Group #32978751 — subgroups

Subgroups — #32978751

Order 18688.

456 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 151 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	#22609784	C₂	1	2	yes	maximal
2	C₂×#17988329	C₂	1	2	yes	maximal
3	#22607812	C₂	1	2	yes	maximal
4	C₄×#125×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596831	—	2	2	no	—
6	C₂×#14556502	C₂²	1	4	yes	—
7	#17988342	—	2	2	no	—
8	#17988329	—	2	4	no	—
9	C₂×#125×C₇₃	C₂²	1	1	yes	characteristic
10	#389×C₇₃	C₄	1	2	yes	—
11	#387×C₇₃	—	2	2	no	—
12	#382×C₇₃	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₇₃	—	2	2	no	—
14	C₄×C₄⋊C₄×C₇₃	C₄	1	2	yes	—
15	C₄³×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×#10596831	—	2	4	no	—
17	C₄⋊#10596831	—	2	8	no	—
18	#14556502	—	2	4	no	—
19	#14556502	—	2	4	no	—
20	#125×C₇₃	—	2	2	no	—
21	#125×C₇₃	D₄	1	2	yes	—
22	#125×C₇₃	Q₈	1	2	yes	—
23	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₃	—	2	2	no	—
28	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₃	D₄	1	2	yes	—
31	C₂×C₄²×C₇₃	D₄	1	2	yes	—
32	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
33	#95×C₇₃	—	2	2	no	—
34	#95×C₇₃	—	2	2	no	—
35	#95×C₇₃	—	2	2	no	—
36	C₂×#10596831	—	2	8	no	—
37	C₂×#10596831	—	4	4	no	—
38	C₂×#10596831	—	4	4	no	—
39	C₂²×C₄×C₇₃	—	2	2	no	—
40	C₂²×C₄×C₇₃	—	2	2	no	—
41	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
46	C₄⋊C₄×C₇₃	—	2	2	no	—
47	C₄⋊C₄×C₇₃	—	2	4	no	—
48	C₄⋊C₄×C₇₃	—	2	4	no	—
49	C₄⋊C₄×C₇₃	—	4	2	no	—
50	C₄⋊C₄×C₇₃	—	4	4	no	—
51	C₄⋊C₄×C₇₃	C₄⋊C₄	1	4	yes	—
52	C₄²×C₇₃	—	2	2	no	—
53	C₄²×C₇₃	—	2	4	no	—
54	C₄²×C₇₃	—	4	1	no	—
55	C₄²×C₇₃	—	4	2	no	—
56	C₄²×C₇₃	—	4	4	no	—
57	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇₃	#41	1	2	yes	—
60	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	—	2	1	no	—
62	C₂×C₄×C₇₃	—	2	1	no	—
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	—	2	4	no	—
66	C₂×C₄×C₇₃	—	2	1	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	4	2	no	—
72	C₂×C₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	—	4	2	no	—
74	C₂×C₄×C₇₃	—	4	2	no	—
75	C₂×C₄×C₇₃	—	4	2	no	—
76	C₂×C₄×C₇₃	#101	1	2	yes	—
77	C₂×C₄×C₇₃	#104	1	4	yes	—
78	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#121	1	2	yes	—
81	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
82	#10596831	—	4	8	no	—
83	C₂²×C₇₃	#380	1	1	yes	characteristic
84	C₂²×C₇₃	C₂×#101	1	1	yes	characteristic
85	C₂²×C₇₃	#413	1	1	yes	characteristic
86	C₂²×C₇₃	C₂×#104	1	2	yes	—
87	C₂²×C₇₃	#421	1	2	yes	—
88	C₄×C₇₃	—	2	1	no	—
89	C₄×C₇₃	—	2	1	no	—
90	C₄×C₇₃	—	2	2	no	—
91	C₄×C₇₃	—	2	2	no	—
92	C₄×C₇₃	—	4	1	no	—
93	C₄×C₇₃	—	4	4	no	—
94	C₄×C₇₃	—	4	2	no	—
95	C₄×C₇₃	—	8	2	no	—
96	C₄×C₇₃	#330	1	4	yes	complemented
97	#8580	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	C₂×#330	1	1	yes	characteristic
99	C₂×C₇₃	#1529	1	2	yes	—
100	C₂×C₇₃	#1537	1	1	yes	characteristic
101	C₂×C₇₃	#1801	1	1	yes	characteristic
102	C₂×C₇₃	#1806	1	2	yes	—
103	C₄×#125	D₇₃	1	1	yes	characteristic
104	#1767	—	73	2	no	—
105	C₂×#330	—	73	2	no	—
106	#1295	—	73	2	no	—
107	C₇₃	#8580	1	1	yes	characteristic, complemented, Sylow
108	C₂×#125	C₂×D₇₃	1	1	yes	characteristic
109	C₂×C₄⋊C₈	—	146	2	no	—
110	C₂×#97	—	73	4	no	—
111	#389	Dic₇₃	1	2	yes	—
112	#387	—	2	2	no	—
113	#382	C₂×D₇₃	1	1	yes	characteristic
114	C₄×C₄⋊C₄	—	2	2	no	—
115	C₄×C₄⋊C₄	Dic₇₃	1	2	yes	—
116	C₄³	C₂×D₇₃	1	1	yes	characteristic
117	#336	—	146	2	no	—
118	#330	—	146	4	no	—
119	C₂²×C₈	—	146	4	no	—
120	#125	—	2	2	no	—
121	#125	Dic₁₄₆	1	2	yes	—
122	#125	D₂₉₂	1	2	yes	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
126	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
127	C₂×C₄²	—	2	2	no	—
128	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
129	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
132	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
133	C₄⋊C₈	—	146	8	no	—
134	#97	—	146	4	no	—
135	#97	—	146	4	no	—
136	#95	—	2	2	no	—
137	#95	—	2	2	no	—
138	#95	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
143	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
146	C₂×C₈	—	292	4	no	—
147	C₂×C₈	—	292	4	no	—
148	C₂×C₈	—	146	8	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	C₄⋊Dic₇₃	1	4	yes	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	4	1	no	—
158	C₄²	—	4	2	no	—
159	C₄²	—	4	4	no	—
160	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
161	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
162	C₄²	#11924028	1	2	yes	—
163	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	#14556510	1	2	yes	—
180	C₂×C₄	#14556518	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
182	C₂×C₄	#14556550	1	1	yes	characteristic
183	C₂×C₄	#14556565	1	2	yes	—
184	C₂×C₄	#14556570	1	2	yes	—
185	C₈	—	292	8	no	—
186	C₂²	#17988447	1	1	yes	characteristic
187	C₂²	C₂×#14556510	1	1	yes	characteristic
188	C₂²	#17988571	1	1	yes	characteristic
189	C₂²	C₂×#14556518	1	2	yes	—
190	C₂²	#17988604	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	4	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	1	no	—
198	C₄	—	8	2	no	—
199	C₄	#17988329	1	4	yes	complemented
200	C₂	C₂×#17988329	1	1	yes	characteristic
201	C₂	#22608538	1	2	yes	—
202	C₂	#22608569	1	1	yes	characteristic
203	C₂	#22609927	1	1	yes	characteristic
204	C₂	#22609951	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22609784

C₂

yes

maximal

C₂×#17988329

C₂

yes

maximal

#22607812

C₂

yes

maximal

C₄×#125×C₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596831

—

C₂×#14556502

C₂²

yes

—

#17988342

—

#17988329

—

C₂×#125×C₇₃

C₂²

yes

characteristic

#389×C₇₃

C₄

yes

—

#387×C₇₃

—

#382×C₇₃

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₇₃

—

C₄×C₄⋊C₄×C₇₃

C₄

yes

—

C₄³×C₇₃

C₂²

yes

—

—

—

—

#125×C₇₃

—

#125×C₇₃

D₄

yes

—

#125×C₇₃

Q₈

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#95×C₇₃

—

#95×C₇₃

—

#95×C₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#101

yes

—

C₂×C₄×C₇₃

#104

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#421

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

100

C₂×C₇₃

#1537

yes

characteristic

101

C₂×C₇₃

#1801

yes

characteristic

102

C₂×C₇₃

#1806

yes

—

103

C₄×#125

D₇₃

yes

characteristic

104

#1767

—

105

C₂×#330

—

106

#1295

—

107

C₇₃

#8580

yes

characteristic, complemented, Sylow

108

C₂×#125

C₂×D₇₃

yes

characteristic

109

C₂×C₄⋊C₈

—

146

—

110

C₂×#97

—

111

#389

Dic₇₃

yes

—

112

#387

—

113

#382

C₂×D₇₃

yes

characteristic

114

C₄×C₄⋊C₄

—

115

C₄×C₄⋊C₄

Dic₇₃

yes

—

116

C₄³

C₂×D₇₃

yes

characteristic

117

#336

—

146

—

118

#330

—

146

—

119

C₂²×C₈

—

146

—

120

#125

—

121

#125

Dic₁₄₆

yes

—

122

#125

D₂₉₂

yes

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

C₂×Dic₇₃

yes

—

126

C₂×C₄⋊C₄

C₂×Dic₇₃

yes

—

127

C₂×C₄²

—

128

C₂×C₄²

C₂×Dic₇₃

yes

characteristic

129

C₂×C₄²

C₂×Dic₇₃

yes

characteristic

130

C₂×C₄²

C₂²⋊D₇₃

yes

—

131

C₂×C₄²

C₂²⋊D₇₃

yes

—

132

C₂×C₄²

C₂²×D₇₃

yes

characteristic

133

C₄⋊C₈

—

146

—

134

#97

—

146

—

135

#97

—

146

—

136

#95

—

137

#95

—

138

#95

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²⋊Dic₇₃

yes

characteristic

142

C₂²×C₄

C₂²⋊Dic₇₃

yes

—

143

C₂²×C₄

C₂²⋊Dic₇₃

yes

characteristic

144

C₂²×C₄

C₂²×Dic₇₃

yes

characteristic

145

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

—

146

C₂×C₈

—

292

—

147

C₂×C₈

—

292

—

148

C₂×C₈

—

146

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

C₄⋊Dic₇₃

yes

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₂×D₂₉₂

yes

characteristic

161

C₄²

C₂×Dic₁₄₆

yes

characteristic

162

C₄²

#11924028

yes

—

163

C₂³

C₂×C₂²⋊Dic₇₃

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#14556510

yes

—

180

C₂×C₄

#14556518

yes

—

181

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

182

C₂×C₄

#14556550

yes

characteristic

183

C₂×C₄

#14556565

yes

—

184

C₂×C₄

#14556570

yes

—

185

C₈

—

292

—

186

C₂²

#17988447

yes

characteristic

187

C₂²

C₂×#14556510

yes

characteristic

188

C₂²

#17988571

yes

characteristic

189

C₂²

C₂×#14556518

yes

—

190

C₂²

#17988604

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

#17988329

yes

complemented

200

C₂

C₂×#17988329

yes

characteristic

201

C₂

#22608538

yes

—

202

C₂

#22608569

yes

characteristic

203

C₂

#22609927

yes

characteristic

204

C₂

#22609951

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₃
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×#125×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×#125	D₇₃
hypocenter	C₇₃	#8580
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₇₃

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₄×#125×C₇₃

C₂

socle, prime radical

C₂³×C₇₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
96. C₄×C₇₃	#330	semidirect
107. C₇₃	#8580	semidirect
199. C₄	#17988329	semidirect

H ≅ G/N

type

96. C₄×C₇₃

#330

semidirect

107. C₇₃

#8580

semidirect

199. C₄

#17988329

semidirect

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

subgroup	index
#22609784	2
C₂×#17988329	2
#22607812	2
C₄×#125×C₇₃	2
#8580	73

subgroup

index

C₄×#125×C₇₃

Subgroups — #32978751

Conjugacy classes of subgroups

Complements

Maximal subgroups