Group #32985602 — subgroups

Subgroups — #32985602

Order 18688.

442 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 139 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	#22610776	C₂	1	2	yes	maximal
2	#22608690	C₂	1	1	yes	maximal, characteristic
3	#22608603	C₂	1	2	yes	maximal
4	C₈×#11924012	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	1	yes	characteristic
7	#17988611	C₂²	1	2	yes	—
8	C₂×#14556510	—	2	4	no	—
9	C₄×#11924012	C₂²	1	1	yes	characteristic
10	#17988340	—	2	2	no	—
11	#17988333	—	2	4	no	—
12	#17988309	—	2	1	no	—
13	C₈×#10596831	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
15	#500×C₇₃	—	2	1	no	—
16	#500×C₇₃	—	2	1	no	—
17	#500×C₇₃	—	2	2	no	—
18	C₂×#107×C₇₃	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×#11924012	—	2	2	no	—
21	C₂×#11924012	D₄	1	2	yes	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₄⋊#10596831	—	2	2	no	—
24	C₄⋊#10596831	D₄	1	2	yes	—
25	#14556510	—	4	8	no	—
26	C₂²⋊#10596831	—	2	1	no	—
27	#14556502	—	2	1	no	—
28	C₄×#10596831	—	2	1	no	—
29	C₄×#10596831	—	2	1	no	—
30	C₂²×Q₈×C₇₃	—	2	2	no	—
31	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₇₃	—	2	2	no	—
33	C₄⋊Q₈×C₇₃	—	2	2	no	—
34	C₄⋊Q₈×C₇₃	—	2	2	no	—
35	C₄⋊Q₈×C₇₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
39	#107×C₇₃	—	2	4	no	—
40	#107×C₇₃	—	2	4	no	—
41	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₃	D₄	1	2	yes	—
44	#11924012	—	2	2	no	—
45	#11924012	—	4	4	no	—
46	#11924012	D₈	1	4	yes	—
47	C₂×#10596831	—	2	2	no	—
48	C₂×#10596831	—	2	2	no	—
49	C₂×#10596831	—	4	2	no	—
50	C₂×Q₈×C₇₃	—	4	2	no	—
51	C₂×Q₈×C₇₃	—	4	2	no	—
52	C₂×Q₈×C₇₃	—	4	4	no	—
53	C₂×Q₈×C₇₃	—	4	4	no	—
54	C₂²×C₄×C₇₃	—	2	4	no	—
55	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
58	C₂×C₈×C₇₃	—	2	4	no	—
59	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₇₃	—	2	4	no	—
62	C₄⋊C₄×C₇₃	—	2	4	no	—
63	C₄⋊C₄×C₇₃	—	4	2	no	—
64	C₄⋊C₄×C₇₃	—	4	2	no	—
65	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₇₃	#41	1	1	yes	characteristic
69	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
70	Q₈×C₇₃	—	8	4	no	—
71	Q₈×C₇₃	—	8	2	no	—
72	Q₈×C₇₃	—	8	2	no	—
73	C₂×C₄×C₇₃	—	2	2	no	—
74	C₂×C₄×C₇₃	—	2	2	no	—
75	C₂×C₄×C₇₃	—	4	4	no	—
76	C₂×C₄×C₇₃	—	4	8	no	—
77	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₇₃	#135	1	2	yes	—
85	C₈×C₇₃	—	2	2	no	—
86	C₈×C₇₃	#101	1	4	yes	complemented
87	#10596831	—	2	4	no	—
88	#10596831	—	4	2	no	—
89	#10596831	—	8	2	no	—
90	C₂²×C₇₃	—	2	1	no	—
91	C₂²×C₇₃	—	2	1	no	—
92	C₂²×C₇₃	#390	1	1	yes	characteristic
93	C₂²×C₇₃	C₄×D₈	1	2	yes	—
94	C₄×C₇₃	—	2	1	no	—
95	C₄×C₇₃	—	2	1	no	—
96	C₄×C₇₃	—	8	4	no	—
97	C₄×C₇₃	C₂×#101	1	2	yes	—
98	C₄×C₇₃	#466	1	2	yes	—
99	C₄×C₇₃	#473	1	2	yes	—
100	C₄×C₇₃	#493	1	1	yes	characteristic
101	C₄×C₇₃	#495	1	1	yes	characteristic
102	#9799	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	—	2	2	no	—
104	C₂×C₇₃	#1966	1	1	yes	characteristic
105	C₂×C₇₃	#1970	1	1	yes	characteristic
106	C₂×C₇₃	#1979	1	1	yes	characteristic
107	C₂×#500	D₇₃	1	1	yes	characteristic
108	#1966	—	73	2	no	—
109	#1567	—	73	1	no	—
110	#1546	—	73	2	no	—
111	C₈×M₁₆	—	73	1	no	—
112	C₇₃	#9799	1	1	yes	characteristic, complemented, Sylow
113	C₂×C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
114	#500	—	2	1	no	—
115	#500	—	2	1	no	—
116	#500	—	2	2	no	—
117	#431	—	73	1	no	—
118	C₂×#107	Dic₇₃	1	4	yes	—
119	#423	—	73	2	no	—
120	C₂×#101	—	146	4	no	—
121	C₄×M₁₆	—	73	1	no	—
122	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
123	#335	—	146	2	no	—
124	#332	—	146	4	no	—
125	#322	—	146	1	no	—
126	C₈²	—	146	1	no	—
127	C₂²×Q₈	—	2	2	no	—
128	C₂×M₁₆	—	73	2	no	—
129	C₂×M₁₆	—	146	2	no	—
130	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
137	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
138	#107	—	2	4	no	—
139	#107	—	2	4	no	—
140	C₄⋊C₈	—	73	2	no	—
141	C₄⋊C₈	—	146	2	no	—
142	C₄⋊C₈	—	146	2	no	—
143	#101	—	292	8	no	—
144	C₂²⋊C₈	—	146	1	no	—
145	#97	—	146	1	no	—
146	C₄×C₈	—	146	1	no	—
147	C₄×C₈	—	146	1	no	—
148	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	4	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
158	C₂²×C₄	#11924027	1	1	yes	characteristic
159	M₁₆	—	292	4	no	—
160	M₁₆	—	73	4	no	—
161	M₁₆	—	146	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	292	2	no	—
164	C₂×C₈	—	146	2	no	—
165	C₂×C₈	—	146	2	no	—
166	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
167	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	D₄×D₇₃	1	1	yes	characteristic
173	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
174	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
175	C₄²	#11924027	1	1	yes	characteristic
176	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
177	Q₈	—	8	2	no	—
178	Q₈	—	8	2	no	—
179	Q₈	—	8	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
185	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
186	C₂×C₄	#14556562	1	1	yes	characteristic
187	C₂×C₄	#14556562	1	1	yes	characteristic
188	C₂×C₄	#14556578	1	1	yes	characteristic
189	C₂×C₄	#14556587	1	1	yes	characteristic
190	C₂×C₄	D₈×D₇₃	1	2	yes	—
191	C₂×C₄	#14556610	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	—	292	2	no	—
194	C₈	—	584	2	no	—
195	C₈	—	146	4	no	—
196	C₈	#14556510	1	4	yes	complemented
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#17988490	1	1	yes	characteristic
200	C₂²	D₈×Dic₇₃	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	8	4	no	—
204	C₄	C₂×#14556510	1	2	yes	—
205	C₄	#17988820	1	2	yes	—
206	C₄	#17988859	1	2	yes	—
207	C₄	#17988962	1	1	yes	characteristic
208	C₄	#17988970	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	#22610776	1	1	yes	characteristic
211	C₂	#22610798	1	1	yes	characteristic
212	C₂	#22610847	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610776

C₂

yes

maximal

#22608690

C₂

yes

maximal, characteristic

#22608603

C₂

yes

maximal

C₈×#11924012

C₂

yes

maximal, characteristic

C₂×#500×C₇₃

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

characteristic

#17988611

C₂²

yes

—

C₂×#14556510

—

C₄×#11924012

C₂²

yes

—

—

—

—

C₂×C₄⋊Q₈×C₇₃

C₂²

yes

—

#500×C₇₃

—

#500×C₇₃

—

#500×C₇₃

—

C₂×#107×C₇₃

C₄

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×#11924012

—

C₂×#11924012

D₄

yes

—

C₄⋊#10596831

—

C₄⋊#10596831

—

C₄⋊#10596831

D₄

yes

—

—

—

—

—

—

C₂²×Q₈×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

—

#11924012

—

#11924012

—

#11924012

D₈

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

#41

yes

characteristic

C₂³×C₇₃

C₄×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×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

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₈×C₇₃

—

C₈×C₇₃

yes

—

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#390

yes

characteristic

C₂²×C₇₃

C₄×D₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#101

yes

—

C₄×C₇₃

#466

yes

—

C₄×C₇₃

#473

yes

—

100

C₄×C₇₃

#493

yes

characteristic

101

C₄×C₇₃

yes

102

—

103

C₂×C₇₃

—

104

C₂×C₇₃

#1966

yes

characteristic

105

C₂×C₇₃

#1970

yes

characteristic

106

C₂×C₇₃

#1979

yes

characteristic

107

C₂×#500

D₇₃

yes

characteristic

108

#1966

—

109

#1567

—

110

#1546

—

111

C₈×M₁₆

—

112

C₇₃

#9799

yes

characteristic, complemented, Sylow

113

C₂×C₄⋊Q₈

C₂×D₇₃

yes

—

114

#500

—

115

#500

—

116

#500

—

117

#431

—

118

C₂×#107

Dic₇₃

yes

—

119

#423

—

120

C₂×#101

—

146

—

121

C₄×M₁₆

—

122

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

123

#335

—

146

—

124

#332

—

146

—

125

#322

—

146

—

126

C₈²

—

146

—

127

C₂²×Q₈

—

128

C₂×M₁₆

—

129

C₂×M₁₆

—

146

—

130

C₂²×C₈

C₂×Dic₇₃

yes

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

133

C₄⋊Q₈

—

134

C₄⋊Q₈

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

C₂×Dic₇₃

yes

—

137

C₂×C₄²

C₂²×D₇₃

yes

characteristic

138

#107

—

139

#107

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

146

—

142

C₄⋊C₈

—

146

—

143

#101

—

292

—

144

C₂²⋊C₈

—

146

—

145

#97

—

146

—

146

C₄×C₈

—

146

—

147

C₄×C₈

—

146

—

148

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

149

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

150

C₄×C₈

C₂²⋊D₇₃

yes

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

C₂²×Dic₇₃

yes

characteristic

157

C₂²×C₄

D₄×D₇₃

yes

characteristic

158

C₂²×C₄

#11924027

yes

characteristic

159

M₁₆

—

292

—

160

M₁₆

—

161

M₁₆

—

146

—

162

C₂×C₈

—

163

C₂×C₈

—

292

—

164

C₂×C₈

—

146

—

165

C₂×C₈

—

146

—

166

C₂×C₈

C₂²⋊Dic₇₃

yes

—

167

C₂×C₈

C₂²⋊Dic₇₃

yes

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄²

D₄×D₇₃

yes

characteristic

173

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

174

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

175

C₄²

#11924027

yes

characteristic

176

C₂³

D₄×Dic₇₃

yes

characteristic

177

Q₈

—

178

Q₈

—

179

Q₈

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₂×C₂²⋊Dic₇₃

yes

characteristic

185

C₂×C₄

D₄×Dic₇₃

yes

characteristic

186

C₂×C₄

#14556562

yes

characteristic

187

C₂×C₄

#14556562

yes

characteristic

188

C₂×C₄

#14556578

yes

characteristic

189

C₂×C₄

#14556587

yes

characteristic

190

C₂×C₄

D₈×D₇₃

yes

—

191

C₂×C₄

#14556610

yes

—

192

C₈

—

193

C₈

—

292

—

194

C₈

—

584

—

195

C₈

—

146

—

196

C₈

#14556510

yes

complemented

197

C₂²

—

198

C₂²

—

199

C₂²

#17988490

yes

characteristic

200

C₂²

D₈×Dic₇₃

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

C₂×#14556510

yes

—

205

C₄

#17988820

yes

—

206

C₄

#17988859

yes

—

207

C₄

#17988962

yes

characteristic

208

C₄

#17988970

yes

characteristic

209

C₂

—

210

C₂

#22610776

yes

characteristic

211

C₂

#22610798

yes

characteristic

212

C₂

#22610847

yes

characteristic

	subgroup	G/H
center	C₂²	#17988490
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#500×C₇₃	C₂
socle	C₂²×C₇₃	#390
prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#500	D₇₃
hypocenter	C₇₃	#9799
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#17988490

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₂×#500×C₇₃

C₂

socle

C₂²×C₇₃

#390

prime radical

C₂³×C₇₃

C₄×D₄

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
86. C₈×C₇₃	#101	semidirect
112. C₇₃	#9799	semidirect
196. C₈	#14556510	semidirect

H ≅ G/N

type

86. C₈×C₇₃

#101

semidirect

112. C₇₃

#9799

semidirect

196. C₈

#14556510

semidirect

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

subgroup	index
#22610776	2
#22608690	2
#22608603	2
C₈×#11924012	2
C₂×#500×C₇₃	2
#9799	73

subgroup

index

C₂×#500×C₇₃

Subgroups — #32985602

Conjugacy classes of subgroups

Complements

Maximal subgroups