Group #33005060 — subgroups

Subgroups — #33005060

Order 18688.

382 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 117 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×#17988390	C₂	1	2	yes	maximal
2	#22610792	C₂	1	1	yes	maximal, characteristic
3	#22608022	C₂	1	2	yes	maximal
4	#22608009	C₂	1	1	yes	maximal, characteristic
5	C₂×#499×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₂	C₂²	1	1	yes	characteristic
7	C₂²×#11924003	C₂²	1	2	yes	—
8	C₂×#14556525	C₂²	1	1	yes	characteristic
9	C₂×#14556516	—	2	1	no	—
10	#17988493	—	2	1	no	—
11	#17988390	—	2	4	no	—
12	C₂×#125×C₇₃	—	2	1	no	—
13	#499×C₇₃	—	2	4	no	—
14	C₂×#107×C₇₃	C₂²	1	2	yes	—
15	C₂×#106×C₇₃	—	2	1	no	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₄₆	—	2	1	no	—
18	C₂×Dic₂₉₂	—	2	2	no	—
19	C₂×Dic₂₉₂	—	2	2	no	—
20	C₂×Dic₂₉₂	—	2	1	no	—
21	C₂×Dic₂₉₂	Q₈	1	2	yes	—
22	C₂×C₄⋊Dic₇₃	—	2	1	no	—
23	C₂×#11924003	—	2	2	no	—
24	C₂×#11924003	—	2	2	no	—
25	C₂×#11924003	—	2	2	no	—
26	#14556525	—	2	2	no	—
27	#14556516	—	4	2	no	—
28	#14556498	—	4	2	no	—
29	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	#125×C₇₃	—	2	4	no	—
32	#125×C₇₃	—	4	2	no	—
33	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₇₃	—	4	2	no	—
35	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
36	#107×C₇₃	—	2	2	no	—
37	#107×C₇₃	D₄	1	4	yes	—
38	#106×C₇₃	—	2	4	no	—
39	C₄×C₈×C₇₃	—	2	2	no	—
40	C₂×Dic₁₄₆	—	2	2	no	—
41	C₂×Dic₁₄₆	—	4	1	no	—
42	C₂×Dic₁₄₆	—	4	1	no	—
43	C₂×Dic₁₄₆	—	4	1	no	—
44	C₂×Dic₁₄₆	—	4	1	no	—
45	C₂×Dic₁₄₆	—	4	1	no	—
46	C₂²×Dic₇₃	—	4	1	no	—
47	C₂²×Dic₇₃	—	4	1	no	—
48	Dic₂₉₂	—	2	4	no	—
49	Dic₂₉₂	—	4	2	no	—
50	C₄⋊Dic₇₃	—	4	1	no	—
51	C₄⋊Dic₇₃	—	4	1	no	—
52	#11924003	—	4	2	no	—
53	C₂²×C₄×C₇₃	—	4	2	no	—
54	C₂²×C₄×C₇₃	D₈	1	2	yes	—
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	—	2	1	no	—
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₇₃	#41	1	2	yes	—
62	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
63	C₄⋊C₄×C₇₃	—	2	4	no	—
64	C₄⋊C₄×C₇₃	—	4	8	no	—
65	C₄⋊C₄×C₇₃	—	4	2	no	—
66	C₄²×C₇₃	—	2	1	no	—
67	C₄²×C₇₃	—	2	1	no	—
68	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	4	4	no	—
73	C₂×C₄×C₇₃	—	8	2	no	—
74	C₂×C₄×C₇₃	—	8	2	no	—
75	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
81	C₂×Dic₇₃	—	4	2	no	—
82	C₂×Dic₇₃	—	8	1	no	—
83	C₂×Dic₇₃	—	8	1	no	—
84	C₂×Dic₇₃	—	8	1	no	—
85	C₂×Dic₇₃	—	8	2	no	—
86	Dic₁₄₆	—	4	2	no	—
87	Dic₁₄₆	—	4	2	no	—
88	Dic₁₄₆	—	8	1	no	—
89	Dic₁₄₆	—	8	1	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	4	no	—
100	C₄×C₇₃	#393	1	1	yes	characteristic
101	C₄×C₇₃	#448	1	1	yes	characteristic
102	C₄×C₇₃	#476	1	1	yes	characteristic
103	C₄×C₇₃	#489	1	1	yes	characteristic
104	Dic₇₃	—	8	2	no	—
105	Dic₇₃	—	16	1	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	D₇₃	1	1	yes	characteristic
113	C₂×#358	—	73	2	no	—
114	#1969	—	73	1	no	—
115	#1380	—	73	2	no	—
116	#1374	—	73	1	no	—
117	C₇₃	#13366	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	—	73	1	no	—
119	C₂×#125	—	2	1	no	—
120	C₂²×C₁₆	—	73	2	no	—
121	#499	—	2	4	no	—
122	C₂×#107	—	73	1	no	—
123	C₂×#107	C₂×D₇₃	1	2	yes	—
124	C₂×#106	—	2	1	no	—
125	C₂×#103	—	146	1	no	—
126	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
127	#391	—	146	1	no	—
128	#358	—	146	4	no	—
129	C₂²×Q₈	—	146	1	no	—
130	C₂×Q₁₆	—	73	2	no	—
131	C₂×Q₁₆	—	146	2	no	—
132	C₂×Q₁₆	—	146	1	no	—
133	C₂×Q₁₆	—	146	2	no	—
134	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
136	#125	—	2	4	no	—
137	#125	—	4	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	4	2	no	—
140	C₂×C₄⋊C₄	—	146	1	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	—	146	2	no	—
147	#107	C₂²⋊D₇₃	1	4	yes	—
148	#106	—	2	4	no	—
149	#103	—	292	2	no	—
150	C₄×C₈	—	2	2	no	—
151	#95	—	292	2	no	—
152	C₂×Q₈	—	292	1	no	—
153	C₂×Q₈	—	292	1	no	—
154	C₂×Q₈	—	292	1	no	—
155	C₂×Q₈	—	292	1	no	—
156	C₂×Q₈	—	292	1	no	—
157	C₂×Q₈	—	146	2	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	—	292	1	no	—
160	C₂²×C₄	—	292	1	no	—
161	C₂²×C₄	#11924013	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
163	Q₁₆	—	292	2	no	—
164	Q₁₆	—	146	4	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	#11924015	1	2	yes	—
168	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
169	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
170	C₂×C₈	#11924029	1	2	yes	—
171	C₂×C₈	#11924029	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	8	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	292	1	no	—
176	C₄⋊C₄	—	292	1	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	—	292	2	no	—
182	Q₈	—	292	2	no	—
183	Q₈	—	584	1	no	—
184	Q₈	—	584	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	292	2	no	—
192	C₂×C₄	—	584	1	no	—
193	C₂×C₄	—	584	1	no	—
194	C₂×C₄	—	584	1	no	—
195	C₂×C₄	—	584	2	no	—
196	C₂×C₄	#14556560	1	1	yes	characteristic
197	C₂×C₄	#14556567	1	1	yes	characteristic
198	C₂×C₄	#14556580	1	1	yes	characteristic
199	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
200	C₂×C₄	#14556617	1	1	yes	characteristic
201	C₂×C₄	#14556622	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	#14556567	1	2	yes	—
204	C₈	#14556580	1	2	yes	—
205	C₂²	#17988721	1	1	yes	characteristic
206	C₂²	#17988864	1	1	yes	characteristic
207	C₂²	#17988935	1	1	yes	characteristic
208	C₂²	#17989017	1	2	yes	—
209	C₂²	#17989027	1	2	yes	—
210	C₄	—	4	2	no	—
211	C₄	—	8	4	no	—
212	C₄	—	584	2	no	—
213	C₄	—	1168	1	no	—
214	C₄	#17988498	1	1	yes	characteristic
215	C₄	#17988726	1	1	yes	characteristic
216	C₄	#17988874	1	1	yes	characteristic
217	C₄	#17988949	1	1	yes	characteristic
218	C₂	#22611064	1	1	yes	characteristic
219	C₂	#22611871	1	2	yes	—
220	C₂	#22612092	1	2	yes	—
221	C₂	#22612165	1	1	yes	characteristic
222	C₂	#22612178	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988390

C₂

yes

maximal

#22610792

C₂

yes

maximal, characteristic

#22608022

C₂

yes

maximal

#22608009

C₂

yes

maximal, characteristic

C₂×#499×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₂

C₂²

yes

characteristic

C₂²×#11924003

C₂²

yes

—

C₂×#14556525

C₂²

yes

—

—

—

C₂×#125×C₇₃

—

#499×C₇₃

—

C₂×#107×C₇₃

C₂²

yes

—

C₂×#106×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂²×Dic₁₄₆

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

Q₈

yes

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

—

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

#125×C₇₃

—

#125×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

#107×C₇₃

—

#107×C₇₃

D₄

yes

—

#106×C₇₃

—

C₄×C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

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

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

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

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

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

#393

yes

characteristic

101

C₄×C₇₃

#448

yes

characteristic

102

C₄×C₇₃

#476

yes

characteristic

103

C₄×C₇₃

#489

yes

characteristic

104

Dic₇₃

—

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

D₇₃

yes

characteristic

113

C₂×#358

—

114

#1969

—

115

#1380

—

116

#1374

—

117

C₇₃

#13366

yes

characteristic, complemented, Sylow

118

C₂²×Q₁₆

—

119

C₂×#125

—

120

C₂²×C₁₆

—

121

#499

—

122

C₂×#107

—

123

C₂×#107

C₂×D₇₃

yes

—

124

C₂×#106

—

125

C₂×#103

—

146

—

126

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

127

#391

—

146

—

128

#358

—

146

—

129

C₂²×Q₈

—

146

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

146

—

132

C₂×Q₁₆

—

146

—

133

C₂×Q₁₆

—

146

—

134

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

135

C₂²×C₈

C₂²×D₇₃

yes

characteristic

136

#125

—

137

#125

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

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

—

146

—

147

#107

C₂²⋊D₇₃

yes

—

148

#106

—

149

#103

—

292

—

150

C₄×C₈

—

151

#95

—

292

—

152

C₂×Q₈

—

292

—

153

C₂×Q₈

—

292

—

154

C₂×Q₈

—

292

—

155

C₂×Q₈

—

292

—

156

C₂×Q₈

—

292

—

157

C₂×Q₈

—

146

—

158

C₂²×C₄

—

159

C₂²×C₄

—

292

—

160

C₂²×C₄

—

292

—

161

C₂²×C₄

#11924013

yes

—

162

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

163

Q₁₆

—

292

—

164

Q₁₆

—

146

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

#11924015

yes

—

168

C₂×C₈

C₂×C₂²⋊D₇₃

yes

—

169

C₂×C₈

Q₈×D₇₃

yes

characteristic

170

C₂×C₈

#11924029

yes

—

171

C₂×C₈

#11924029

yes

characteristic

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

292

—

176

C₄⋊C₄

—

292

—

177

C₄²

—

178

C₄²

—

179

C₁₆

—

292

—

180

C₂³

C₂×#11924013

yes

characteristic

181

Q₈

—

292

—

182

Q₈

—

292

—

183

Q₈

—

584

—

184

Q₈

—

584

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

292

—

192

C₂×C₄

—

584

—

193

C₂×C₄

—

584

—

194

C₂×C₄

—

584

—

195

C₂×C₄

—

584

—

196

C₂×C₄

#14556560

yes

characteristic

197

C₂×C₄

#14556567

yes

characteristic

198

C₂×C₄

#14556580

yes

characteristic

199

C₂×C₄

C₂×#11924015

yes

characteristic

200

C₂×C₄

#14556617

yes

characteristic

201

C₂×C₄

#14556622

yes

characteristic

202

C₈

—

203

C₈

#14556567

yes

—

204

C₈

#14556580

yes

—

205

C₂²

#17988721

yes

characteristic

206

C₂²

#17988864

yes

characteristic

207

C₂²

#17988935

yes

characteristic

208

C₂²

#17989017

yes

—

209

C₂²

#17989027

yes

—

210

C₄

—

211

C₄

—

212

C₄

—

584

—

213

C₄

—

1168

—

214

C₄

#17988498

yes

characteristic

215

C₄

#17988726

yes

characteristic

216

C₄

#17988874

yes

characteristic

217

C₄

#17988949

yes

characteristic

218

C₂

#22611064

yes

characteristic

219

C₂

#22611871

yes

—

220

C₂

#22612092

yes

—

221

C₂

#22612165

yes

characteristic

222

C₂

#22612178

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11924013
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×#499×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	C₂×#499	D₇₃
hypocenter	C₇₃	#13366
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂³

C₂×#11924013

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×#499×C₇₃

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 ◁ C₂×#499

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×#17988390	2
#22610792	2
#22608022	2
#22608009	2
C₂×#499×C₇₃	2
#13366	73

subgroup

index

C₂×#499×C₇₃

Subgroups — #33005060

Conjugacy classes of subgroups

Complements

Maximal subgroups