Group #34000497 — subgroups

Subgroups — #34000497

Order 18688.

198 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 64 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	#22625494	C₂	1	1	yes	maximal, characteristic
3	#22608685	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556664	C₄	1	2	yes	—
5	#17989609	—	2	1	no	—
6	#17989508	—	2	1	no	—
7	C₄⋊#11924031	—	2	1	no	—
8	#17989451	C₄	1	2	yes	—
9	#17989449	—	2	1	no	—
10	#17988609	—	2	1	no	—
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	#17988338	C₄	1	1	yes	characteristic
13	#17988337	—	2	1	no	—
14	#334×C₇₃	C₄	1	1	yes	characteristic
15	#14556664	—	4	2	no	—
16	C₂×#11924031	—	2	2	no	—
17	C₄⋊#10596833	—	2	2	no	—
18	#14556648	—	4	1	no	—
19	#14556646	—	4	1	no	—
20	C₂×#11924011	—	4	1	no	—
21	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
23	C₄²×D₇₃	D₄	1	1	yes	characteristic
24	C₄⋊#10596831	—	2	1	no	—
25	#14556520	—	4	1	no	—
26	C₄×#10596831	D₄	1	1	yes	characteristic
27	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₇₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	—	2	1	no	—
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#10596833	—	4	2	no	—
32	#11924031	—	4	2	no	—
33	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
36	#11924011	—	8	2	no	—
37	C₂×#10596831	—	4	1	no	—
38	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
40	C₈×D₇₃	—	2	1	no	—
41	C₈×D₇₃	—	2	1	no	—
42	C₈×D₇₃	—	4	1	no	—
43	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×Dic₇₃	D₈	1	1	yes	characteristic
45	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₃	—	4	1	no	—
47	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×D₇₃	#106	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
54	C₂×Dic₇₃	#102	1	1	yes	characteristic
55	C₂×Dic₇₃	#103	1	1	yes	characteristic
56	C₂×Dic₇₃	#107	1	1	yes	characteristic
57	C₄×D₇₃	—	2	2	no	—
58	C₄×D₇₃	—	2	1	no	—
59	C₄×D₇₃	—	2	1	no	—
60	C₄×D₇₃	—	2	2	no	—
61	#10596833	—	8	2	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	—	8	1	no	—
65	#10596831	—	2	1	no	—
66	#10596831	—	8	1	no	—
67	#10596831	#100	1	1	yes	characteristic
68	#10596831	#101	1	1	yes	characteristic
69	C₂²×C₇₃	#340	1	1	yes	characteristic
70	C₂×D₇₃	—	2	1	no	—
71	C₂×D₇₃	—	2	1	no	—
72	C₂×D₇₃	#365	1	2	yes	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	#338	1	1	yes	characteristic
76	C₄×C₇₃	#343	1	1	yes	characteristic
77	Dic₇₃	—	2	1	no	—
78	Dic₇₃	—	2	1	no	—
79	Dic₇₃	#359	1	2	yes	—
80	#7243	—	73	1	no	maximal, Sylow
81	C₂×C₇₃	#1292	1	1	yes	characteristic
82	C₂×C₇₃	#1382	1	1	yes	characteristic
83	C₂×C₇₃	#1383	1	1	yes	characteristic
84	D₇₃	—	2	2	no	—
85	C₂×C₄⋊C₁₆	—	73	1	no	—
86	#1568	—	73	1	no	—
87	#1566	—	73	1	no	—
88	C₇₃	#7243	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₁₆	—	73	2	no	—
90	#432	—	146	1	no	—
91	#423	—	146	1	no	—
92	C₂×C₄×C₈	—	73	1	no	—
93	C₄⋊C₁₆	—	146	1	no	—
94	C₄⋊C₁₆	—	146	1	no	—
95	#335	—	73	2	no	—
96	#334	—	73	1	no	—
97	#334	—	146	1	no	—
98	#334	—	146	1	no	—
99	#334	#64408	1	1	yes	characteristic
100	C₂×M₁₆	—	292	1	no	—
101	C₂²×C₈	—	73	1	no	—
102	C₂²×C₈	—	73	1	no	—
103	C₂×C₄²	—	73	1	no	—
104	C₂×C₁₆	—	292	2	no	—
105	C₂×C₁₆	—	146	2	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	292	1	no	—
108	C₄⋊C₈	—	146	2	no	—
109	C₄⋊C₈	—	146	1	no	—
110	C₂²⋊C₈	—	292	1	no	—
111	#97	—	292	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₄×C₈	—	73	1	no	—
114	C₄×C₈	—	73	1	no	—
115	C₄×C₈	C₂×#64408	1	1	yes	characteristic
116	C₂²×C₄	—	73	1	no	—
117	C₂²×C₄	—	73	1	no	—
118	C₂²×C₄	—	73	1	no	—
119	M₁₆	—	584	2	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	292	1	no	—
122	C₂×C₈	—	292	2	no	—
123	C₂×C₈	—	292	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	—	146	1	no	—
127	C₂×C₈	—	146	1	no	—
128	C₂×C₈	C₄×#64408	1	1	yes	characteristic
129	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
130	C₄²	—	73	1	no	—
131	C₄²	—	73	1	no	—
132	C₄²	—	73	1	no	—
133	C₄²	C₂²⋊#64408	1	1	yes	characteristic
134	C₁₆	—	292	2	no	—
135	C₂³	—	73	1	no	—
136	C₂×C₄	—	73	1	no	—
137	C₂×C₄	—	73	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	146	2	no	—
140	C₂×C₄	—	146	2	no	—
141	C₂×C₄	—	146	1	no	—
142	C₂×C₄	—	146	1	no	—
143	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#14556653	1	1	yes	characteristic
145	C₂×C₄	#14556654	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	8	1	no	—
149	C₈	—	584	1	no	—
150	C₈	—	584	2	no	—
151	C₈	—	73	1	no	—
152	C₈	—	73	1	no	—
153	C₈	—	146	1	no	—
154	C₂²	—	73	2	no	—
155	C₂²	—	146	1	no	—
156	C₂²	—	146	1	no	—
157	C₂²	#17989460	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	73	2	no	—
161	C₄	—	146	1	no	—
162	C₄	—	146	1	no	—
163	C₄	#17989456	1	1	yes	characteristic
164	C₄	#17989466	1	1	yes	characteristic
165	C₂	—	146	2	no	—
166	C₂	#22624800	1	1	yes	characteristic
167	C₂	#22624994	1	1	yes	characteristic
168	C₂	#22624995	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626261

C₂

yes

maximal, characteristic

#22625494

C₂

yes

maximal, characteristic

#22608685

C₂

yes

maximal, characteristic

C₂×#14556664

C₄

yes

—

—

—

—

C₄

yes

—

#17989449

—

#17988609

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988338

C₄

yes

characteristic

#17988337

—

#334×C₇₃

C₄

yes

—

—

—

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×#10596833

—

#11924031

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

D₈

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

—

C₄×D₇₃

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

—

—

yes

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#365

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#359

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

C₂×C₄⋊C₁₆

—

#1568

—

#1566

—

C₇₃

#7243

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#432

—

146

—

#423

—

146

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

C₄⋊C₁₆

—

146

—

#335

—

#334

—

#334

—

146

—

#334

—

146

—

#334

#64408

yes

characteristic

100

C₂×M₁₆

—

292

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂×C₄²

—

104

C₂×C₁₆

—

292

—

105

C₂×C₁₆

—

146

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

292

—

108

C₄⋊C₈

—

146

—

109

C₄⋊C₈

—

146

—

110

C₂²⋊C₈

—

292

—

111

#97

—

292

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

C₂×#64408

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

584

—

120

C₂×C₈

—

121

C₂×C₈

—

292

—

122

C₂×C₈

—

292

—

123

C₂×C₈

—

292

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

146

—

127

C₂×C₈

—

146

—

128

C₂×C₈

C₄×#64408

yes

characteristic

129

C₂×C₈

C₄⋊#64408

yes

characteristic

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂²⋊#64408

yes

characteristic

134

C₁₆

—

292

—

135

C₂³

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

—

146

—

142

C₂×C₄

—

146

—

143

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

144

C₂×C₄

#14556653

yes

characteristic

145

C₂×C₄

#14556654

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

584

—

150

C₈

—

584

—

151

C₈

—

152

C₈

—

153

C₈

—

146

—

154

C₂²

—

155

C₂²

—

146

—

156

C₂²

—

146

—

157

C₂²

#17989460

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

146

—

162

C₄

—

146

—

163

C₄

#17989456

yes

characteristic

164

C₄

#17989466

yes

characteristic

165

C₂

—

146

—

166

C₂

#22624800

yes

characteristic

167

C₂

#22624994

yes

characteristic

168

C₂

#22624995

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	#106
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#334	#64408
hypocenter	C₇₃	#7243
LCS	G ▷ C₂×C₈×C₇₃ ▷ 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 ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

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

H ≅ G/N

type

88. C₇₃

#7243

semidirect

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

subgroup	index
#22626261	2
#22625494	2
#22608685	2
#7243	73

subgroup

index

Subgroups — #34000497

Conjugacy classes of subgroups

Complements

Maximal subgroups