Group #32985821 — subgroups

Subgroups — #32985821

Order 18688.

710 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 191 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	C₂×#17988964	C₂	1	1	yes	maximal, characteristic
2	#22610736	C₂	1	4	yes	maximal, complemented
3	C₂×#17988308	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₈₄	—	2	2	no	—
6	C₂×#14556588	C₂²	1	2	yes	—
7	#17988964	—	2	2	no	—
8	#17988964	C₄	1	4	yes	complemented
9	C₂×#14556511	—	2	8	no	—
10	C₂×C₄×#10596831	C₂²	1	2	yes	—
11	#17988308	—	2	2	no	—
12	C₂×#106×C₇₃	—	2	2	no	—
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	#384×C₇₃	C₂²	1	2	yes	—
15	C₂²×D₂₉₂	—	2	4	no	—
16	C₂²×D₂₉₂	—	2	2	no	—
17	C₂×D₅₈₄	—	2	8	no	—
18	C₂×D₅₈₄	—	4	4	no	—
19	C₂×D₅₈₄	—	4	4	no	—
20	C₂²×#10596831	D₄	1	4	yes	—
21	#14556588	—	2	2	no	—
22	#14556588	—	2	4	no	—
23	#14556588	C₂×C₄	1	4	yes	—
24	#14556511	—	4	16	no	—
25	C₄×#10596831	—	2	2	no	—
26	C₄×#10596831	—	2	2	no	—
27	C₂²×C₈×C₇₃	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
30	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
31	#106×C₇₃	—	4	4	no	—
32	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₇₃	D₄	1	2	yes	—
34	C₂³×D₇₃	—	4	2	no	—
35	C₂×D₂₉₂	—	2	4	no	—
36	C₂×D₂₉₂	—	2	8	no	—
37	C₂×D₂₉₂	—	4	4	no	—
38	C₂×D₂₉₂	—	4	4	no	—
39	C₂×D₂₉₂	—	4	8	no	—
40	C₂×D₂₉₂	—	4	8	no	—
41	C₂×D₂₉₂	—	4	4	no	—
42	C₂×D₂₉₂	—	4	2	no	—
43	D₅₈₄	—	4	16	no	—
44	C₂×#10596831	—	2	4	no	—
45	C₂×#10596831	—	2	8	no	—
46	C₂²×C₄×C₇₃	—	4	2	no	—
47	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂²⋊C₄	1	4	yes	—
50	C₂×C₈×C₇₃	D₈	1	4	yes	—
51	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
52	C₄⋊C₄×C₇₃	—	4	4	no	—
53	C₄⋊C₄×C₇₃	—	4	8	no	—
54	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₇₃	#41	1	2	yes	—
57	C₂³×C₇₃	#127	1	1	yes	characteristic
58	C₂²×D₇₃	—	4	4	no	—
59	C₂²×D₇₃	—	8	2	no	—
60	C₂²×D₇₃	—	8	2	no	—
61	C₂²×D₇₃	—	8	4	no	—
62	C₂²×D₇₃	—	8	4	no	—
63	C₂×C₄×C₇₃	—	8	2	no	—
64	C₂×C₄×C₇₃	—	8	4	no	—
65	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₇₃	#121	1	2	yes	—
68	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	8	yes	—
72	D₂₉₂	—	4	8	no	—
73	D₂₉₂	—	4	16	no	—
74	D₂₉₂	—	8	8	no	—
75	D₂₉₂	—	8	4	no	—
76	C₈×C₇₃	#102	1	8	yes	complemented
77	#10596831	—	4	4	no	—
78	C₂²×C₇₃	#390	1	1	yes	characteristic
79	C₂²×C₇₃	#492	1	1	yes	characteristic
80	C₂²×C₇₃	#493	1	1	yes	characteristic
81	C₂²×C₇₃	#496	1	4	yes	—
82	C₂×D₇₃	—	8	4	no	—
83	C₂×D₇₃	—	8	8	no	—
84	C₂×D₇₃	—	16	2	no	—
85	C₂×D₇₃	—	16	2	no	—
86	C₂×D₇₃	—	16	4	no	—
87	C₄×C₇₃	—	16	2	no	—
88	C₄×C₇₃	C₂×#102	1	2	yes	—
89	C₄×C₇₃	#442	1	4	yes	—
90	C₄×C₇₃	#469	1	4	yes	—
91	C₄×C₇₃	#486	1	2	yes	—
92	#9837	—	73	1	no	maximal, Sylow
93	C₂×C₇₃	#1715	1	2	yes	—
94	C₂×C₇₃	#1721	1	2	yes	—
95	C₂×C₇₃	#1957	1	1	yes	characteristic
96	C₂×C₇₃	#1972	1	2	yes	—
97	D₇₃	—	16	4	no	—
98	C₂×#493	—	73	1	no	—
99	#1957	—	73	4	no	—
100	#1844	D₇₃	1	1	yes	characteristic, complemented
101	C₂×#322	—	73	1	no	—
102	C₇₃	#9837	1	1	yes	characteristic, complemented, Sylow
103	C₂²×D₈	—	146	2	no	—
104	C₂×#127	—	73	2	no	—
105	#493	—	73	4	no	—
106	#493	—	146	2	no	—
107	C₂×#106	—	2	2	no	—
108	C₂×#102	—	146	8	no	—
109	C₂×C₄×C₈	—	73	2	no	—
110	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
111	#384	C₂×D₇₃	1	2	yes	—
112	#322	—	146	2	no	—
113	C₂²×D₄	—	146	4	no	—
114	C₂²×D₄	—	146	2	no	—
115	C₂×D₈	—	292	4	no	—
116	C₂×D₈	—	292	4	no	—
117	C₂×D₈	—	146	8	no	—
118	C₂²×C₈	—	73	4	no	—
119	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
120	#127	—	73	4	no	—
121	#127	—	146	2	no	—
122	#127	—	146	4	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
126	#106	—	4	4	no	—
127	#102	—	292	16	no	—
128	C₄×C₈	—	146	2	no	—
129	C₄×C₈	—	146	2	no	—
130	C₄×C₈	C₄×D₇₃	1	2	yes	—
131	C₄×C₈	D₂₉₂	1	2	yes	—
132	C₂⁴	—	292	2	no	—
133	C₂×D₄	—	292	4	no	—
134	C₂×D₄	—	292	2	no	—
135	C₂×D₄	—	292	4	no	—
136	C₂×D₄	—	292	4	no	—
137	C₂×D₄	—	292	8	no	—
138	C₂×D₄	—	292	8	no	—
139	C₂×D₄	—	146	4	no	—
140	C₂×D₄	—	146	8	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	D₄×D₇₃	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
144	D₈	—	292	16	no	—
145	C₂×C₈	—	146	4	no	—
146	C₂×C₈	—	146	8	no	—
147	C₂×C₈	#11924005	1	4	yes	—
148	C₂×C₈	#11924013	1	4	yes	—
149	C₂×C₈	#11924015	1	4	yes	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
153	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
154	C₄²	#11924029	1	2	yes	—
155	C₂³	—	292	4	no	—
156	C₂³	—	584	2	no	—
157	C₂³	—	584	2	no	—
158	C₂³	—	584	4	no	—
159	C₂³	—	584	4	no	—
160	C₂³	#14556587	1	1	yes	characteristic
161	D₄	—	292	16	no	—
162	D₄	—	292	8	no	—
163	D₄	—	584	4	no	—
164	D₄	—	584	8	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
168	C₂×C₄	#14556553	1	2	yes	—
169	C₂×C₄	#14556566	1	2	yes	—
170	C₂×C₄	#14556580	1	1	yes	characteristic
171	C₂×C₄	C₂×#11924013	1	2	yes	—
172	C₂×C₄	C₂×#11924015	1	2	yes	—
173	C₂×C₄	#14556618	1	8	yes	—
174	C₈	—	292	4	no	—
175	C₈	#14556511	1	8	yes	complemented
176	C₂²	—	584	4	no	—
177	C₂²	—	584	8	no	—
178	C₂²	—	1168	2	no	—
179	C₂²	—	1168	2	no	—
180	C₂²	—	1168	4	no	—
181	C₂²	#17988488	1	1	yes	characteristic
182	C₂²	#17988961	1	1	yes	characteristic
183	C₂²	#17988963	1	1	yes	characteristic
184	C₂²	#17988973	1	4	yes	—
185	C₄	—	16	2	no	—
186	C₄	C₂×#14556511	1	2	yes	—
187	C₄	#17988698	1	4	yes	—
188	C₄	#17988841	1	4	yes	—
189	C₄	#17988935	1	2	yes	—
190	C₂	—	1168	4	no	—
191	C₂	#22609560	1	2	yes	—
192	C₂	#22609595	1	2	yes	—
193	C₂	#22610736	1	1	yes	characteristic
194	C₂	#22610805	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988964

C₂

yes

maximal, characteristic

#22610736

C₂

yes

maximal, complemented

C₂×#17988308

C₂

yes

maximal, characteristic, complemented

C₇₃×#1844

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₈₄

—

C₂×#14556588

C₂²

yes

—

#17988964

—

#17988964

C₄

yes

complemented

C₂×#14556511

—

C₂×C₄×#10596831

C₂²

yes

—

#17988308

—

C₂×#106×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#384×C₇₃

C₂²

yes

—

C₂²×D₂₉₂

—

C₂²×D₂₉₂

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂²×#10596831

D₄

yes

—

#14556588

—

#14556588

—

#14556588

C₂×C₄

yes

—

#14556511

—

C₄×#10596831

—

C₄×#10596831

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#106×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

C₂×#10596831

—

C₂×#10596831

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

#127

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

#102

yes

complemented

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#102

yes

—

C₄×C₇₃

#442

yes

—

C₄×C₇₃

#469

yes

—

C₄×C₇₃

#486

yes

—

#9837

—

maximal, Sylow

C₂×C₇₃

#1715

yes

—

C₂×C₇₃

#1721

yes

—

C₂×C₇₃

#1957

yes

characteristic

C₂×C₇₃

#1972

yes

—

D₇₃

—

C₂×#493

—

#1957

—

100

#1844

D₇₃

yes

characteristic, complemented

101

C₂×#322

—

102

C₇₃

#9837

yes

characteristic, complemented, Sylow

103

C₂²×D₈

—

146

—

104

C₂×#127

—

105

#493

—

106

#493

—

146

—

107

C₂×#106

—

108

C₂×#102

—

146

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

111

#384

C₂×D₇₃

yes

—

112

#322

—

146

—

113

C₂²×D₄

—

146

—

114

C₂²×D₄

—

146

—

115

C₂×D₈

—

292

—

116

C₂×D₈

—

292

—

117

C₂×D₈

—

146

—

118

C₂²×C₈

—

119

C₂²×C₈

C₂²⋊D₇₃

yes

—

120

#127

—

121

#127

—

146

—

122

#127

—

146

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

C₂²×D₇₃

yes

characteristic

126

#106

—

127

#102

—

292

—

128

C₄×C₈

—

146

—

129

C₄×C₈

—

146

—

130

C₄×C₈

C₄×D₇₃

yes

—

131

C₄×C₈

D₂₉₂

yes

—

132

C₂⁴

—

292

—

133

C₂×D₄

—

292

—

134

C₂×D₄

—

292

—

135

C₂×D₄

—

292

—

136

C₂×D₄

—

292

—

137

C₂×D₄

—

292

—

138

C₂×D₄

—

292

—

139

C₂×D₄

—

146

—

140

C₂×D₄

—

146

—

141

C₂²×C₄

—

142

C₂²×C₄

D₄×D₇₃

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

144

D₈

—

292

—

145

C₂×C₈

—

146

—

146

C₂×C₈

—

146

—

147

C₂×C₈

#11924005

yes

—

148

C₂×C₈

#11924013

yes

—

149

C₂×C₈

#11924015

yes

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂×C₄×D₇₃

yes

characteristic

153

C₄²

C₂×D₂₉₂

yes

characteristic

154

C₄²

#11924029

yes

—

155

C₂³

—

292

—

156

C₂³

—

584

—

157

C₂³

—

584

—

158

C₂³

—

584

—

159

C₂³

—

584

—

160

C₂³

#14556587

yes

characteristic

161

D₄

—

292

—

162

D₄

—

292

—

163

D₄

—

584

—

164

D₄

—

584

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

C₂×#11924005

yes

characteristic

168

C₂×C₄

#14556553

yes

—

169

C₂×C₄

#14556566

yes

—

170

C₂×C₄

#14556580

yes

characteristic

171

C₂×C₄

C₂×#11924013

yes

—

172

C₂×C₄

C₂×#11924015

yes

—

173

C₂×C₄

#14556618

yes

—

174

C₈

—

292

—

175

C₈

#14556511

yes

complemented

176

C₂²

—

584

—

177

C₂²

—

584

—

178

C₂²

—

1168

—

179

C₂²

—

1168

—

180

C₂²

—

1168

—

181

C₂²

#17988488

yes

characteristic

182

C₂²

#17988961

yes

characteristic

183

C₂²

#17988963

yes

characteristic

184

C₂²

#17988973

yes

—

185

C₄

—

186

C₄

C₂×#14556511

yes

—

187

C₄

#17988698

yes

—

188

C₄

#17988841

yes

—

189

C₄

#17988935

yes

—

190

C₂

—

1168

—

191

C₂

#22609560

yes

—

192

C₂

#22609595

yes

—

193

C₂

#22610736

yes

characteristic

194

C₂

#22610805

yes

—

	subgroup	G/H
center	C₂³	#14556587
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1844	C₂
socle	C₂³×C₇₃	#127
prime radical	C₂×#17988964	C₂
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1844	D₇₃
hypocenter	C₇₃	#9837
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14556587

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1844

C₂

C₂³×C₇₃

C₂

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
2. #22610736	C₂	semidirect
3. C₂×#17988308	C₂	semidirect
4. C₇₃×#1844	C₂	semidirect
8. #17988964	C₄	semidirect
76. C₈×C₇₃	#102	semidirect
100. #1844	D₇₃	semidirect
102. C₇₃	#9837	semidirect
175. C₈	#14556511	semidirect

H ≅ G/N

type

2. #22610736

C₂

semidirect

3. C₂×#17988308

C₂

semidirect

4. C₇₃×#1844

C₂

semidirect

8. #17988964

C₄

semidirect

76. C₈×C₇₃

#102

semidirect

100. #1844

D₇₃

semidirect

102. C₇₃

#9837

semidirect

175. C₈

#14556511

semidirect

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

subgroup	index
C₂×#17988964	2
#22610736	2
C₂×#17988308	2
C₇₃×#1844	2
#9837	73

subgroup

index

C₇₃×#1844

Subgroups — #32985821

Conjugacy classes of subgroups

Complements

Maximal subgroups