Group #32986106 — subgroups

Subgroups — #32986106

Order 18688.

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

normal subgroups only

This group has 430 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	#22610048	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
3	#22607814	C₂	1	4	yes	maximal
4	C₇₃×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
6	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
7	C₂×C₄×#10596831	—	2	1	no	—
8	#17988342	—	2	1	no	—
9	#17988342	—	2	2	no	—
10	C₂×#125×C₇₃	—	2	1	no	—
11	#389×C₇₃	—	2	2	no	—
12	#389×C₇₃	C₄	1	4	yes	—
13	#384×C₇₃	C₂²	1	2	yes	—
14	#382×C₇₃	—	2	1	no	—
15	C₄³×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×#10596831	—	2	2	no	—
17	C₂²×#10596831	—	2	2	no	—
18	C₄⋊#10596831	—	2	2	no	—
19	C₄⋊#10596831	—	2	4	no	—
20	C₄⋊#10596831	Q₈	1	4	yes	—
21	C₄×#10596831	—	4	2	no	—
22	#125×C₇₃	—	4	4	no	—
23	C₂×C₄⋊C₄×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	4	yes	—
27	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇₃	D₄	1	2	yes	—
29	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	#95×C₇₃	—	2	4	no	—
33	C₂×#10596831	—	2	4	no	—
34	C₂×#10596831	—	4	2	no	—
35	C₂×#10596831	—	4	2	no	—
36	C₂×#10596831	—	4	2	no	—
37	C₂×#10596831	—	4	4	no	—
38	C₂²×C₄×C₇₃	—	2	4	no	—
39	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₇₃	—	4	4	no	—
45	C₄⋊C₄×C₇₃	—	4	8	no	—
46	C₄⋊C₄×C₇₃	—	4	4	no	—
47	C₄⋊C₄×C₇₃	—	4	4	no	—
48	C₄²×C₇₃	—	2	4	no	—
49	C₄²×C₇₃	—	2	2	no	—
50	C₄²×C₇₃	—	4	2	no	—
51	C₄²×C₇₃	SD₁₆	1	2	yes	—
52	C₄²×C₇₃	Q₁₆	1	2	yes	—
53	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
54	C₄²×C₇₃	#41	1	2	yes	—
55	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	—	2	2	no	—
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	—	4	4	no	—
63	C₂×C₄×C₇₃	—	4	4	no	—
64	C₂×C₄×C₇₃	—	4	4	no	—
65	C₂×C₄×C₇₃	#103	1	4	yes	—
66	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#135	1	4	yes	—
73	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
74	#10596831	—	4	4	no	—
75	#10596831	—	8	2	no	—
76	C₂²×C₇₃	#391	1	1	yes	characteristic
77	C₂²×C₇₃	#413	1	2	yes	—
78	C₂²×C₇₃	C₂×#103	1	1	yes	characteristic
79	C₂²×C₇₃	#416	1	2	yes	—
80	C₂²×C₇₃	#418	1	1	yes	characteristic
81	C₄×C₇₃	—	2	4	no	—
82	C₄×C₇₃	—	2	2	no	—
83	C₄×C₇₃	—	4	2	no	—
84	C₄×C₇₃	—	8	4	no	—
85	C₄×C₇₃	#476	1	2	yes	—
86	C₄×C₇₃	#477	1	2	yes	—
87	C₄×C₇₃	#488	1	2	yes	—
88	C₄×C₇₃	#499	1	2	yes	—
89	#9897	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	#1542	1	2	yes	—
91	C₂×C₇₃	#1553	1	2	yes	—
92	C₂×C₇₃	#1987	1	1	yes	characteristic
93	C₂×C₇₃	#1991	1	1	yes	characteristic
94	C₂×C₇₃	#1998	1	1	yes	characteristic
95	#2859	D₇₃	1	1	yes	characteristic
96	#1831	—	73	1	no	—
97	C₄×C₄⋊C₈	—	73	1	no	—
98	#1296	—	73	4	no	—
99	C₇₃	#9897	1	1	yes	characteristic, complemented, Sylow
100	C₂×#125	—	2	1	no	—
101	C₂×C₄⋊C₈	—	73	2	no	—
102	C₂×C₄⋊C₈	—	73	2	no	—
103	C₂×C₄×C₈	—	146	1	no	—
104	#389	—	2	2	no	—
105	#389	Dic₇₃	1	4	yes	—
106	#384	C₂×D₇₃	1	2	yes	—
107	#382	—	2	1	no	—
108	C₄³	C₂×D₇₃	1	1	yes	characteristic
109	#336	—	146	2	no	—
110	#336	—	146	1	no	—
111	C₂²×C₈	—	146	2	no	—
112	C₂²×C₈	—	146	2	no	—
113	#125	—	4	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
118	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
119	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
120	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
121	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
123	C₄⋊C₈	—	73	4	no	—
124	C₄⋊C₈	—	146	4	no	—
125	C₄⋊C₈	—	146	2	no	—
126	C₄×C₈	—	292	2	no	—
127	#95	—	2	4	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
130	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
131	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
134	C₂×C₈	—	292	2	no	—
135	C₂×C₈	—	292	2	no	—
136	C₂×C₈	—	292	2	no	—
137	C₂×C₈	—	292	4	no	—
138	C₂×C₈	—	146	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#11924015	1	2	yes	—
147	C₄²	#11924018	1	2	yes	—
148	C₄²	Q₈×D₇₃	1	2	yes	—
149	C₄²	#11924029	1	2	yes	—
150	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	#14556515	1	4	yes	—
161	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
162	C₂×C₄	#14556567	1	2	yes	—
163	C₂×C₄	#14556580	1	1	yes	characteristic
164	C₂×C₄	#14556589	1	1	yes	characteristic
165	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
166	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
167	C₂×C₄	#14556611	1	4	yes	—
168	C₂×C₄	#14556617	1	2	yes	—
169	C₈	—	292	4	no	—
170	C₈	—	584	2	no	—
171	C₂²	#17988494	1	1	yes	characteristic
172	C₂²	#17988571	1	2	yes	—
173	C₂²	C₂×#14556515	1	1	yes	characteristic
174	C₂²	#17988583	1	2	yes	—
175	C₂²	#17988592	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	4	no	—
180	C₄	#17988874	1	2	yes	—
181	C₄	#17988879	1	2	yes	—
182	C₄	#17988944	1	2	yes	—
183	C₄	#17988986	1	2	yes	—
184	C₂	#22608588	1	2	yes	—
185	C₂	#22608636	1	2	yes	—
186	C₂	#22610880	1	1	yes	characteristic
187	C₂	#22610897	1	1	yes	characteristic
188	C₂	#22610925	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610048

C₂

yes

maximal, characteristic

C₄×C₄⋊#10596831

C₂

yes

maximal, characteristic

#22607814

C₂

yes

maximal

C₇₃×#2859

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596831

C₂²

yes

—

C₂×C₄⋊#10596831

C₂²

yes

—

C₂×C₄×#10596831

—

#17988342

—

#17988342

—

C₂×#125×C₇₃

—

#389×C₇₃

—

#389×C₇₃

C₄

yes

—

#384×C₇₃

C₂²

yes

—

#382×C₇₃

—

C₄³×C₇₃

C₂²

yes

—

—

—

—

Q₈

yes

—

C₄×#10596831

—

#125×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#95×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

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

—

C₄²×C₇₃

Q₁₆

yes

—

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

#103

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×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

#391

yes

characteristic

C₂²×C₇₃

#413

yes

—

C₂²×C₇₃

C₂×#103

yes

characteristic

C₂²×C₇₃

#416

yes

—

C₂²×C₇₃

#418

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#476

yes

—

C₄×C₇₃

#477

yes

—

C₄×C₇₃

#488

yes

—

C₄×C₇₃

#499

yes

—

#9897

—

maximal, Sylow

C₂×C₇₃

#1542

yes

—

C₂×C₇₃

#1553

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

characteristic

#1831

—

C₄×C₄⋊C₈

—

#1296

—

C₇₃

#9897

yes

characteristic, complemented, Sylow

100

C₂×#125

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄×C₈

—

146

—

104

#389

—

105

#389

Dic₇₃

yes

—

106

#384

C₂×D₇₃

yes

—

107

#382

—

108

C₄³

C₂×D₇₃

yes

characteristic

109

#336

—

146

—

110

#336

—

146

—

111

C₂²×C₈

—

146

—

112

C₂²×C₈

—

146

—

113

#125

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

C₂×Dic₇₃

yes

—

118

C₂×C₄²

C₂×Dic₇₃

yes

—

119

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

120

C₂×C₄²

C₂²⋊D₇₃

yes

—

121

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

122

C₂×C₄²

C₂²×D₇₃

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

146

—

125

C₄⋊C₈

—

146

—

126

C₄×C₈

—

292

—

127

#95

—

128

C₂²×C₄

—

129

C₂²×C₄

C₂²⋊Dic₇₃

yes

—

130

C₂²×C₄

C₂²⋊Dic₇₃

yes

—

131

C₂²×C₄

C₂²×Dic₇₃

yes

characteristic

132

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

133

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

134

C₂×C₈

—

292

—

135

C₂×C₈

—

292

—

136

C₂×C₈

—

292

—

137

C₂×C₈

—

292

—

138

C₂×C₈

—

146

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#11924015

yes

—

147

C₄²

#11924018

yes

—

148

C₄²

Q₈×D₇₃

yes

—

149

C₄²

#11924029

yes

—

150

C₂³

C₂×C₂²⋊Dic₇₃

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

#14556515

yes

—

161

C₂×C₄

Q₈×Dic₇₃

yes

—

162

C₂×C₄

#14556567

yes

—

163

C₂×C₄

#14556580

yes

characteristic

164

C₂×C₄

#14556589

yes

characteristic

165

C₂×C₄

C₂×#11924015

yes

characteristic

166

C₂×C₄

C₂×#11924018

yes

characteristic

167

C₂×C₄

#14556611

yes

—

168

C₂×C₄

#14556617

yes

—

169

C₈

—

292

—

170

C₈

—

584

—

171

C₂²

#17988494

yes

characteristic

172

C₂²

#17988571

yes

—

173

C₂²

C₂×#14556515

yes

characteristic

174

C₂²

#17988583

yes

—

175

C₂²

#17988592

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#17988874

yes

—

181

C₄

#17988879

yes

—

182

C₄

#17988944

yes

—

183

C₄

#17988986

yes

—

184

C₂

#22608588

yes

—

185

C₂

#22608636

yes

—

186

C₂

#22610880

yes

characteristic

187

C₂

#22610897

yes

characteristic

188

C₂

#22610925

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₃
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#2859	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#2859	D₇₃
hypocenter	C₇₃	#9897
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₇₃

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#2859

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 ◁ #2859

N	H ≅ G/N	type
99. C₇₃	#9897	semidirect

H ≅ G/N

type

99. C₇₃

#9897

semidirect

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

subgroup	index
#22610048	2
C₄×C₄⋊#10596831	2
#22607814	2
C₇₃×#2859	2
#9897	73

subgroup

index

C₇₃×#2859

Subgroups — #32986106

Conjugacy classes of subgroups

Complements

Maximal subgroups