Group #34000496 — subgroups

Subgroups — #34000496

Order 18688.

198 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 68 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	#22625491	C₂	1	1	yes	maximal, characteristic
2	#22625485	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988405	C₂	1	1	yes	maximal, characteristic
4	#17989608	—	2	1	no	—
5	#17989590	—	2	1	no	—
6	#17989451	C₄	1	2	yes	—
7	#17989449	C₄	1	2	yes	—
8	#17989448	—	2	1	no	—
9	#17989448	—	2	1	no	—
10	C₂²×#11924003	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
12	#17988405	—	2	1	no	—
13	#17988405	—	2	1	no	—
14	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924037	—	4	1	no	—
16	#14556658	—	4	1	no	—
17	C₄⋊#10596833	—	2	2	no	—
18	C₄⋊#10596833	—	2	2	no	—
19	C₂²⋊#10596833	—	4	1	no	—
20	#14556646	—	4	1	no	—
21	C₂²×#10596831	Q₈	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
23	C₂×#11924003	—	2	1	no	—
24	C₂×#11924003	—	4	1	no	—
25	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₃	D₄	1	1	yes	characteristic
28	C₈×Dic₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	—	2	1	no	—
31	C₂×C₁₆×C₇₃	—	4	1	no	—
32	#11924037	—	8	1	no	—
33	#11924037	—	8	1	no	—
34	C₂×#10596833	—	4	2	no	—
35	C₂×#10596833	—	4	2	no	—
36	C₂²×Dic₇₃	SD₁₆	1	1	yes	characteristic
37	C₂²×Dic₇₃	SD₁₆	1	1	yes	characteristic
38	C₂×#10596831	—	4	1	no	—
39	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₇₃	D₈	1	1	yes	characteristic
44	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
45	#11924003	—	4	1	no	—
46	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₃	—	2	1	no	—
48	C₂×C₈×C₇₃	—	2	1	no	—
49	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₁₆×C₇₃	—	4	1	no	—
52	C₂³×C₇₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	—	2	2	no	—
60	C₂×Dic₇₃	#102	1	1	yes	characteristic
61	C₂×Dic₇₃	#102	1	1	yes	characteristic
62	C₂×Dic₇₃	#103	1	1	yes	characteristic
63	C₂×Dic₇₃	#103	1	1	yes	characteristic
64	#10596833	—	8	2	no	—
65	#10596833	—	8	2	no	—
66	C₈×C₇₃	—	2	1	no	—
67	C₈×C₇₃	#100	1	1	yes	characteristic
68	C₈×C₇₃	#101	1	1	yes	characteristic
69	#10596831	—	2	1	no	—
70	#10596831	—	2	1	no	—
71	C₂²×C₇₃	—	2	1	no	—
72	C₂²×C₇₃	—	2	1	no	—
73	C₂²×C₇₃	#340	1	1	yes	characteristic
74	C₂²×C₇₃	#365	1	1	yes	characteristic
75	C₂²×C₇₃	#365	1	1	yes	characteristic
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	#338	1	1	yes	characteristic
78	C₄×C₇₃	#343	1	1	yes	characteristic
79	Dic₇₃	—	2	1	no	—
80	Dic₇₃	—	2	1	no	—
81	Dic₇₃	—	2	1	no	—
82	Dic₇₃	#359	1	2	yes	—
83	#7243	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	—	2	1	no	—
85	C₂×C₇₃	—	2	1	no	—
86	C₂×C₇₃	#1292	1	1	yes	characteristic
87	C₂×C₇₃	#1382	1	1	yes	characteristic
88	C₂×C₇₃	#1383	1	1	yes	characteristic
89	C₂×C₄⋊C₁₆	—	73	1	no	—
90	#1568	—	73	1	no	—
91	#1566	—	73	1	no	—
92	C₇₃	#7243	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₁₆	—	73	1	no	—
94	C₂²×C₁₆	#64408	1	1	yes	characteristic
95	#432	—	146	1	no	—
96	#423	—	146	1	no	—
97	C₂×C₄×C₈	—	73	1	no	—
98	C₄⋊C₁₆	—	146	1	no	—
99	C₄⋊C₁₆	—	146	1	no	—
100	#335	—	73	2	no	—
101	#334	—	73	2	no	—
102	#334	—	146	1	no	—
103	#334	—	146	1	no	—
104	C₂×M₁₆	—	292	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
107	C₂×C₄²	—	73	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	4	1	no	—
110	C₂×C₁₆	—	292	1	no	—
111	C₂×C₁₆	—	146	1	no	—
112	C₄⋊C₈	—	292	1	no	—
113	C₄⋊C₈	—	146	2	no	—
114	C₄⋊C₈	—	146	2	no	—
115	C₂²⋊C₈	—	292	1	no	—
116	#97	—	292	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	—	73	1	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₂²×C₄	—	73	1	no	—
122	C₂²×C₄	—	73	1	no	—
123	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
124	M₁₆	—	584	1	no	—
125	M₁₆	—	584	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	292	2	no	—
129	C₂×C₈	—	292	2	no	—
130	C₂×C₈	—	292	1	no	—
131	C₂×C₈	—	73	1	no	—
132	C₂×C₈	—	73	1	no	—
133	C₂×C₈	C₄×#64408	1	1	yes	characteristic
134	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
135	C₄²	—	73	1	no	—
136	C₄²	—	73	1	no	—
137	C₄²	—	73	1	no	—
138	C₄²	—	73	1	no	—
139	C₁₆	—	4	1	no	—
140	C₁₆	—	292	1	no	—
141	C₂³	#14556659	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	146	2	no	—
149	C₂×C₄	—	146	1	no	—
150	C₂×C₄	—	146	1	no	—
151	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
152	C₂×C₄	#14556660	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	584	2	no	—
155	C₈	—	584	2	no	—
156	C₈	—	146	1	no	—
157	C₈	—	146	1	no	—
158	C₈	#14556651	1	1	yes	characteristic
159	C₈	#14556652	1	1	yes	characteristic
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	#17989461	1	1	yes	characteristic
163	C₂²	#17989511	1	1	yes	characteristic
164	C₂²	#17989512	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	73	2	no	—
167	C₄	—	146	1	no	—
168	C₄	—	146	1	no	—
169	C₄	—	146	1	no	—
170	C₄	#17989456	1	1	yes	characteristic
171	C₄	#17989467	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#22624801	1	1	yes	characteristic
175	C₂	#22624993	1	1	yes	characteristic
176	C₂	#22624996	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625491

C₂

yes

maximal, characteristic

#22625485

C₂

yes

maximal, characteristic

C₂×#17988405

C₂

yes

maximal, characteristic

#17989608

—

#17989590

—

#17989451

C₄

yes

—

#17989449

C₄

yes

—

#17989448

—

#17989448

—

C₂²×#11924003

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988405

—

#17988405

—

C₂²×C₁₆×C₇₃

C₄

yes

—

—

—

—

—

—

Q₈

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

characteristic

C₂×#11924003

—

C₂×#11924003

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

—

—

—

—

C₂²×Dic₇₃

SD₁₆

yes

characteristic

C₂²×Dic₇₃

SD₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

Q₁₆

yes

characteristic

#11924003

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

—

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#359

yes

—

#7243

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₄⋊C₁₆

—

#1568

—

#1566

—

C₇₃

#7243

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

#64408

yes

characteristic

#432

—

146

—

#423

—

146

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

C₄⋊C₁₆

—

146

—

100

#335

—

101

#334

—

102

#334

—

146

—

103

#334

—

146

—

104

C₂×M₁₆

—

292

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×#64408

yes

characteristic

107

C₂×C₄²

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

292

—

111

C₂×C₁₆

—

146

—

112

C₄⋊C₈

—

292

—

113

C₄⋊C₈

—

146

—

114

C₄⋊C₈

—

146

—

115

C₂²⋊C₈

—

292

—

116

#97

—

292

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₄⋊#64408

yes

characteristic

124

M₁₆

—

584

—

125

M₁₆

—

584

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

292

—

129

C₂×C₈

—

292

—

130

C₂×C₈

—

292

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

C₄×#64408

yes

characteristic

134

C₂×C₈

C₂²⋊#64408

yes

characteristic

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₁₆

—

140

C₁₆

—

292

—

141

C₂³

#14556659

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

146

—

149

C₂×C₄

—

146

—

150

C₂×C₄

—

146

—

151

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

152

C₂×C₄

#14556660

yes

characteristic

153

C₈

—

154

C₈

—

584

—

155

C₈

—

584

—

156

C₈

—

146

—

157

C₈

—

146

—

158

C₈

#14556651

yes

characteristic

159

C₈

#14556652

yes

characteristic

160

C₂²

—

161

C₂²

—

162

C₂²

#17989461

yes

characteristic

163

C₂²

#17989511

yes

characteristic

164

C₂²

#17989512

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

146

—

168

C₄

—

146

—

169

C₄

—

146

—

170

C₄

#17989456

yes

characteristic

171

C₄

#17989467

yes

characteristic

172

C₂

—

173

C₂

—

174

C₂

#22624801

yes

characteristic

175

C₂

#22624993

yes

characteristic

176

C₂

#22624996

yes

characteristic

	subgroup	G/H
center	C₂²	#17989461
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#106
cosocle	C₂×C₈×Dic₇₃	C₂²
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	#7243
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

#17989461

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂²×C₈

C₂×#64408

Fitting

C₂²×C₁₆×C₇₃

C₄

C₂²×C₇₃

C₂³×C₇₃

C₂×C₈×Dic₇₃

C₂²

C₂²×C₁₆

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
92. C₇₃	#7243	semidirect

H ≅ G/N

type

92. C₇₃

#7243

semidirect

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

subgroup	index
#22625491	2
#22625485	2
C₂×#17988405	2
#7243	73

subgroup

index

Subgroups — #34000496

Conjugacy classes of subgroups

Complements

Maximal subgroups