Group #33004507 — subgroups

Subgroups — #33004507

Order 18688.

386 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 159 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×C₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic
3	#22608008	C₂	1	4	yes	maximal
4	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
6	C₂²×#11924003	C₂²	1	2	yes	—
7	C₈×Dic₁₄₆	—	2	2	no	—
8	C₈×Dic₁₄₆	—	2	2	no	—
9	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
10	C₂×#14556520	C₄	1	2	yes	—
11	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
12	C₄⋊#11924003	—	2	2	no	—
13	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	#346×C₇₃	—	2	2	no	—
16	C₂²×Dic₁₄₆	C₈	1	1	yes	characteristic
17	C₂²×#10596831	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₄₆	—	2	2	no	—
19	C₄×Dic₁₄₆	—	2	2	no	—
20	C₂×C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
21	C₂×#11924007	C₈	1	2	yes	—
22	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
23	C₂×#11924003	—	2	2	no	—
24	C₂×#11924003	—	2	4	no	—
25	C₄⋊#10596831	—	2	2	no	—
26	#14556520	—	2	4	no	—
27	C₈×Dic₇₃	—	2	2	no	—
28	C₈×Dic₇₃	—	2	2	no	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₃	—	2	2	no	—
33	C₂×C₁₆×C₇₃	—	2	2	no	—
34	C₂×C₁₆×C₇₃	Q₈	1	4	yes	complemented
35	C₄×C₈×C₇₃	D₄	1	2	yes	—
36	C₄×C₈×C₇₃	D₄	1	2	yes	—
37	C₂×Dic₁₄₆	—	2	2	no	—
38	C₂×Dic₁₄₆	—	2	4	no	—
39	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
40	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
41	C₂×#10596831	—	2	2	no	—
42	C₂×#10596831	—	2	4	no	—
43	C₄⋊Dic₇₃	—	2	2	no	—
44	#11924007	—	2	4	no	—
45	C₄×Dic₇₃	—	2	2	no	—
46	C₄×Dic₇₃	—	2	2	no	—
47	#11924003	—	4	2	no	—
48	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	—	2	2	no	—
52	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₇₃	#41	1	2	yes	—
57	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₇₃	M₁₆	1	2	yes	—
59	C₁₆×C₇₃	—	2	4	no	—
60	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	—	2	2	no	—
63	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#131	1	2	yes	—
69	C₂×Dic₇₃	—	2	2	no	—
70	C₂×Dic₇₃	—	2	2	no	—
71	C₂×Dic₇₃	—	2	2	no	—
72	C₂×Dic₇₃	—	2	4	no	—
73	C₂×Dic₇₃	M₃₂	1	4	yes	—
74	Dic₁₄₆	—	2	8	no	—
75	C₈×C₇₃	—	2	2	no	—
76	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₇₃	#124	1	1	yes	characteristic
78	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
79	#10596831	—	4	2	no	—
80	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₂²×C₇₃	C₈×Q₈	1	2	yes	—
82	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
83	C₂²×C₇₃	#504	1	2	yes	—
84	C₄×C₇₃	—	2	2	no	—
85	C₄×C₇₃	—	2	2	no	—
86	C₄×C₇₃	#391	1	1	yes	characteristic
87	C₄×C₇₃	#408	1	1	yes	characteristic
88	C₄×C₇₃	#446	1	2	yes	—
89	Dic₇₃	—	2	4	no	—
90	Dic₇₃	—	4	2	no	—
91	#13264	—	73	1	no	maximal, Sylow
92	C₂×C₇₃	#1984	1	1	yes	characteristic
93	C₂×C₇₃	#2112	1	1	yes	characteristic
94	C₂×C₇₃	#2113	1	1	yes	characteristic
95	C₂×C₇₃	#2183	1	4	yes	—
96	C₂×C₈×Q₈	—	73	1	no	—
97	C₂×C₄⋊C₁₆	—	73	1	no	—
98	C₂×#346	D₇₃	1	1	yes	characteristic
99	#1374	—	73	4	no	—
100	C₇₃	#13264	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄×Q₈	—	73	1	no	—
102	C₂²×C₁₆	—	73	2	no	—
103	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
104	C₈×Q₈	—	146	2	no	—
105	C₈×Q₈	—	146	2	no	—
106	C₂×C₄⋊C₈	—	73	2	no	—
107	C₂×C₄⋊C₈	—	73	1	no	—
108	C₂×C₄×C₈	—	73	2	no	—
109	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
110	C₄⋊C₁₆	—	146	2	no	—
111	#346	—	2	2	no	—
112	C₂²×Q₈	—	73	1	no	—
113	C₂²×C₈	—	73	2	no	—
114	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
116	C₄×Q₈	—	146	2	no	—
117	C₄×Q₈	—	146	2	no	—
118	C₂×C₄⋊C₄	—	73	2	no	—
119	C₂×C₄⋊C₄	—	73	1	no	—
120	C₂×C₄²	—	73	2	no	—
121	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	146	2	no	—
125	C₂×C₁₆	—	146	4	no	—
126	C₂×C₁₆	Dic₁₄₆	1	4	yes	complemented
127	C₄⋊C₈	—	146	4	no	—
128	C₄⋊C₈	—	146	2	no	—
129	C₄×C₈	—	146	2	no	—
130	C₄×C₈	—	146	2	no	—
131	C₄×C₈	D₂₉₂	1	2	yes	—
132	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
133	C₂×Q₈	—	146	4	no	—
134	C₂×Q₈	—	146	2	no	—
135	C₂²×C₄	—	73	2	no	—
136	C₂²×C₄	—	73	2	no	—
137	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
138	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	146	2	no	—
142	C₂×C₈	—	146	4	no	—
143	C₂×C₈	#11924005	1	2	yes	—
144	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
145	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
146	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
147	C₂×C₈	#11924028	1	2	yes	—
148	C₄⋊C₄	—	146	4	no	—
149	C₄⋊C₄	—	146	2	no	—
150	C₄²	—	146	2	no	—
151	C₄²	—	146	2	no	—
152	C₄²	#11924005	1	2	yes	—
153	C₄²	#11924011	1	2	yes	—
154	C₁₆	—	2	4	no	—
155	C₁₆	—	292	2	no	—
156	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
157	Q₈	—	146	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	73	4	no	—
161	C₂×C₄	—	146	4	no	—
162	C₂×C₄	—	146	2	no	—
163	C₂×C₄	—	146	2	no	—
164	C₂×C₄	—	146	2	no	—
165	C₂×C₄	#14556503	1	2	yes	—
166	C₂×C₄	#14556503	1	2	yes	—
167	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
169	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
170	C₂×C₄	#14556597	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	292	2	no	—
173	C₈	#14556570	1	2	yes	—
174	C₈	#14556579	1	1	yes	characteristic
175	C₈	#14556591	1	1	yes	characteristic
176	C₂²	C₂×#14556503	1	1	yes	characteristic
177	C₂²	C₈×Dic₁₄₆	1	2	yes	—
178	C₂²	M₃₂×D₇₃	1	2	yes	—
179	C₂²	#17989005	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	292	2	no	—
183	C₄	—	146	4	no	—
184	C₄	#17988493	1	1	yes	characteristic
185	C₄	#17988553	1	1	yes	characteristic
186	C₄	#17988717	1	2	yes	—
187	C₂	#22610867	1	1	yes	characteristic
188	C₂	#22611541	1	1	yes	characteristic
189	C₂	#22611546	1	1	yes	characteristic
190	C₂	#22611837	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₄₆

C₂

yes

maximal, characteristic

C₂×C₄⋊#11924003

C₂

yes

maximal, characteristic

#22608008

C₂

yes

maximal

C₂×#346×C₇₃

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₄₆

C₄

yes

characteristic

C₂²×#11924003

C₂²

yes

—

C₈×Dic₁₄₆

—

C₈×Dic₁₄₆

—

C₂×C₄⋊#10596831

C₄

yes

characteristic

C₂×#14556520

C₄

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

—

C₄⋊#11924003

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#346×C₇₃

—

C₂²×Dic₁₄₆

C₈

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

C₂×C₄⋊Dic₇₃

C₈

yes

characteristic

C₂×#11924007

C₈

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

Q₈

yes

complemented

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

M₁₆

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×Q₈

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

#131

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

M₃₂

yes

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊Q₈

yes

—

C₈×C₇₃

#124

yes

characteristic

C₈×C₇₃

C₄⋊Q₈

yes

characteristic

#10596831

—

C₂²×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₃

C₈×Q₈

yes

—

C₂²×C₇₃

C₂×M₃₂

yes

—

C₂²×C₇₃

#504

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₈×Q₈

—

C₂×C₄⋊C₁₆

—

C₂×#346

D₇₃

yes

characteristic

#1374

—

100

C₇₃

#13264

yes

characteristic, complemented, Sylow

101

C₂×C₄×Q₈

—

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₇₃

yes

—

104

C₈×Q₈

—

146

—

105

C₈×Q₈

—

146

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

110

C₄⋊C₁₆

—

146

—

111

#346

—

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₇₃

yes

characteristic

115

C₂²×C₈

C₂²×D₇₃

yes

characteristic

116

C₄×Q₈

—

146

—

117

C₄×Q₈

—

146

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

—

121

C₂×C₄²

C₄×D₇₃

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

146

—

125

C₂×C₁₆

—

146

—

126

C₂×C₁₆

Dic₁₄₆

yes

complemented

127

C₄⋊C₈

—

146

—

128

C₄⋊C₈

—

146

—

129

C₄×C₈

—

146

—

130

C₄×C₈

—

146

—

131

C₄×C₈

D₂₉₂

yes

—

132

C₄×C₈

C₂²⋊D₇₃

yes

—

133

C₂×Q₈

—

146

—

134

C₂×Q₈

—

146

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₈×D₇₃

yes

characteristic

138

C₂²×C₄

C₈×D₇₃

yes

characteristic

139

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

146

—

142

C₂×C₈

—

146

—

143

C₂×C₈

#11924005

yes

—

144

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

145

C₂×C₈

C₂×D₂₉₂

yes

characteristic

146

C₂×C₈

C₂×Dic₁₄₆

yes

—

147

C₂×C₈

#11924028

yes

—

148

C₄⋊C₄

—

146

—

149

C₄⋊C₄

—

146

—

150

C₄²

—

146

—

151

C₄²

—

146

—

152

C₄²

#11924005

yes

—

153

C₄²

#11924011

yes

—

154

C₁₆

—

155

C₁₆

—

292

—

156

C₂³

C₂×C₈×D₇₃

yes

characteristic

157

Q₈

—

146

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

146

—

162

C₂×C₄

—

146

—

163

C₂×C₄

—

146

—

164

C₂×C₄

—

146

—

165

C₂×C₄

#14556503

yes

—

166

C₂×C₄

#14556503

yes

—

167

C₂×C₄

C₂×#11924005

yes

characteristic

168

C₂×C₄

C₄×Dic₁₄₆

yes

—

169

C₂×C₄

C₂×#11924011

yes

characteristic

170

C₂×C₄

#14556597

yes

—

171

C₈

—

172

C₈

—

292

—

173

C₈

#14556570

yes

—

174

C₈

#14556579

yes

characteristic

175

C₈

#14556591

yes

characteristic

176

C₂²

C₂×#14556503

yes

characteristic

177

C₂²

C₈×Dic₁₄₆

yes

—

178

C₂²

M₃₂×D₇₃

yes

—

179

C₂²

#17989005

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

292

—

183

C₄

—

146

—

184

C₄

#17988493

yes

characteristic

185

C₄

#17988553

yes

characteristic

186

C₄

#17988717

yes

—

187

C₂

#22610867

yes

characteristic

188

C₂

#22611541

yes

characteristic

189

C₂

#22611546

yes

characteristic

190

C₂

#22611837

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator, socle, prime radical	C₂³×C₇₃	C₂²×C₈
Fitting	C₂×#346×C₇₃	C₂
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	#13264
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₇₃

commutator, socle, prime radical

C₂³×C₇₃

C₂²×C₈

Fitting

C₂×#346×C₇₃

C₂

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₈ ◁ C₂×#346

N	H ≅ G/N	type
34. C₂×C₁₆×C₇₃	Q₈	semidirect
100. C₇₃	#13264	semidirect
126. C₂×C₁₆	Dic₁₄₆	semidirect

H ≅ G/N

type

34. C₂×C₁₆×C₇₃

Q₈

semidirect

100. C₇₃

#13264

semidirect

126. C₂×C₁₆

Dic₁₄₆

semidirect

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

subgroup	index
C₂×C₈×Dic₁₄₆	2
C₂×C₄⋊#11924003	2
#22608008	2
C₂×#346×C₇₃	2
#13264	73

subgroup

index

C₂×C₈×Dic₁₄₆

C₂×C₄⋊#11924003

#22608008

C₂×#346×C₇₃

#13264

Subgroups — #33004507

Conjugacy classes of subgroups

Complements

Maximal subgroups