Group #32979637 — subgroups

Subgroups — #32979637

Order 18688.

262 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 107 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	#22609682	C₂	1	1	yes	maximal, characteristic
2	#22608679	C₂	1	1	yes	maximal, characteristic
3	#22607775	C₂	1	2	yes	maximal
4	#22607759	C₂	1	2	yes	maximal
5	C₇₃×#1564	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×#14556502	C₂²	1	1	yes	characteristic
8	#17988433	C₂²	1	1	yes	characteristic
9	#17988341	C₄	1	2	yes	—
10	#17988341	C₄	1	2	yes	—
11	#17988340	—	2	1	no	—
12	#17988338	—	2	1	no	—
13	#431×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	#335×C₇₃	—	2	1	no	—
16	#334×C₇₃	—	2	1	no	—
17	C₂²×#10596831	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
19	C₄⋊#10596831	—	2	2	no	—
20	C₂²⋊#10596831	—	2	2	no	—
21	#14556502	—	2	1	no	—
22	#14556502	—	2	1	no	—
23	C₄×#10596831	—	2	2	no	—
24	#14556498	—	2	1	no	—
25	#14556498	—	2	1	no	—
26	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	—	2	2	no	—
30	C₂²⋊C₈×C₇₃	—	2	2	no	—
31	C₄×C₈×C₇₃	—	2	2	no	—
32	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
36	C₂²×Dic₇₃	—	2	2	no	—
37	C₂×#10596831	—	2	2	no	—
38	C₂×#10596831	—	2	2	no	—
39	C₂×#10596831	—	2	2	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₄×Dic₇₃	—	2	2	no	—
42	C₄×Dic₇₃	—	2	2	no	—
43	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
46	C₂×C₈×C₇₃	—	2	2	no	—
47	C₂×C₈×C₇₃	—	2	2	no	—
48	C₂×C₈×C₇₃	—	4	2	no	—
49	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₇₃	#41	1	1	yes	characteristic
56	C₄²×C₇₃	#41	1	1	yes	characteristic
57	C₂³×C₇₃	#117	1	1	yes	characteristic
58	C₂×C₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#125	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#125	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#131	1	2	yes	—
67	C₂×C₄×C₇₃	#131	1	2	yes	—
68	C₂×Dic₇₃	—	4	2	no	—
69	C₂×Dic₇₃	—	4	2	no	—
70	C₂×Dic₇₃	—	4	2	no	—
71	C₈×C₇₃	—	2	1	no	—
72	C₈×C₇₃	—	2	1	no	—
73	C₈×C₇₃	—	2	1	no	—
74	C₈×C₇₃	—	2	1	no	—
75	C₈×C₇₃	—	8	2	no	—
76	#10596831	—	4	2	no	—
77	#10596831	—	8	2	no	—
78	C₂²×C₇₃	—	2	1	no	—
79	C₂²×C₇₃	—	2	1	no	—
80	C₂²×C₇₃	#382	1	1	yes	characteristic
81	C₂²×C₇₃	#433	1	2	yes	—
82	C₄×C₇₃	—	2	1	no	—
83	C₄×C₇₃	—	2	1	no	—
84	C₄×C₇₃	—	2	1	no	—
85	C₄×C₇₃	—	2	1	no	—
86	C₄×C₇₃	#424	1	1	yes	characteristic
87	C₄×C₇₃	#424	1	1	yes	characteristic
88	C₄×C₇₃	#433	1	2	yes	—
89	Dic₇₃	—	8	2	no	—
90	#8782	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	—	2	2	no	—
92	C₂×C₇₃	#1832	1	1	yes	characteristic
93	C₂×C₇₃	#1833	1	1	yes	characteristic
94	C₂×C₇₃	#1833	1	1	yes	characteristic
95	#1746	—	73	1	no	—
96	#1564	—	73	1	no	—
97	#1564	D₇₃	1	1	yes	characteristic
98	#1281	—	73	2	no	—
99	#1274	—	73	2	no	—
100	C₇₃	#8782	1	1	yes	characteristic, complemented, Sylow
101	#431	—	73	2	no	—
102	#431	C₂×D₇₃	1	2	yes	—
103	C₂×#97	—	73	1	no	—
104	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
105	#376	—	73	1	no	—
106	#336	—	73	2	no	—
107	#336	—	73	2	no	—
108	#335	—	2	1	no	—
109	#335	—	146	1	no	—
110	#334	—	2	1	no	—
111	#334	—	146	1	no	—
112	C₂²×C₈	—	73	2	no	—
113	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
115	C₂×C₄²	—	73	2	no	—
116	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	146	2	no	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	146	2	no	—
121	#97	—	146	1	no	—
122	#97	—	146	1	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	146	2	no	—
125	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
126	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
129	#95	—	146	1	no	—
130	#95	—	146	1	no	—
131	C₂²×C₄	—	146	2	no	—
132	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
133	C₂²×C₄	#11924027	1	1	yes	characteristic
134	C₂²×C₄	#11924029	1	1	yes	characteristic
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	292	2	no	—
139	C₂×C₈	—	146	2	no	—
140	C₂×C₈	—	146	2	no	—
141	C₂×C₈	—	146	2	no	—
142	C₂×C₈	#11924007	1	1	yes	characteristic
143	C₂×C₈	#11924007	1	1	yes	characteristic
144	C₂×C₈	#11924007	1	1	yes	characteristic
145	C₂×C₈	#11924007	1	1	yes	characteristic
146	C₄²	—	146	2	no	—
147	C₄²	—	146	2	no	—
148	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
149	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
150	C₄²	#11924027	1	1	yes	characteristic
151	C₄²	#11924029	1	1	yes	characteristic
152	C₂³	#14556549	1	1	yes	characteristic
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	292	2	no	—
156	C₂×C₄	—	292	2	no	—
157	C₂×C₄	—	292	2	no	—
158	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
159	C₂×C₄	#14556549	1	1	yes	characteristic
160	C₂×C₄	#14556578	1	1	yes	characteristic
161	C₂×C₄	#14556580	1	1	yes	characteristic
162	C₂×C₄	#14556582	1	1	yes	characteristic
163	C₂×C₄	#14556582	1	1	yes	characteristic
164	C₂×C₄	#14556597	1	2	yes	—
165	C₂×C₄	#14556599	1	2	yes	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	8	2	no	—
171	C₈	—	292	2	no	—
172	C₈	—	584	2	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	#17988452	1	1	yes	characteristic
176	C₂²	#17988647	1	2	yes	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	584	2	no	—
182	C₄	#17988613	1	1	yes	characteristic
183	C₄	#17988615	1	1	yes	characteristic
184	C₄	#17988647	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	#22610050	1	1	yes	characteristic
187	C₂	#22610054	1	1	yes	characteristic
188	C₂	#22610056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22609682

C₂

yes

maximal, characteristic

#22608679

C₂

yes

maximal, characteristic

#22607775

C₂

yes

maximal

#22607759

C₂

yes

maximal

C₇₃×#1564

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

—

C₂×#14556502

C₂²

yes

characteristic

#17988433

C₂²

yes

characteristic

#17988341

C₄

yes

—

#17988341

C₄

yes

—

#17988340

—

#17988338

—

#431×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#335×C₇₃

—

#334×C₇₃

—

C₂²×#10596831

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

Q₈

yes

characteristic

C₄×C₈×C₇₃

Q₈

yes

characteristic

C₂²×Dic₇₃

—

—

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

C₂²×C₄×C₇₃

yes

C₂²×C₄×C₇₃

yes

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

C₄²×C₇₃

yes

C₄²×C₇₃

yes

C₂³×C₇₃

yes

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#382

yes

characteristic

C₂²×C₇₃

#433

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#8782

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

D₇₃

yes

characteristic

#1281

—

#1274

—

100

C₇₃

#8782

yes

characteristic, complemented, Sylow

101

#431

—

102

#431

C₂×D₇₃

yes

—

103

C₂×#97

—

104

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

105

#376

—

106

#336

—

107

#336

—

108

#335

—

109

#335

—

146

—

110

#334

—

111

#334

—

146

—

112

C₂²×C₈

—

113

C₂²×C₈

C₄×D₇₃

yes

characteristic

114

C₂²×C₈

C₄×D₇₃

yes

characteristic

115

C₂×C₄²

—

116

C₂×C₄²

C₂²×D₇₃

yes

characteristic

117

C₄⋊C₈

—

118

C₄⋊C₈

—

146

—

119

C₂²⋊C₈

—

120

C₂²⋊C₈

—

146

—

121

#97

—

146

—

122

#97

—

146

—

123

C₄×C₈

—

124

C₄×C₈

—

146

—

125

C₄×C₈

Dic₁₄₆

yes

characteristic

126

C₄×C₈

Dic₁₄₆

yes

characteristic

127

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

128

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

129

#95

—

146

—

130

#95

—

146

—

131

C₂²×C₄

—

146

—

132

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

133

C₂²×C₄

#11924027

yes

characteristic

134

C₂²×C₄

#11924029

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

292

—

139

C₂×C₈

—

146

—

140

C₂×C₈

—

146

—

141

C₂×C₈

—

146

—

142

C₂×C₈

#11924007

yes

characteristic

143

C₂×C₈

#11924007

yes

characteristic

144

C₂×C₈

#11924007

yes

characteristic

145

C₂×C₈

#11924007

yes

characteristic

146

C₄²

—

146

—

147

C₄²

—

146

—

148

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

149

C₄²

C₂×Dic₁₄₆

yes

characteristic

150

C₄²

#11924027

yes

characteristic

151

C₄²

#11924029

yes

characteristic

152

C₂³

#14556549

yes

characteristic

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

292

—

156

C₂×C₄

—

292

—

157

C₂×C₄

—

292

—

158

C₂×C₄

C₂×#11924007

yes

characteristic

159

C₂×C₄

#14556549

yes

characteristic

160

C₂×C₄

#14556578

yes

characteristic

161

C₂×C₄

#14556580

yes

characteristic

162

C₂×C₄

#14556582

yes

characteristic

163

C₂×C₄

#14556582

yes

characteristic

164

C₂×C₄

#14556597

yes

—

165

C₂×C₄

#14556599

yes

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

292

—

172

C₈

—

584

—

173

C₂²

—

174

C₂²

—

175

C₂²

#17988452

yes

characteristic

176

C₂²

#17988647

yes

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

584

—

182

C₄

#17988613

yes

characteristic

183

C₄

#17988615

yes

characteristic

184

C₄

#17988647

yes

—

185

C₂

—

186

C₂

#22610050

yes

characteristic

187

C₂

#22610054

yes

characteristic

188

C₂

#22610056

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556549
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1564	C₂
socle	C₂²×C₇₃	#382
prime radical	C₂³×C₇₃	#117
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1564	D₇₃
hypocenter	C₇₃	#8782
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

subgroup

G/H

center

C₂×C₄

#14556549

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1564

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

N	H ≅ G/N	type
100. C₇₃	#8782	semidirect

H ≅ G/N

type

100. C₇₃

#8782

semidirect

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

subgroup	index
#22609682	2
#22608679	2
#22607775	2
#22607759	2
C₇₃×#1564	2
#8782	73

subgroup

index

C₇₃×#1564

Subgroups — #32979637

Conjugacy classes of subgroups

Complements

Maximal subgroups