Group #32991100 — subgroups

Subgroups — #32991100

Order 18688.

442 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 165 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	#22610892	C₂	1	2	yes	maximal
2	#22610733	C₂	1	1	yes	maximal, characteristic
3	#22610088	C₂	1	1	yes	maximal, characteristic
4	#22610088	C₂	1	2	yes	maximal
5	C₇₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	C₂²	1	1	yes	characteristic
7	C₂×#14556523	—	2	2	no	—
8	C₂×#14556523	—	2	4	no	—
9	C₂×#14556516	—	2	2	no	—
10	#17988457	C₂²	1	1	yes	characteristic
11	#17988457	C₂²	1	2	yes	—
12	#17988350	—	2	4	no	—
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
15	C₂²×Dic₁₄₆	—	2	1	no	—
16	#14556591	—	2	2	no	—
17	#14556591	—	2	1	no	—
18	#14556591	Q₈	1	2	yes	—
19	C₂×C₄⋊Dic₇₃	—	2	2	no	—
20	C₂×C₄⋊Dic₇₃	—	2	2	no	—
21	C₂×C₄⋊Dic₇₃	—	2	2	no	—
22	C₂×C₄⋊Dic₇₃	—	2	2	no	—
23	C₂×C₄⋊Dic₇₃	—	2	1	no	—
24	C₂×C₄⋊Dic₇₃	—	2	2	no	—
25	#14556523	—	4	4	no	—
26	#14556523	—	4	8	no	—
27	#14556516	—	4	4	no	—
28	C₂²×C₈×C₇₃	D₄	1	2	yes	—
29	C₂²×C₈×C₇₃	D₄	1	4	yes	—
30	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₇₃	—	2	2	no	—
32	C₄×C₈×C₇₃	—	2	2	no	—
33	C₄×C₈×C₇₃	—	2	2	no	—
34	C₂×Dic₁₄₆	—	2	2	no	—
35	C₂×Dic₁₄₆	—	4	2	no	—
36	C₂×Dic₁₄₆	—	4	2	no	—
37	C₂×Dic₁₄₆	—	4	1	no	—
38	C₂²×Dic₇₃	—	4	1	no	—
39	C₂²×Dic₇₃	—	4	1	no	—
40	C₂²×Dic₇₃	—	4	2	no	—
41	C₄⋊Dic₇₃	—	2	4	no	—
42	C₄⋊Dic₇₃	—	4	2	no	—
43	C₄⋊Dic₇₃	—	4	2	no	—
44	C₄⋊Dic₇₃	—	4	4	no	—
45	C₄⋊Dic₇₃	—	4	4	no	—
46	C₄⋊Dic₇₃	—	4	4	no	—
47	C₄⋊Dic₇₃	—	4	4	no	—
48	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₇₃	—	2	2	no	—
51	C₂×C₈×C₇₃	—	2	4	no	—
52	C₂×C₈×C₇₃	—	2	2	no	—
53	C₂×C₈×C₇₃	—	2	4	no	—
54	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
55	C₂×C₈×C₇₃	SD₁₆	1	8	yes	—
56	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₇₃	#41	1	2	yes	—
58	C₄²×C₇₃	#41	1	1	yes	characteristic
59	C₂³×C₇₃	#127	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#123	1	2	yes	—
64	C₂×C₄×C₇₃	C₂×SD₁₆	1	4	yes	—
65	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
66	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
67	C₂×C₄×C₇₃	#137	1	4	yes	—
68	C₂×Dic₇₃	—	4	2	no	—
69	C₂×Dic₇₃	—	8	1	no	—
70	C₂×Dic₇₃	—	8	1	no	—
71	C₂×Dic₇₃	—	8	1	no	—
72	C₂×Dic₇₃	—	8	2	no	—
73	C₂×Dic₇₃	—	8	2	no	—
74	C₂×Dic₇₃	—	8	2	no	—
75	C₂×Dic₇₃	—	8	2	no	—
76	Dic₁₄₆	—	4	4	no	—
77	Dic₁₄₆	—	8	2	no	—
78	C₈×C₇₃	—	2	4	no	—
79	C₈×C₇₃	—	2	8	no	—
80	C₂²×C₇₃	#399	1	1	yes	characteristic
81	C₂²×C₇₃	#492	1	1	yes	characteristic
82	C₂²×C₇₃	#492	1	2	yes	—
83	C₂²×C₇₃	#496	1	1	yes	characteristic
84	C₂²×C₇₃	#497	1	2	yes	—
85	C₄×C₇₃	#475	1	4	yes	—
86	C₄×C₇₃	#476	1	2	yes	—
87	C₄×C₇₃	#481	1	2	yes	—
88	C₄×C₇₃	#483	1	4	yes	—
89	Dic₇₃	—	8	2	no	—
90	Dic₇₃	—	16	1	no	—
91	Dic₇₃	—	16	2	no	—
92	#10685	—	73	1	no	maximal, Sylow
93	C₂×C₇₃	#1705	1	1	yes	characteristic
94	C₂×C₇₃	#1714	1	2	yes	—
95	C₂×C₇₃	#1716	1	1	yes	characteristic
96	C₂×C₇₃	#2094	1	1	yes	characteristic
97	C₂×C₇₃	#2096	1	2	yes	—
98	#1990	—	73	2	no	—
99	#1956	—	73	1	no	—
100	#1844	—	73	2	no	—
101	#1844	—	73	1	no	—
102	#1273	D₇₃	1	1	yes	characteristic
103	C₇₃	#10685	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₄⋊Q₈	—	73	1	no	—
105	C₂×#106	—	146	2	no	—
106	C₂×#106	—	146	4	no	—
107	C₂×#103	—	146	2	no	—
108	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
110	#384	—	73	1	no	—
111	#384	—	73	2	no	—
112	#340	—	146	4	no	—
113	C₂²×Q₈	—	146	1	no	—
114	C₂²×C₈	D₂₉₂	1	4	yes	—
115	C₂²×C₈	D₂₉₂	1	2	yes	—
116	C₄⋊Q₈	—	73	2	no	—
117	C₄⋊Q₈	—	146	1	no	—
118	C₄⋊Q₈	—	146	2	no	—
119	C₂×C₄⋊C₄	—	146	2	no	—
120	C₂×C₄⋊C₄	—	146	1	no	—
121	C₂×C₄⋊C₄	—	146	2	no	—
122	C₂×C₄⋊C₄	—	146	2	no	—
123	C₂×C₄⋊C₄	—	146	2	no	—
124	C₂×C₄⋊C₄	—	146	2	no	—
125	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
126	#106	—	292	4	no	—
127	#106	—	292	8	no	—
128	#103	—	292	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₂×Q₈	—	292	1	no	—
133	C₂×Q₈	—	292	2	no	—
134	C₂×Q₈	—	292	2	no	—
135	C₂×Q₈	—	146	2	no	—
136	C₂²×C₄	—	292	1	no	—
137	C₂²×C₄	—	292	1	no	—
138	C₂²×C₄	—	292	2	no	—
139	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	#11924017	1	4	yes	—
146	C₂×C₈	#11924017	1	8	yes	—
147	C₄⋊C₄	—	292	2	no	—
148	C₄⋊C₄	—	292	2	no	—
149	C₄⋊C₄	—	292	4	no	—
150	C₄⋊C₄	—	292	4	no	—
151	C₄⋊C₄	—	292	4	no	—
152	C₄⋊C₄	—	292	4	no	—
153	C₄⋊C₄	—	146	4	no	—
154	C₄²	Q₈×D₇₃	1	1	yes	characteristic
155	C₄²	#11924027	1	2	yes	—
156	C₄²	#11924027	1	1	yes	characteristic
157	C₂³	#14556588	1	1	yes	characteristic
158	Q₈	—	292	4	no	—
159	Q₈	—	584	2	no	—
160	C₂×C₄	—	292	2	no	—
161	C₂×C₄	—	584	1	no	—
162	C₂×C₄	—	584	1	no	—
163	C₂×C₄	—	584	1	no	—
164	C₂×C₄	—	584	2	no	—
165	C₂×C₄	—	584	2	no	—
166	C₂×C₄	—	584	2	no	—
167	C₂×C₄	—	584	2	no	—
168	C₂×C₄	#14556571	1	2	yes	—
169	C₂×C₄	#14556571	1	1	yes	characteristic
170	C₂×C₄	#14556575	1	1	yes	characteristic
171	C₂×C₄	#14556575	1	2	yes	—
172	C₂×C₄	C₂×#11924017	1	4	yes	—
173	C₂×C₄	C₂×#11924017	1	2	yes	—
174	C₂×C₄	#14556619	1	2	yes	—
175	C₂×C₄	#14556624	1	4	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	2	8	no	—
178	C₂²	#17988524	1	1	yes	characteristic
179	C₂²	#17988960	1	1	yes	characteristic
180	C₂²	#17988960	1	2	yes	—
181	C₂²	#17988976	1	1	yes	characteristic
182	C₂²	#17988981	1	2	yes	—
183	C₄	—	584	2	no	—
184	C₄	—	1168	1	no	—
185	C₄	—	1168	2	no	—
186	C₄	#17988873	1	4	yes	—
187	C₄	#17988878	1	2	yes	—
188	C₄	#17988906	1	2	yes	—
189	C₄	#17988918	1	4	yes	—
190	C₂	#22609518	1	1	yes	characteristic
191	C₂	#22609553	1	2	yes	—
192	C₂	#22609564	1	1	yes	characteristic
193	C₂	#22611463	1	1	yes	characteristic
194	C₂	#22611473	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610892

C₂

yes

maximal

#22610733

C₂

yes

maximal, characteristic

#22610088

C₂

yes

maximal, characteristic

#22610088

C₂

yes

maximal

C₇₃×#1273

C₂

yes

maximal, characteristic

C₂×#14556591

C₂²

yes

—

—

—

C₂²

yes

characteristic

#17988457

C₂²

yes

—

#17988350

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₃

C₂²

yes

—

C₂²×Dic₁₄₆

—

#14556591

—

#14556591

—

#14556591

Q₈

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

#14556523

—

#14556523

—

#14556516

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₄²×C₇₃

yes

C₂³×C₇₃

yes

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#496

yes

characteristic

C₂²×C₇₃

#497

yes

—

C₄×C₇₃

#475

yes

—

C₄×C₇₃

#476

yes

—

C₄×C₇₃

#481

yes

—

C₄×C₇₃

#483

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

#1990

—

#1956

—

100

#1844

—

101

#1844

—

102

#1273

D₇₃

yes

characteristic

103

C₇₃

#10685

yes

characteristic, complemented, Sylow

104

C₂×C₄⋊Q₈

—

105

C₂×#106

—

146

—

106

C₂×#106

—

146

—

107

C₂×#103

—

146

—

108

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

109

C₂×C₄×C₈

C₂×D₇₃

yes

—

110

#384

—

111

#384

—

112

#340

—

146

—

113

C₂²×Q₈

—

146

—

114

C₂²×C₈

D₂₉₂

yes

—

115

C₂²×C₈

D₂₉₂

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

146

—

118

C₄⋊Q₈

—

146

—

119

C₂×C₄⋊C₄

—

146

—

120

C₂×C₄⋊C₄

—

146

—

121

C₂×C₄⋊C₄

—

146

—

122

C₂×C₄⋊C₄

—

146

—

123

C₂×C₄⋊C₄

—

146

—

124

C₂×C₄⋊C₄

—

146

—

125

C₂×C₄²

C₂²×D₇₃

yes

characteristic

126

#106

—

292

—

127

#106

—

292

—

128

#103

—

292

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

292

—

133

C₂×Q₈

—

292

—

134

C₂×Q₈

—

292

—

135

C₂×Q₈

—

146

—

136

C₂²×C₄

—

292

—

137

C₂²×C₄

—

292

—

138

C₂²×C₄

—

292

—

139

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

140

C₂²×C₄

C₂×D₂₉₂

yes

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

#11924017

yes

—

146

C₂×C₈

#11924017

yes

—

147

C₄⋊C₄

—

292

—

148

C₄⋊C₄

—

292

—

149

C₄⋊C₄

—

292

—

150

C₄⋊C₄

—

292

—

151

C₄⋊C₄

—

292

—

152

C₄⋊C₄

—

292

—

153

C₄⋊C₄

—

146

—

154

C₄²

Q₈×D₇₃

yes

characteristic

155

C₄²

#11924027

yes

—

156

C₄²

#11924027

yes

characteristic

157

C₂³

#14556588

yes

characteristic

158

Q₈

—

292

—

159

Q₈

—

584

—

160

C₂×C₄

—

292

—

161

C₂×C₄

—

584

—

162

C₂×C₄

—

584

—

163

C₂×C₄

—

584

—

164

C₂×C₄

—

584

—

165

C₂×C₄

—

584

—

166

C₂×C₄

—

584

—

167

C₂×C₄

—

584

—

168

C₂×C₄

#14556571

yes

—

169

C₂×C₄

#14556571

yes

characteristic

170

C₂×C₄

#14556575

yes

characteristic

171

C₂×C₄

#14556575

yes

—

172

C₂×C₄

C₂×#11924017

yes

—

173

C₂×C₄

C₂×#11924017

yes

—

174

C₂×C₄

#14556619

yes

—

175

C₂×C₄

#14556624

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#17988524

yes

characteristic

179

C₂²

#17988960

yes

characteristic

180

C₂²

#17988960

yes

—

181

C₂²

#17988976

yes

characteristic

182

C₂²

#17988981

yes

—

183

C₄

—

584

—

184

C₄

—

1168

—

185

C₄

—

1168

—

186

C₄

#17988873

yes

—

187

C₄

#17988878

yes

—

188

C₄

#17988906

yes

—

189

C₄

#17988918

yes

—

190

C₂

#22609518

yes

characteristic

191

C₂

#22609553

yes

—

192

C₂

#22609564

yes

characteristic

193

C₂

#22611463

yes

characteristic

194

C₂

#22611473

yes

—

	subgroup	G/H
center	C₂³	#14556588
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1273	C₂
socle, prime radical	C₂³×C₇₃	#127
hypercenter	#1273	D₇₃
hypocenter	C₇₃	#10685
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#14556588

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1273

C₂

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
103. C₇₃	#10685	semidirect

H ≅ G/N

type

103. C₇₃

#10685

semidirect

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

subgroup	index
#22610892	2
#22610733	2
#22610088	2
C₇₃×#1273	2
#10685	73

subgroup

index

C₇₃×#1273

Subgroups — #32991100

Conjugacy classes of subgroups

Complements

Maximal subgroups