Group #33005361 — subgroups

Subgroups — #33005361

Order 18688.

540 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 101 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×#17988400	C₂	1	2	yes	maximal, complemented
2	#22611647	C₂	1	2	yes	maximal, complemented
3	#22610679	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608013	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#493×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	C₂×#14556527	C₂²	1	1	yes	characteristic
8	#17988625	C₂²	1	1	yes	characteristic
9	#17988613	—	2	1	no	—
10	#17988400	—	2	4	no	—
11	#17988388	—	2	4	no	—
12	#17988347	—	2	1	no	—
13	C₂²×D₈×C₇₃	—	2	1	no	—
14	C₂²×D₈×C₇₃	C₂²	1	2	yes	—
15	C₂×#127×C₇₃	—	2	1	no	—
16	#493×C₇₃	—	2	4	no	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×#11924012	—	2	1	no	—
19	#14556550	—	2	1	no	—
20	#14556531	—	2	2	no	—
21	#14556531	—	2	2	no	—
22	C₂×#11924003	—	2	2	no	—
23	#14556527	—	2	2	no	—
24	#14556525	D₄	1	2	yes	complemented
25	#14556523	—	2	1	no	—
26	#14556520	—	4	2	no	—
27	C₂²×D₄×C₇₃	—	2	2	no	—
28	C₂²×D₄×C₇₃	—	4	2	no	—
29	C₂×D₈×C₇₃	—	2	4	no	—
30	C₂×D₈×C₇₃	—	2	2	no	—
31	C₂×D₈×C₇₃	—	2	2	no	—
32	C₂×D₈×C₇₃	—	2	2	no	—
33	C₂×D₈×C₇₃	—	2	2	no	—
34	C₂×D₈×C₇₃	—	2	2	no	—
35	C₂×D₈×C₇₃	—	4	2	no	—
36	C₂×D₈×C₇₃	—	4	2	no	—
37	C₂×D₈×C₇₃	D₄	1	4	yes	—
38	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
40	#127×C₇₃	—	2	4	no	—
41	#127×C₇₃	—	4	2	no	—
42	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₃	—	2	2	no	—
44	#11924012	—	4	1	no	—
45	#11924012	—	4	1	no	—
46	C₂×#10596831	—	4	1	no	—
47	C₄⋊Dic₇₃	—	2	2	no	—
48	C₂²⋊Dic₇₃	—	4	1	no	—
49	C₄×Dic₇₃	—	4	1	no	—
50	#11924003	—	4	2	no	—
51	C₂⁴×C₇₃	—	4	2	no	—
52	C₂×D₄×C₇₃	—	2	4	no	—
53	C₂×D₄×C₇₃	—	4	8	no	—
54	C₂×D₄×C₇₃	—	4	2	no	—
55	C₂×D₄×C₇₃	—	4	2	no	—
56	C₂×D₄×C₇₃	—	4	2	no	—
57	C₂×D₄×C₇₃	—	4	2	no	—
58	C₂×D₄×C₇₃	—	4	2	no	—
59	C₂×D₄×C₇₃	—	8	8	no	—
60	C₂²×C₄×C₇₃	—	2	1	no	—
61	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	D₈×C₇₃	—	2	4	no	—
63	D₈×C₇₃	—	2	4	no	—
64	D₈×C₇₃	—	4	8	no	—
65	D₈×C₇₃	—	4	2	no	—
66	D₈×C₇₃	—	4	2	no	—
67	C₂×C₈×C₇₃	—	2	1	no	—
68	C₂×C₈×C₇₃	—	2	1	no	—
69	C₂×C₈×C₇₃	D₈	1	2	yes	—
70	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₇₃	#41	1	2	yes	—
73	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
74	C₄²×C₇₃	—	2	1	no	—
75	C₄²×C₇₃	SD₁₆	1	2	yes	—
76	C₂³×C₇₃	—	4	4	no	—
77	C₂³×C₇₃	—	8	8	no	—
78	C₂³×C₇₃	—	8	2	no	—
79	C₂³×C₇₃	—	8	2	no	—
80	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	D₄×C₇₃	—	4	4	no	—
82	D₄×C₇₃	—	4	4	no	—
83	D₄×C₇₃	—	8	16	no	—
84	D₄×C₇₃	—	8	2	no	—
85	D₄×C₇₃	—	8	2	no	—
86	C₂×C₄×C₇₃	—	2	4	no	—
87	C₂×C₄×C₇₃	—	2	2	no	—
88	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
89	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×Dic₇₃	—	4	2	no	—
95	C₈×C₇₃	—	2	2	no	—
96	C₈×C₇₃	#121	1	2	yes	—
97	C₈×C₇₃	#123	1	2	yes	—
98	#10596831	—	8	1	no	—
99	C₂²×C₇₃	—	2	2	no	—
100	C₂²×C₇₃	—	8	8	no	—
101	C₂²×C₇₃	—	8	4	no	—
102	C₂²×C₇₃	—	16	4	no	—
103	C₂²×C₇₃	—	16	4	no	—
104	C₂²×C₇₃	#450	1	1	yes	characteristic
105	C₂²×C₇₃	#459	1	1	yes	characteristic
106	C₂²×C₇₃	#490	1	1	yes	characteristic
107	C₄×C₇₃	—	2	4	no	—
108	C₄×C₇₃	#394	1	1	yes	characteristic
109	C₄×C₇₃	#447	1	1	yes	characteristic
110	C₄×C₇₃	#460	1	1	yes	characteristic
111	C₄×C₇₃	#486	1	1	yes	characteristic
112	Dic₇₃	—	8	2	no	—
113	#13413	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	—	2	1	no	—
115	C₂×C₇₃	—	2	1	no	—
116	C₂×C₇₃	—	16	4	no	—
117	C₂×C₇₃	#2041	1	1	yes	characteristic
118	C₂×C₇₃	#2213	1	2	yes	—
119	C₂×#493	D₇₃	1	1	yes	characteristic
120	C₂×#361	—	73	2	no	—
121	#2140	—	73	2	no	—
122	#1945	—	73	1	no	—
123	#1376	—	73	1	no	—
124	C₇₃	#13413	1	1	yes	characteristic, complemented, Sylow
125	C₂²×D₈	—	2	1	no	—
126	C₂²×D₈	C₂×D₇₃	1	2	yes	—
127	C₂×#127	—	2	1	no	—
128	C₂×M₃₂	—	73	2	no	—
129	#493	—	2	4	no	—
130	C₂×#108	—	73	1	no	—
131	#427	—	73	1	no	—
132	#424	—	146	1	no	—
133	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
134	#361	—	146	4	no	—
135	#357	—	146	4	no	—
136	#339	—	146	1	no	—
137	C₂²×D₄	—	2	2	no	—
138	C₂²×D₄	—	4	2	no	—
139	C₂×D₈	—	2	4	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	—	4	2	no	—
146	C₂×D₈	—	4	2	no	—
147	C₂×D₈	C₂²⋊D₇₃	1	4	yes	—
148	C₂×M₁₆	—	146	1	no	—
149	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
151	#127	—	2	4	no	—
152	#127	—	4	2	no	—
153	#117	—	146	1	no	—
154	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
155	M₃₂	—	146	2	no	—
156	M₃₂	—	146	2	no	—
157	C₂×C₁₆	—	146	2	no	—
158	#108	—	146	2	no	—
159	#107	—	73	2	no	—
160	#106	—	146	1	no	—
161	C₄⋊C₈	—	292	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂⁴	—	4	2	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	4	8	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	8	8	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
174	D₈	—	2	4	no	—
175	D₈	—	2	4	no	—
176	D₈	—	4	8	no	—
177	D₈	—	4	2	no	—
178	D₈	—	4	2	no	—
179	M₁₆	—	292	1	no	—
180	M₁₆	—	292	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	292	1	no	—
184	C₂×C₈	#11924013	1	2	yes	—
185	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
187	C₂×C₈	#11924027	1	2	yes	—
188	C₂×C₈	#11924027	1	1	yes	characteristic
189	C₄⋊C₄	—	146	2	no	—
190	C₂²⋊C₄	—	292	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	292	1	no	—
193	C₄²	#11924016	1	2	yes	—
194	C₁₆	—	292	2	no	—
195	C₂³	—	4	4	no	—
196	C₂³	—	8	8	no	—
197	C₂³	—	8	2	no	—
198	C₂³	—	8	2	no	—
199	C₂³	#14556617	1	1	yes	characteristic
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	8	16	no	—
203	D₄	—	8	2	no	—
204	D₄	—	8	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	292	2	no	—
208	C₂×C₄	#14556560	1	1	yes	characteristic
209	C₂×C₄	#14556562	1	1	yes	characteristic
210	C₂×C₄	#14556578	1	1	yes	characteristic
211	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
212	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
213	C₂×C₄	#14556617	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	584	1	no	—
216	C₈	#14556563	1	2	yes	—
217	C₈	#14556572	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	8	no	—
220	C₂²	—	8	4	no	—
221	C₂²	—	16	4	no	—
222	C₂²	—	16	4	no	—
223	C₂²	#17988738	1	1	yes	characteristic
224	C₂²	#17988779	1	1	yes	characteristic
225	C₂²	#17988951	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	584	2	no	—
228	C₄	#17988501	1	1	yes	characteristic
229	C₄	#17988721	1	1	yes	characteristic
230	C₄	#17988784	1	1	yes	characteristic
231	C₄	#17988933	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	16	4	no	—
235	C₂	#22611162	1	1	yes	characteristic
236	C₂	#22612004	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988400

C₂

yes

maximal, complemented

#22611647

C₂

yes

maximal, complemented

#22610679

C₂

yes

maximal, characteristic, complemented

#22608013

C₂

yes

maximal, characteristic, complemented

C₂×#493×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556531

C₂²

yes

—

C₂×#14556527

C₂²

yes

characteristic

#17988625

C₂²

yes

—

—

—

—

C₂²×D₈×C₇₃

—

C₂²×D₈×C₇₃

C₂²

yes

—

C₂×#127×C₇₃

—

#493×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

—

—

—

—

—

D₄

yes

complemented

#14556523

—

#14556520

—

C₂²×D₄×C₇₃

—

C₂²×D₄×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

#127×C₇₃

—

#127×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂⁴×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

AGL(1,ℤ/8)

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

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

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#121

yes

—

C₈×C₇₃

#123

yes

—

#10596831

—

C₂²×C₇₃

—

100

C₂²×C₇₃

—

101

C₂²×C₇₃

—

102

C₂²×C₇₃

—

103

C₂²×C₇₃

—

104

C₂²×C₇₃

#450

yes

characteristic

105

C₂²×C₇₃

#459

yes

characteristic

106

C₂²×C₇₃

#490

yes

characteristic

107

C₄×C₇₃

—

108

C₄×C₇₃

#394

yes

characteristic

109

C₄×C₇₃

#447

yes

characteristic

110

C₄×C₇₃

#460

yes

characteristic

111

C₄×C₇₃

#486

yes

characteristic

112

Dic₇₃

—

113

#13413

—

maximal, Sylow

114

C₂×C₇₃

—

115

C₂×C₇₃

—

116

C₂×C₇₃

—

117

C₂×C₇₃

#2041

yes

characteristic

118

C₂×C₇₃

#2213

yes

—

119

C₂×#493

D₇₃

yes

characteristic

120

C₂×#361

—

121

#2140

—

122

#1945

—

123

#1376

—

124

C₇₃

#13413

yes

characteristic, complemented, Sylow

125

C₂²×D₈

—

126

C₂²×D₈

C₂×D₇₃

yes

—

127

C₂×#127

—

128

C₂×M₃₂

—

129

#493

—

130

C₂×#108

—

131

#427

—

132

#424

—

146

—

133

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

134

#361

—

146

—

135

#357

—

146

—

136

#339

—

146

—

137

C₂²×D₄

—

138

C₂²×D₄

—

139

C₂×D₈

—

140

C₂×D₈

—

141

C₂×D₈

—

142

C₂×D₈

—

143

C₂×D₈

—

144

C₂×D₈

—

145

C₂×D₈

—

146

C₂×D₈

—

147

C₂×D₈

C₂²⋊D₇₃

yes

—

148

C₂×M₁₆

—

146

—

149

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

150

C₂²×C₈

C₂²×D₇₃

yes

characteristic

151

#127

—

152

#127

—

153

#117

—

146

—

154

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

155

M₃₂

—

146

—

156

M₃₂

—

146

—

157

C₂×C₁₆

—

146

—

158

#108

—

146

—

159

#107

—

160

#106

—

146

—

161

C₄⋊C₈

—

292

—

162

C₄×C₈

—

163

C₂⁴

—

164

C₂×D₄

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂×D₄

—

170

C₂×D₄

—

171

C₂×D₄

—

172

C₂²×C₄

—

173

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

174

D₈

—

175

D₈

—

176

D₈

—

177

D₈

—

178

D₈

—

179

M₁₆

—

292

—

180

M₁₆

—

292

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

292

—

184

C₂×C₈

#11924013

yes

—

185

C₂×C₈

D₄×D₇₃

yes

characteristic

186

C₂×C₈

C₂×C₂²⋊D₇₃

yes

—

187

C₂×C₈

#11924027

yes

—

188

C₂×C₈

#11924027

yes

characteristic

189

C₄⋊C₄

—

146

—

190

C₂²⋊C₄

—

292

—

191

C₄²

—

192

C₄²

—

292

—

193

C₄²

#11924016

yes

—

194

C₁₆

—

292

—

195

C₂³

—

196

C₂³

—

197

C₂³

—

198

C₂³

—

199

C₂³

#14556617

yes

characteristic

200

D₄

—

201

D₄

—

202

D₄

—

203

D₄

—

204

D₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

292

—

208

C₂×C₄

#14556560

yes

characteristic

209

C₂×C₄

#14556562

yes

characteristic

210

C₂×C₄

#14556578

yes

characteristic

211

C₂×C₄

C₂×#11924013

yes

characteristic

212

C₂×C₄

C₂×#11924016

yes

characteristic

213

C₂×C₄

#14556617

yes

characteristic

214

C₈

—

215

C₈

—

584

—

216

C₈

#14556563

yes

—

217

C₈

#14556572

yes

—

218

C₂²

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

#17988738

yes

characteristic

224

C₂²

#17988779

yes

characteristic

225

C₂²

#17988951

yes

characteristic

226

C₄

—

227

C₄

—

584

—

228

C₄

#17988501

yes

characteristic

229

C₄

#17988721

yes

characteristic

230

C₄

#17988784

yes

characteristic

231

C₄

#17988933

yes

characteristic

232

C₂

—

233

C₂

—

234

C₂

—

235

C₂

#22611162

yes

characteristic

236

C₂

#22612004

yes

—

	subgroup	G/H
center	C₂²	#17988779
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting, prime radical	C₂×#493×C₇₃	C₂
socle	C₂²×C₇₃	#459
hypercenter	C₂×#493	D₇₃
hypocenter	C₇₃	#13413
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#493

subgroup

G/H

center

C₂²

#17988779

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting, prime radical

C₂×#493×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#493

N	H ≅ G/N	type
1. C₂×#17988400	C₂	semidirect
2. #22611647	C₂	semidirect
3. #22610679	C₂	semidirect
4. #22608013	C₂	semidirect
24. #14556525	D₄	semidirect
124. C₇₃	#13413	semidirect

H ≅ G/N

type

1. C₂×#17988400

C₂

semidirect

2. #22611647

C₂

semidirect

3. #22610679

C₂

semidirect

4. #22608013

C₂

semidirect

24. #14556525

D₄

semidirect

124. C₇₃

#13413

semidirect

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

subgroup	index
C₂×#17988400	2
#22611647	2
#22610679	2
#22608013	2
C₂×#493×C₇₃	2
#13413	73

subgroup

index

C₂×#493×C₇₃

Subgroups — #33005361

Conjugacy classes of subgroups

Complements

Maximal subgroups