Group #34000279 — subgroups

Subgroups — #34000279

Order 18688.

274 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 88 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	C₄×#14556660	C₂	1	1	yes	maximal, characteristic
2	#22625473	C₂	1	1	yes	maximal, characteristic
3	#346×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556660	C₄	1	2	yes	—
5	#17989588	—	2	1	no	—
6	C₄×C₄⋊#64408	—	2	1	no	—
7	#17989461	—	2	1	no	—
8	#17989449	—	2	1	no	—
9	#17989448	C₄	1	2	yes	complemented
10	C₂×C₁₆×D₇₃	—	2	1	no	—
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	#17988363	C₄	1	1	yes	characteristic, complemented
13	#17988362	—	2	1	no	—
14	#346×C₇₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#11924036	—	4	1	no	—
16	C₂×C₄⋊#64408	—	2	2	no	—
17	C₂×C₄×#64408	—	4	1	no	—
18	#14556660	—	2	2	no	—
19	#14556660	C₈	1	4	yes	complemented
20	#14556658	—	4	1	no	—
21	C₄⋊#10596833	—	2	2	no	—
22	(C₂×#64408)⋊C₄	—	4	1	no	—
23	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
25	C₄²×D₇₃	D₄	1	1	yes	characteristic
26	C₂×#11924003	—	2	1	no	—
27	C₁₆×D₇₃	—	2	4	no	—
28	C₄×#10596831	Q₈	1	1	yes	characteristic
29	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₇₃	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	—	2	1	no	—
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×#64408	—	4	2	no	—
34	#11924036	—	8	2	no	—
35	C₂×#10596833	—	4	2	no	—
36	C₄⋊#64408	—	2	4	no	—
37	C₄⋊#64408	—	4	2	no	—
38	C₄×#64408	—	8	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	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
44	C₈×D₇₃	—	2	2	no	—
45	C₈×D₇₃	C₂×C₈	1	2	yes	—
46	C₈×D₇₃	M₁₆	1	2	yes	—
47	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
49	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
50	#11924003	—	2	2	no	—
51	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₁₆×C₇₃	—	2	2	no	—
55	C₂²×D₇₃	#102	1	1	yes	characteristic
56	C₂×#64408	—	4	4	no	—
57	C₂×#64408	—	8	2	no	—
58	C₂×#64408	—	8	2	no	—
59	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
62	C₂×Dic₇₃	#102	1	1	yes	characteristic
63	C₂×Dic₇₃	#103	1	1	yes	characteristic
64	C₂×Dic₇₃	#103	1	1	yes	characteristic
65	C₄×D₇₃	—	2	2	no	—
66	C₄×D₇₃	—	2	1	no	—
67	C₄×D₇₃	—	2	1	no	—
68	C₄×D₇₃	C₂²⋊C₈	1	2	yes	—
69	C₄×D₇₃	#104	1	2	yes	—
70	#10596833	—	8	2	no	—
71	C₈×C₇₃	—	2	1	no	—
72	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
73	C₈×C₇₃	#97	1	1	yes	characteristic
74	#10596831	—	2	1	no	—
75	#10596831	C₄⋊C₈	1	2	yes	—
76	C₂²×C₇₃	#340	1	1	yes	characteristic
77	C₂×D₇₃	D₄⋊C₈	1	2	yes	—
78	C₂×D₇₃	#360	1	2	yes	—
79	C₂×D₇₃	#361	1	1	yes	characteristic
80	C₂×D₇₃	#362	1	1	yes	characteristic
81	#64408	—	8	4	no	—
82	C₄×C₇₃	—	2	1	no	—
83	C₄×C₇₃	—	2	1	no	—
84	C₄×C₇₃	#336	1	1	yes	characteristic
85	C₄×C₇₃	#339	1	1	yes	characteristic
86	Dic₇₃	—	2	1	no	—
87	Dic₇₃	—	2	1	no	—
88	Dic₇₃	Q₈⋊C₈	1	2	yes	—
89	#7178	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	#1279	1	1	yes	characteristic
91	C₂×C₇₃	#1386	1	1	yes	characteristic
92	C₂×C₇₃	#1387	1	1	yes	characteristic
93	D₇₃	#1333	1	2	yes	—
94	D₇₃	#1334	1	2	yes	—
95	C₂×#346	—	73	1	no	—
96	C₄×#107	—	73	1	no	—
97	C₂×#334	—	73	1	no	—
98	C₇₃	#7178	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₁₆	—	146	1	no	—
100	C₂×#107	—	73	2	no	—
101	C₂×C₄⋊C₈	—	146	1	no	—
102	C₂×C₄×C₈	—	73	1	no	—
103	C₄×C₄⋊C₄	—	146	1	no	—
104	#346	—	73	1	no	—
105	#346	—	146	1	no	—
106	#346	#64408	1	1	yes	characteristic, complemented
107	#340	—	146	1	no	—
108	#334	—	73	2	no	—
109	#334	—	146	1	no	—
110	C₂²×C₈	—	292	1	no	—
111	C₂²×C₈	—	73	1	no	—
112	C₂²×C₈	—	73	1	no	—
113	C₂×C₄⋊C₄	—	146	2	no	—
114	C₂×C₄²	—	292	1	no	—
115	C₂×C₄²	—	73	1	no	—
116	C₂×C₁₆	—	2	1	no	—
117	C₂×C₁₆	—	146	4	no	—
118	C₂×C₁₆	—	146	1	no	—
119	#107	—	73	4	no	—
120	#107	—	146	2	no	—
121	C₄⋊C₈	—	292	1	no	—
122	C₄⋊C₈	—	146	2	no	—
123	C₄×C₈	—	73	1	no	—
124	C₄×C₈	—	73	1	no	—
125	C₄×C₈	—	73	1	no	—
126	C₄×C₈	C₂×#64408	1	1	yes	characteristic
127	#95	—	292	1	no	—
128	C₂²×C₄	—	292	2	no	—
129	C₂²×C₄	—	73	1	no	—
130	C₂²×C₄	—	73	1	no	—
131	C₂²×C₄	—	73	1	no	—
132	C₂×C₈	—	292	2	no	—
133	C₂×C₈	—	584	2	no	—
134	C₂×C₈	—	73	1	no	—
135	C₂×C₈	—	73	2	no	—
136	C₂×C₈	—	73	2	no	—
137	C₂×C₈	—	73	1	no	—
138	C₂×C₈	—	146	2	no	—
139	C₂×C₈	C₄×#64408	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
141	C₄⋊C₄	—	292	2	no	—
142	C₄⋊C₄	—	146	4	no	—
143	C₄²	—	584	2	no	—
144	C₄²	—	73	1	no	—
145	C₄²	—	73	1	no	—
146	C₄²	—	73	1	no	—
147	C₄²	C₄⋊#64408	1	1	yes	characteristic
148	C₁₆	—	2	2	no	—
149	C₁₆	—	146	2	no	—
150	C₂³	—	73	1	no	—
151	C₂×C₄	—	292	4	no	—
152	C₂×C₄	—	584	2	no	—
153	C₂×C₄	—	584	2	no	—
154	C₂×C₄	—	73	1	no	—
155	C₂×C₄	—	73	2	no	—
156	C₂×C₄	—	73	2	no	—
157	C₂×C₄	—	73	1	no	—
158	C₂×C₄	—	73	1	no	—
159	C₂×C₄	—	146	1	no	—
160	C₂×C₄	—	146	2	no	—
161	C₂×C₄	—	146	1	no	—
162	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
163	C₂×C₄	#14556659	1	1	yes	characteristic
164	C₂×C₄	#14556660	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	584	2	no	—
167	C₈	—	73	2	no	—
168	C₈	—	146	1	no	—
169	C₈	C₈×#64408	1	1	yes	characteristic
170	C₈	#14556645	1	1	yes	characteristic
171	C₂²	—	73	2	no	—
172	C₂²	—	73	1	no	—
173	C₂²	—	73	1	no	—
174	C₂²	—	73	2	no	—
175	C₂²	#17989461	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	584	4	no	—
179	C₄	—	73	2	no	—
180	C₄	—	146	1	no	—
181	C₄	—	146	1	no	—
182	C₄	#17989452	1	1	yes	characteristic
183	C₄	#17989458	1	1	yes	characteristic
184	C₂	—	73	2	no	—
185	C₂	—	73	2	no	—
186	C₂	#22624766	1	1	yes	characteristic
187	C₂	#22625003	1	1	yes	characteristic
188	C₂	#22625006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556660

C₂

yes

maximal, characteristic

#22625473

C₂

yes

maximal, characteristic

#346×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556660

C₄

yes

—

#17989588

—

C₄×C₄⋊#64408

—

#17989461

—

#17989449

—

#17989448

C₄

yes

complemented

C₂×C₁₆×D₇₃

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988363

C₄

yes

characteristic, complemented

#17988362

—

#346×C₇₃

C₄

yes

characteristic, complemented

C₂×#11924036

—

C₂×C₄⋊#64408

—

C₂×C₄×#64408

—

#14556660

—

#14556660

C₈

yes

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₄²×D₇₃

D₄

yes

characteristic

C₂×#11924003

—

C₁₆×D₇₃

—

C₄×#10596831

Q₈

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

—

—

—

—

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

C₂×C₈

yes

—

C₈×D₇₃

M₁₆

yes

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

characteristic

C₄×Dic₇₃

Q₁₆

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

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

—

C₄×D₇₃

C₂²⋊C₈

yes

—

C₄×D₇₃

#104

yes

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

C₄×C₈

yes

C₈×C₇₃

yes

—

C₄⋊C₈

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

D₄⋊C₈

yes

—

C₂×D₇₃

#360

yes

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

Q₈⋊C₈

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

—

D₇₃

#1334

yes

—

C₂×#346

—

C₄×#107

—

C₂×#334

—

C₇₃

#7178

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

146

—

100

C₂×#107

—

101

C₂×C₄⋊C₈

—

146

—

102

C₂×C₄×C₈

—

103

C₄×C₄⋊C₄

—

146

—

104

#346

—

105

#346

—

146

—

106

#346

#64408

yes

characteristic, complemented

107

#340

—

146

—

108

#334

—

109

#334

—

146

—

110

C₂²×C₈

—

292

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₂×C₄⋊C₄

—

146

—

114

C₂×C₄²

—

292

—

115

C₂×C₄²

—

116

C₂×C₁₆

—

117

C₂×C₁₆

—

146

—

118

C₂×C₁₆

—

146

—

119

#107

—

120

#107

—

146

—

121

C₄⋊C₈

—

292

—

122

C₄⋊C₈

—

146

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

C₂×#64408

yes

characteristic

127

#95

—

292

—

128

C₂²×C₄

—

292

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂×C₈

—

292

—

133

C₂×C₈

—

584

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

146

—

139

C₂×C₈

C₄×#64408

yes

characteristic

140

C₂×C₈

C₂²⋊#64408

yes

characteristic

141

C₄⋊C₄

—

292

—

142

C₄⋊C₄

—

146

—

143

C₄²

—

584

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

C₄⋊#64408

yes

characteristic

148

C₁₆

—

149

C₁₆

—

146

—

150

C₂³

—

151

C₂×C₄

—

292

—

152

C₂×C₄

—

584

—

153

C₂×C₄

—

584

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

146

—

160

C₂×C₄

—

146

—

161

C₂×C₄

—

146

—

162

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

163

C₂×C₄

#14556659

yes

characteristic

164

C₂×C₄

#14556660

yes

characteristic

165

C₈

—

166

C₈

—

584

—

167

C₈

—

168

C₈

—

146

—

169

C₈

C₈×#64408

yes

characteristic

170

C₈

#14556645

yes

characteristic

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#17989461

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

584

—

179

C₄

—

180

C₄

—

146

—

181

C₄

—

146

—

182

C₄

#17989452

yes

characteristic

183

C₄

#17989458

yes

characteristic

184

C₂

—

185

C₂

—

186

C₂

#22624766

yes

characteristic

187

C₂

#22625003

yes

characteristic

188

C₂

#22625006

yes

characteristic

	subgroup	G/H
center	C₂²	#17989461
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₄×C₈	C₂×#64408
Fitting	#346×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂²×D₇₃	#102
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#346	#64408
hypocenter	C₇₃	#7178
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

#17989461

commutator

C₈×C₇₃

C₄×C₈

Frattini

C₄×C₈

C₂×#64408

Fitting

#346×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄×C₈×D₇₃

C₂²

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
9. #17989448	C₄	semidirect
12. #17988363	C₄	semidirect
14. #346×C₇₃	C₄	semidirect
19. #14556660	C₈	semidirect
98. C₇₃	#7178	semidirect
106. #346	#64408	semidirect

H ≅ G/N

type

9. #17989448

C₄

semidirect

12. #17988363

C₄

semidirect

14. #346×C₇₃

C₄

semidirect

19. #14556660

C₈

semidirect

98. C₇₃

#7178

semidirect

106. #346

#64408

semidirect

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

subgroup	index
C₄×#14556660	2
#22625473	2
#346×D₇₃	2
#7178	73

subgroup

index

C₄×#14556660

#22625473

#346×D₇₃

#7178

Subgroups — #34000279

Conjugacy classes of subgroups

Complements

Maximal subgroups