Group #33002196 — subgroups

Subgroups — #33002196

Order 18688.

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

normal subgroups only

This group has 454 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	C₂×#17988542	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#11924003	C₂	1	1	yes	maximal, characteristic
3	#22608008	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	C₄	1	1	yes	characteristic
6	C₂×#14556550	C₄	1	1	yes	characteristic
7	C₂²×#11924003	C₂²	1	2	yes	—
8	#17988542	—	2	2	no	—
9	#17988542	—	2	2	no	—
10	C₂×#14556501	C₄	1	2	yes	—
11	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
12	C₂²⋊#11924003	—	2	2	no	—
13	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
15	C₂²⋊C₁₆×C₇₃	—	2	2	no	—
16	C₂×#11924012	—	2	4	no	—
17	C₂×#11924012	—	2	2	no	—
18	C₂²×#10596831	C₂×C₄	1	2	yes	—
19	#14556550	—	2	2	no	—
20	#14556550	—	2	2	no	—
21	C₂×C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
22	C₂×C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₇₃	C₈	1	2	yes	—
24	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
25	C₂×#11924003	—	2	2	no	—
26	C₂×#11924003	—	2	4	no	—
27	#14556501	—	2	4	no	—
28	C₈×Dic₇₃	—	2	2	no	—
29	C₈×Dic₇₃	—	2	2	no	—
30	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	—	2	4	no	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	D₄	1	2	yes	—
34	C₂²×C₈×C₇₃	D₄	1	2	yes	—
35	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₁₆×C₇₃	—	2	2	no	—
37	C₂×C₁₆×C₇₃	—	2	4	no	—
38	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
39	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
40	#11924012	—	2	8	no	—
41	C₂×#10596831	—	2	2	no	—
42	C₂×#10596831	—	2	2	no	—
43	C₂×#10596831	M₁₆	1	4	yes	—
44	C₄⋊Dic₇₃	—	2	2	no	—
45	C₄⋊Dic₇₃	—	2	2	no	—
46	C₂²⋊Dic₇₃	—	2	4	no	—
47	C₄×Dic₇₃	—	2	2	no	—
48	C₄×Dic₇₃	—	2	2	no	—
49	#11924003	—	4	2	no	—
50	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	—	2	2	no	—
52	C₂²×C₄×C₇₃	—	2	2	no	—
53	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
56	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	—	2	2	no	—
58	C₂×C₈×C₇₃	—	2	4	no	—
59	C₂×C₈×C₇₃	—	2	4	no	—
60	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	#41	1	4	yes	—
64	C₁₆×C₇₃	—	4	2	no	—
65	C₂³×C₇₃	—	2	2	no	—
66	C₂³×C₇₃	—	2	2	no	—
67	C₂³×C₇₃	—	2	2	no	—
68	C₂³×C₇₃	C₂²⋊C₈	1	2	yes	—
69	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	—	2	4	no	—
74	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#117	1	2	yes	—
77	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×Dic₇₃	—	2	2	no	—
80	C₂×Dic₇₃	—	2	2	no	—
81	C₂×Dic₇₃	—	2	4	no	—
82	C₂×Dic₇₃	—	2	4	no	—
83	C₈×C₇₃	—	2	2	no	—
84	C₈×C₇₃	#123	1	2	yes	—
85	C₈×C₇₃	#123	1	2	yes	—
86	#10596831	—	2	4	no	—
87	C₂²×C₇₃	—	2	2	no	—
88	C₂²×C₇₃	—	2	2	no	—
89	C₂²×C₇₃	—	2	2	no	—
90	C₂²×C₇₃	—	4	4	no	—
91	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
92	C₂²×C₇₃	#431	1	2	yes	—
93	C₂²×C₇₃	#504	1	2	yes	—
94	C₂²×C₇₃	#504	1	2	yes	—
95	C₄×C₇₃	—	2	2	no	—
96	C₄×C₇₃	#381	1	1	yes	characteristic
97	C₄×C₇₃	#407	1	1	yes	characteristic
98	C₄×C₇₃	#432	1	2	yes	—
99	Dic₇₃	—	4	2	no	—
100	Dic₇₃	—	4	2	no	—
101	#12845	—	73	1	no	maximal, Sylow
102	C₂×C₇₃	—	4	2	no	—
103	C₂×C₇₃	#1817	1	1	yes	characteristic
104	C₂×C₇₃	#2113	1	1	yes	characteristic
105	C₂×C₇₃	#2113	1	1	yes	characteristic
106	C₂×C₇₃	#2164	1	4	yes	—
107	C₂×#405	—	73	1	no	—
108	C₂×C₂²⋊C₁₆	—	73	1	no	—
109	C₂×C₂²⋊C₁₆	D₇₃	1	1	yes	characteristic
110	#1374	—	73	4	no	—
111	C₇₃	#12845	1	1	yes	characteristic, complemented, Sylow
112	C₂²×M₁₆	—	73	1	no	—
113	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
114	C₂×#117	—	73	1	no	—
115	C₂²×C₁₆	—	73	2	no	—
116	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
117	#405	—	146	2	no	—
118	#405	—	146	2	no	—
119	C₂×#97	—	73	2	no	—
120	C₂×C₄×C₈	—	73	2	no	—
121	C₂²⋊C₁₆	—	2	2	no	—
122	C₂²⋊C₁₆	—	146	2	no	—
123	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
124	C₂×M₁₆	—	146	4	no	—
125	C₂×M₁₆	—	146	2	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	73	2	no	—
128	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
129	C₂²×C₈	D₂₉₂	1	2	yes	—
130	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
131	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
132	#117	—	146	2	no	—
133	#117	—	146	2	no	—
134	C₂×C₄⋊C₄	—	73	1	no	—
135	C₂×C₄⋊C₄	—	73	1	no	—
136	C₂×C₂²⋊C₄	—	73	2	no	—
137	C₂×C₄²	—	73	2	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	2	4	no	—
140	C₂×C₁₆	—	146	2	no	—
141	C₂×C₁₆	—	146	4	no	—
142	#97	—	146	4	no	—
143	C₄×C₈	—	146	2	no	—
144	C₄×C₈	—	146	2	no	—
145	C₂⁴	C₈×D₇₃	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	73	2	no	—
149	C₂²×C₄	—	73	2	no	—
150	C₂²×C₄	#11924005	1	2	yes	—
151	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
152	C₂²×C₄	#11924011	1	2	yes	—
153	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
154	M₁₆	—	146	8	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	73	4	no	—
159	C₂×C₈	—	146	2	no	—
160	C₂×C₈	—	146	2	no	—
161	C₂×C₈	#11924005	1	2	yes	—
162	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
163	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
164	C₂×C₈	#11924027	1	4	yes	—
165	C₄⋊C₄	—	146	2	no	—
166	C₄⋊C₄	—	146	2	no	—
167	C₂²⋊C₄	—	146	4	no	—
168	C₄²	—	146	2	no	—
169	C₄²	—	146	2	no	—
170	C₁₆	—	4	2	no	—
171	C₁₆	—	292	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	#14556503	1	2	yes	—
176	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	146	4	no	—
182	C₂×C₄	—	146	4	no	—
183	C₂×C₄	—	146	2	no	—
184	C₂×C₄	—	146	2	no	—
185	C₂×C₄	#14556503	1	2	yes	—
186	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
187	C₂×C₄	#14556547	1	2	yes	—
188	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
189	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	—	146	4	no	—
192	C₈	#14556572	1	2	yes	—
193	C₈	#14556575	1	2	yes	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	C₂×#14556503	1	1	yes	characteristic
199	C₂²	#17988637	1	2	yes	—
200	C₂²	#17989003	1	2	yes	—
201	C₂²	#17989005	1	2	yes	—
202	C₄	—	2	2	no	—
203	C₄	—	292	2	no	—
204	C₄	—	292	2	no	—
205	C₄	#17988448	1	1	yes	characteristic
206	C₄	#17988550	1	1	yes	characteristic
207	C₄	#17988641	1	2	yes	—
208	C₂	—	4	2	no	—
209	C₂	#22609997	1	1	yes	characteristic
210	C₂	#22611544	1	1	yes	characteristic
211	C₂	#22611546	1	1	yes	characteristic
212	C₂	#22611735	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988542

C₂

yes

maximal, characteristic

C₂×C₂²⋊#11924003

C₂

yes

maximal, characteristic

#22608008

C₂

yes

maximal, complemented

C₂×C₂²⋊C₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂²×#11924012

C₄

yes

characteristic

C₂×#14556550

C₄

yes

characteristic

C₂²×#11924003

C₂²

yes

—

#17988542

—

#17988542

—

C₂×#14556501

C₄

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

—

C₂²⋊#11924003

—

C₂³×C₈×C₇₃

C₂²

yes

characteristic

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂²⋊C₁₆×C₇₃

—

C₂×#11924012

—

C₂×#11924012

—

C₂²×#10596831

C₂×C₄

yes

—

#14556550

—

#14556550

—

C₂×C₄⋊Dic₇₃

C₈

yes

characteristic

C₂×C₄⋊Dic₇₃

C₈

yes

characteristic

C₂×C₂²⋊Dic₇₃

C₈

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₂×#11924003

—

C₂×#11924003

—

#14556501

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂³×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂²×Dic₇₃

C₂×C₈

yes

—

—

—

—

M₁₆

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂⁴×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

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

#41

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂²⋊C₈

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#123

yes

—

C₈×C₇₃

#123

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₃

#431

yes

—

C₂²×C₇₃

#504

yes

—

C₂²×C₇₃

#504

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

100

Dic₇₃

—

101

#12845

—

maximal, Sylow

102

C₂×C₇₃

—

103

C₂×C₇₃

#1817

yes

characteristic

104

C₂×C₇₃

#2113

yes

characteristic

105

C₂×C₇₃

#2113

yes

characteristic

106

C₂×C₇₃

#2164

yes

—

107

C₂×#405

—

108

C₂×C₂²⋊C₁₆

—

109

C₂×C₂²⋊C₁₆

D₇₃

yes

characteristic

110

#1374

—

111

C₇₃

#12845

yes

characteristic, complemented, Sylow

112

C₂²×M₁₆

—

113

C₂³×C₈

C₂×D₇₃

yes

characteristic

114

C₂×#117

—

115

C₂²×C₁₆

—

116

C₂²×C₁₆

C₂×D₇₃

yes

—

117

#405

—

146

—

118

#405

—

146

—

119

C₂×#97

—

120

C₂×C₄×C₈

—

121

C₂²⋊C₁₆

—

122

C₂²⋊C₁₆

—

146

—

123

C₂³×C₄

C₄×D₇₃

yes

characteristic

124

C₂×M₁₆

—

146

—

125

C₂×M₁₆

—

146

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

C₄×D₇₃

yes

characteristic

129

C₂²×C₈

D₂₉₂

yes

—

130

C₂²×C₈

C₂²⋊D₇₃

yes

—

131

C₂²×C₈

C₂²×D₇₃

yes

characteristic

132

#117

—

146

—

133

#117

—

146

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₂²⋊C₄

—

137

C₂×C₄²

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

C₂×C₁₆

—

146

—

141

C₂×C₁₆

—

146

—

142

#97

—

146

—

143

C₄×C₈

—

146

—

144

C₄×C₈

—

146

—

145

C₂⁴

C₈×D₇₃

yes

characteristic

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

#11924005

yes

—

151

C₂²×C₄

C₈×D₇₃

yes

characteristic

152

C₂²×C₄

#11924011

yes

—

153

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

154

M₁₆

—

146

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

146

—

160

C₂×C₈

—

146

—

161

C₂×C₈

#11924005

yes

—

162

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

163

C₂×C₈

C₂×D₂₉₂

yes

characteristic

164

C₂×C₈

#11924027

yes

—

165

C₄⋊C₄

—

146

—

166

C₄⋊C₄

—

146

—

167

C₂²⋊C₄

—

146

—

168

C₄²

—

146

—

169

C₄²

—

146

—

170

C₁₆

—

171

C₁₆

—

292

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

#14556503

yes

—

176

C₂³

C₂×C₈×D₇₃

yes

characteristic

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

146

—

182

C₂×C₄

—

146

—

183

C₂×C₄

—

146

—

184

C₂×C₄

—

146

—

185

C₂×C₄

#14556503

yes

—

186

C₂×C₄

C₂×#11924005

yes

characteristic

187

C₂×C₄

#14556547

yes

—

188

C₂×C₄

M₁₆×D₇₃

yes

—

189

C₂×C₄

C₂×#11924011

yes

characteristic

190

C₈

—

191

C₈

—

146

—

192

C₈

#14556572

yes

—

193

C₈

#14556575

yes

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

C₂×#14556503

yes

characteristic

199

C₂²

#17988637

yes

—

200

C₂²

#17989003

yes

—

201

C₂²

#17989005

yes

—

202

C₄

—

203

C₄

—

292

—

204

C₄

—

292

—

205

C₄

#17988448

yes

characteristic

206

C₄

#17988550

yes

characteristic

207

C₄

#17988641

yes

—

208

C₂

—

209

C₂

#22609997

yes

characteristic

210

C₂

#22611544

yes

characteristic

211

C₂

#22611546

yes

characteristic

212

C₂

#22611735

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator, socle	C₂³×C₇₃	C₂²×C₈
Fitting	C₂×C₂²⋊C₁₆×C₇₃	C₂
prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₇₃
hypocenter	C₇₃	#12845
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₇₃

commutator, socle

C₂³×C₇₃

C₂²×C₈

Fitting

C₂×C₂²⋊C₁₆×C₇₃

C₂

prime radical

C₂⁴×C₇₃

C₂×C₈

cosocle

C₂²×C₈×C₇₃

C₂³

hypercenter

C₂×C₂²⋊C₁₆

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

N	H ≅ G/N	type
3. #22608008	C₂	semidirect
111. C₇₃	#12845	semidirect

H ≅ G/N

type

3. #22608008

C₂

semidirect

111. C₇₃

#12845

semidirect

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

subgroup	index
C₂×#17988542	2
C₂×C₂²⋊#11924003	2
#22608008	2
C₂×C₂²⋊C₁₆×C₇₃	2
#12845	73

subgroup

index

C₂×#17988542

C₂×C₂²⋊#11924003

#22608008

C₂×C₂²⋊C₁₆×C₇₃

#12845

Subgroups — #33002196

Conjugacy classes of subgroups

Complements

Maximal subgroups