Group #32985476 — subgroups

Subgroups — #32985476

Order 18688.

506 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22610069	C₂	1	1	yes	maximal, characteristic, complemented
2	#22608603	C₂	1	2	yes	maximal, complemented
3	#22608599	C₂	1	2	yes	maximal
4	#22608223	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#492×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	#17988611	C₂²	1	2	yes	—
8	#17988352	—	2	2	no	—
9	#17988333	—	2	4	no	—
10	#17988331	—	2	4	no	—
11	#17988309	—	2	2	no	—
12	C₂²×SD₁₆×C₇₃	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#127×C₇₃	C₂²	1	1	yes	characteristic
15	#492×C₇₃	—	2	2	no	—
16	#492×C₇₃	C₄	1	4	yes	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×#11924012	—	2	2	no	—
19	C₄⋊#10596831	—	2	2	no	—
20	C₄⋊#10596831	—	2	2	no	—
21	C₄⋊#10596831	D₄	1	4	yes	—
22	C₂²⋊#10596831	—	2	2	no	—
23	C₄×#10596831	—	2	2	no	—
24	C₂²×Q₈×C₇₃	—	2	1	no	—
25	C₂²×D₄×C₇₃	—	2	1	no	—
26	C₂²×D₄×C₇₃	—	2	2	no	—
27	C₂×SD₁₆×C₇₃	—	2	8	no	—
28	C₂×SD₁₆×C₇₃	—	4	8	no	—
29	C₂²×C₈×C₇₃	D₄	1	2	yes	—
30	C₄⋊Q₈×C₇₃	—	2	2	no	—
31	C₄⋊Q₈×C₇₃	—	2	1	no	—
32	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
33	#127×C₇₃	—	2	1	no	—
34	#127×C₇₃	—	2	2	no	—
35	#127×C₇₃	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
37	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₃	D₄	1	2	yes	—
40	#11924012	—	4	4	no	—
41	C₂×#10596831	—	2	4	no	—
42	C₂×#10596831	—	4	2	no	—
43	C₂⁴×C₇₃	—	4	1	no	—
44	C₂×Q₈×C₇₃	—	2	2	no	—
45	C₂×Q₈×C₇₃	—	4	1	no	—
46	C₂×Q₈×C₇₃	—	4	4	no	—
47	C₂×D₄×C₇₃	—	2	2	no	—
48	C₂×D₄×C₇₃	—	2	4	no	—
49	C₂×D₄×C₇₃	—	4	2	no	—
50	C₂×D₄×C₇₃	—	4	1	no	—
51	C₂×D₄×C₇₃	—	4	4	no	—
52	C₂×D₄×C₇₃	—	8	4	no	—
53	C₂²×C₄×C₇₃	—	4	1	no	—
54	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	#41	1	2	yes	—
56	SD₁₆×C₇₃	—	4	16	no	—
57	C₂×C₈×C₇₃	—	2	4	no	—
58	C₂×C₈×C₇₃	C₂²⋊C₄	1	4	yes	—
59	C₄⋊C₄×C₇₃	—	2	4	no	—
60	C₄⋊C₄×C₇₃	—	4	2	no	—
61	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇₃	C₂×D₄	1	2	yes	—
63	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂³×C₇₃	—	4	2	no	—
65	C₂³×C₇₃	—	8	4	no	—
66	C₂³×C₇₃	—	8	2	no	—
67	C₂³×C₇₃	#124	1	1	yes	characteristic
68	Q₈×C₇₃	—	4	4	no	—
69	Q₈×C₇₃	—	8	2	no	—
70	D₄×C₇₃	—	4	4	no	—
71	D₄×C₇₃	—	8	4	no	—
72	D₄×C₇₃	—	8	4	no	—
73	D₄×C₇₃	—	8	2	no	—
74	C₂×C₄×C₇₃	—	2	4	no	—
75	C₂×C₄×C₇₃	—	4	2	no	—
76	C₂×C₄×C₇₃	—	8	2	no	—
77	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₇₃	#121	1	2	yes	—
80	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
81	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
82	C₈×C₇₃	—	2	4	no	—
83	#10596831	—	4	4	no	—
84	#10596831	—	8	2	no	—
85	C₂²×C₇₃	—	2	2	no	—
86	C₂²×C₇₃	—	8	2	no	—
87	C₂²×C₇₃	—	8	4	no	—
88	C₂²×C₇₃	—	16	4	no	—
89	C₂²×C₇₃	#390	1	1	yes	characteristic
90	C₂²×C₇₃	#490	1	2	yes	—
91	C₄×C₇₃	—	2	2	no	—
92	C₄×C₇₃	—	8	2	no	—
93	C₄×C₇₃	#418	1	2	yes	—
94	C₄×C₇₃	#471	1	4	yes	—
95	C₄×C₇₃	#496	1	2	yes	—
96	#9777	—	73	1	no	maximal, Sylow
97	C₂×C₇₃	—	2	2	no	—
98	C₂×C₇₃	—	16	2	no	—
99	C₂×C₇₃	#1961	1	1	yes	characteristic
100	C₂×C₇₃	#1977	1	2	yes	—
101	C₂×#492	D₇₃	1	1	yes	characteristic
102	#1837	—	73	1	no	—
103	#1546	—	73	2	no	—
104	#1545	—	73	2	no	—
105	#1452	—	73	1	no	—
106	C₇₃	#9777	1	1	yes	characteristic, complemented, Sylow
107	C₂²×SD₁₆	—	2	2	no	—
108	C₂×C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
109	C₂×#127	C₂×D₇₃	1	1	yes	characteristic
110	#492	—	2	2	no	—
111	#492	Dic₇₃	1	4	yes	—
112	#431	—	73	2	no	—
113	#423	—	73	2	no	—
114	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
115	#341	—	146	2	no	—
116	#332	—	146	4	no	—
117	#331	—	146	4	no	—
118	#322	—	146	2	no	—
119	C₂²×Q₈	—	2	1	no	—
120	C₂²×D₄	—	2	1	no	—
121	C₂²×D₄	—	2	2	no	—
122	C₂×SD₁₆	—	2	8	no	—
123	C₂×SD₁₆	—	4	8	no	—
124	C₂×M₁₆	—	146	2	no	—
125	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	1	no	—
128	C₄⋊Q₈	C₂×Dic₇₃	1	2	yes	—
129	#127	—	2	2	no	—
130	#127	—	2	1	no	—
131	#127	C₂×Dic₇₃	1	2	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
134	C₄⋊C₈	—	73	4	no	—
135	C₄⋊C₈	—	146	2	no	—
136	C₄⋊C₈	—	146	2	no	—
137	C₂²⋊C₈	—	146	2	no	—
138	C₄×C₈	—	146	2	no	—
139	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
140	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
141	C₂⁴	—	4	1	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂×D₄	—	4	4	no	—
150	C₂×D₄	—	8	4	no	—
151	C₂²×C₄	—	4	1	no	—
152	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
153	C₂²×C₄	#11924027	1	2	yes	—
154	SD₁₆	—	4	16	no	—
155	M₁₆	—	292	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	292	2	no	—
158	C₂×C₈	—	146	4	no	—
159	C₂×C₈	C₂²⋊Dic₇₃	1	4	yes	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
163	C₄²	D₄×D₇₃	1	2	yes	—
164	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
165	C₂³	—	4	2	no	—
166	C₂³	—	8	4	no	—
167	C₂³	—	8	2	no	—
168	C₂³	#14556578	1	1	yes	characteristic
169	Q₈	—	4	4	no	—
170	Q₈	—	8	2	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	4	no	—
173	D₄	—	8	2	no	—
174	D₄	—	8	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
180	C₂×C₄	#14556562	1	2	yes	—
181	C₂×C₄	#14556587	1	1	yes	characteristic
182	C₂×C₄	#14556617	1	4	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	292	4	no	—
185	C₈	—	584	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	8	2	no	—
188	C₂²	—	8	4	no	—
189	C₂²	—	16	4	no	—
190	C₂²	#17988490	1	1	yes	characteristic
191	C₂²	#17988951	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	8	2	no	—
194	C₄	#17988592	1	2	yes	—
195	C₄	#17988847	1	4	yes	—
196	C₄	#17988974	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	—	16	2	no	—
199	C₂	#22610756	1	1	yes	characteristic
200	C₂	#22610836	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610069

C₂

yes

maximal, characteristic, complemented

#22608603

C₂

yes

maximal, complemented

#22608599

C₂

yes

maximal

#22608223

C₂

yes

maximal, characteristic, complemented

C₂×#492×C₇₃

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

—

#17988611

C₂²

yes

—

—

—

—

—

C₂²×SD₁₆×C₇₃

—

C₂×C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₂×#127×C₇₃

C₂²

yes

characteristic

#492×C₇₃

—

#492×C₇₃

C₄

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

—

—

D₄

yes

—

C₂²⋊#10596831

—

C₄×#10596831

—

C₂²×Q₈×C₇₃

—

C₂²×D₄×C₇₃

—

C₂²×D₄×C₇₃

—

C₂×SD₁₆×C₇₃

—

C₂×SD₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

C₂×C₄

yes

—

#127×C₇₃

—

#127×C₇₃

—

#127×C₇₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

#11924012

—

C₂×#10596831

—

C₂×#10596831

—

C₂⁴×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

SD₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#124

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

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

#127

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#390

yes

characteristic

C₂²×C₇₃

#490

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#418

yes

—

C₄×C₇₃

#471

yes

—

C₄×C₇₃

#496

yes

—

#9777

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1961

yes

characteristic

100

C₂×C₇₃

#1977

yes

—

101

C₂×#492

D₇₃

yes

characteristic

102

#1837

—

103

#1546

—

104

#1545

—

105

#1452

—

106

C₇₃

#9777

yes

characteristic, complemented, Sylow

107

C₂²×SD₁₆

—

108

C₂×C₄⋊Q₈

C₂×D₇₃

yes

characteristic

109

C₂×#127

C₂×D₇₃

yes

characteristic

110

#492

—

111

#492

Dic₇₃

yes

—

112

#431

—

113

#423

—

114

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

115

#341

—

146

—

116

#332

—

146

—

117

#331

—

146

—

118

#322

—

146

—

119

C₂²×Q₈

—

120

C₂²×D₄

—

121

C₂²×D₄

—

122

C₂×SD₁₆

—

123

C₂×SD₁₆

—

124

C₂×M₁₆

—

146

—

125

C₂²×C₈

C₂²⋊D₇₃

yes

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂×Dic₇₃

yes

—

129

#127

—

130

#127

—

131

#127

C₂×Dic₇₃

yes

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂²×D₇₃

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

146

—

136

C₄⋊C₈

—

146

—

137

C₂²⋊C₈

—

146

—

138

C₄×C₈

—

146

—

139

C₄×C₈

C₂×Dic₇₃

yes

—

140

C₄×C₈

C₂²⋊D₇₃

yes

—

141

C₂⁴

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

153

C₂²×C₄

#11924027

yes

—

154

SD₁₆

—

155

M₁₆

—

292

—

156

C₂×C₈

—

157

C₂×C₈

—

292

—

158

C₂×C₈

—

146

—

159

C₂×C₈

C₂²⋊Dic₇₃

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄²

C₂²×Dic₇₃

yes

characteristic

163

C₄²

D₄×D₇₃

yes

—

164

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

#14556578

yes

characteristic

169

Q₈

—

170

Q₈

—

171

D₄

—

172

D₄

—

173

D₄

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₂²⋊Dic₇₃

yes

characteristic

179

C₂×C₄

D₄×Dic₇₃

yes

—

180

C₂×C₄

#14556562

yes

—

181

C₂×C₄

#14556587

yes

characteristic

182

C₂×C₄

#14556617

yes

—

183

C₈

—

184

C₈

—

292

—

185

C₈

—

584

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

#17988490

yes

characteristic

191

C₂²

#17988951

yes

—

192

C₄

—

193

C₄

—

194

C₄

#17988592

yes

—

195

C₄

#17988847

yes

—

196

C₄

#17988974

yes

—

197

C₂

—

198

C₂

—

199

C₂

#22610756

yes

characteristic

200

C₂

#22610836

yes

—

	subgroup	G/H
center	C₂²	#17988490
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#492×C₇₃	C₂
socle	C₂²×C₇₃	#390
prime radical	C₂×#127×C₇₃	C₂²
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#492	D₇₃
hypocenter	C₇₃	#9777
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#492

subgroup

G/H

center

C₂²

#17988490

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₂×#492×C₇₃

C₂

socle

C₂²×C₇₃

#390

prime radical

C₂×#127×C₇₃

C₂²

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#492

N	H ≅ G/N	type
1. #22610069	C₂	semidirect
2. #22608603	C₂	semidirect
4. #22608223	C₂	semidirect
106. C₇₃	#9777	semidirect

H ≅ G/N

type

1. #22610069

C₂

semidirect

2. #22608603

C₂

semidirect

4. #22608223

C₂

semidirect

106. C₇₃

#9777

semidirect

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

subgroup	index
#22610069	2
#22608603	2
#22608599	2
#22608223	2
C₂×#492×C₇₃	2
#9777	73

subgroup

index

C₂×#492×C₇₃

Subgroups — #32985476

Conjugacy classes of subgroups

Complements

Maximal subgroups