Group #32974258 — subgroups

Subgroups — #32974258

Order 18688.

468 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22609680	C₂	1	2	yes	maximal
2	#22607757	C₂	1	4	yes	maximal
3	C₄×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556502	—	2	2	no	—
5	#17988342	—	2	2	no	—
6	#17988342	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₇₃	C₄	1	2	yes	—
9	#391×C₇₃	C₂²	1	2	yes	—
10	#389×C₇₃	—	2	2	no	—
11	#384×C₇₃	C₄	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₇₃	—	2	2	no	—
13	C₄³×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×#10596831	—	2	4	no	—
15	#14556502	—	2	8	no	—
16	C₂²×Q₈×C₇₃	C₂×C₄	1	2	yes	—
17	C₄⋊Q₈×C₇₃	—	2	4	no	—
18	C₄×Q₈×C₇₃	—	4	4	no	—
19	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
20	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇₃	—	2	2	no	—
24	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₃	D₄	1	2	yes	—
27	C₂×C₄²×C₇₃	D₄	1	2	yes	—
28	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
29	#95×C₇₃	—	2	4	no	—
30	C₂×#10596831	—	2	8	no	—
31	C₂×#10596831	—	4	4	no	—
32	C₂×#10596831	—	4	4	no	—
33	C₂×Q₈×C₇₃	—	2	2	no	—
34	C₂×Q₈×C₇₃	—	4	2	no	—
35	C₂×Q₈×C₇₃	—	4	2	no	—
36	C₂×Q₈×C₇₃	M₁₆	1	4	yes	—
37	C₂²×C₄×C₇₃	—	2	2	no	—
38	C₂²×C₄×C₇₃	—	2	2	no	—
39	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
44	C₄⋊C₄×C₇₃	—	2	8	no	—
45	C₄⋊C₄×C₇₃	—	4	2	no	—
46	C₄⋊C₄×C₇₃	—	4	4	no	—
47	C₄⋊C₄×C₇₃	—	4	4	no	—
48	C₄²×C₇₃	—	2	2	no	—
49	C₄²×C₇₃	—	2	4	no	—
50	C₄²×C₇₃	—	2	2	no	—
51	C₄²×C₇₃	—	2	4	no	—
52	C₄²×C₇₃	—	4	1	no	—
53	C₄²×C₇₃	—	4	4	no	—
54	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	Q₈×C₇₃	—	4	4	no	—
56	Q₈×C₇₃	—	4	4	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	2	1	no	—
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	—	2	2	no	—
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	4	no	—
65	C₂×C₄×C₇₃	—	2	1	no	—
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	4	2	no	—
68	C₂×C₄×C₇₃	—	4	2	no	—
69	C₂×C₄×C₇₃	—	4	2	no	—
70	C₂×C₄×C₇₃	—	4	4	no	—
71	C₂×C₄×C₇₃	#100	1	2	yes	—
72	C₂×C₄×C₇₃	#104	1	4	yes	—
73	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₇₃	#137	1	4	yes	—
75	#10596831	—	4	8	no	—
76	C₂²×C₇₃	#407	1	1	yes	characteristic
77	C₂²×C₇₃	#410	1	1	yes	characteristic
78	C₂²×C₇₃	C₂×#100	1	1	yes	characteristic
79	C₂²×C₇₃	#419	1	2	yes	—
80	C₂²×C₇₃	C₂×#104	1	2	yes	—
81	C₄×C₇₃	—	2	2	no	—
82	C₄×C₇₃	—	2	2	no	—
83	C₄×C₇₃	—	2	4	no	—
84	C₄×C₇₃	—	2	4	no	—
85	C₄×C₇₃	—	4	4	no	—
86	C₄×C₇₃	—	4	1	no	—
87	C₄×C₇₃	—	8	2	no	—
88	#7639	—	73	1	no	maximal, Sylow
89	C₂×C₇₃	#1466	1	1	yes	characteristic
90	C₂×C₇₃	#1491	1	2	yes	—
91	C₂×C₇₃	#1506	1	2	yes	—
92	C₂×C₇₃	#1525	1	1	yes	characteristic
93	C₂×C₇₃	#1558	1	1	yes	characteristic
94	C₄×C₄⋊Q₈	D₇₃	1	1	yes	characteristic
95	#1745	—	73	2	no	—
96	#1272	—	73	4	no	—
97	C₇₃	#7639	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄⋊Q₈	Dic₇₃	1	1	yes	characteristic
99	C₂×C₄×Q₈	Dic₇₃	1	2	yes	—
100	C₂×#97	—	146	2	no	—
101	#391	C₂×D₇₃	1	2	yes	—
102	#389	—	2	2	no	—
103	#384	Dic₇₃	1	1	yes	characteristic
104	C₄×C₄⋊C₄	—	2	2	no	—
105	C₄³	C₂×D₇₃	1	1	yes	characteristic
106	#336	—	73	4	no	—
107	#336	—	146	2	no	—
108	C₂²×Q₈	C₂×Dic₇₃	1	2	yes	—
109	C₂²×C₈	—	146	4	no	—
110	C₄⋊Q₈	—	2	4	no	—
111	C₄×Q₈	—	4	4	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
116	C₂×C₄²	—	2	2	no	—
117	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
118	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
119	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
120	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
121	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
122	#97	—	146	8	no	—
123	#95	—	2	4	no	—
124	C₂×Q₈	—	2	2	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	#11924012	1	4	yes	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
131	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
133	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
135	C₂×C₈	—	292	4	no	—
136	C₂×C₈	—	292	4	no	—
137	C₂×C₈	—	146	8	no	—
138	C₄⋊C₄	—	2	8	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	4	1	no	—
147	C₄²	—	4	4	no	—
148	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
149	Q₈	—	4	4	no	—
150	Q₈	—	4	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	#14556508	1	2	yes	—
166	C₂×C₄	#14556518	1	4	yes	—
167	C₂×C₄	C₂×#11924012	1	2	yes	—
168	C₂×C₄	#14556622	1	4	yes	—
169	C₈	—	292	8	no	—
170	C₂²	#17988551	1	1	yes	characteristic
171	C₂²	#17988561	1	1	yes	characteristic
172	C₂²	C₂×#14556508	1	1	yes	characteristic
173	C₂²	#17988596	1	2	yes	—
174	C₂²	C₂×#14556518	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	4	no	—
179	C₄	—	4	1	no	—
180	C₄	—	4	4	no	—
181	C₄	—	8	2	no	—
182	C₂	#22608272	1	1	yes	characteristic
183	C₂	#22608378	1	2	yes	—
184	C₂	#22608443	1	2	yes	—
185	C₂	#22608523	1	1	yes	characteristic
186	C₂	#22608657	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22609680

C₂

yes

maximal

#22607757

C₂

yes

maximal

C₄×C₄⋊Q₈×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556502

—

#17988342

—

#17988342

C₂²

yes

—

C₂×C₄⋊Q₈×C₇₃

C₄

yes

characteristic

C₂×C₄×Q₈×C₇₃

C₄

yes

—

#391×C₇₃

C₂²

yes

—

#389×C₇₃

—

#384×C₇₃

C₄

yes

characteristic

C₄×C₄⋊C₄×C₇₃

—

C₄³×C₇₃

C₂²

yes

characteristic

C₂²×#10596831

—

#14556502

—

C₂²×Q₈×C₇₃

C₂×C₄

yes

—

C₄⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

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

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

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

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×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

#100

yes

—

C₂×C₄×C₇₃

#104

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

#137

yes

—

#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

—

C₄×C₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#1506

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₄×C₄⋊Q₈

D₇₃

yes

characteristic

#1745

—

#1272

—

C₇₃

#7639

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

Dic₇₃

yes

characteristic

C₂×C₄×Q₈

Dic₇₃

yes

—

100

C₂×#97

—

146

—

101

#391

C₂×D₇₃

yes

—

102

#389

—

103

#384

Dic₇₃

yes

characteristic

104

C₄×C₄⋊C₄

—

105

C₄³

C₂×D₇₃

yes

characteristic

106

#336

—

107

#336

—

146

—

108

C₂²×Q₈

C₂×Dic₇₃

yes

—

109

C₂²×C₈

—

146

—

110

C₄⋊Q₈

—

111

C₄×Q₈

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

C₂×Dic₇₃

yes

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂×Dic₇₃

yes

characteristic

118

C₂×C₄²

C₂×Dic₇₃

yes

characteristic

119

C₂×C₄²

C₂²⋊D₇₃

yes

—

120

C₂×C₄²

C₂²⋊D₇₃

yes

—

121

C₂×C₄²

C₂²×D₇₃

yes

characteristic

122

#97

—

146

—

123

#95

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

#11924012

yes

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂²⋊Dic₇₃

yes

characteristic

131

C₂²×C₄

C₂²⋊Dic₇₃

yes

characteristic

132

C₂²×C₄

C₂²⋊Dic₇₃

yes

—

133

C₂²×C₄

C₂²×Dic₇₃

yes

characteristic

134

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

—

135

C₂×C₈

—

292

—

136

C₂×C₈

—

292

—

137

C₂×C₈

—

146

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₂³

C₂×C₂²⋊Dic₇₃

yes

characteristic

149

Q₈

—

150

Q₈

—

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

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#14556508

yes

—

166

C₂×C₄

#14556518

yes

—

167

C₂×C₄

C₂×#11924012

yes

—

168

C₂×C₄

#14556622

yes

—

169

C₈

—

292

—

170

C₂²

#17988551

yes

characteristic

171

C₂²

#17988561

yes

characteristic

172

C₂²

C₂×#14556508

yes

characteristic

173

C₂²

#17988596

yes

—

174

C₂²

C₂×#14556518

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₂

#22608272

yes

characteristic

183

C₂

#22608378

yes

—

184

C₂

#22608443

yes

—

185

C₂

#22608523

yes

characteristic

186

C₂

#22608657

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₃
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×C₄⋊Q₈×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₇₃
hypocenter	C₇₃	#7639
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₇₃

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₄×C₄⋊Q₈×C₇₃

C₂

socle, prime radical

C₂³×C₇₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇₃

C₂³

hypercenter

C₄×C₄⋊Q₈

D₇₃

hypocenter

C₇₃

#7639

LCS

G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

N	H ≅ G/N	type
97. C₇₃	#7639	semidirect

H ≅ G/N

type

97. C₇₃

#7639

semidirect

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

subgroup	index
#22609680	2
#22607757	2
C₄×C₄⋊Q₈×C₇₃	2
#7639	73

subgroup

index

#22609680

#22607757

C₄×C₄⋊Q₈×C₇₃

#7639

Subgroups — #32974258

Conjugacy classes of subgroups

Complements

Maximal subgroups