Group #33002459 — subgroups

Subgroups — #33002459

Order 18688.

374 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 135 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	#22611691	C₂	1	2	yes	maximal, complemented
2	#22611545	C₂	1	1	yes	maximal, characteristic
3	#22609731	C₂	1	1	yes	maximal, characteristic
4	#22608016	C₂	1	2	yes	maximal, complemented
5	C₇₃×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	1	yes	characteristic
7	C₂²×#11924003	C₂²	1	1	yes	characteristic
8	#17988571	C₄	1	1	yes	characteristic
9	#17988561	C₄	1	1	yes	characteristic
10	#17988542	C₂²	1	2	yes	—
11	#17988405	—	2	4	no	—
12	C₂²⋊#11924003	—	2	2	no	—
13	#17988358	C₄	1	2	yes	—
14	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
15	#504×C₇₃	—	2	4	no	—
16	C₂×M₃₂×C₇₃	—	2	2	no	—
17	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
18	C₂×#11924012	C₂×C₄	1	2	yes	—
19	#14556550	C₂×C₄	1	2	yes	—
20	#14556531	—	2	2	no	—
21	C₂×#11924003	—	2	1	no	—
22	C₂×#11924003	—	2	1	no	—
23	C₂×#11924003	—	2	2	no	—
24	#14556510	—	2	1	no	—
25	#14556508	—	2	1	no	—
26	#14556506	C₈	1	4	yes	—
27	#14556501	—	2	2	no	—
28	C₈×Dic₇₃	—	2	2	no	—
29	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
30	#131×C₇₃	—	2	2	no	—
31	#131×C₇₃	—	2	2	no	—
32	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₇₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₇₃	D₄	1	2	yes	—
35	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
36	M₃₂×C₇₃	—	2	4	no	—
37	M₃₂×C₇₃	—	2	2	no	—
38	M₃₂×C₇₃	—	2	2	no	—
39	C₂×C₁₆×C₇₃	—	2	4	no	—
40	C₂×C₁₆×C₇₃	D₄	1	4	yes	—
41	#11924012	—	2	2	no	—
42	#11924012	—	2	2	no	—
43	C₂×#10596831	—	2	2	no	—
44	C₄⋊Dic₇₃	—	2	2	no	—
45	C₂²⋊Dic₇₃	C₂×C₈	1	4	yes	—
46	C₄×Dic₇₃	—	2	2	no	—
47	#11924003	—	4	1	no	—
48	#11924003	—	4	1	no	—
49	#41×C₇₃	—	2	2	no	—
50	#41×C₇₃	—	2	2	no	—
51	C₂×Q₈×C₇₃	M₁₆	1	1	yes	characteristic
52	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
53	C₂×D₄×C₇₃	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₇₃	—	2	2	no	—
57	M₁₆×C₇₃	—	2	4	no	—
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₂×C₈×C₇₃	—	2	2	no	—
60	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₇₃	#41	1	2	yes	—
65	C₁₆×C₇₃	—	2	4	no	—
66	C₂³×C₇₃	C₄⋊C₈	1	2	yes	—
67	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₇₃	—	2	2	no	—
69	D₄×C₇₃	—	2	4	no	—
70	D₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#131	1	2	yes	—
78	C₂×Dic₇₃	—	2	4	no	—
79	C₈×C₇₃	—	2	2	no	—
80	C₈×C₇₃	#120	1	1	yes	characteristic
81	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
82	C₈×C₇₃	#123	1	1	yes	characteristic
83	#10596831	—	4	2	no	—
84	C₂²×C₇₃	—	2	2	no	—
85	C₂²×C₇₃	—	4	1	no	—
86	C₂²×C₇₃	—	4	2	no	—
87	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₇₃	C₈×D₄	1	2	yes	—
89	C₄×C₇₃	—	2	2	no	—
90	C₄×C₇₃	#385	1	1	yes	characteristic
91	C₄×C₇₃	#424	1	1	yes	characteristic
92	C₄×C₇₃	#435	1	2	yes	—
93	Dic₇₃	—	4	4	no	—
94	#12915	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	—	2	2	no	—
96	C₂×C₇₃	—	2	1	no	—
97	C₂×C₇₃	—	4	2	no	—
98	C₂×C₇₃	#1850	1	1	yes	characteristic
99	#3407	D₇₃	1	1	yes	characteristic
100	#2151	—	73	2	no	—
101	#2113	—	73	1	no	—
102	#1757	—	73	1	no	—
103	#1378	—	73	2	no	—
104	C₇₃	#12915	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	C₂×D₇₃	1	1	yes	characteristic
106	#504	—	2	4	no	—
107	C₂×M₃₂	—	2	2	no	—
108	C₂×M₃₂	—	73	1	no	—
109	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
110	C₂²×C₁₆	—	73	1	no	—
111	#413	—	73	1	no	—
112	#410	—	73	1	no	—
113	#405	—	73	2	no	—
114	C₄⋊C₁₆	—	146	4	no	—
115	C₂²⋊C₁₆	—	146	2	no	—
116	#344	—	73	2	no	—
117	C₂×#41	C₄×D₇₃	1	1	yes	characteristic
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	C₂×M₁₆	—	73	2	no	—
121	C₂×M₁₆	Dic₁₄₆	1	2	yes	—
122	C₂×M₁₆	C₄×D₇₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
124	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
125	#117	—	73	2	no	—
126	M₃₂	—	2	4	no	—
127	M₃₂	—	2	2	no	—
128	M₃₂	—	2	2	no	—
129	M₃₂	—	146	2	no	—
130	C₂×C₁₆	—	2	4	no	—
131	C₂×C₁₆	—	146	1	no	—
132	C₂×C₁₆	—	146	1	no	—
133	C₂×C₁₆	—	146	2	no	—
134	C₂×C₁₆	C₂²⋊D₇₃	1	4	yes	—
135	#101	—	146	1	no	—
136	#100	—	146	1	no	—
137	#99	—	73	4	no	—
138	#97	—	146	2	no	—
139	C₄×C₈	—	146	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	#11924011	1	1	yes	characteristic
143	C₂×D₄	C₈×D₇₃	1	2	yes	—
144	C₂×D₄	#11924011	1	1	yes	characteristic
145	C₂²×C₄	#11924007	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	146	2	no	—
150	M₁₆	—	146	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	146	2	no	—
154	C₂×C₈	#11924007	1	2	yes	—
155	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
156	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
157	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
158	C₂×C₈	#11924028	1	2	yes	—
159	C₄⋊C₄	—	146	2	no	—
160	C₂²⋊C₄	—	73	4	no	—
161	C₄²	—	146	2	no	—
162	C₁₆	—	2	4	no	—
163	C₁₆	—	292	1	no	—
164	C₁₆	—	292	1	no	—
165	C₂³	#14556520	1	2	yes	—
166	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	146	4	no	—
173	C₂×C₄	#14556520	1	2	yes	—
174	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
175	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
176	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
177	C₂×C₄	#14556597	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	292	2	no	—
180	C₈	#14556560	1	1	yes	characteristic
181	C₈	#14556570	1	2	yes	—
182	C₈	#14556574	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×#14556520	1	1	yes	characteristic
187	C₂²	C₈×C₂²⋊D₇₃	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	292	4	no	—
190	C₄	#17988460	1	1	yes	characteristic
191	C₄	#17988613	1	1	yes	characteristic
192	C₄	#17988658	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#22610109	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611691

C₂

yes

maximal, complemented

#22611545

C₂

yes

maximal, characteristic

#22609731

C₂

yes

maximal, characteristic

#22608016

C₂

yes

maximal, complemented

C₇₃×#3407

C₂

yes

maximal, characteristic

C₂×#14556531

C₂²

yes

characteristic

C₂²×#11924003

C₂²

yes

characteristic

#17988571

C₄

yes

characteristic

#17988561

C₄

yes

characteristic

#17988542

C₂²

yes

—

#17988405

—

C₂²⋊#11924003

—

#17988358

C₄

yes

—

C₂×#131×C₇₃

C₂²

yes

characteristic

#504×C₇₃

—

C₂×M₃₂×C₇₃

—

C₂×M₃₂×C₇₃

C₂²

yes

—

C₂×#11924012

C₂×C₄

yes

—

#14556550

C₂×C₄

yes

—

—

—

—

—

—

—

C₈

yes

—

#14556501

—

C₈×Dic₇₃

—

C₂×#41×C₇₃

C₂×C₄

yes

characteristic

#131×C₇₃

—

#131×C₇₃

—

C₂×M₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₃

Q₈

yes

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

M₃₂×C₇₃

—

M₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

—

#11924003

—

#11924003

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

M₁₆

yes

characteristic

C₂×D₄×C₇₃

C₂×C₈

yes

—

C₂×D₄×C₇₃

M₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₄⋊C₈

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

#131

yes

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#120

yes

characteristic

C₈×C₇₃

C₂²⋊Q₈

yes

—

C₈×C₇₃

#123

yes

characteristic

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₃

C₈×D₄

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12915

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1850

yes

characteristic

#3407

D₇₃

yes

characteristic

100

#2151

—

101

#2113

—

102

#1757

—

103

#1378

—

104

C₇₃

#12915

yes

characteristic, complemented, Sylow

105

C₂×#131

C₂×D₇₃

yes

characteristic

106

#504

—

107

C₂×M₃₂

—

108

C₂×M₃₂

—

109

C₂×M₃₂

C₂×D₇₃

yes

—

110

C₂²×C₁₆

—

111

#413

—

112

#410

—

113

#405

—

114

C₄⋊C₁₆

—

146

—

115

C₂²⋊C₁₆

—

146

—

116

#344

—

117

C₂×#41

C₄×D₇₃

yes

characteristic

118

#131

—

119

#131

—

120

C₂×M₁₆

—

121

C₂×M₁₆

Dic₁₄₆

yes

—

122

C₂×M₁₆

C₄×D₇₃

yes

characteristic

123

C₂²×C₈

C₂²⋊D₇₃

yes

—

124

C₂²×C₈

C₂²×D₇₃

yes

characteristic

125

#117

—

126

M₃₂

—

127

M₃₂

—

128

M₃₂

—

129

M₃₂

—

146

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

146

—

132

C₂×C₁₆

—

146

—

133

C₂×C₁₆

—

146

—

134

C₂×C₁₆

C₂²⋊D₇₃

yes

—

135

#101

—

146

—

136

#100

—

146

—

137

#99

—

138

#97

—

146

—

139

C₄×C₈

—

146

—

140

#41

—

141

#41

—

142

C₂×Q₈

#11924011

yes

characteristic

143

C₂×D₄

C₈×D₇₃

yes

—

144

C₂×D₄

#11924011

yes

characteristic

145

C₂²×C₄

#11924007

yes

—

146

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

146

—

150

M₁₆

—

146

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

146

—

154

C₂×C₈

#11924007

yes

—

155

C₂×C₈

C₂×C₂²⋊D₇₃

yes

—

156

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

157

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

158

C₂×C₈

#11924028

yes

—

159

C₄⋊C₄

—

146

—

160

C₂²⋊C₄

—

161

C₄²

—

146

—

162

C₁₆

—

163

C₁₆

—

292

—

164

C₁₆

—

292

—

165

C₂³

#14556520

yes

—

166

C₂³

C₂×C₈×D₇₃

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

146

—

173

C₂×C₄

#14556520

yes

—

174

C₂×C₄

C₂×#11924007

yes

characteristic

175

C₂×C₄

C₄×C₂²⋊D₇₃

yes

—

176

C₂×C₄

C₂×#11924011

yes

characteristic

177

C₂×C₄

#14556597

yes

—

178

C₈

—

179

C₈

—

292

—

180

C₈

#14556560

yes

characteristic

181

C₈

#14556570

yes

—

182

C₈

#14556574

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×#14556520

yes

characteristic

187

C₂²

C₈×C₂²⋊D₇₃

yes

—

188

C₄

—

189

C₄

—

292

—

190

C₄

#17988460

yes

characteristic

191

C₄

#17988613

yes

characteristic

192

C₄

#17988658

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#22610109

yes

characteristic

	subgroup	G/H
center	C₈	#14556560
commutator	C₂³×C₇₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#3407	C₂
socle	C₂×C₇₃	#1850
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#3407	D₇₃
hypocenter	C₇₃	#12915
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

subgroup

G/H

center

C₈

#14556560

commutator

C₂³×C₇₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#3407

C₂

socle

C₂×C₇₃

#1850

prime radical

C₂×#41×C₇₃

C₂×C₄

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

N	H ≅ G/N	type
1. #22611691	C₂	semidirect
4. #22608016	C₂	semidirect
104. C₇₃	#12915	semidirect

H ≅ G/N

type

1. #22611691

C₂

semidirect

4. #22608016

C₂

semidirect

104. C₇₃

#12915

semidirect

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

subgroup	index
#22611691	2
#22611545	2
#22609731	2
#22608016	2
C₇₃×#3407	2
#12915	73

subgroup

index

C₇₃×#3407

Subgroups — #33002459

Conjugacy classes of subgroups

Complements

Maximal subgroups