Group #32986016 — subgroups

Subgroups — #32986016

Order 18688.

486 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 167 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	#22610083	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556502	C₂	1	1	yes	maximal, characteristic
3	#22607772	C₂	1	4	yes	maximal
4	C₄×#125×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596831	—	2	2	no	—
6	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
7	C₂×#14556502	—	2	2	no	—
8	C₂×C₄×#10596831	C₂²	1	2	yes	—
9	C₂×#125×C₇₃	C₄	1	1	yes	characteristic
10	#389×C₇₃	C₄	1	2	yes	—
11	#387×C₇₃	—	2	2	no	—
12	#382×C₇₃	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₇₃	—	2	2	no	—
14	C₄×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
15	C₄³×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×#10596831	—	2	2	no	—
17	C₂²×#10596831	—	2	2	no	—
18	C₄⋊#10596831	—	2	4	no	—
19	C₄⋊#10596831	—	4	4	no	—
20	#14556502	—	2	8	no	—
21	C₄×#10596831	—	2	2	no	—
22	C₄×#10596831	Q₈	1	4	yes	—
23	#125×C₇₃	—	2	4	no	—
24	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇₃	—	2	2	no	—
29	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	D₄	1	2	yes	—
32	C₂×C₄²×C₇₃	D₄	1	2	yes	—
33	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
34	#95×C₇₃	—	2	2	no	—
35	#95×C₇₃	—	2	2	no	—
36	#95×C₇₃	—	2	2	no	—
37	C₂×#10596831	—	2	8	no	—
38	C₂×#10596831	—	2	4	no	—
39	C₂×#10596831	—	4	2	no	—
40	C₂×#10596831	—	4	4	no	—
41	C₂²×C₄×C₇₃	—	2	2	no	—
42	C₂²×C₄×C₇₃	—	2	2	no	—
43	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₄⋊C₄×C₇₃	—	2	8	no	—
49	C₄⋊C₄×C₇₃	—	2	2	no	—
50	C₄⋊C₄×C₇₃	—	2	4	no	—
51	C₄⋊C₄×C₇₃	—	4	4	no	—
52	C₄⋊C₄×C₇₃	M₁₆	1	4	yes	—
53	C₄²×C₇₃	—	2	4	no	—
54	C₄²×C₇₃	—	2	2	no	—
55	C₄²×C₇₃	—	2	2	no	—
56	C₄²×C₇₃	—	2	2	no	—
57	C₄²×C₇₃	—	2	2	no	—
58	C₄²×C₇₃	—	4	2	no	—
59	C₄²×C₇₃	—	4	2	no	—
60	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
61	C₄²×C₇₃	#41	1	2	yes	—
62	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	2	4	no	—
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	—	4	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	#104	1	4	yes	—
76	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
77	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₇₃	#131	1	2	yes	—
82	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₇₃	#137	1	2	yes	—
84	#10596831	—	2	8	no	—
85	#10596831	—	8	2	no	—
86	C₂²×C₇₃	#391	1	1	yes	characteristic
87	C₂²×C₇₃	#407	1	1	yes	characteristic
88	C₂²×C₇₃	#408	1	1	yes	characteristic
89	C₂²×C₇₃	#417	1	2	yes	—
90	C₂²×C₇₃	C₂×#104	1	2	yes	—
91	C₄×C₇₃	—	2	2	no	—
92	C₄×C₇₃	—	2	4	no	—
93	C₄×C₇₃	—	4	2	no	—
94	C₄×C₇₃	—	4	4	no	—
95	C₄×C₇₃	—	8	2	no	—
96	C₄×C₇₃	#446	1	4	yes	—
97	C₄×C₇₃	#490	1	1	yes	characteristic
98	C₄×C₇₃	#491	1	1	yes	characteristic
99	C₄×C₇₃	#501	1	2	yes	—
100	#9876	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	#1483	1	2	yes	—
102	C₂×C₇₃	#1489	1	2	yes	—
103	C₂×C₇₃	#1985	1	1	yes	characteristic
104	C₂×C₇₃	#1992	1	1	yes	characteristic
105	C₂×C₇₃	#1994	1	1	yes	characteristic
106	C₄×#125	D₇₃	1	1	yes	characteristic
107	#1842	—	73	1	no	—
108	C₄×#97	—	73	1	no	—
109	#1279	—	73	4	no	—
110	C₇₃	#9876	1	1	yes	characteristic, complemented, Sylow
111	C₂×#125	Dic₇₃	1	1	yes	characteristic
112	C₂×C₄⋊C₈	—	73	2	no	—
113	C₂×C₄⋊C₈	—	146	2	no	—
114	C₂×#97	—	146	2	no	—
115	C₂×C₄×C₈	—	73	2	no	—
116	#389	Dic₇₃	1	2	yes	—
117	#387	—	2	2	no	—
118	#382	Dic₇₃	1	1	yes	characteristic
119	C₄×C₄⋊C₄	—	2	2	no	—
120	C₄×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
121	C₄³	C₂×D₇₃	1	1	yes	characteristic
122	C₂²×C₈	—	146	2	no	—
123	C₂²×C₈	—	146	2	no	—
124	#125	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
128	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
131	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
134	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
135	C₄⋊C₈	—	292	4	no	—
136	C₄⋊C₈	—	146	4	no	—
137	#97	—	146	8	no	—
138	C₄×C₈	—	73	4	no	—
139	C₄×C₈	—	146	2	no	—
140	#95	—	2	2	no	—
141	#95	—	2	2	no	—
142	#95	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
146	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
147	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
150	C₂×C₈	—	292	2	no	—
151	C₂×C₈	—	292	4	no	—
152	C₂×C₈	—	146	8	no	—
153	C₂×C₈	—	146	4	no	—
154	C₄⋊C₄	—	2	8	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	#11924012	1	4	yes	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	4	2	no	—
166	C₄²	Q₈×D₇₃	1	2	yes	—
167	C₄²	#11924029	1	2	yes	—
168	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	#14556518	1	4	yes	—
182	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
183	C₂×C₄	#14556567	1	2	yes	—
184	C₂×C₄	#14556580	1	1	yes	characteristic
185	C₂×C₄	#14556589	1	1	yes	characteristic
186	C₂×C₄	C₂×#11924012	1	2	yes	—
187	C₂×C₄	#14556598	1	2	yes	—
188	C₂×C₄	#14556616	1	2	yes	—
189	C₂×C₄	#14556621	1	2	yes	—
190	C₈	—	584	2	no	—
191	C₈	—	146	8	no	—
192	C₂²	#17988494	1	1	yes	characteristic
193	C₂²	#17988551	1	1	yes	characteristic
194	C₂²	#17988554	1	1	yes	characteristic
195	C₂²	#17988588	1	2	yes	—
196	C₂²	C₂×#14556518	1	2	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	4	no	—
201	C₄	—	8	2	no	—
202	C₄	#17988718	1	4	yes	—
203	C₄	#17988953	1	1	yes	characteristic
204	C₄	#17988957	1	1	yes	characteristic
205	C₄	#17988997	1	2	yes	—
206	C₂	#22608344	1	2	yes	—
207	C₂	#22608370	1	2	yes	—
208	C₂	#22610872	1	1	yes	characteristic
209	C₂	#22610901	1	1	yes	characteristic
210	C₂	#22610909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610083

C₂

yes

maximal, characteristic

C₄×#14556502

C₂

yes

maximal, characteristic

#22607772

C₂

yes

maximal

C₄×#125×C₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596831

—

C₂×C₄⋊#10596831

C₂²

yes

—

C₂×#14556502

—

C₂×C₄×#10596831

C₂²

yes

—

C₂×#125×C₇₃

C₄

yes

characteristic

#389×C₇₃

C₄

yes

—

#387×C₇₃

—

#382×C₇₃

C₄

yes

characteristic

C₄×C₄⋊C₄×C₇₃

—

C₄×C₄⋊C₄×C₇₃

C₂²

yes

—

C₄³×C₇₃

C₂²

yes

—

—

—

—

—

—

Q₈

yes

—

#125×C₇₃

—

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

—

—

—

—

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

M₁₆

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

—

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

#104

yes

—

C₂×C₄×C₇₃

C₄×Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₃

#137

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

C₂×#104

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#446

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

100

#9876

—

maximal, Sylow

101

C₂×C₇₃

#1483

yes

—

102

C₂×C₇₃

#1489

yes

—

103

C₂×C₇₃

#1985

yes

characteristic

104

C₂×C₇₃

#1992

yes

characteristic

105

C₂×C₇₃

#1994

yes

characteristic

106

C₄×#125

D₇₃

yes

characteristic

107

#1842

—

108

C₄×#97

—

109

#1279

—

110

C₇₃

#9876

yes

characteristic, complemented, Sylow

111

C₂×#125

Dic₇₃

yes

characteristic

112

C₂×C₄⋊C₈

—

113

C₂×C₄⋊C₈

—

146

—

114

C₂×#97

—

146

—

115

C₂×C₄×C₈

—

116

#389

Dic₇₃

yes

—

117

#387

—

118

#382

Dic₇₃

yes

characteristic

119

C₄×C₄⋊C₄

—

120

C₄×C₄⋊C₄

C₂×D₇₃

yes

—

121

C₄³

C₂×D₇₃

yes

characteristic

122

C₂²×C₈

—

146

—

123

C₂²×C₈

—

146

—

124

#125

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

C₂×Dic₇₃

yes

—

128

C₂×C₄⋊C₄

C₂×Dic₇₃

yes

—

129

C₂×C₄²

—

130

C₂×C₄²

C₂×Dic₇₃

yes

characteristic

131

C₂×C₄²

C₂×Dic₇₃

yes

characteristic

132

C₂×C₄²

C₂²⋊D₇₃

yes

—

133

C₂×C₄²

C₂²⋊D₇₃

yes

—

134

C₂×C₄²

C₂²×D₇₃

yes

characteristic

135

C₄⋊C₈

—

292

—

136

C₄⋊C₈

—

146

—

137

#97

—

146

—

138

C₄×C₈

—

139

C₄×C₈

—

146

—

140

#95

—

141

#95

—

142

#95

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂²⋊Dic₇₃

yes

—

146

C₂²×C₄

C₂²⋊Dic₇₃

yes

—

147

C₂²×C₄

C₂²×Dic₇₃

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

149

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

150

C₂×C₈

—

292

—

151

C₂×C₈

—

292

—

152

C₂×C₈

—

146

—

153

C₂×C₈

—

146

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

#11924012

yes

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

Q₈×D₇₃

yes

—

167

C₄²

#11924029

yes

—

168

C₂³

C₂×C₂²⋊Dic₇₃

yes

characteristic

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

—

180

C₂×C₄

—

181

C₂×C₄

#14556518

yes

—

182

C₂×C₄

Q₈×Dic₇₃

yes

—

183

C₂×C₄

#14556567

yes

—

184

C₂×C₄

#14556580

yes

characteristic

185

C₂×C₄

#14556589

yes

characteristic

186

C₂×C₄

C₂×#11924012

yes

—

187

C₂×C₄

#14556598

yes

—

188

C₂×C₄

#14556616

yes

—

189

C₂×C₄

#14556621

yes

—

190

C₈

—

584

—

191

C₈

—

146

—

192

C₂²

#17988494

yes

characteristic

193

C₂²

#17988551

yes

characteristic

194

C₂²

#17988554

yes

characteristic

195

C₂²

#17988588

yes

—

196

C₂²

C₂×#14556518

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

#17988718

yes

—

203

C₄

#17988953

yes

characteristic

204

C₄

#17988957

yes

characteristic

205

C₄

#17988997

yes

—

206

C₂

#22608344

yes

—

207

C₂

#22608370

yes

—

208

C₂

#22610872

yes

characteristic

209

C₂

#22610901

yes

characteristic

210

C₂

#22610909

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
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	#9876
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₇₃

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 ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
110. C₇₃	#9876	semidirect

H ≅ G/N

type

110. C₇₃

#9876

semidirect

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

subgroup	index
#22610083	2
C₄×#14556502	2
#22607772	2
C₄×#125×C₇₃	2
#9876	73

subgroup

index

C₄×#125×C₇₃

Subgroups — #32986016

Conjugacy classes of subgroups

Complements

Maximal subgroups