Group #32991114 — subgroups

Subgroups — #32991114

Order 18688.

264 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22610106	C₂	1	1	yes	maximal, characteristic
2	#22607834	C₂	1	2	yes	maximal
3	#22607828	C₂	1	2	yes	maximal
4	#22607762	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1940	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×#14556527	—	2	1	no	—
8	C₂×#14556527	—	2	1	no	—
9	#17988611	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
11	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
14	#336×C₇₃	—	2	1	no	—
15	C₂×#11924012	—	2	2	no	—
16	#14556527	—	4	2	no	—
17	#14556527	—	4	2	no	—
18	C₄⋊#10596831	—	2	4	no	—
19	C₄⋊#10596831	—	2	2	no	—
20	C₂²⋊#10596831	—	2	2	no	—
21	C₄×#10596831	—	2	2	no	—
22	C₂²×C₈×C₇₃	—	2	1	no	—
23	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
26	C₂×C₄²×C₇₃	—	2	1	no	—
27	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	—	2	2	no	—
29	C₄⋊C₈×C₇₃	—	4	2	no	—
30	#97×C₇₃	—	2	2	no	—
31	#95×C₇₃	Q₈	1	1	yes	characteristic
32	#95×C₇₃	Q₈	1	1	yes	characteristic
33	#11924012	—	4	2	no	—
34	#11924012	—	4	2	no	—
35	C₂×#10596831	—	4	2	no	—
36	C₂×#10596831	—	4	2	no	—
37	C₂²×C₄×C₇₃	—	2	1	no	—
38	C₂²×C₄×C₇₃	—	2	1	no	—
39	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₂×C₈×C₇₃	—	2	1	no	—
45	C₂×C₈×C₇₃	—	2	1	no	—
46	C₂×C₈×C₇₃	—	4	1	no	—
47	C₂×C₈×C₇₃	—	4	1	no	—
48	C₂×C₈×C₇₃	—	4	1	no	—
49	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
50	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
51	C₄⋊C₄×C₇₃	—	4	2	no	—
52	C₄²×C₇₃	—	4	1	no	—
53	C₄²×C₇₃	—	4	1	no	—
54	C₄²×C₇₃	#41	1	2	yes	—
55	C₄²×C₇₃	#41	1	2	yes	—
56	C₂³×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	—	4	1	no	—
61	C₂×C₄×C₇₃	—	4	1	no	—
62	C₂×C₄×C₇₃	—	4	1	no	—
63	C₂×C₄×C₇₃	—	4	1	no	—
64	C₂×C₄×C₇₃	—	4	1	no	—
65	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#125	1	2	yes	—
68	C₂×C₄×C₇₃	#126	1	2	yes	—
69	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₈×C₇₃	—	2	2	no	—
74	C₈×C₇₃	—	2	2	no	—
75	C₈×C₇₃	—	8	1	no	—
76	#10596831	—	8	2	no	—
77	#10596831	—	8	2	no	—
78	C₂²×C₇₃	—	2	1	no	—
79	C₂²×C₇₃	—	2	1	no	—
80	C₂²×C₇₃	#400	1	1	yes	characteristic
81	C₂²×C₇₃	#476	1	1	yes	characteristic
82	C₂²×C₇₃	#477	1	1	yes	characteristic
83	C₄×C₇₃	—	2	2	no	—
84	C₄×C₇₃	—	2	2	no	—
85	C₄×C₇₃	—	4	2	no	—
86	C₄×C₇₃	—	8	1	no	—
87	C₄×C₇₃	#481	1	1	yes	characteristic
88	C₄×C₇₃	#484	1	1	yes	characteristic
89	C₄×C₇₃	#487	1	1	yes	characteristic
90	C₄×C₇₃	#490	1	1	yes	characteristic
91	#10688	—	73	1	no	maximal, Sylow
92	C₂×C₇₃	—	2	1	no	—
93	C₂×C₇₃	—	2	1	no	—
94	C₂×C₇₃	#2100	1	1	yes	characteristic
95	C₂×C₇₃	#2102	1	1	yes	characteristic
96	C₂×C₇₃	#2103	1	1	yes	characteristic
97	#1940	D₇₃	1	1	yes	characteristic
98	#1849	—	73	1	no	—
99	#1303	—	73	2	no	—
100	#1301	—	73	2	no	—
101	#1275	—	73	1	no	—
102	C₇₃	#10688	1	1	yes	characteristic, complemented, Sylow
103	#431	—	73	2	no	—
104	C₂×#108	—	146	1	no	—
105	C₂×#108	—	146	1	no	—
106	#423	—	73	2	no	—
107	C₂×C₄⋊C₈	—	2	1	no	—
108	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
109	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
111	#336	—	2	1	no	—
112	C₂×M₁₆	—	146	2	no	—
113	C₂²×C₈	—	2	1	no	—
114	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	2	1	no	—
117	C₂×C₄²	—	2	1	no	—
118	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
119	#108	—	292	2	no	—
120	#108	—	292	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	4	2	no	—
123	C₄⋊C₈	—	146	2	no	—
124	C₄⋊C₈	—	146	4	no	—
125	C₂²⋊C₈	—	146	2	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	146	2	no	—
128	#95	Dic₁₄₆	1	1	yes	characteristic
129	#95	Dic₁₄₆	1	1	yes	characteristic
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
133	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
134	C₂²×C₄	#11924028	1	1	yes	characteristic
135	M₁₆	—	292	2	no	—
136	M₁₆	—	292	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	—	292	2	no	—
145	C₂×C₈	—	292	2	no	—
146	C₂×C₈	#11924016	1	2	yes	—
147	C₂×C₈	#11924018	1	2	yes	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄²	—	4	1	no	—
150	C₄²	—	4	1	no	—
151	C₄²	#11924027	1	2	yes	—
152	C₄²	#11924029	1	2	yes	—
153	C₂³	#14556570	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	#14556572	1	1	yes	characteristic
163	C₂×C₄	#14556580	1	1	yes	characteristic
164	C₂×C₄	#14556582	1	2	yes	—
165	C₂×C₄	#14556586	1	2	yes	—
166	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
167	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
168	C₂×C₄	#14556616	1	1	yes	characteristic
169	C₂×C₄	#14556616	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	8	1	no	—
173	C₈	—	584	2	no	—
174	C₈	—	584	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	#17988526	1	1	yes	characteristic
178	C₂²	#17988877	1	1	yes	characteristic
179	C₂²	#17988882	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	#17988903	1	1	yes	characteristic
185	C₄	#17988920	1	1	yes	characteristic
186	C₄	#17988939	1	1	yes	characteristic
187	C₄	#17988953	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	#22611494	1	1	yes	characteristic
191	C₂	#22611506	1	1	yes	characteristic
192	C₂	#22611511	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610106

C₂

yes

maximal, characteristic

#22607834

C₂

yes

maximal

#22607828

C₂

yes

maximal

#22607762

C₂

yes

maximal, characteristic

C₇₃×#1940

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

—

C₂×#14556527

—

C₂×#14556527

—

#17988611

C₂²

yes

—

C₂×C₄⋊C₈×C₇₃

—

C₂×C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂×#97×C₇₃

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

#336×C₇₃

—

—

—

—

—

—

—

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

#97×C₇₃

—

#95×C₇₃

Q₈

yes

characteristic

#95×C₇₃

Q₈

yes

—

—

—

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

Q₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

#41

yes

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

#126

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

characteristic

#1849

—

#1303

—

100

#1301

—

101

#1275

—

102

C₇₃

#10688

yes

characteristic, complemented, Sylow

103

#431

—

104

C₂×#108

—

146

—

105

C₂×#108

—

146

—

106

#423

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

C₂×D₇₃

yes

characteristic

109

C₂×#97

C₂×D₇₃

yes

characteristic

110

C₄×C₄⋊C₄

C₂×D₇₃

yes

characteristic

111

#336

—

112

C₂×M₁₆

—

146

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

115

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

116

C₂×C₄⋊C₄

—

117

C₂×C₄²

—

118

C₂×C₄²

C₂²×D₇₃

yes

characteristic

119

#108

—

292

—

120

#108

—

292

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

146

—

124

C₄⋊C₈

—

146

—

125

C₂²⋊C₈

—

146

—

126

#97

—

127

C₄×C₈

—

146

—

128

#95

Dic₁₄₆

yes

characteristic

129

#95

Dic₁₄₆

yes

characteristic

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

133

C₂²×C₄

C₂×Dic₁₄₆

yes

characteristic

134

C₂²×C₄

#11924028

yes

characteristic

135

M₁₆

—

292

—

136

M₁₆

—

292

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

292

—

145

C₂×C₈

—

292

—

146

C₂×C₈

#11924016

yes

—

147

C₂×C₈

#11924018

yes

—

148

C₄⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

#11924027

yes

—

152

C₄²

#11924029

yes

—

153

C₂³

#14556570

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#14556572

yes

characteristic

163

C₂×C₄

#14556580

yes

characteristic

164

C₂×C₄

#14556582

yes

—

165

C₂×C₄

#14556586

yes

—

166

C₂×C₄

C₂×#11924016

yes

characteristic

167

C₂×C₄

C₂×#11924018

yes

characteristic

168

C₂×C₄

#14556616

yes

characteristic

169

C₂×C₄

#14556616

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

584

—

174

C₈

—

584

—

175

C₂²

—

176

C₂²

—

177

C₂²

#17988526

yes

characteristic

178

C₂²

#17988877

yes

characteristic

179

C₂²

#17988882

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#17988903

yes

characteristic

185

C₄

#17988920

yes

characteristic

186

C₄

#17988939

yes

characteristic

187

C₄

#17988953

yes

characteristic

188

C₂

—

189

C₂

—

190

C₂

#22611494

yes

characteristic

191

C₂

#22611506

yes

characteristic

192

C₂

#22611511

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556580
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1940	C₂
socle	C₂²×C₇₃	#400
prime radical	C₂³×C₇₃	C₂²⋊Q₈
hypercenter	#1940	D₇₃
hypocenter	C₇₃	#10688
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1940

subgroup

G/H

center

C₂×C₄

#14556580

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1940

C₂

socle

C₂²×C₇₃

#400

prime radical

C₂³×C₇₃

C₂²⋊Q₈

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1940

N	H ≅ G/N	type
102. C₇₃	#10688	semidirect

H ≅ G/N

type

102. C₇₃

#10688

semidirect

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

subgroup	index
#22610106	2
#22607834	2
#22607828	2
#22607762	2
C₇₃×#1940	2
#10688	73

subgroup

index

C₇₃×#1940

Subgroups — #32991114

Conjugacy classes of subgroups

Complements

Maximal subgroups