Group #32978933 — subgroups

Subgroups — #32978933

Order 18688.

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

normal subgroups only

This group has 362 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×#17988711	C₂	1	1	yes	maximal, characteristic
2	#22608692	C₂	1	2	yes	maximal
3	#22608274	C₂	1	2	yes	maximal
4	#22607801	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1290	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
7	#17988711	—	2	2	no	—
8	#17988711	—	2	2	no	—
9	#17988644	C₂²	1	2	yes	—
10	C₂×#14556520	C₄	1	2	yes	—
11	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
13	#17988339	—	2	2	no	—
14	#17988337	—	2	2	no	—
15	#17988312	—	2	4	no	—
16	#432×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₄₆	Q₈	1	1	yes	characteristic
19	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₄₆	—	2	2	no	—
22	C₄×Dic₁₄₆	—	2	2	no	—
23	C₂×C₄⋊Dic₇₃	Q₈	1	1	yes	characteristic
24	C₂×#11924007	D₄	1	2	yes	—
25	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
26	C₄⋊#10596831	—	2	2	no	—
27	#14556520	—	2	4	no	—
28	C₂²⋊#10596831	—	2	2	no	—
29	#14556502	—	2	2	no	—
30	C₄×#10596831	D₄	1	2	yes	—
31	C₄×#10596831	D₄	1	2	yes	—
32	C₈×Dic₇₃	—	2	2	no	—
33	C₈×Dic₇₃	—	2	2	no	—
34	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₇₃	—	2	2	no	—
37	C₄⋊C₈×C₇₃	—	2	2	no	—
38	C₂²⋊C₈×C₇₃	—	2	2	no	—
39	#97×C₇₃	—	2	2	no	—
40	C₂×Dic₁₄₆	—	2	2	no	—
41	C₂×Dic₁₄₆	—	2	4	no	—
42	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
43	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
44	C₂×#10596831	—	2	2	no	—
45	C₂×#10596831	—	4	1	no	—
46	C₂×#10596831	—	4	2	no	—
47	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
49	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
50	C₄⋊Dic₇₃	—	2	2	no	—
51	#11924007	—	2	4	no	—
52	C₄×Dic₇₃	—	2	2	no	—
53	C₄×Dic₇₃	—	2	2	no	—
54	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	—	2	2	no	—
58	C₂×C₈×C₇₃	—	4	2	no	—
59	C₂×C₈×C₇₃	—	4	2	no	—
60	C₄²×C₇₃	C₂×D₄	1	2	yes	—
61	C₄²×C₇₃	#41	1	2	yes	—
62	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	1	no	—
65	C₂×C₄×C₇₃	—	2	1	no	—
66	C₂×C₄×C₇₃	—	4	1	no	—
67	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#121	1	2	yes	—
70	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×Dic₇₃	—	2	2	no	—
72	C₂×Dic₇₃	—	2	2	no	—
73	C₂×Dic₇₃	—	2	2	no	—
74	C₂×Dic₇₃	—	4	2	no	—
75	C₂×Dic₇₃	#108	1	4	yes	—
76	Dic₁₄₆	—	2	8	no	—
77	C₈×C₇₃	—	4	2	no	—
78	C₈×C₇₃	—	8	2	no	—
79	#10596831	—	2	1	no	—
80	#10596831	—	2	2	no	—
81	#10596831	—	8	2	no	—
82	#10596831	#101	1	2	yes	—
83	C₂²×C₇₃	—	2	1	no	—
84	C₂²×C₇₃	—	2	1	no	—
85	C₂²×C₇₃	#380	1	1	yes	characteristic
86	C₂²×C₇₃	C₂×#108	1	2	yes	—
87	C₄×C₇₃	—	2	1	no	—
88	C₄×C₇₃	—	2	2	no	—
89	C₄×C₇₃	—	2	2	no	—
90	C₄×C₇₃	C₂×#101	1	1	yes	characteristic
91	C₄×C₇₃	#413	1	1	yes	characteristic
92	Dic₇₃	—	2	4	no	—
93	Dic₇₃	—	4	2	no	—
94	#8621	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	—	2	2	no	—
96	C₂×C₇₃	#1801	1	1	yes	characteristic
97	C₂×C₇₃	#1810	1	1	yes	characteristic
98	C₂×C₇₃	#1812	1	1	yes	characteristic
99	C₂×C₈×Q₈	—	73	1	no	—
100	#1568	—	73	2	no	—
101	#1467	—	73	2	no	—
102	#1290	—	73	1	no	—
103	#1290	D₇₃	1	1	yes	characteristic
104	C₇₃	#8621	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄×Q₈	—	73	1	no	—
106	C₈×Q₈	—	146	2	no	—
107	C₈×Q₈	—	146	2	no	—
108	#432	—	73	2	no	—
109	#432	C₂×D₇₃	1	2	yes	—
110	C₂×C₄⋊C₈	—	73	2	no	—
111	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
112	C₂×C₄×C₈	—	73	1	no	—
113	C₂×C₄×C₈	—	73	2	no	—
114	#335	—	146	2	no	—
115	#334	—	146	2	no	—
116	#324	—	146	4	no	—
117	C₂²×Q₈	—	73	1	no	—
118	C₂²×C₈	—	73	1	no	—
119	C₂²×C₈	—	73	1	no	—
120	C₂²×C₈	C₄×D₇₃	1	2	yes	—
121	C₄×Q₈	—	146	2	no	—
122	C₄×Q₈	—	146	2	no	—
123	C₂×C₄⋊C₄	—	73	2	no	—
124	C₂×C₄⋊C₄	—	73	1	no	—
125	C₂×C₄²	—	73	2	no	—
126	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	146	4	no	—
130	C₄⋊C₈	—	146	2	no	—
131	C₂²⋊C₈	—	2	2	no	—
132	C₂²⋊C₈	—	146	2	no	—
133	#97	—	2	2	no	—
134	#97	—	146	2	no	—
135	C₄×C₈	—	73	2	no	—
136	C₄×C₈	—	73	2	no	—
137	C₄×C₈	—	146	2	no	—
138	C₄×C₈	—	146	2	no	—
139	C₂×Q₈	—	146	4	no	—
140	C₂×Q₈	—	146	2	no	—
141	C₂²×C₄	—	73	2	no	—
142	C₂²×C₄	—	73	2	no	—
143	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
144	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
145	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	292	2	no	—
150	C₂×C₈	—	292	1	no	—
151	C₂×C₈	—	73	1	no	—
152	C₂×C₈	—	73	1	no	—
153	C₂×C₈	—	73	2	no	—
154	C₂×C₈	—	146	2	no	—
155	C₄⋊C₄	—	146	4	no	—
156	C₄⋊C₄	—	146	2	no	—
157	C₄²	—	146	2	no	—
158	C₄²	—	146	2	no	—
159	C₄²	D₄×D₇₃	1	2	yes	—
160	C₄²	#11924028	1	2	yes	—
161	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
162	Q₈	—	146	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	292	2	no	—
168	C₂×C₄	—	73	4	no	—
169	C₂×C₄	—	146	2	no	—
170	C₂×C₄	—	146	2	no	—
171	C₂×C₄	—	146	2	no	—
172	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
173	C₂×C₄	#14556548	1	1	yes	characteristic
174	C₂×C₄	#14556564	1	2	yes	—
175	C₂×C₄	#14556569	1	2	yes	—
176	C₈	—	4	2	no	—
177	C₈	—	8	2	no	—
178	C₈	—	584	2	no	—
179	C₈	—	73	2	no	—
180	C₈	—	146	1	no	—
181	C₈	—	146	2	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#17988446	1	1	yes	characteristic
185	C₂²	#108×D₇₃	1	2	yes	—
186	C₄	—	2	1	no	—
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	292	2	no	—
190	C₄	—	146	4	no	—
191	C₄	#101×D₇₃	1	1	yes	characteristic
192	C₄	#17988568	1	1	yes	characteristic
193	C₂	—	2	2	no	—
194	C₂	#22609926	1	1	yes	characteristic
195	C₂	#22609968	1	1	yes	characteristic
196	C₂	#22609976	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988711

C₂

yes

maximal, characteristic

#22608692

C₂

yes

maximal

#22608274

C₂

yes

maximal

#22607801

C₂

yes

maximal, characteristic

C₇₃×#1290

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₄₆

C₂²

yes

—

—

C₂²

yes

—

C₂×#14556520

C₄

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

—

#17988339

—

#17988337

—

#17988312

—

#432×C₇₃

C₂²

yes

—

C₂×C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂²×Dic₁₄₆

Q₈

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

C₂×C₄⋊Dic₇₃

Q₈

yes

characteristic

C₂×#11924007

D₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#10596831

D₄

yes

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

#97×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂²⋊C₄

yes

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#108

yes

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

—

—

—

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#380

yes

characteristic

C₂²×C₇₃

C₂×#108

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#8621

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₈×Q₈

—

100

#1568

—

101

#1467

—

102

#1290

—

103

#1290

D₇₃

yes

characteristic

104

C₇₃

#8621

yes

characteristic, complemented, Sylow

105

C₂×C₄×Q₈

—

106

C₈×Q₈

—

146

—

107

C₈×Q₈

—

146

—

108

#432

—

109

#432

C₂×D₇₃

yes

—

110

C₂×C₄⋊C₈

—

111

C₂×C₄⋊C₈

C₂×D₇₃

yes

characteristic

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

#335

—

146

—

115

#334

—

146

—

116

#324

—

146

—

117

C₂²×Q₈

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

C₄×D₇₃

yes

—

121

C₄×Q₈

—

146

—

122

C₄×Q₈

—

146

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

—

126

C₂×C₄²

C₂²×D₇₃

yes

characteristic

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

146

—

130

C₄⋊C₈

—

146

—

131

C₂²⋊C₈

—

132

C₂²⋊C₈

—

146

—

133

#97

—

134

#97

—

146

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

146

—

138

C₄×C₈

—

146

—

139

C₂×Q₈

—

146

—

140

C₂×Q₈

—

146

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

144

C₂²×C₄

D₄×D₇₃

yes

characteristic

145

C₂²×C₄

Q₈×D₇₃

yes

characteristic

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

292

—

150

C₂×C₈

—

292

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

146

—

155

C₄⋊C₄

—

146

—

156

C₄⋊C₄

—

146

—

157

C₄²

—

146

—

158

C₄²

—

146

—

159

C₄²

D₄×D₇₃

yes

—

160

C₄²

#11924028

yes

—

161

C₂³

C₄⋊C₄×D₇₃

yes

characteristic

162

Q₈

—

146

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

292

—

168

C₂×C₄

—

169

C₂×C₄

—

146

—

170

C₂×C₄

—

146

—

171

C₂×C₄

—

146

—

172

C₂×C₄

C₂²⋊C₄×D₇₃

yes

characteristic

173

C₂×C₄

#14556548

yes

characteristic

174

C₂×C₄

#14556564

yes

—

175

C₂×C₄

#14556569

yes

—

176

C₈

—

177

C₈

—

178

C₈

—

584

—

179

C₈

—

180

C₈

—

146

—

181

C₈

—

146

—

182

C₂²

—

183

C₂²

—

184

C₂²

#17988446

yes

characteristic

185

C₂²

#108×D₇₃

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

292

—

190

C₄

—

146

—

191

C₄

#101×D₇₃

yes

characteristic

192

C₄

#17988568

yes

characteristic

193

C₂

—

194

C₂

#22609926

yes

characteristic

195

C₂

#22609968

yes

characteristic

196

C₂

#22609976

yes

characteristic

	subgroup	G/H
center	C₂²	#17988446
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1290	C₂
socle	C₂²×C₇₃	#380
prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1290	D₇₃
hypocenter	C₇₃	#8621
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1290

subgroup

G/H

center

C₂²

#17988446

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1290

C₂

socle

C₂²×C₇₃

#380

prime radical

C₂³×C₇₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1290

N	H ≅ G/N	type
104. C₇₃	#8621	semidirect

H ≅ G/N

type

104. C₇₃

#8621

semidirect

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

subgroup	index
C₂×#17988711	2
#22608692	2
#22608274	2
#22607801	2
C₇₃×#1290	2
#8621	73

subgroup

index

C₇₃×#1290

Subgroups — #32978933

Conjugacy classes of subgroups

Complements

Maximal subgroups