Group #34005428 — subgroups

Subgroups — #34005428

Order 18688.

378 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 144 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	#22625863	C₂	1	1	yes	maximal, characteristic
2	#22625846	C₂	1	1	yes	maximal, characteristic
3	#22624761	C₂	1	4	yes	maximal
4	#97×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596833	—	2	2	no	—
6	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	#17989453	—	2	2	no	—
9	C₂×#14556502	C₄	1	1	yes	characteristic
10	C₂×#14556501	C₄	1	2	yes	—
11	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
12	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
14	C₂²×#10596833	—	2	2	no	—
15	C₂²×#10596833	—	2	2	no	—
16	C₄⋊#10596833	—	2	4	no	—
17	C₄⋊#10596833	—	2	4	no	—
18	C₄⋊#10596833	—	4	4	no	—
19	C₂²×#10596831	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
22	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
23	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
24	#14556502	C₈	1	4	yes	complemented
25	#14556501	—	2	4	no	—
26	C₈×Dic₇₃	—	2	2	no	—
27	C₈×Dic₇₃	—	2	2	no	—
28	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
30	#97×C₇₃	—	2	2	no	—
31	C₂×#10596833	—	4	2	no	—
32	C₂×#10596833	—	4	2	no	—
33	C₂×#10596833	—	4	2	no	—
34	C₂×#10596833	—	4	4	no	—
35	C₂×#10596833	—	4	2	no	—
36	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
37	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×#10596831	—	2	2	no	—
40	C₂×#10596831	—	2	2	no	—
41	C₂×#10596831	C₂×C₈	1	4	yes	—
42	C₄×Dic₇₃	—	2	2	no	—
43	C₄×Dic₇₃	—	2	2	no	—
44	C₄×Dic₇₃	—	2	2	no	—
45	C₄×Dic₇₃	—	4	2	no	—
46	C₄×Dic₇₃	#41	1	2	yes	—
47	C₄×Dic₇₃	#41	1	2	yes	—
48	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	—	2	2	no	—
51	C₂×C₈×C₇₃	—	2	2	no	—
52	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
53	C₄²×C₇₃	C₂×C₈	1	2	yes	—
54	C₄²×C₇₃	M₁₆	1	2	yes	—
55	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
59	C₂×C₄×C₇₃	#117	1	2	yes	—
60	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
63	C₂×Dic₇₃	—	2	2	no	—
64	C₂×Dic₇₃	—	2	2	no	—
65	C₂×Dic₇₃	—	2	2	no	—
66	C₂×Dic₇₃	—	2	2	no	—
67	C₂×Dic₇₃	#124	1	1	yes	characteristic
68	C₂×Dic₇₃	#124	1	1	yes	characteristic
69	C₂×Dic₇₃	#125	1	2	yes	—
70	C₂×Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₇₃	#137	1	2	yes	—
72	#10596833	—	8	2	no	—
73	#10596833	—	8	2	no	—
74	C₈×C₇₃	—	2	4	no	—
75	#10596831	—	2	4	no	—
76	C₂²×C₇₃	#382	1	1	yes	characteristic
77	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₂²×C₇₃	#423	1	1	yes	characteristic
79	C₂²×C₇₃	#428	1	2	yes	—
80	C₂²×C₇₃	#430	1	2	yes	—
81	C₄×C₇₃	—	2	4	no	—
82	C₄×C₇₃	#431	1	2	yes	—
83	C₄×C₇₃	#432	1	2	yes	—
84	Dic₇₃	—	2	2	no	—
85	Dic₇₃	—	4	2	no	—
86	Dic₇₃	#489	1	2	yes	—
87	Dic₇₃	#490	1	1	yes	characteristic
88	Dic₇₃	#491	1	1	yes	characteristic
89	#8786	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	#1565	1	2	yes	—
91	C₂×C₇₃	#1569	1	2	yes	—
92	C₂×C₇₃	#1831	1	1	yes	characteristic
93	C₂×C₇₃	#1836	1	1	yes	characteristic
94	C₂×C₇₃	#1838	1	1	yes	characteristic
95	#1842	—	73	1	no	—
96	#1831	—	73	1	no	—
97	C₄×#97	—	73	1	no	—
98	#1277	—	73	4	no	—
99	C₇₃	#8786	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊C₈	—	73	2	no	—
101	C₂×C₄⋊C₈	—	73	2	no	—
102	C₂×C₄⋊C₈	—	146	2	no	—
103	C₂×#97	—	73	1	no	—
104	C₂×#97	—	73	2	no	—
105	C₂×#97	#64408	1	1	yes	characteristic
106	C₂×C₄×C₈	—	73	2	no	—
107	C₄³	—	73	1	no	—
108	#336	—	146	2	no	—
109	C₂²×C₈	—	73	2	no	—
110	C₂²×C₈	—	146	2	no	—
111	C₂²×C₈	—	146	2	no	—
112	C₂²×C₈	C₂×#64408	1	2	yes	—
113	C₂×C₄²	—	73	2	no	—
114	C₂×C₄²	—	73	1	no	—
115	C₂×C₄²	—	73	2	no	—
116	C₂×C₄²	—	73	1	no	—
117	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
118	C₄⋊C₈	—	292	4	no	—
119	C₄⋊C₈	—	146	4	no	—
120	C₄⋊C₈	—	146	4	no	—
121	#97	—	2	2	no	—
122	#97	—	73	4	no	—
123	#97	—	146	4	no	—
124	C₄×C₈	—	146	2	no	—
125	C₄×C₈	—	146	2	no	—
126	C₂²×C₄	—	73	1	no	—
127	C₂²×C₄	—	73	1	no	—
128	C₂²×C₄	—	73	2	no	—
129	C₂²×C₄	C₄⋊#64408	1	2	yes	—
130	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	292	4	no	—
134	C₂×C₈	—	292	2	no	—
135	C₂×C₈	—	292	2	no	—
136	C₂×C₈	—	292	2	no	—
137	C₂×C₈	—	292	2	no	—
138	C₂×C₈	—	73	4	no	—
139	C₂×C₈	—	146	2	no	—
140	C₂×C₈	—	146	2	no	—
141	C₂×C₈	#11924038	1	4	yes	—
142	C₄²	—	292	2	no	—
143	C₄²	—	73	2	no	—
144	C₄²	—	73	2	no	—
145	C₄²	—	146	2	no	—
146	C₄²	—	146	2	no	—
147	C₄²	—	146	2	no	—
148	C₄²	#11924036	1	2	yes	—
149	C₄²	#11924037	1	2	yes	—
150	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	73	1	no	—
154	C₂×C₄	—	73	1	no	—
155	C₂×C₄	—	73	2	no	—
156	C₂×C₄	—	73	2	no	—
157	C₂×C₄	—	73	2	no	—
158	C₂×C₄	—	146	2	no	—
159	C₂×C₄	—	146	2	no	—
160	C₂×C₄	—	146	2	no	—
161	C₂×C₄	—	146	2	no	—
162	C₂×C₄	#14556658	1	4	yes	—
163	C₂×C₄	#14556670	1	2	yes	—
164	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
165	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
166	C₂×C₄	C₂×#11924038	1	2	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	584	2	no	—
169	C₈	—	584	2	no	—
170	C₈	—	146	4	no	—
171	C₂²	#17989531	1	1	yes	characteristic
172	C₂²	C₂×#14556658	1	1	yes	characteristic
173	C₂²	#17989591	1	1	yes	characteristic
174	C₂²	#17989598	1	2	yes	—
175	C₂²	#17989602	1	2	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	292	2	no	—
178	C₄	—	73	1	no	—
179	C₄	—	73	2	no	—
180	C₄	—	73	1	no	—
181	C₄	—	146	2	no	—
182	C₄	#17989606	1	2	yes	—
183	C₄	#17989608	1	2	yes	—
184	C₂	#22625484	1	2	yes	—
185	C₂	#22625496	1	2	yes	—
186	C₂	#22625848	1	1	yes	characteristic
187	C₂	#22625854	1	1	yes	characteristic
188	C₂	#22625857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625863

C₂

yes

maximal, characteristic

#22625846

C₂

yes

maximal, characteristic

#22624761

C₂

yes

maximal

#97×Dic₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596833

—

C₂×C₄⋊#10596833

C₂²

yes

—

C₂×C₄⋊#10596833

C₂²

yes

—

#17989453

—

C₂×#14556502

C₄

yes

characteristic

C₂×#14556501

C₄

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

—

C₄²×Dic₇₃

C₂²

yes

characteristic

C₂×#97×C₇₃

C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

—

C₂×C₄×Dic₇₃

Q₈

yes

—

C₂×C₄×Dic₇₃

C₂³

yes

characteristic

#14556502

C₈

yes

complemented

#14556501

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

#97×C₇₃

—

—

—

—

—

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

C₂²×Dic₇₃

C₂×D₄

yes

characteristic

C₂²×Dic₇₃

C₂×Q₈

yes

—

—

C₂×C₈

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

#41

yes

—

C₄×Dic₇₃

#41

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

M₁₆

yes

—

C₄²×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

M₁₆

yes

—

C₂³×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

—

C₂×Dic₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

#137

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#382

yes

characteristic

C₂²×C₇₃

C₂×C₄⋊C₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#430

yes

—

C₄×C₇₃

—

C₄×C₇₃

#431

yes

—

C₄×C₇₃

#432

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#489

yes

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

—

C₂×C₇₃

#1569

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

C₄×#97

—

#1277

—

C₇₃

#8786

yes

characteristic, complemented, Sylow

100

C₂×C₄⋊C₈

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄⋊C₈

—

146

—

103

C₂×#97

—

104

C₂×#97

—

105

C₂×#97

#64408

yes

characteristic

106

C₂×C₄×C₈

—

107

C₄³

—

108

#336

—

146

—

109

C₂²×C₈

—

110

C₂²×C₈

—

146

—

111

C₂²×C₈

—

146

—

112

C₂²×C₈

C₂×#64408

yes

—

113

C₂×C₄²

—

114

C₂×C₄²

—

115

C₂×C₄²

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂×#64408

yes

characteristic

118

C₄⋊C₈

—

292

—

119

C₄⋊C₈

—

146

—

120

C₄⋊C₈

—

146

—

121

#97

—

122

#97

—

123

#97

—

146

—

124

C₄×C₈

—

146

—

125

C₄×C₈

—

146

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₄⋊#64408

yes

—

130

C₂²×C₄

C₂²×#64408

yes

characteristic

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

292

—

134

C₂×C₈

—

292

—

135

C₂×C₈

—

292

—

136

C₂×C₈

—

292

—

137

C₂×C₈

—

292

—

138

C₂×C₈

—

139

C₂×C₈

—

146

—

140

C₂×C₈

—

146

—

141

C₂×C₈

#11924038

yes

—

142

C₄²

—

292

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

—

146

C₄²

—

146

—

147

C₄²

—

146

—

148

C₄²

#11924036

yes

—

149

C₄²

#11924037

yes

—

150

C₂³

C₂×C₄⋊#64408

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

146

—

159

C₂×C₄

—

146

—

160

C₂×C₄

—

146

—

161

C₂×C₄

—

146

—

162

C₂×C₄

#14556658

yes

—

163

C₂×C₄

#14556670

yes

—

164

C₂×C₄

C₂×#11924036

yes

characteristic

165

C₂×C₄

C₂×#11924037

yes

characteristic

166

C₂×C₄

C₂×#11924038

yes

—

167

C₈

—

168

C₈

—

584

—

169

C₈

—

584

—

170

C₈

—

146

—

171

C₂²

#17989531

yes

characteristic

172

C₂²

C₂×#14556658

yes

characteristic

173

C₂²

#17989591

yes

characteristic

174

C₂²

#17989598

yes

—

175

C₂²

#17989602

yes

—

176

C₄

—

177

C₄

—

292

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

146

—

182

C₄

#17989606

yes

—

183

C₄

#17989608

yes

—

184

C₂

#22625484

yes

—

185

C₂

#22625496

yes

—

186

C₂

#22625848

yes

characteristic

187

C₂

#22625854

yes

characteristic

188

C₂

#22625857

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#97×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	#8786
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×C₄⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

C₂×#97×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₇₃

C₂³

C₂×#97

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
24. #14556502	C₈	semidirect
99. C₇₃	#8786	semidirect

H ≅ G/N

type

24. #14556502

C₈

semidirect

99. C₇₃

#8786

semidirect

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

subgroup	index
#22625863	2
#22625846	2
#22624761	2
#97×Dic₇₃	2
#8786	73

subgroup

index

#97×Dic₇₃

Subgroups — #34005428

Conjugacy classes of subgroups

Complements

Maximal subgroups