Group #34000598 — subgroups

Subgroups — #34000598

Order 18688.

246 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 84 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	#22626261	C₂	1	1	yes	maximal, characteristic
2	#22625868	C₂	1	1	yes	maximal, characteristic
3	#334×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556664	—	2	1	no	—
5	C₂×#14556660	C₄	1	2	yes	—
6	C₂×#14556659	—	2	1	no	—
7	#17989531	—	2	1	no	—
8	#17989508	C₄	1	2	yes	complemented
9	C₄⋊#11924031	—	2	1	no	—
10	C₄⋊C₈×D₇₃	—	2	1	no	—
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	#17988338	C₄	1	1	yes	characteristic, complemented
13	#17988337	—	2	1	no	—
14	#334×C₇₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#64408	—	2	2	no	—
16	#14556664	—	4	1	no	—
17	#14556664	—	4	1	no	—
18	C₂×#11924031	—	2	2	no	—
19	#14556660	—	4	2	no	—
20	#14556659	—	4	1	no	—
21	#14556659	—	4	1	no	—
22	(C₂×#64408)⋊C₄	—	4	1	no	—
23	(C₂×#64408)⋊C₄	—	4	1	no	—
24	C₂×C₈×D₇₃	—	4	1	no	—
25	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
27	C₄²×D₇₃	D₄	1	1	yes	characteristic
28	C₄⋊#10596831	—	2	1	no	—
29	#14556520	—	4	1	no	—
30	C₄×#10596831	D₄	1	1	yes	characteristic
31	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
33	C₄⋊C₈×C₇₃	—	2	1	no	—
34	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×#64408	—	4	2	no	—
36	C₄⋊#64408	—	4	2	no	—
37	C₄⋊#64408	—	4	2	no	—
38	#11924031	—	4	2	no	—
39	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
41	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
42	C₂×#10596831	—	4	1	no	—
43	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
45	C₈×D₇₃	—	2	1	no	—
46	C₈×D₇₃	—	2	1	no	—
47	C₈×D₇₃	—	8	2	no	—
48	C₈×D₇₃	D₈	1	1	yes	characteristic
49	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
50	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
51	C₈×D₇₃	Q₁₆	1	1	yes	characteristic
52	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
54	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
55	C₂×C₈×C₇₃	—	4	1	no	—
56	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×D₇₃	#102	1	1	yes	characteristic
60	C₂×#64408	—	8	2	no	—
61	C₂×#64408	—	8	2	no	—
62	C₂×#64408	—	8	2	no	—
63	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
66	C₂×Dic₇₃	#103	1	1	yes	characteristic
67	C₂×Dic₇₃	#106	1	1	yes	characteristic
68	C₂×Dic₇₃	#107	1	1	yes	characteristic
69	C₄×D₇₃	—	2	2	no	—
70	C₄×D₇₃	—	2	2	no	—
71	C₄×D₇₃	#100	1	1	yes	characteristic
72	C₄×D₇₃	#101	1	1	yes	characteristic
73	C₄×D₇₃	#102	1	1	yes	characteristic
74	C₄×D₇₃	#103	1	1	yes	characteristic
75	C₈×C₇₃	—	2	1	no	—
76	C₈×C₇₃	—	8	1	no	—
77	C₈×C₇₃	#106	1	1	yes	characteristic
78	C₈×C₇₃	#107	1	1	yes	characteristic
79	#10596831	—	2	1	no	—
80	#10596831	—	8	1	no	—
81	#10596831	#102	1	1	yes	characteristic
82	#10596831	#103	1	1	yes	characteristic
83	C₂²×C₇₃	#340	1	1	yes	characteristic
84	C₂×D₇₃	#331	1	1	yes	characteristic
85	C₂×D₇₃	#332	1	1	yes	characteristic
86	C₂×D₇₃	#359	1	2	yes	—
87	C₂×D₇₃	#360	1	2	yes	—
88	#64408	—	16	2	no	—
89	C₄×C₇₃	—	2	1	no	—
90	C₄×C₇₃	—	2	1	no	—
91	C₄×C₇₃	#340	1	1	yes	characteristic
92	C₄×C₇₃	#343	1	1	yes	characteristic
93	Dic₇₃	—	2	1	no	—
94	Dic₇₃	—	2	1	no	—
95	Dic₇₃	#332	1	1	yes	characteristic
96	Dic₇₃	#333	1	1	yes	characteristic
97	#7275	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	#1296	1	1	yes	characteristic
99	C₂×C₇₃	#1382	1	1	yes	characteristic
100	C₂×C₇₃	#1386	1	1	yes	characteristic
101	D₇₃	#1360	1	2	yes	—
102	D₇₃	#1363	1	2	yes	—
103	C₂×C₄⋊C₁₆	—	73	1	no	—
104	#1846	—	73	1	no	—
105	C₂×#334	—	73	1	no	—
106	C₇₃	#7275	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₁₆	—	146	1	no	—
108	C₂×#107	—	73	2	no	—
109	C₂×#106	—	146	1	no	—
110	C₂×C₄⋊C₈	—	146	1	no	—
111	C₂×C₄×C₈	—	73	1	no	—
112	#382	—	146	1	no	—
113	C₄⋊C₁₆	—	73	2	no	—
114	C₄⋊C₁₆	—	146	1	no	—
115	#334	—	73	1	no	—
116	#334	—	146	1	no	—
117	#334	#64408	1	1	yes	characteristic, complemented
118	C₂²×C₈	—	292	1	no	—
119	C₂²×C₈	—	73	1	no	—
120	C₂²×C₈	—	73	1	no	—
121	C₂×C₄⋊C₄	—	146	2	no	—
122	C₂×C₄²	—	73	1	no	—
123	C₂×C₁₆	—	292	1	no	—
124	C₂×C₁₆	—	292	1	no	—
125	C₂×C₁₆	—	146	2	no	—
126	#107	—	292	2	no	—
127	#106	—	292	1	no	—
128	#106	—	292	1	no	—
129	C₄⋊C₈	—	2	1	no	—
130	C₄⋊C₈	—	292	1	no	—
131	C₄⋊C₈	—	146	1	no	—
132	C₄×C₈	—	73	1	no	—
133	C₄×C₈	—	73	1	no	—
134	C₄×C₈	—	73	1	no	—
135	C₄×C₈	C₂×#64408	1	1	yes	characteristic
136	#95	—	292	1	no	—
137	#95	—	292	1	no	—
138	C₂²×C₄	—	292	2	no	—
139	C₂²×C₄	—	73	1	no	—
140	C₂²×C₄	—	73	1	no	—
141	C₂²×C₄	—	73	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	292	1	no	—
144	C₂×C₈	—	584	2	no	—
145	C₂×C₈	—	73	1	no	—
146	C₂×C₈	—	73	1	no	—
147	C₂×C₈	—	73	1	no	—
148	C₂×C₈	—	73	1	no	—
149	C₂×C₈	—	73	1	no	—
150	C₂×C₈	—	73	1	no	—
151	C₂×C₈	—	146	1	no	—
152	C₂×C₈	—	146	1	no	—
153	C₂×C₈	C₄×#64408	1	1	yes	characteristic
154	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
155	C₄⋊C₄	—	292	2	no	—
156	C₄⋊C₄	—	292	2	no	—
157	C₄²	—	73	1	no	—
158	C₄²	—	73	1	no	—
159	C₄²	—	73	1	no	—
160	C₄²	C₂²⋊#64408	1	1	yes	characteristic
161	C₁₆	—	292	2	no	—
162	C₂³	—	73	1	no	—
163	C₂×C₄	—	584	2	no	—
164	C₂×C₄	—	584	2	no	—
165	C₂×C₄	—	584	2	no	—
166	C₂×C₄	—	73	1	no	—
167	C₂×C₄	—	73	1	no	—
168	C₂×C₄	—	73	1	no	—
169	C₂×C₄	—	73	1	no	—
170	C₂×C₄	—	73	1	no	—
171	C₂×C₄	—	73	1	no	—
172	C₂×C₄	—	73	1	no	—
173	C₂×C₄	—	146	2	no	—
174	C₂×C₄	—	146	2	no	—
175	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	#14556653	1	1	yes	characteristic
177	C₂×C₄	#14556654	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	8	1	no	—
180	C₈	—	584	1	no	—
181	C₈	—	73	1	no	—
182	C₈	—	73	1	no	—
183	C₈	—	146	1	no	—
184	C₈	#14556659	1	1	yes	characteristic
185	C₈	#14556660	1	1	yes	characteristic
186	C₂²	—	73	1	no	—
187	C₂²	—	73	1	no	—
188	C₂²	—	73	2	no	—
189	C₂²	—	73	2	no	—
190	C₂²	#17989460	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	73	1	no	—
194	C₄	—	73	1	no	—
195	C₄	—	1168	2	no	—
196	C₄	—	146	1	no	—
197	C₄	—	146	1	no	—
198	C₄	#17989461	1	1	yes	characteristic
199	C₄	#17989466	1	1	yes	characteristic
200	C₂	—	73	2	no	—
201	C₂	—	73	2	no	—
202	C₂	#22624810	1	1	yes	characteristic
203	C₂	#22624994	1	1	yes	characteristic
204	C₂	#22625004	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626261

C₂

yes

maximal, characteristic

#22625868

C₂

yes

maximal, characteristic

#334×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556664

—

C₂×#14556660

C₄

yes

—

C₂×#14556659

—

#17989531

—

#17989508

C₄

yes

complemented

C₄⋊#11924031

—

C₄⋊C₈×D₇₃

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988338

C₄

yes

characteristic, complemented

#17988337

—

#334×C₇₃

C₄

yes

characteristic, complemented

C₂×C₄⋊#64408

—

—

—

—

—

—

—

—

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₄²×D₇₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

C₄⋊#64408

—

C₄⋊#64408

—

#11924031

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

D₈

yes

characteristic

C₈×D₇₃

SD₁₆

yes

characteristic

C₈×D₇₃

SD₁₆

yes

characteristic

C₈×D₇₃

Q₁₆

yes

characteristic

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

characteristic

C₄×Dic₇₃

Q₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

C₂²×D₇₃

yes

—

—

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

—

yes

yes

C₂²×C₇₃

yes

C₂×D₇₃

yes

C₂×D₇₃

yes

C₂×D₇₃

yes

—

C₂×D₇₃

#360

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

100

C₂×C₇₃

#1386

yes

characteristic

101

D₇₃

#1360

yes

—

102

D₇₃

#1363

yes

—

103

C₂×C₄⋊C₁₆

—

104

#1846

—

105

C₂×#334

—

106

C₇₃

#7275

yes

characteristic, complemented, Sylow

107

C₂²×C₁₆

—

146

—

108

C₂×#107

—

109

C₂×#106

—

146

—

110

C₂×C₄⋊C₈

—

146

—

111

C₂×C₄×C₈

—

112

#382

—

146

—

113

C₄⋊C₁₆

—

114

C₄⋊C₁₆

—

146

—

115

#334

—

116

#334

—

146

—

117

#334

#64408

yes

characteristic, complemented

118

C₂²×C₈

—

292

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₂×C₄⋊C₄

—

146

—

122

C₂×C₄²

—

123

C₂×C₁₆

—

292

—

124

C₂×C₁₆

—

292

—

125

C₂×C₁₆

—

146

—

126

#107

—

292

—

127

#106

—

292

—

128

#106

—

292

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

292

—

131

C₄⋊C₈

—

146

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

C₂×#64408

yes

characteristic

136

#95

—

292

—

137

#95

—

292

—

138

C₂²×C₄

—

292

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂×C₈

—

143

C₂×C₈

—

292

—

144

C₂×C₈

—

584

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

146

—

152

C₂×C₈

—

146

—

153

C₂×C₈

C₄×#64408

yes

characteristic

154

C₂×C₈

C₄⋊#64408

yes

characteristic

155

C₄⋊C₄

—

292

—

156

C₄⋊C₄

—

292

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₂²⋊#64408

yes

characteristic

161

C₁₆

—

292

—

162

C₂³

—

163

C₂×C₄

—

584

—

164

C₂×C₄

—

584

—

165

C₂×C₄

—

584

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

146

—

174

C₂×C₄

—

146

—

175

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

176

C₂×C₄

#14556653

yes

characteristic

177

C₂×C₄

#14556654

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

584

—

181

C₈

—

182

C₈

—

183

C₈

—

146

—

184

C₈

#14556659

yes

characteristic

185

C₈

#14556660

yes

characteristic

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

#17989460

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

1168

—

196

C₄

—

146

—

197

C₄

—

146

—

198

C₄

#17989461

yes

characteristic

199

C₄

#17989466

yes

characteristic

200

C₂

—

201

C₂

—

202

C₂

#22624810

yes

characteristic

203

C₂

#22624994

yes

characteristic

204

C₂

#22625004

yes

characteristic

	subgroup	G/H
center	C₂²	#17989460
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₄×C₈	C₂×#64408
Fitting	#334×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂²×D₇₃	#102
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#334	#64408
hypocenter	C₇₃	#7275
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#17989460

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₄×C₈

C₂×#64408

Fitting

#334×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄×C₈×D₇₃

C₂²

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
8. #17989508	C₄	semidirect
12. #17988338	C₄	semidirect
14. #334×C₇₃	C₄	semidirect
106. C₇₃	#7275	semidirect
117. #334	#64408	semidirect

H ≅ G/N

type

8. #17989508

C₄

semidirect

12. #17988338

C₄

semidirect

14. #334×C₇₃

C₄

semidirect

106. C₇₃

#7275

semidirect

117. #334

#64408

semidirect

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

subgroup	index
#22626261	2
#22625868	2
#334×D₇₃	2
#7275	73

subgroup

index

#22626261

#22625868

#334×D₇₃

#7275

Subgroups — #34000598

Conjugacy classes of subgroups

Complements

Maximal subgroups