Group #32989093 — subgroups

Subgroups — #32989093

Order 18688.

402 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 89 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	#22618318	C₂	1	1	yes	maximal, characteristic, complemented
2	#22610759	C₂	1	1	yes	maximal, characteristic
3	#22608492	C₂	1	2	yes	maximal
4	#22607790	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989137	C₂²	1	1	yes	characteristic
7	#17988611	C₂²	1	1	yes	characteristic
8	C₂×#14556519	—	2	2	no	—
9	#17988485	—	2	2	no	—
10	#17988471	—	2	4	no	—
11	#17988453	C₂²	1	2	yes	—
12	#17988322	—	2	4	no	—
13	#431×C₇₃	C₂²	1	2	yes	—
14	C₄³×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#11924028	—	2	1	no	—
16	C₂×#11924012	—	2	1	no	—
17	#14556591	Q₈	1	1	yes	characteristic
18	#14556588	Q₈	1	1	yes	characteristic
19	#14556579	—	2	1	no	—
20	#14556565	—	2	2	no	—
21	C₄×D₂₉₂	—	2	2	no	—
22	C₂×C₄⋊Dic₇₃	—	2	2	no	—
23	C₂×#11924007	—	2	4	no	—
24	C₄⋊#10596831	—	2	2	no	—
25	#14556519	—	4	4	no	—
26	#14556498	—	2	4	no	—
27	#14556498	—	2	2	no	—
28	#14556498	D₄	1	4	yes	—
29	C₂×C₄²×C₇₃	—	2	2	no	—
30	C₂×C₄²×C₇₃	D₄	1	2	yes	—
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₇₃	—	2	2	no	—
33	C₂²⋊C₈×C₇₃	—	2	2	no	—
34	C₄×C₈×C₇₃	—	2	2	no	—
35	#11924028	—	4	2	no	—
36	#11924028	—	4	2	no	—
37	C₂×Dic₁₄₆	—	2	1	no	—
38	C₂×D₂₉₂	—	2	1	no	—
39	C₂×D₂₉₂	—	2	2	no	—
40	C₂×C₂²⋊D₇₃	—	4	1	no	—
41	C₂²×Dic₇₃	—	2	4	no	—
42	C₂×C₄×D₇₃	—	4	1	no	—
43	#11924012	—	4	2	no	—
44	C₂×#10596831	—	4	1	no	—
45	C₄⋊Dic₇₃	—	2	2	no	—
46	C₄⋊Dic₇₃	—	4	2	no	—
47	C₄⋊Dic₇₃	—	4	2	no	—
48	#11924007	—	4	8	no	—
49	#11924005	—	4	2	no	—
50	C₂²×C₄×C₇₃	—	2	2	no	—
51	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	—	4	2	no	—
54	C₄²×C₇₃	—	2	2	no	—
55	C₄²×C₇₃	—	2	2	no	—
56	C₄²×C₇₃	—	4	4	no	—
57	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₇₃	#41	1	1	yes	characteristic
59	C₄²×C₇₃	#41	1	2	yes	—
60	C₂³×C₇₃	#120	1	1	yes	characteristic
61	C₂²×D₇₃	—	4	1	no	—
62	C₂×C₄×C₇₃	—	2	4	no	—
63	C₂×C₄×C₇₃	—	4	2	no	—
64	C₂×C₄×C₇₃	—	4	2	no	—
65	C₂×C₄×C₇₃	—	4	2	no	—
66	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#124	1	2	yes	—
69	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#135	1	2	yes	—
71	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
72	C₂²⋊D₇₃	—	8	2	no	—
73	C₂×Dic₇₃	—	4	8	no	—
74	C₂×Dic₇₃	—	4	1	no	—
75	C₂×Dic₇₃	—	4	4	no	—
76	D₂₉₂	—	4	2	no	—
77	D₂₉₂	—	8	2	no	—
78	C₄×D₇₃	—	8	1	no	—
79	C₄×D₇₃	—	8	1	no	—
80	Dic₁₄₆	—	4	2	no	—
81	C₈×C₇₃	—	8	2	no	—
82	#10596831	—	8	1	no	—
83	C₂²×C₇₃	—	2	2	no	—
84	C₂²×C₇₃	#393	1	1	yes	characteristic
85	C₂²×C₇₃	#449	1	2	yes	—
86	C₂×D₇₃	—	8	1	no	—
87	C₂×D₇₃	—	8	2	no	—
88	C₄×C₇₃	—	2	2	no	—
89	C₄×C₇₃	—	2	2	no	—
90	C₄×C₇₃	—	4	4	no	—
91	C₄×C₇₃	#478	1	2	yes	—
92	C₄×C₇₃	#488	1	1	yes	characteristic
93	C₄×C₇₃	#490	1	1	yes	characteristic
94	Dic₇₃	—	8	4	no	—
95	Dic₇₃	—	8	1	no	—
96	#10366	—	73	1	no	maximal, Sylow
97	C₂×C₇₃	—	2	2	no	—
98	C₂×C₇₃	#2025	1	1	yes	characteristic
99	C₂×C₇₃	#2031	1	2	yes	—
100	D₇₃	—	16	1	no	—
101	#2864	—	73	1	no	—
102	#1962	—	73	1	no	—
103	#1517	—	73	2	no	—
104	#1286	—	73	2	no	—
105	#1276	D₇₃	1	1	yes	characteristic, complemented
106	C₇₃	#10366	1	1	yes	characteristic, complemented, Sylow
107	#532	—	73	1	no	—
108	#431	C₂×D₇₃	1	2	yes	—
109	#423	—	73	1	no	—
110	C₂×#104	—	146	2	no	—
111	#389	—	146	2	no	—
112	#387	—	146	4	no	—
113	#382	—	73	2	no	—
114	C₄³	C₂×D₇₃	1	1	yes	characteristic
115	#327	—	146	4	no	—
116	C₂×#41	—	146	1	no	—
117	C₂×M₁₆	—	146	1	no	—
118	C₄⋊Q₈	—	73	1	no	—
119	#127	—	73	1	no	—
120	#124	—	146	1	no	—
121	#121	—	146	2	no	—
122	C₄×D₄	—	146	2	no	—
123	C₂×C₄⋊C₄	—	146	4	no	—
124	C₂×C₄⋊C₄	—	146	2	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	D₂₉₂	1	2	yes	—
127	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	146	2	no	—
130	#104	—	292	4	no	—
131	C₂²⋊C₈	—	2	2	no	—
132	C₄×C₈	—	2	2	no	—
133	#95	—	73	4	no	—
134	#95	—	146	4	no	—
135	#95	—	146	2	no	—
136	#41	—	292	2	no	—
137	#41	—	292	2	no	—
138	C₂×Q₈	—	146	1	no	—
139	C₂×D₄	—	292	1	no	—
140	C₂×D₄	—	146	2	no	—
141	C₂×D₄	—	146	1	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	292	1	no	—
144	C₂²×C₄	—	146	4	no	—
145	C₂²×C₄	D₄×D₇₃	1	2	yes	—
146	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
147	M₁₆	—	292	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	292	1	no	—
150	C₄⋊C₄	—	292	8	no	—
151	C₄⋊C₄	—	292	2	no	—
152	C₄⋊C₄	—	292	2	no	—
153	C₄⋊C₄	—	146	2	no	—
154	C₂²⋊C₄	—	292	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	4	no	—
158	C₄²	Q₈×D₇₃	1	1	yes	characteristic
159	C₄²	#11924027	1	1	yes	characteristic
160	C₄²	#11924028	1	2	yes	—
161	C₂³	—	292	1	no	—
162	C₂³	#14556561	1	1	yes	characteristic
163	Q₈	—	292	2	no	—
164	D₄	—	292	2	no	—
165	D₄	—	584	2	no	—
166	D₄	—	584	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	292	8	no	—
172	C₂×C₄	—	292	4	no	—
173	C₂×C₄	—	292	1	no	—
174	C₂×C₄	—	584	1	no	—
175	C₂×C₄	—	584	1	no	—
176	C₂×C₄	#14556569	1	2	yes	—
177	C₂×C₄	#14556571	1	1	yes	characteristic
178	C₂×C₄	#14556577	1	2	yes	—
179	C₂×C₄	#14556579	1	1	yes	characteristic
180	C₂×C₄	#14556613	1	2	yes	—
181	C₂×C₄	#14556619	1	2	yes	—
182	C₈	—	8	2	no	—
183	C₈	—	584	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	584	1	no	—
186	C₂²	—	584	2	no	—
187	C₂²	#17988500	1	1	yes	characteristic
188	C₂²	#17988734	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	—	584	4	no	—
193	C₄	—	584	1	no	—
194	C₄	#17988888	1	2	yes	—
195	C₄	#17988942	1	1	yes	characteristic
196	C₄	#17988952	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	—	1168	1	no	—
199	C₂	#22611078	1	1	yes	characteristic
200	C₂	#22611104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22618318

C₂

yes

maximal, characteristic, complemented

#22610759

C₂

yes

maximal, characteristic

#22608492

C₂

yes

maximal

#22607790

C₂

yes

maximal, complemented

C₇₃×#1276

C₂

yes

maximal, characteristic, complemented

#17989137

C₂²

yes

characteristic

#17988611

C₂²

yes

—

—

—

C₂²

yes

—

#17988322

—

#431×C₇₃

C₂²

yes

—

C₄³×C₇₃

C₂²

yes

—

—

Q₈

yes

characteristic

#14556588

Q₈

yes

characteristic

#14556579

—

#14556565

—

C₄×D₂₉₂

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

—

D₄

yes

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924005

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

C₄²×C₇₃

yes

C₄²×C₇₃

yes

—

C₂³×C₇₃

#120

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

#124

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#393

yes

characteristic

C₂²×C₇₃

#449

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#478

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#10366

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2025

yes

characteristic

C₂×C₇₃

#2031

yes

—

100

D₇₃

—

101

#2864

—

102

#1962

—

103

#1517

—

104

#1286

—

105

#1276

D₇₃

yes

characteristic, complemented

106

C₇₃

#10366

yes

characteristic, complemented, Sylow

107

#532

—

108

#431

C₂×D₇₃

yes

—

109

#423

—

110

C₂×#104

—

146

—

111

#389

—

146

—

112

#387

—

146

—

113

#382

—

114

C₄³

C₂×D₇₃

yes

characteristic

115

#327

—

146

—

116

C₂×#41

—

146

—

117

C₂×M₁₆

—

146

—

118

C₄⋊Q₈

—

119

#127

—

120

#124

—

146

—

121

#121

—

146

—

122

C₄×D₄

—

146

—

123

C₂×C₄⋊C₄

—

146

—

124

C₂×C₄⋊C₄

—

146

—

125

C₂×C₄²

—

126

C₂×C₄²

D₂₉₂

yes

—

127

C₂×C₄²

C₂²×D₇₃

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

146

—

130

#104

—

292

—

131

C₂²⋊C₈

—

132

C₄×C₈

—

133

#95

—

134

#95

—

146

—

135

#95

—

146

—

136

#41

—

292

—

137

#41

—

292

—

138

C₂×Q₈

—

146

—

139

C₂×D₄

—

292

—

140

C₂×D₄

—

146

—

141

C₂×D₄

—

146

—

142

C₂²×C₄

—

143

C₂²×C₄

—

292

—

144

C₂²×C₄

—

146

—

145

C₂²×C₄

D₄×D₇₃

yes

—

146

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

147

M₁₆

—

292

—

148

C₂×C₈

—

149

C₂×C₈

—

292

—

150

C₄⋊C₄

—

292

—

151

C₄⋊C₄

—

292

—

152

C₄⋊C₄

—

292

—

153

C₄⋊C₄

—

146

—

154

C₂²⋊C₄

—

292

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

Q₈×D₇₃

yes

characteristic

159

C₄²

#11924027

yes

characteristic

160

C₄²

#11924028

yes

—

161

C₂³

—

292

—

162

C₂³

#14556561

yes

characteristic

163

Q₈

—

292

—

164

D₄

—

292

—

165

D₄

—

584

—

166

D₄

—

584

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

292

—

172

C₂×C₄

—

292

—

173

C₂×C₄

—

292

—

174

C₂×C₄

—

584

—

175

C₂×C₄

—

584

—

176

C₂×C₄

#14556569

yes

—

177

C₂×C₄

#14556571

yes

characteristic

178

C₂×C₄

#14556577

yes

—

179

C₂×C₄

#14556579

yes

characteristic

180

C₂×C₄

#14556613

yes

—

181

C₂×C₄

#14556619

yes

—

182

C₈

—

183

C₈

—

584

—

184

C₂²

—

185

C₂²

—

584

—

186

C₂²

—

584

—

187

C₂²

#17988500

yes

characteristic

188

C₂²

#17988734

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

584

—

193

C₄

—

584

—

194

C₄

#17988888

yes

—

195

C₄

#17988942

yes

characteristic

196

C₄

#17988952

yes

characteristic

197

C₂

—

198

C₂

—

1168

—

199

C₂

#22611078

yes

characteristic

200

C₂

#22611104

yes

—

	subgroup	G/H
center	C₂²	#17988500
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1276	C₂
socle	C₂²×C₇₃	#393
prime radical	#17989137	C₂²
hypercenter	#1276	D₇₃
hypocenter	C₇₃	#10366
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#17988500

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1276

C₂

C₂²×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
1. #22618318	C₂	semidirect
4. #22607790	C₂	semidirect
5. C₇₃×#1276	C₂	semidirect
105. #1276	D₇₃	semidirect
106. C₇₃	#10366	semidirect

H ≅ G/N

type

1. #22618318

C₂

semidirect

4. #22607790

C₂

semidirect

5. C₇₃×#1276

C₂

semidirect

105. #1276

D₇₃

semidirect

106. C₇₃

#10366

semidirect

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

subgroup	index
#22618318	2
#22610759	2
#22608492	2
#22607790	2
C₇₃×#1276	2
#10366	73

subgroup

index

C₇₃×#1276

Subgroups — #32989093

Conjugacy classes of subgroups

Complements

Maximal subgroups