Group #33004215 — subgroups

Subgroups — #33004215

Order 18688.

590 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 159 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×#17988964	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988387	C₂	1	4	yes	maximal, complemented
3	C₂×#17988363	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₈₄	—	2	1	no	—
6	C₂²×D₅₈₄	C₂²	1	2	yes	—
7	C₂×#14556588	—	2	1	no	—
8	C₂²×#11924003	C₂²	1	2	yes	—
9	#17988964	—	2	2	no	—
10	#17988964	C₄	1	4	yes	complemented
11	#17988387	—	2	8	no	—
12	#17988363	—	2	2	no	—
13	C₂×#107×C₇₃	C₂²	1	2	yes	—
14	C₂×#106×C₇₃	—	2	1	no	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#382×C₇₃	—	2	1	no	—
17	C₂²×D₂₉₂	—	2	2	no	—
18	C₂²×D₂₉₂	—	4	2	no	—
19	C₂×D₅₈₄	—	2	4	no	—
20	C₂×D₅₈₄	—	2	4	no	—
21	C₂×D₅₈₄	—	2	2	no	—
22	C₂×D₅₈₄	—	2	4	no	—
23	C₂×D₅₈₄	—	4	2	no	—
24	C₂×D₅₈₄	—	4	2	no	—
25	C₂×D₅₈₄	C₂×C₄	1	4	yes	—
26	#14556588	—	2	4	no	—
27	#14556588	—	4	2	no	—
28	C₂×#11924003	—	2	2	no	—
29	C₂×#11924003	—	2	2	no	—
30	C₂×#11924003	D₄	1	4	yes	complemented
31	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
34	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
35	#107×C₇₃	—	2	4	no	—
36	#106×C₇₃	—	4	2	no	—
37	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₃	D₄	1	2	yes	—
39	#95×C₇₃	—	4	2	no	—
40	C₂³×D₇₃	—	4	2	no	—
41	C₂×D₂₉₂	—	2	4	no	—
42	C₂×D₂₉₂	—	4	2	no	—
43	C₂×D₂₉₂	—	4	2	no	—
44	C₂×D₂₉₂	—	4	2	no	—
45	C₂×D₂₉₂	—	4	2	no	—
46	C₂×D₂₉₂	—	4	8	no	—
47	C₂×D₂₉₂	—	4	2	no	—
48	C₂×D₂₉₂	—	8	8	no	—
49	D₅₈₄	—	2	8	no	—
50	D₅₈₄	—	4	8	no	—
51	D₅₈₄	—	4	4	no	—
52	#11924003	—	2	4	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	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₃	D₈	1	2	yes	—
58	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	#41	1	2	yes	—
63	C₄⋊C₄×C₇₃	—	4	2	no	—
64	C₄⋊C₄×C₇₃	—	4	2	no	—
65	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₇₃	SD₁₆	1	2	yes	—
67	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
68	C₂²×D₇₃	—	4	4	no	—
69	C₂²×D₇₃	—	8	2	no	—
70	C₂²×D₇₃	—	8	2	no	—
71	C₂²×D₇₃	—	8	8	no	—
72	C₂×C₄×C₇₃	—	2	4	no	—
73	C₂×C₄×C₇₃	—	8	2	no	—
74	C₂×C₄×C₇₃	—	8	4	no	—
75	C₂×C₄×C₇₃	#102	1	4	yes	—
76	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
82	D₂₉₂	—	4	4	no	—
83	D₂₉₂	—	4	4	no	—
84	D₂₉₂	—	8	16	no	—
85	D₂₉₂	—	8	2	no	—
86	D₂₉₂	—	8	2	no	—
87	C₈×C₇₃	#102	1	4	yes	—
88	C₈×C₇₃	C₄×D₄	1	2	yes	—
89	C₈×C₇₃	#121	1	2	yes	—
90	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
91	C₂²×C₇₃	#486	1	1	yes	characteristic
92	C₂²×C₇₃	#493	1	1	yes	characteristic
93	C₂²×C₇₃	#509	1	4	yes	—
94	C₂×D₇₃	—	8	8	no	—
95	C₂×D₇₃	—	8	4	no	—
96	C₂×D₇₃	—	16	4	no	—
97	C₂×D₇₃	—	16	4	no	—
98	C₄×C₇₃	—	2	4	no	—
99	C₄×C₇₃	—	16	2	no	—
100	C₄×C₇₃	#390	1	1	yes	characteristic
101	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
102	C₄×C₇₃	#486	1	1	yes	characteristic
103	C₄×C₇₃	#492	1	1	yes	characteristic
104	#13206	—	73	1	no	maximal, Sylow
105	C₂×C₇₃	#1957	1	1	yes	characteristic
106	C₂×C₇₃	#2140	1	2	yes	—
107	C₂×C₇₃	#2244	1	2	yes	—
108	C₂×C₇₃	#2250	1	2	yes	—
109	D₇₃	—	16	4	no	—
110	C₂×#493	—	73	1	no	—
111	C₂×#357	—	73	4	no	—
112	C₂×#346	—	73	1	no	—
113	#1846	D₇₃	1	1	yes	characteristic, complemented
114	C₇₃	#13206	1	1	yes	characteristic, complemented, Sylow
115	C₂²×D₈	—	73	2	no	—
116	C₂²×D₈	—	146	1	no	—
117	C₂×#127	—	146	1	no	—
118	C₂²×C₁₆	—	73	2	no	—
119	#493	—	73	4	no	—
120	#493	—	146	2	no	—
121	C₂×#107	C₂×D₇₃	1	2	yes	—
122	C₂×#106	—	2	1	no	—
123	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
124	#382	—	2	1	no	—
125	#357	—	146	8	no	—
126	#346	—	146	2	no	—
127	C₂²×D₄	—	292	2	no	—
128	C₂²×D₄	—	146	2	no	—
129	C₂×D₈	—	292	2	no	—
130	C₂×D₈	—	292	2	no	—
131	C₂×D₈	—	73	4	no	—
132	C₂×D₈	—	146	4	no	—
133	C₂×D₈	—	146	4	no	—
134	C₂×D₈	—	146	2	no	—
135	C₂×D₈	—	146	4	no	—
136	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
138	#127	—	292	2	no	—
139	#127	—	146	4	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
142	C₂×C₁₆	—	73	4	no	—
143	C₂×C₁₆	—	146	2	no	—
144	C₂×C₁₆	—	146	2	no	—
145	#107	—	2	4	no	—
146	#106	—	4	2	no	—
147	C₄×C₈	C₄×D₇₃	1	2	yes	—
148	C₄×C₈	D₂₉₂	1	2	yes	—
149	#95	—	4	2	no	—
150	C₂⁴	—	292	2	no	—
151	C₂×D₄	—	292	8	no	—
152	C₂×D₄	—	292	2	no	—
153	C₂×D₄	—	292	2	no	—
154	C₂×D₄	—	292	2	no	—
155	C₂×D₄	—	292	2	no	—
156	C₂×D₄	—	292	2	no	—
157	C₂×D₄	—	584	8	no	—
158	C₂×D₄	—	146	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	#11924013	1	2	yes	—
161	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
162	D₈	—	292	8	no	—
163	D₈	—	292	4	no	—
164	D₈	—	146	8	no	—
165	C₂×C₈	#11924005	1	2	yes	—
166	C₂×C₈	#11924013	1	2	yes	—
167	C₂×C₈	#11924015	1	2	yes	—
168	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
169	C₂×C₈	D₄×D₇₃	1	2	yes	—
170	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
171	C₂×C₈	#11924029	1	2	yes	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	#11924005	1	2	yes	—
175	C₄²	#11924015	1	2	yes	—
176	C₁₆	—	146	4	no	—
177	C₂³	—	292	4	no	—
178	C₂³	—	584	2	no	—
179	C₂³	—	584	2	no	—
180	C₂³	—	584	8	no	—
181	C₂³	C₂×#11924013	1	1	yes	characteristic
182	D₄	—	292	4	no	—
183	D₄	—	292	4	no	—
184	D₄	—	584	16	no	—
185	D₄	—	584	2	no	—
186	D₄	—	584	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	—	8	4	no	—
190	C₂×C₄	#14556511	1	4	yes	—
191	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
192	C₂×C₄	#14556580	1	1	yes	characteristic
193	C₂×C₄	#14556587	1	1	yes	characteristic
194	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
195	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
196	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
197	C₈	#14556511	1	4	yes	—
198	C₈	#14556553	1	2	yes	—
199	C₈	#14556566	1	2	yes	—
200	C₂²	—	584	4	no	—
201	C₂²	—	584	8	no	—
202	C₂²	—	1168	4	no	—
203	C₂²	—	1168	4	no	—
204	C₂²	C₂×#14556511	1	1	yes	characteristic
205	C₂²	#17988935	1	1	yes	characteristic
206	C₂²	#17988963	1	1	yes	characteristic
207	C₂²	#17989022	1	4	yes	—
208	C₄	—	2	4	no	—
209	C₄	—	16	2	no	—
210	C₄	#17988488	1	1	yes	characteristic
211	C₄	C₂×#14556511	1	1	yes	characteristic
212	C₄	#17988935	1	1	yes	characteristic
213	C₄	#17988961	1	1	yes	characteristic
214	C₂	—	1168	4	no	—
215	C₂	#22610736	1	1	yes	characteristic
216	C₂	#22611646	1	2	yes	—
217	C₂	#22612174	1	2	yes	—
218	C₂	#22612198	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988964

C₂

yes

maximal, characteristic

C₂×#17988387

C₂

yes

maximal, complemented

C₂×#17988363

C₂

yes

maximal, characteristic, complemented

C₇₃×#1846

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₈₄

—

C₂²×D₅₈₄

C₂²

yes

—

C₂×#14556588

—

C₂²×#11924003

C₂²

yes

—

#17988964

—

#17988964

C₄

yes

complemented

#17988387

—

#17988363

—

C₂×#107×C₇₃

C₂²

yes

—

C₂×#106×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#382×C₇₃

—

C₂²×D₂₉₂

—

C₂²×D₂₉₂

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

C₂×C₄

yes

—

—

—

—

—

D₄

yes

complemented

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

#107×C₇₃

—

#106×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

#95×C₇₃

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

#11924003

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

SD₁₆

yes

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

#102

yes

—

C₈×C₇₃

C₄×D₄

yes

—

C₈×C₇₃

#121

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

100

C₄×C₇₃

#390

yes

characteristic

101

C₄×C₇₃

C₂×#102

yes

characteristic

102

C₄×C₇₃

#486

yes

characteristic

103

C₄×C₇₃

yes

104

—

105

C₂×C₇₃

#1957

yes

characteristic

106

C₂×C₇₃

#2140

yes

—

107

C₂×C₇₃

#2244

yes

—

108

C₂×C₇₃

#2250

yes

—

109

D₇₃

—

110

C₂×#493

—

111

C₂×#357

—

112

C₂×#346

—

113

#1846

D₇₃

yes

characteristic, complemented

114

C₇₃

#13206

yes

characteristic, complemented, Sylow

115

C₂²×D₈

—

116

C₂²×D₈

—

146

—

117

C₂×#127

—

146

—

118

C₂²×C₁₆

—

119

#493

—

120

#493

—

146

—

121

C₂×#107

C₂×D₇₃

yes

—

122

C₂×#106

—

123

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

124

#382

—

125

#357

—

146

—

126

#346

—

146

—

127

C₂²×D₄

—

292

—

128

C₂²×D₄

—

146

—

129

C₂×D₈

—

292

—

130

C₂×D₈

—

292

—

131

C₂×D₈

—

132

C₂×D₈

—

146

—

133

C₂×D₈

—

146

—

134

C₂×D₈

—

146

—

135

C₂×D₈

—

146

—

136

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

137

C₂²×C₈

C₂²×D₇₃

yes

characteristic

138

#127

—

292

—

139

#127

—

146

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

142

C₂×C₁₆

—

143

C₂×C₁₆

—

146

—

144

C₂×C₁₆

—

146

—

145

#107

—

146

#106

—

147

C₄×C₈

C₄×D₇₃

yes

—

148

C₄×C₈

D₂₉₂

yes

—

149

#95

—

150

C₂⁴

—

292

—

151

C₂×D₄

—

292

—

152

C₂×D₄

—

292

—

153

C₂×D₄

—

292

—

154

C₂×D₄

—

292

—

155

C₂×D₄

—

292

—

156

C₂×D₄

—

292

—

157

C₂×D₄

—

584

—

158

C₂×D₄

—

146

—

159

C₂²×C₄

—

160

C₂²×C₄

#11924013

yes

—

161

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

162

D₈

—

292

—

163

D₈

—

292

—

164

D₈

—

146

—

165

C₂×C₈

#11924005

yes

—

166

C₂×C₈

#11924013

yes

—

167

C₂×C₈

#11924015

yes

—

168

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

169

C₂×C₈

D₄×D₇₃

yes

—

170

C₂×C₈

C₂×D₂₉₂

yes

characteristic

171

C₂×C₈

#11924029

yes

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄²

#11924005

yes

—

175

C₄²

#11924015

yes

—

176

C₁₆

—

146

—

177

C₂³

—

292

—

178

C₂³

—

584

—

179

C₂³

—

584

—

180

C₂³

—

584

—

181

C₂³

C₂×#11924013

yes

characteristic

182

D₄

—

292

—

183

D₄

—

292

—

184

D₄

—

584

—

185

D₄

—

584

—

186

D₄

—

584

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

#14556511

yes

—

191

C₂×C₄

C₂×#11924005

yes

characteristic

192

C₂×C₄

#14556580

yes

characteristic

193

C₂×C₄

#14556587

yes

characteristic

194

C₂×C₄

C₂×#11924013

yes

characteristic

195

C₂×C₄

C₂×#11924015

yes

characteristic

196

C₂×C₄

C₂×#11924015

yes

characteristic

197

C₈

#14556511

yes

—

198

C₈

#14556553

yes

—

199

C₈

#14556566

yes

—

200

C₂²

—

584

—

201

C₂²

—

584

—

202

C₂²

—

1168

—

203

C₂²

—

1168

—

204

C₂²

C₂×#14556511

yes

characteristic

205

C₂²

#17988935

yes

characteristic

206

C₂²

#17988963

yes

characteristic

207

C₂²

#17989022

yes

—

208

C₄

—

209

C₄

—

210

C₄

#17988488

yes

characteristic

211

C₄

C₂×#14556511

yes

characteristic

212

C₄

#17988935

yes

characteristic

213

C₄

#17988961

yes

characteristic

214

C₂

—

1168

—

215

C₂

#22610736

yes

characteristic

216

C₂

#22611646

yes

—

217

C₂

#22612174

yes

—

218

C₂

#22612198

yes

—

	subgroup	G/H
center	C₂³	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1846	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂×#17988964	C₂
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1846	D₇₃
hypocenter	C₇₃	#13206
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1846

C₂

socle

C₂³×C₇₃

C₂×D₈

prime radical

C₂×#17988964

C₂

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
2. C₂×#17988387	C₂	semidirect
3. C₂×#17988363	C₂	semidirect
4. C₇₃×#1846	C₂	semidirect
10. #17988964	C₄	semidirect
30. C₂×#11924003	D₄	semidirect
113. #1846	D₇₃	semidirect
114. C₇₃	#13206	semidirect

H ≅ G/N

type

2. C₂×#17988387

C₂

semidirect

3. C₂×#17988363

C₂

semidirect

4. C₇₃×#1846

C₂

semidirect

10. #17988964

C₄

semidirect

30. C₂×#11924003

D₄

semidirect

113. #1846

D₇₃

semidirect

114. C₇₃

#13206

semidirect

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

subgroup	index
C₂×#17988964	2
C₂×#17988387	2
C₂×#17988363	2
C₇₃×#1846	2
#13206	73

subgroup

index

C₇₃×#1846

Subgroups — #33004215

Conjugacy classes of subgroups

Complements

Maximal subgroups