Group #34005485 — subgroups

Subgroups — #34005485

Order 18688.

382 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 140 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	#22625863	C₂	1	1	yes	maximal, characteristic
2	#22625848	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989451	C₂	1	2	yes	maximal
4	C₂×#17989448	C₂	1	2	yes	maximal
5	#22609675	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	—	2	2	no	—
7	C₂×C₄⋊#10596833	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#10596833	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
10	#17989453	—	2	2	no	—
11	#17989451	—	2	4	no	—
12	#17989448	—	2	4	no	—
13	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
14	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
15	#17988342	C₄	1	1	yes	characteristic
16	#17988341	C₄	1	2	yes	—
17	#336×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	—	2	1	no	—
19	C₂²×#10596833	—	2	1	no	—
20	C₂²×#10596833	—	2	2	no	—
21	C₄⋊#10596833	—	2	2	no	—
22	C₄⋊#10596833	—	2	2	no	—
23	C₄⋊#10596833	—	2	2	no	—
24	C₄⋊#10596833	—	2	2	no	—
25	C₄⋊#10596833	—	4	4	no	—
26	C₂²×#10596831	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
29	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
30	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
31	C₈×Dic₇₃	D₄	1	4	yes	—
32	C₈×Dic₇₃	Q₈	1	4	yes	—
33	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#10596833	—	4	1	no	—
36	C₂×#10596833	—	4	2	no	—
37	C₂×#10596833	—	4	1	no	—
38	C₂×#10596833	—	4	2	no	—
39	C₂×#10596833	—	4	4	no	—
40	C₂×#10596833	—	4	2	no	—
41	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
42	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×#10596831	—	2	2	no	—
45	C₂×#10596831	—	2	2	no	—
46	C₂×#10596831	C₄⋊C₄	1	4	yes	—
47	C₄×Dic₇₃	—	2	4	no	—
48	C₄×Dic₇₃	—	2	1	no	—
49	C₄×Dic₇₃	—	2	1	no	—
50	C₄×Dic₇₃	—	2	2	no	—
51	C₄×Dic₇₃	—	2	2	no	—
52	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
53	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	—	2	2	no	—
57	C₂×C₈×C₇₃	—	2	2	no	—
58	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
59	C₄²×C₇₃	—	2	2	no	—
60	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	—	2	2	no	—
63	C₂×C₄×C₇₃	#108	1	4	yes	—
64	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₇₃	#131	1	2	yes	—
66	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
68	C₂×Dic₇₃	—	2	2	no	—
69	C₂×Dic₇₃	—	2	2	no	—
70	C₂×Dic₇₃	—	2	2	no	—
71	C₂×Dic₇₃	—	2	2	no	—
72	C₂×Dic₇₃	#127	1	1	yes	characteristic
73	C₂×Dic₇₃	C₄⋊Q₈	1	2	yes	—
74	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×Dic₇₃	#137	1	2	yes	—
77	#10596833	—	8	1	no	—
78	#10596833	—	8	2	no	—
79	#10596833	—	8	1	no	—
80	C₈×C₇₃	—	2	4	no	—
81	#10596831	—	2	4	no	—
82	C₂²×C₇₃	#384	1	1	yes	characteristic
83	C₂²×C₇₃	#424	1	1	yes	characteristic
84	C₂²×C₇₃	#424	1	1	yes	characteristic
85	C₂²×C₇₃	#428	1	2	yes	—
86	C₂²×C₇₃	C₂×#108	1	2	yes	—
87	C₄×C₇₃	—	2	4	no	—
88	C₄×C₇₃	#424	1	2	yes	—
89	C₄×C₇₃	#424	1	2	yes	—
90	Dic₇₃	—	2	2	no	—
91	Dic₇₃	—	4	2	no	—
92	Dic₇₃	#496	1	1	yes	characteristic
93	Dic₇₃	#497	1	1	yes	characteristic
94	Dic₇₃	#501	1	2	yes	—
95	#8812	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	#1573	1	2	yes	—
97	C₂×C₇₃	#1574	1	2	yes	—
98	C₂×C₇₃	#1843	1	1	yes	characteristic
99	C₂×C₇₃	#1847	1	1	yes	characteristic
100	C₂×C₇₃	#1848	1	1	yes	characteristic
101	#1842	—	73	1	no	—
102	#1831	—	73	1	no	—
103	#1744	—	73	1	no	—
104	C₂×#335	—	73	2	no	—
105	C₂×#334	—	73	2	no	—
106	C₇₃	#8812	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊C₈	—	73	1	no	—
108	C₂×C₄⋊C₈	—	73	2	no	—
109	C₂×C₄⋊C₈	—	73	1	no	—
110	C₂×C₄⋊C₈	—	146	2	no	—
111	C₂×C₄×C₈	—	73	2	no	—
112	C₄³	—	73	1	no	—
113	#336	—	73	2	no	—
114	#336	—	73	1	no	—
115	#336	—	146	2	no	—
116	#336	#64408	1	1	yes	characteristic
117	#335	—	146	4	no	—
118	#334	—	146	4	no	—
119	C₂²×C₈	—	73	2	no	—
120	C₂²×C₈	—	146	1	no	—
121	C₂²×C₈	—	146	1	no	—
122	C₂²×C₈	—	146	2	no	—
123	C₂²×C₈	C₂×#64408	1	2	yes	—
124	C₂×C₄²	—	73	1	no	—
125	C₂×C₄²	—	73	2	no	—
126	C₂×C₄²	—	73	2	no	—
127	C₂×C₄²	—	73	1	no	—
128	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
129	C₄⋊C₈	—	292	4	no	—
130	C₄⋊C₈	—	146	2	no	—
131	C₄⋊C₈	—	146	2	no	—
132	C₄⋊C₈	—	146	2	no	—
133	C₄⋊C₈	—	146	2	no	—
134	C₄×C₈	—	73	4	no	—
135	C₄×C₈	—	73	4	no	—
136	C₂²×C₄	—	73	1	no	—
137	C₂²×C₄	—	73	1	no	—
138	C₂²×C₄	—	73	2	no	—
139	C₂²×C₄	C₄⋊#64408	1	2	yes	—
140	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	292	4	no	—
144	C₂×C₈	—	292	1	no	—
145	C₂×C₈	—	292	2	no	—
146	C₂×C₈	—	292	1	no	—
147	C₂×C₈	—	292	2	no	—
148	C₂×C₈	—	292	2	no	—
149	C₂×C₈	—	73	4	no	—
150	C₂×C₈	—	146	2	no	—
151	C₂×C₈	—	146	2	no	—
152	C₂×C₈	C₄⋊#64408	1	4	yes	—
153	C₄²	—	2	2	no	—
154	C₄²	—	73	2	no	—
155	C₄²	—	73	2	no	—
156	C₄²	—	146	2	no	—
157	C₄²	—	146	2	no	—
158	C₄²	—	146	1	no	—
159	C₄²	—	146	1	no	—
160	C₄²	—	146	4	no	—
161	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	73	1	no	—
165	C₂×C₄	—	73	1	no	—
166	C₂×C₄	—	73	2	no	—
167	C₂×C₄	—	73	2	no	—
168	C₂×C₄	—	73	2	no	—
169	C₂×C₄	—	146	2	no	—
170	C₂×C₄	—	146	2	no	—
171	C₂×C₄	—	146	2	no	—
172	C₂×C₄	—	146	2	no	—
173	C₂×C₄	#14556661	1	4	yes	—
174	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
175	C₂×C₄	#14556678	1	1	yes	characteristic
176	C₂×C₄	#14556678	1	2	yes	—
177	C₂×C₄	#14556679	1	1	yes	characteristic
178	C₈	—	2	4	no	—
179	C₈	—	584	2	no	—
180	C₈	—	584	1	no	—
181	C₈	—	584	1	no	—
182	C₈	—	146	4	no	—
183	C₂²	#17989533	1	1	yes	characteristic
184	C₂²	#17989593	1	1	yes	characteristic
185	C₂²	#17989594	1	1	yes	characteristic
186	C₂²	#17989598	1	2	yes	—
187	C₂²	C₂×#14556661	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	292	2	no	—
190	C₄	—	73	1	no	—
191	C₄	—	73	1	no	—
192	C₄	—	73	2	no	—
193	C₄	—	146	2	no	—
194	C₄	#17989593	1	2	yes	—
195	C₄	#17989594	1	2	yes	—
196	C₂	#22625504	1	2	yes	—
197	C₂	#22625508	1	2	yes	—
198	C₂	#22625865	1	1	yes	characteristic
199	C₂	#22625869	1	1	yes	characteristic
200	C₂	#22625870	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625863

C₂

yes

maximal, characteristic

#22625848

C₂

yes

maximal, characteristic

C₂×#17989451

C₂

yes

maximal

C₂×#17989448

C₂

yes

maximal

#22609675

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596833

—

C₂×C₄⋊#10596833

C₂²

yes

characteristic

C₂×C₄⋊#10596833

C₂²

yes

characteristic

C₂×C₄⋊#10596833

C₂²

yes

—

#17989453

—

#17989451

—

#17989448

—

C₂×C₈×Dic₇₃

C₂²

yes

—

C₄²×Dic₇₃

C₂²

yes

characteristic

#17988342

C₄

yes

characteristic

#17988341

C₄

yes

—

#336×C₇₃

C₄

yes

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

—

C₂×C₄×Dic₇₃

Q₈

yes

—

C₂×C₄×Dic₇₃

C₂³

yes

characteristic

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

C₂²×Dic₇₃

C₂×D₄

yes

characteristic

C₂²×Dic₇₃

C₂×Q₈

yes

—

—

C₄⋊C₄

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×D₄

yes

—

C₄×Dic₇₃

C₂×Q₈

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#108

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#127

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

—

C₂×Dic₇₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

#137

yes

—

#10596833

—

#10596833

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

C₂×#108

yes

—

C₄×C₇₃

—

C₄×C₇₃

#424

yes

—

C₄×C₇₃

#424

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

—

#8812

—

maximal, Sylow

C₂×C₇₃

#1573

yes

—

C₂×C₇₃

#1574

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

100

C₂×C₇₃

#1848

yes

characteristic

101

#1842

—

102

#1831

—

103

#1744

—

104

C₂×#335

—

105

C₂×#334

—

106

C₇₃

#8812

yes

characteristic, complemented, Sylow

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

—

109

C₂×C₄⋊C₈

—

110

C₂×C₄⋊C₈

—

146

—

111

C₂×C₄×C₈

—

112

C₄³

—

113

#336

—

114

#336

—

115

#336

—

146

—

116

yes

117

—

146

—

118

#334

—

146

—

119

C₂²×C₈

—

120

C₂²×C₈

—

146

—

121

C₂²×C₈

—

146

—

122

C₂²×C₈

—

146

—

123

C₂²×C₈

C₂×#64408

yes

—

124

C₂×C₄²

—

125

C₂×C₄²

—

126

C₂×C₄²

—

127

C₂×C₄²

—

128

C₂×C₄²

C₂×#64408

yes

characteristic

129

C₄⋊C₈

—

292

—

130

C₄⋊C₈

—

146

—

131

C₄⋊C₈

—

146

—

132

C₄⋊C₈

—

146

—

133

C₄⋊C₈

—

146

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₄⋊#64408

yes

—

140

C₂²×C₄

C₂²×#64408

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

292

—

144

C₂×C₈

—

292

—

145

C₂×C₈

—

292

—

146

C₂×C₈

—

292

—

147

C₂×C₈

—

292

—

148

C₂×C₈

—

292

—

149

C₂×C₈

—

150

C₂×C₈

—

146

—

151

C₂×C₈

—

146

—

152

C₂×C₈

C₄⋊#64408

yes

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

146

—

157

C₄²

—

146

—

158

C₄²

—

146

—

159

C₄²

—

146

—

160

C₄²

—

146

—

161

C₂³

C₂×C₄⋊#64408

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

146

—

170

C₂×C₄

—

146

—

171

C₂×C₄

—

146

—

172

C₂×C₄

—

146

—

173

C₂×C₄

#14556661

yes

—

174

C₂×C₄

C₂×C₄⋊#64408

yes

—

175

C₂×C₄

#14556678

yes

characteristic

176

C₂×C₄

#14556678

yes

—

177

C₂×C₄

#14556679

yes

characteristic

178

C₈

—

179

C₈

—

584

—

180

C₈

—

584

—

181

C₈

—

584

—

182

C₈

—

146

—

183

C₂²

#17989533

yes

characteristic

184

C₂²

#17989593

yes

characteristic

185

C₂²

#17989594

yes

characteristic

186

C₂²

#17989598

yes

—

187

C₂²

C₂×#14556661

yes

—

188

C₄

—

189

C₄

—

292

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

146

—

194

C₄

#17989593

yes

—

195

C₄

#17989594

yes

—

196

C₂

#22625504

yes

—

197

C₂

#22625508

yes

—

198

C₂

#22625865

yes

characteristic

199

C₂

#22625869

yes

characteristic

200

C₂

#22625870

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#336×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	#336	#64408
hypocenter	C₇₃	#8812
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×C₄⋊#64408

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#336×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₇₃

C₂³

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
106. C₇₃	#8812	semidirect

H ≅ G/N

type

106. C₇₃

#8812

semidirect

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

subgroup	index
#22625863	2
#22625848	2
C₂×#17989451	2
C₂×#17989448	2
#22609675	2
#8812	73

subgroup

index

Subgroups — #34005485

Conjugacy classes of subgroups

Complements

Maximal subgroups