃

C₂²

yes

characteristic

C₂×#103×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#391×C₇₃

—

#14556583

—

#14556583

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

—

D₄

yes

—

#14556523

—

C₂²×Q₈×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

Q₈

yes

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

#107×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

—

#95×C₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924003

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

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

—

C₂×C₄×C₇₃

#120

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊Q₈

yes

—

C₈×C₇₃

#124

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#510

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

100

C₄×C₇₃

—

101

C₄×C₇₃

#393

yes

characteristic

102

C₄×C₇₃

#448

yes

characteristic

103

C₄×C₇₃

#476

yes

characteristic

104

C₄×C₇₃

#489

yes

characteristic

105

Dic₇₃

—

106

#13366

—

maximal, Sylow

107

C₂×C₇₃

#2023

yes

characteristic

108

C₂×C₇₃

#2189

yes

—

109

C₂×C₇₃

#2229

yes

—

110

C₂×C₇₃

#2242

yes

characteristic

111

C₂×C₇₃

#2245

yes

characteristic

112

C₂×#499

—

113

C₂×#358

—

114

#1969

D₇₃

yes

characteristic

115

#1380

—

116

#1374

—

117

C₇₃

#13366

yes

characteristic, complemented, Sylow

118

C₂²×Q₁₆

C₂×D₇₃

yes

characteristic

119

C₂×#125

—

146

—

120

C₂²×C₁₆

—

121

#499

—

146

—

122

C₂×#107

—

123

C₂×#107

C₂×D₇₃

yes

characteristic

124

C₂×#106

—

146

—

125

C₂×#103

—

126

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

127

#391

—

128

#358

—

146

—

129

C₂²×Q₈

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

Dic₁₄₆

yes

—

134

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

135

C₂²×C₈

C₂²×D₇₃

yes

characteristic

136

#125

—

292

—

137

#125

—

146

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

292

—

140

C₂×C₄⋊C₄

—

146

—

141

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

142

C₂×C₁₆

—

146

—

143

C₂×C₁₆

—

146

—

144

C₂×C₁₆

—

146

—

145

#107

—

146

#107

—

147

#107

—

146

—

148

#106

—

146

—

149

#103

—

150

C₄×C₈

—

151

#95

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

292

—

161

C₂²×C₄

#11924013

yes

—

162

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

163

Q₁₆

—

164

Q₁₆

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

#11924016

yes

—

168

C₂×C₈

D₄×D₇₃

yes

—

169

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

170

C₂×C₈

#11924027

yes

—

171

C₂×C₈

#11924028

yes

characteristic

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

292

—

175

C₄⋊C₄

—

292

—

176

C₄⋊C₄

—

146

—

177

C₄²

—

178

C₄²

—

179

C₁₆

—

292

—

180

C₂³

C₂×#11924013

yes

characteristic

181

Q₈

—

182

Q₈

—

183

Q₈

—

184

Q₈

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

292

—

194

C₂×C₄

—

584

—

195

C₂×C₄

—

584

—

196

C₂×C₄

#14556559

yes

characteristic

197

C₂×C₄

#14556570

yes

characteristic

198

C₂×C₄

#14556578

yes

characteristic

199

C₂×C₄

C₂×#11924016

yes

characteristic

200

C₂×C₄

#14556616

yes

characteristic

201

C₂×C₄

#14556621

yes

characteristic

202

C₈

—

203

C₈

#14556568

yes

—

204

C₈

#14556577

yes

—

205

C₂²

#17988720

yes

characteristic

206

C₂²

#17988867

yes

characteristic

207

C₂²

#17988933

yes

characteristic

208

C₂²

#17989017

yes

—

209

C₂²

#17989028

yes

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

584

—

214

C₄

#17988499

yes

characteristic

215

C₄

#17988725

yes

characteristic

216

C₄

#17988877

yes

characteristic

217

C₄

#17988946

yes

characteristic

218

C₂

#22611067

yes

characteristic

219

C₂

#22611869

yes

—

220

C₂

#22612095

yes

—

221

C₂

#22612162

yes

characteristic

222

C₂

#22612176

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11924013
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1969	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	#1969	D₇₃
hypocenter	C₇₃	#13366
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

subgroup

G/H

center

C₂³

C₂×#11924013

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1969

C₂

socle, prime radical

C₂³×C₇₃

C₂×D₈

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

N	H ≅ G/N	type
117. C₇₃	#13366	semidirect

H ≅ G/N

type

117. C₇₃

#13366

semidirect

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

subgroup	index
C₂×#17988989	2
C₂×#17988391	2
#22608022	2
#22608009	2
C₇₃×#1969	2
#13366	73

subgroup

index

C₇₃×#1969

Subgroups — #33005063

Conjugacy classes of subgroups

Complements

Maximal subgroups