Group #33073691 — subgroups

Subgroups — #33073691

Order 18688.

706 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 187 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	#22620516	C₂	1	4	yes	maximal
2	#22620442	C₂	1	2	yes	maximal
3	#22609548	C₂	1	2	yes	maximal, complemented
4	#22609541	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609527	C₂	1	4	yes	maximal, complemented
6	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₇₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
8	#17989142	C₂²	1	4	yes	—
9	#17988992	C₂²	1	2	yes	—
10	#17988990	C₂²	1	4	yes	—
11	#17988988	—	2	4	no	—
12	#17988983	C₂²	1	4	yes	—
13	#17988966	C₂²	1	4	yes	—
14	#17988938	C₂²	1	4	yes	—
15	#17988827	—	2	8	no	—
16	#17988679	—	2	8	no	—
17	#17988624	—	2	4	no	—
18	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₇₃	C₂²	1	2	yes	—
20	C₈×#10596831	C₂²	1	2	yes	—
21	C₈×#10596831	C₂²	1	1	yes	characteristic
22	#500×C₇₃	C₂²	1	2	yes	—
23	#487×C₇₃	C₂²	1	4	yes	—
24	C₈²×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×#11924018	—	2	8	no	—
26	C₂×C₈×D₇₃	D₄	1	4	yes	—
27	C₂×C₈×D₇₃	D₄	1	2	yes	—
28	#14556591	C₂³	1	4	yes	—
29	#14556582	—	2	4	no	—
30	#14556571	—	2	4	no	—
31	#14556567	—	2	4	no	—
32	Q₈×Dic₇₃	—	2	4	no	—
33	#14556556	—	2	4	no	—
34	#14556549	—	2	4	no	—
35	C₄²×D₇₃	C₂³	1	1	yes	characteristic
36	#14556525	—	2	4	no	—
37	#14556524	—	2	8	no	—
38	#14556522	—	2	8	no	—
39	#14556516	—	2	8	no	—
40	C₄×#10596831	C₂³	1	2	yes	—
41	C₄×#10596831	C₂³	1	1	yes	characteristic
42	C₈×Dic₇₃	D₄	1	4	yes	—
43	C₈×Dic₇₃	D₄	1	2	yes	—
44	C₄⋊Q₈×C₇₃	C₂³	1	4	yes	—
45	#107×C₇₃	—	2	4	no	—
46	#103×C₇₃	—	2	8	no	—
47	C₄×C₈×C₇₃	C₂³	1	2	yes	—
48	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₄₆	—	2	4	no	—
50	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
52	#11924018	—	4	16	no	—
53	C₂×#10596831	C₂×D₄	1	2	yes	—
54	C₂×#10596831	C₂×D₄	1	4	yes	—
55	C₈×D₇₃	—	2	8	no	—
56	C₈×D₇₃	Q₁₆	1	8	yes	complemented
57	C₄⋊Dic₇₃	—	2	4	no	—
58	C₄⋊Dic₇₃	—	2	4	no	—
59	C₄⋊Dic₇₃	—	4	4	no	—
60	#11924007	—	4	4	no	—
61	#11924005	—	4	4	no	—
62	C₄×Dic₇₃	—	4	4	no	—
63	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
64	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₇₃	—	2	4	no	—
66	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
67	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
68	C₄⋊C₄×C₇₃	—	2	4	no	—
69	C₄⋊C₄×C₇₃	—	2	4	no	—
70	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
71	C₂²×D₇₃	#127	1	1	yes	characteristic
72	Q₈×C₇₃	—	4	8	no	—
73	C₂×C₄×C₇₃	—	4	4	no	—
74	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
76	C₂×Dic₇₃	—	4	4	no	—
77	C₂×Dic₇₃	#127	1	2	yes	—
78	C₂×Dic₇₃	#127	1	1	yes	characteristic
79	C₄×D₇₃	—	2	4	no	—
80	C₄×D₇₃	C₂×Q₁₆	1	4	yes	—
81	Dic₁₄₆	—	4	8	no	—
82	C₈×C₇₃	—	2	4	no	—
83	C₈×C₇₃	C₂×Q₁₆	1	4	yes	—
84	#10596831	C₂×Q₁₆	1	4	yes	—
85	#10596831	#135	1	8	yes	—
86	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
87	C₂×D₇₃	—	2	2	no	—
88	C₂×D₇₃	#494	1	2	yes	—
89	C₄×C₇₃	—	8	4	no	—
90	C₄×C₇₃	C₂²×Q₁₆	1	2	yes	—
91	C₄×C₇₃	C₂×#135	1	4	yes	—
92	Dic₇₃	—	2	2	no	—
93	Dic₇₃	—	8	4	no	—
94	Dic₇₃	#494	1	2	yes	—
95	#19604	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	C₂×#494	1	1	yes	characteristic
97	C₂×C₇₃	#3147	1	2	yes	—
98	D₇₃	—	2	2	no	—
99	#3170	—	73	4	no	—
100	#3160	—	73	2	no	—
101	#1713	—	73	2	no	—
102	#1710	—	73	1	no	—
103	#1710	D₇₃	1	1	yes	characteristic, complemented
104	#1707	—	73	4	no	—
105	C₂×C₈²	—	73	1	no	—
106	C₇₃	#19604	1	1	yes	characteristic, complemented, Sylow
107	#533	—	73	4	no	—
108	#500	—	73	4	no	—
109	#500	—	73	2	no	—
110	#500	C₂×D₇₃	1	2	yes	—
111	#499	—	146	4	no	—
112	#498	—	73	4	no	—
113	#494	—	73	4	no	—
114	#487	—	73	4	no	—
115	#487	C₂×D₇₃	1	4	yes	—
116	#467	—	146	8	no	—
117	C₄×Q₁₆	—	146	8	no	—
118	#427	—	146	4	no	—
119	C₂×C₄×C₈	—	73	1	no	—
120	C₂×C₄×C₈	—	73	2	no	—
121	C₈²	—	73	2	no	—
122	C₈²	—	73	1	no	—
123	C₈²	C₂×D₇₃	1	1	yes	characteristic
124	C₂×Q₁₆	—	146	8	no	—
125	C₂²×C₈	—	73	2	no	—
126	C₂²×C₈	—	73	4	no	—
127	C₄⋊Q₈	—	73	4	no	—
128	C₄⋊Q₈	C₂²×D₇₃	1	4	yes	—
129	#125	—	146	4	no	—
130	C₂²⋊Q₈	—	146	4	no	—
131	C₂²⋊Q₈	—	146	4	no	—
132	C₄×Q₈	—	146	4	no	—
133	C₄×Q₈	—	146	4	no	—
134	#117	—	146	4	no	—
135	C₂×C₄²	—	73	1	no	—
136	#107	—	2	4	no	—
137	#107	—	146	8	no	—
138	#107	—	146	4	no	—
139	#106	—	146	8	no	—
140	#103	—	2	8	no	—
141	#103	—	146	8	no	—
142	C₄×C₈	—	73	4	no	—
143	C₄×C₈	—	73	2	no	—
144	C₄×C₈	—	73	2	no	—
145	C₄×C₈	—	73	1	no	—
146	C₄×C₈	C₂²×D₇₃	1	2	yes	—
147	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	146	4	no	—
150	C₂²×C₄	—	73	1	no	—
151	C₂²×C₄	—	73	2	no	—
152	Q₁₆	—	292	16	no	—
153	C₂×C₈	—	73	2	no	—
154	C₂×C₈	—	73	4	no	—
155	C₂×C₈	—	73	8	no	—
156	C₂×C₈	—	146	8	no	—
157	C₂×C₈	D₄×D₇₃	1	4	yes	—
158	C₂×C₈	D₄×D₇₃	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	292	4	no	—
162	C₄⋊C₄	—	292	4	no	—
163	C₄⋊C₄	—	146	4	no	—
164	C₄⋊C₄	—	146	4	no	—
165	C₂²⋊C₄	—	292	4	no	—
166	C₄²	—	292	4	no	—
167	C₄²	—	73	2	no	—
168	C₄²	—	73	1	no	—
169	C₄²	C₂³×D₇₃	1	1	yes	characteristic
170	C₂³	—	73	1	no	—
171	Q₈	—	4	8	no	—
172	Q₈	—	292	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	292	4	no	—
175	C₂×C₄	—	73	1	no	—
176	C₂×C₄	—	73	4	no	—
177	C₂×C₄	—	73	2	no	—
178	C₂×C₄	—	146	4	no	—
179	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	73	4	no	—
183	C₈	—	73	8	no	—
184	C₈	Q₁₆×D₇₃	1	4	yes	—
185	C₂²	—	73	2	no	—
186	C₂²	—	146	2	no	—
187	C₂²	#127×D₇₃	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	584	4	no	—
190	C₄	—	73	2	no	—
191	C₄	—	146	2	no	—
192	C₄	C₂×Q₁₆×D₇₃	1	2	yes	—
193	C₄	#135×D₇₃	1	4	yes	—
194	C₂	—	146	2	no	—
195	C₂	#494×D₇₃	1	1	yes	characteristic
196	C₂	#22620356	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22620516

C₂

yes

maximal

#22620442

C₂

yes

maximal

#22609548

C₂

yes

maximal, complemented

#22609541

C₂

yes

maximal, characteristic, complemented

#22609527

C₂

yes

maximal, complemented

C₈²×D₇₃

C₂

yes

maximal, characteristic

C₇₃×#1710

C₂

yes

maximal, characteristic, complemented

#17989142

C₂²

yes

—

#17988992

C₂²

yes

—

#17988990

C₂²

yes

—

#17988988

—

#17988983

C₂²

yes

—

#17988966

C₂²

yes

—

#17988938

C₂²

yes

—

#17988827

—

#17988679

—

#17988624

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

—

C₈×#10596831

C₂²

yes

—

C₈×#10596831

C₂²

yes

characteristic

#500×C₇₃

C₂²

yes

—

#487×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×#11924018

—

C₂×C₈×D₇₃

D₄

yes

—

C₂×C₈×D₇₃

D₄

yes

—

#14556591

C₂³

yes

—

#14556582

—

#14556571

—

#14556567

—

Q₈×Dic₇₃

—

#14556556

—

#14556549

—

C₄²×D₇₃

C₂³

yes

—

—

—

—

C₂³

yes

—

C₄×#10596831

C₂³

yes

characteristic

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

C₂³

yes

—

#107×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×C₄×D₇₃

C₂×D₄

yes

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

#11924018

—

C₂×#10596831

C₂×D₄

yes

—

C₂×#10596831

C₂×D₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

Q₁₆

yes

complemented

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×D₄

yes

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂⁴

yes

characteristic

C₂²×D₇₃

#127

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×D₄

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#127

yes

—

C₂×Dic₇₃

#127

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

C₂×Q₁₆

yes

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×Q₁₆

yes

—

#10596831

C₂×Q₁₆

yes

—

#10596831

#135

yes

—

C₂²×C₇₃

C₂×#127

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#494

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂²×Q₁₆

yes

—

C₄×C₇₃

C₂×#135

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#494

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#3170

—

100

#3160

—

101

#1713

—

102

#1710

—

103

#1710

D₇₃

yes

characteristic, complemented

104

#1707

—

105

C₂×C₈²

—

106

C₇₃

#19604

yes

characteristic, complemented, Sylow

107

#533

—

108

#500

—

109

#500

—

110

#500

C₂×D₇₃

yes

—

111

#499

—

146

—

112

#498

—

113

#494

—

114

#487

—

115

#487

C₂×D₇₃

yes

—

116

#467

—

146

—

117

C₄×Q₁₆

—

146

—

118

#427

—

146

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₈²

—

122

C₈²

—

123

C₈²

C₂×D₇₃

yes

characteristic

124

C₂×Q₁₆

—

146

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂²×D₇₃

yes

—

129

#125

—

146

—

130

C₂²⋊Q₈

—

146

—

131

C₂²⋊Q₈

—

146

—

132

C₄×Q₈

—

146

—

133

C₄×Q₈

—

146

—

134

#117

—

146

—

135

C₂×C₄²

—

136

#107

—

137

#107

—

146

—

138

#107

—

146

—

139

#106

—

146

—

140

#103

—

141

#103

—

146

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₂²×D₇₃

yes

—

147

C₄×C₈

C₂²×D₇₃

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

146

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

Q₁₆

—

292

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

146

—

157

C₂×C₈

D₄×D₇₃

yes

—

158

C₂×C₈

D₄×D₇₃

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

292

—

162

C₄⋊C₄

—

292

—

163

C₄⋊C₄

—

146

—

164

C₄⋊C₄

—

146

—

165

C₂²⋊C₄

—

292

—

166

C₄²

—

292

—

167

C₄²

—

168

C₄²

—

169

C₄²

C₂³×D₇₃

yes

characteristic

170

C₂³

—

171

Q₈

—

172

Q₈

—

292

—

173

C₂×C₄

—

174

C₂×C₄

—

292

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

146

—

179

C₂×C₄

C₂×D₄×D₇₃

yes

characteristic

180

C₂×C₄

C₂×D₄×D₇₃

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

Q₁₆×D₇₃

yes

—

185

C₂²

—

186

C₂²

—

146

—

187

C₂²

#127×D₇₃

yes

characteristic

188

C₄

—

189

C₄

—

584

—

190

C₄

—

191

C₄

—

146

—

192

C₄

C₂×Q₁₆×D₇₃

yes

—

193

C₄

#135×D₇₃

yes

—

194

C₂

—

146

—

195

C₂

#494×D₇₃

yes

characteristic

196

C₂

#22620356

yes

—

	subgroup	G/H
center	C₂²	#127×D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1710	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	C₂²×D₇₃	#127
hypercenter	#1710	D₇₃
hypocenter	C₇₃	#19604
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1710

subgroup

G/H

center

C₂²

#127×D₇₃

commutator, cosocle

C₄²×C₇₃

C₂⁴

Frattini

C₄²

C₂³×D₇₃

Fitting

C₇₃×#1710

C₂

C₂²×C₇₃

C₂²×D₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #1710

N	H ≅ G/N	type
3. #22609548	C₂	semidirect
4. #22609541	C₂	semidirect
5. #22609527	C₂	semidirect
7. C₇₃×#1710	C₂	semidirect
56. C₈×D₇₃	Q₁₆	semidirect
103. #1710	D₇₃	semidirect
106. C₇₃	#19604	semidirect

H ≅ G/N

type

3. #22609548

C₂

semidirect

4. #22609541

C₂

semidirect

5. #22609527

C₂

semidirect

7. C₇₃×#1710

C₂

semidirect

56. C₈×D₇₃

Q₁₆

semidirect

103. #1710

D₇₃

semidirect

106. C₇₃

#19604

semidirect

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

subgroup	index
#22620516	2
#22620442	2
#22609548	2
#22609541	2
#22609527	2
C₈²×D₇₃	2
C₇₃×#1710	2
#19604	73

subgroup

index

C₈²×D₇₃

C₇₃×#1710

#19604

Subgroups — #33073691

Conjugacy classes of subgroups

Complements

Maximal subgroups