Group #34000498 — subgroups

Subgroups — #34000498

Order 18688.

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

normal subgroups only

This group has 192 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	#22626264	C₂	1	1	yes	maximal, characteristic
2	#22625487	C₂	1	1	yes	maximal, characteristic
3	#22608693	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556666	C₄	1	2	yes	—
5	#17989591	—	2	1	no	—
6	#17989508	—	2	1	no	—
7	C₄⋊#11924031	—	2	1	no	—
8	#17989449	—	2	1	no	—
9	#17989449	C₄	1	2	yes	—
10	#17988643	—	2	1	no	—
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	#17988340	C₄	1	1	yes	characteristic
13	#17988337	—	2	1	no	—
14	#335×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924038	—	4	1	no	—
16	#14556666	—	4	2	no	—
17	C₂×#11924031	—	2	2	no	—
18	C₄⋊#10596833	—	2	2	no	—
19	C₄⋊#10596833	—	4	1	no	—
20	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
22	C₄²×D₇₃	D₄	1	1	yes	characteristic
23	C₄⋊#10596831	—	2	1	no	—
24	#14556503	—	4	1	no	—
25	#14556501	—	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	#11924038	—	8	1	no	—
32	#11924038	—	8	1	no	—
33	C₂×#10596833	—	4	2	no	—
34	#11924031	—	4	2	no	—
35	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
37	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
38	C₂×#10596831	—	4	1	no	—
39	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
41	C₈×D₇₃	—	2	1	no	—
42	C₈×D₇₃	—	2	1	no	—
43	C₈×D₇₃	—	4	1	no	—
44	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
46	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₇₃	—	4	1	no	—
48	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×D₇₃	#107	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
55	C₂×Dic₇₃	#102	1	1	yes	characteristic
56	C₂×Dic₇₃	#103	1	1	yes	characteristic
57	C₂×Dic₇₃	#106	1	1	yes	characteristic
58	C₄×D₇₃	—	2	1	no	—
59	C₄×D₇₃	—	2	1	no	—
60	C₄×D₇₃	—	2	1	no	—
61	C₄×D₇₃	—	2	1	no	—
62	C₄×D₇₃	—	4	1	no	—
63	#10596833	—	8	2	no	—
64	C₈×C₇₃	—	2	1	no	—
65	C₈×C₇₃	—	2	1	no	—
66	C₈×C₇₃	—	8	1	no	—
67	#10596831	—	2	1	no	—
68	#10596831	—	8	1	no	—
69	#10596831	#100	1	1	yes	characteristic
70	#10596831	#101	1	1	yes	characteristic
71	C₂²×C₇₃	#340	1	1	yes	characteristic
72	C₂×D₇₃	—	2	1	no	—
73	C₂×D₇₃	—	2	1	no	—
74	C₂×D₇₃	—	2	1	no	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	#338	1	1	yes	characteristic
77	C₄×C₇₃	#343	1	1	yes	characteristic
78	C₄×C₇₃	#360	1	1	yes	characteristic
79	C₄×C₇₃	#360	1	1	yes	characteristic
80	Dic₇₃	—	2	1	no	—
81	Dic₇₃	—	2	1	no	—
82	Dic₇₃	—	2	1	no	—
83	#7244	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	#1292	1	1	yes	characteristic
85	C₂×C₇₃	#1386	1	1	yes	characteristic
86	C₂×C₇₃	#1388	1	1	yes	characteristic
87	D₇₃	—	4	1	no	—
88	#2150	—	73	1	no	—
89	#1568	—	73	1	no	—
90	#1566	—	73	1	no	—
91	C₇₃	#7244	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₃₂	—	73	2	no	—
93	#432	—	146	1	no	—
94	#423	—	146	1	no	—
95	C₂×C₄×C₈	—	73	1	no	—
96	C₄⋊C₁₆	—	146	1	no	—
97	C₄⋊C₁₆	—	146	1	no	—
98	#335	—	73	1	no	—
99	#335	#64408	1	1	yes	characteristic
100	#334	—	73	2	no	—
101	#334	—	146	1	no	—
102	#334	—	146	1	no	—
103	C₂×M₁₆	—	292	1	no	—
104	C₂²×C₈	—	73	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂×C₄²	—	73	1	no	—
107	M₃₂	—	292	2	no	—
108	C₂×C₁₆	—	146	2	no	—
109	C₄⋊C₈	—	2	1	no	—
110	C₄⋊C₈	—	292	1	no	—
111	C₄⋊C₈	—	146	2	no	—
112	C₄⋊C₈	—	146	1	no	—
113	C₂²⋊C₈	—	292	1	no	—
114	#97	—	292	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₄×C₈	—	73	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	C₂×#64408	1	1	yes	characteristic
119	C₂²×C₄	—	73	1	no	—
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	73	1	no	—
122	M₁₆	—	584	1	no	—
123	M₁₆	—	584	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	292	1	no	—
126	C₂×C₈	—	292	2	no	—
127	C₂×C₈	—	292	1	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	146	1	no	—
131	C₂×C₈	—	146	1	no	—
132	C₂×C₈	C₄×#64408	1	1	yes	characteristic
133	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
134	C₄²	—	73	1	no	—
135	C₄²	—	73	1	no	—
136	C₄²	—	73	1	no	—
137	C₄²	C₂²⋊#64408	1	1	yes	characteristic
138	C₁₆	—	292	2	no	—
139	C₂³	—	73	1	no	—
140	C₂×C₄	—	292	1	no	—
141	C₂×C₄	—	73	1	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	146	1	no	—
145	C₂×C₄	—	146	1	no	—
146	C₂×C₄	—	146	1	no	—
147	C₂×C₄	—	146	1	no	—
148	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#14556653	1	1	yes	characteristic
150	C₂×C₄	#14556654	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	8	1	no	—
154	C₈	—	584	2	no	—
155	C₈	—	584	1	no	—
156	C₈	—	73	1	no	—
157	C₈	—	73	1	no	—
158	C₈	—	146	1	no	—
159	C₂²	—	146	1	no	—
160	C₂²	—	146	1	no	—
161	C₂²	—	146	1	no	—
162	C₂²	#17989460	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	146	1	no	—
165	C₄	—	146	1	no	—
166	C₄	—	146	1	no	—
167	C₄	#17989456	1	1	yes	characteristic
168	C₄	#17989466	1	1	yes	characteristic
169	C₄	#17989503	1	1	yes	characteristic
170	C₄	#17989504	1	1	yes	characteristic
171	C₂	—	292	1	no	—
172	C₂	#22624800	1	1	yes	characteristic
173	C₂	#22625002	1	1	yes	characteristic
174	C₂	#22625010	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626264

C₂

yes

maximal, characteristic

#22625487

C₂

yes

maximal, characteristic

#22608693

C₂

yes

maximal, characteristic

C₂×#14556666

C₄

yes

—

—

—

—

—

C₄

yes

—

#17988643

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988340

C₄

yes

characteristic

#17988337

—

#335×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

—

—

—

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

Q₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

characteristic

C₄×Dic₇₃

SD₁₆

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

—

C₄×D₇₃

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

—

—

yes

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2150

—

#1568

—

#1566

—

C₇₃

#7244

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

#432

—

146

—

#423

—

146

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

C₄⋊C₁₆

—

146

—

—

yes

100

—

101

#334

—

146

—

102

#334

—

146

—

103

C₂×M₁₆

—

292

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

M₃₂

—

292

—

108

C₂×C₁₆

—

146

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

292

—

111

C₄⋊C₈

—

146

—

112

C₄⋊C₈

—

146

—

113

C₂²⋊C₈

—

292

—

114

#97

—

292

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

C₂×#64408

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

M₁₆

—

584

—

123

M₁₆

—

584

—

124

C₂×C₈

—

125

C₂×C₈

—

292

—

126

C₂×C₈

—

292

—

127

C₂×C₈

—

292

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

146

—

131

C₂×C₈

—

146

—

132

C₂×C₈

C₄×#64408

yes

characteristic

133

C₂×C₈

C₄⋊#64408

yes

characteristic

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂²⋊#64408

yes

characteristic

138

C₁₆

—

292

—

139

C₂³

—

140

C₂×C₄

—

292

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

—

146

—

147

C₂×C₄

—

146

—

148

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

149

C₂×C₄

#14556653

yes

characteristic

150

C₂×C₄

#14556654

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

584

—

155

C₈

—

584

—

156

C₈

—

157

C₈

—

158

C₈

—

146

—

159

C₂²

—

146

—

160

C₂²

—

146

—

161

C₂²

—

146

—

162

C₂²

#17989460

yes

characteristic

163

C₄

—

164

C₄

—

146

—

165

C₄

—

146

—

166

C₄

—

146

—

167

C₄

#17989456

yes

characteristic

168

C₄

#17989466

yes

characteristic

169

C₄

#17989503

yes

characteristic

170

C₄

#17989504

yes

characteristic

171

C₂

—

292

—

172

C₂

#22624800

yes

characteristic

173

C₂

#22625002

yes

characteristic

174

C₂

#22625010

yes

characteristic

	subgroup	G/H
center	C₂²	#17989460
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₄×C₈	C₂×#64408
Fitting	#335×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂²×D₇₃	#107
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#335	#64408
hypocenter	C₇₃	#7244
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂²

#17989460

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₄×C₈

C₂×#64408

Fitting

#335×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 ◁ #335

N	H ≅ G/N	type
91. C₇₃	#7244	semidirect

H ≅ G/N

type

91. C₇₃

#7244

semidirect

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

subgroup	index
#22626264	2
#22625487	2
#22608693	2
#7244	73

subgroup

index

Subgroups — #34000498

Conjugacy classes of subgroups

Complements

Maximal subgroups