Group #32991110 — subgroups

Subgroups — #32991110

Order 18688.

264 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22610654	C₂	1	1	yes	maximal, characteristic
2	#22610106	C₂	1	1	yes	maximal, characteristic
3	#22607833	C₂	1	2	yes	maximal
4	#22607827	C₂	1	2	yes	maximal
5	C₇₃×#1275	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556527	—	2	1	no	—
7	C₂×#14556527	—	2	1	no	—
8	#17988611	C₂²	1	2	yes	—
9	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
10	C₂×#14556520	—	2	1	no	—
11	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	#17988341	—	2	1	no	—
13	#431×C₇₃	C₂²	1	2	yes	—
14	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#11924012	—	2	2	no	—
16	C₂²×#10596831	—	2	1	no	—
17	C₂×C₄⋊Dic₇₃	—	2	1	no	—
18	C₂×C₄×Dic₇₃	—	2	1	no	—
19	#14556527	—	4	2	no	—
20	#14556527	—	4	2	no	—
21	C₄⋊#10596831	—	2	2	no	—
22	C₄⋊#10596831	—	2	4	no	—
23	#14556520	—	4	2	no	—
24	#14556498	Q₈	1	1	yes	characteristic
25	#14556498	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	—	2	2	no	—
30	C₂²⋊C₈×C₇₃	—	2	2	no	—
31	#97×C₇₃	—	2	2	no	—
32	C₄×C₈×C₇₃	—	2	2	no	—
33	C₂²×Dic₇₃	—	2	1	no	—
34	C₂²×Dic₇₃	—	2	1	no	—
35	#11924012	—	4	2	no	—
36	#11924012	—	4	2	no	—
37	C₂×#10596831	—	4	1	no	—
38	C₂×#10596831	—	4	2	no	—
39	C₂×#10596831	—	4	1	no	—
40	C₂×#10596831	—	4	1	no	—
41	C₄⋊Dic₇₃	—	4	2	no	—
42	C₄×Dic₇₃	—	4	1	no	—
43	C₄×Dic₇₃	—	4	1	no	—
44	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
47	C₂×C₈×C₇₃	—	2	1	no	—
48	C₂×C₈×C₇₃	—	2	1	no	—
49	C₂×C₈×C₇₃	—	2	1	no	—
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₂×C₈×C₇₃	—	4	2	no	—
52	C₂×C₈×C₇₃	D₈	1	2	yes	—
53	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
54	C₄²×C₇₃	#41	1	2	yes	—
55	C₄²×C₇₃	#41	1	2	yes	—
56	C₂³×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₇₃	—	4	1	no	—
58	C₂×C₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#125	1	2	yes	—
62	C₂×C₄×C₇₃	#126	1	2	yes	—
63	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
67	C₂×Dic₇₃	—	2	2	no	—
68	C₂×Dic₇₃	—	4	1	no	—
69	C₂×Dic₇₃	—	4	1	no	—
70	C₂×Dic₇₃	—	4	1	no	—
71	C₂×Dic₇₃	—	4	1	no	—
72	C₂×Dic₇₃	—	4	1	no	—
73	C₈×C₇₃	—	2	2	no	—
74	C₈×C₇₃	—	2	2	no	—
75	C₈×C₇₃	—	8	2	no	—
76	#10596831	—	8	1	no	—
77	#10596831	—	8	2	no	—
78	C₂²×C₇₃	—	2	1	no	—
79	C₂²×C₇₃	—	2	1	no	—
80	C₂²×C₇₃	#400	1	1	yes	characteristic
81	C₂²×C₇₃	#474	1	1	yes	characteristic
82	C₂²×C₇₃	#475	1	1	yes	characteristic
83	C₄×C₇₃	—	2	2	no	—
84	C₄×C₇₃	—	2	2	no	—
85	C₄×C₇₃	#480	1	1	yes	characteristic
86	C₄×C₇₃	#483	1	1	yes	characteristic
87	C₄×C₇₃	#486	1	1	yes	characteristic
88	C₄×C₇₃	#491	1	1	yes	characteristic
89	Dic₇₃	—	4	2	no	—
90	Dic₇₃	—	8	1	no	—
91	#10687	—	73	1	no	maximal, Sylow
92	C₂×C₇₃	—	2	1	no	—
93	C₂×C₇₃	—	2	1	no	—
94	C₂×C₇₃	#2099	1	1	yes	characteristic
95	C₂×C₇₃	#2102	1	1	yes	characteristic
96	C₂×C₇₃	#2103	1	1	yes	characteristic
97	#1940	—	73	1	no	—
98	#1849	—	73	1	no	—
99	#1303	—	73	2	no	—
100	#1301	—	73	2	no	—
101	#1275	D₇₃	1	1	yes	characteristic
102	C₇₃	#10687	1	1	yes	characteristic, complemented, Sylow
103	#431	C₂×D₇₃	1	2	yes	—
104	C₂×#108	—	146	1	no	—
105	C₂×#108	—	146	1	no	—
106	#423	—	73	2	no	—
107	C₂×C₄⋊C₈	—	73	1	no	—
108	C₂×C₄⋊C₈	—	146	1	no	—
109	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	73	1	no	—
111	#336	—	146	1	no	—
112	C₂×M₁₆	—	146	2	no	—
113	C₂²×C₈	—	146	1	no	—
114	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
115	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	146	1	no	—
117	C₂×C₄²	—	146	1	no	—
118	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
119	#108	—	292	2	no	—
120	#108	—	292	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	292	2	no	—
123	C₄⋊C₈	—	146	2	no	—
124	C₄⋊C₈	—	146	4	no	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	#95	—	73	1	no	—
129	#95	—	73	1	no	—
130	C₂²×C₄	—	146	1	no	—
131	C₂²×C₄	—	146	1	no	—
132	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
134	C₂²×C₄	#11924027	1	1	yes	characteristic
135	M₁₆	—	292	2	no	—
136	M₁₆	—	292	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	292	1	no	—
143	C₂×C₈	—	292	2	no	—
144	C₂×C₈	—	292	1	no	—
145	C₂×C₈	—	292	1	no	—
146	C₂×C₈	D₅₈₄	1	2	yes	—
147	C₂×C₈	#11924017	1	2	yes	—
148	C₄⋊C₄	—	292	2	no	—
149	C₄²	—	292	1	no	—
150	C₄²	—	292	1	no	—
151	C₄²	#11924027	1	2	yes	—
152	C₄²	#11924028	1	2	yes	—
153	C₂³	#14556571	1	1	yes	characteristic
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	292	1	no	—
157	C₂×C₄	—	292	1	no	—
158	C₂×C₄	—	292	1	no	—
159	C₂×C₄	—	292	1	no	—
160	C₂×C₄	—	292	1	no	—
161	C₂×C₄	—	146	2	no	—
162	C₂×C₄	#14556575	1	1	yes	characteristic
163	C₂×C₄	#14556579	1	1	yes	characteristic
164	C₂×C₄	#14556581	1	2	yes	—
165	C₂×C₄	#14556584	1	2	yes	—
166	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
167	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
168	C₂×C₄	#14556624	1	1	yes	characteristic
169	C₂×C₄	#14556624	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	8	2	no	—
173	C₈	—	584	1	no	—
174	C₈	—	584	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	#17988527	1	1	yes	characteristic
178	C₂²	#17988868	1	1	yes	characteristic
179	C₂²	#17988873	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	292	2	no	—
183	C₄	—	584	1	no	—
184	C₄	#17988901	1	1	yes	characteristic
185	C₄	#17988918	1	1	yes	characteristic
186	C₄	#17988934	1	1	yes	characteristic
187	C₄	#17988956	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	#22611490	1	1	yes	characteristic
191	C₂	#22611507	1	1	yes	characteristic
192	C₂	#22611512	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610654

C₂

yes

maximal, characteristic

#22610106

C₂

yes

maximal, characteristic

#22607833

C₂

yes

maximal

#22607827

C₂

yes

maximal

C₇₃×#1275

C₂

yes

maximal, characteristic

C₂×#14556527

—

C₂×#14556527

—

#17988611

C₂²

yes

—

C₂×C₄⋊#10596831

C₂²

yes

characteristic

C₂×#14556520

—

C₄×C₄⋊Dic₇₃

C₂²

yes

characteristic

#17988341

—

#431×C₇₃

C₂²

yes

—

C₂×#97×C₇₃

C₂²

yes

characteristic

C₂×#11924012

—

C₂²×#10596831

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄×Dic₇₃

—

—

—

—

—

—

Q₈

yes

characteristic

#14556498

Q₈

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

#97×C₇₃

—

C₄×C₈×C₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

—

—

—

—

—

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄²×C₇₃

#41

yes

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

#126

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#10687

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

100

#1301

—

101

#1275

D₇₃

yes

characteristic

102

C₇₃

#10687

yes

characteristic, complemented, Sylow

103

#431

C₂×D₇₃

yes

—

104

C₂×#108

—

146

—

105

C₂×#108

—

146

—

106

#423

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

—

146

—

109

C₂×#97

C₂×D₇₃

yes

characteristic

110

C₄×C₄⋊C₄

—

111

#336

—

146

—

112

C₂×M₁₆

—

146

—

113

C₂²×C₈

—

146

—

114

C₂²×C₈

D₂₉₂

yes

characteristic

115

C₂²×C₈

D₂₉₂

yes

characteristic

116

C₂×C₄⋊C₄

—

146

—

117

C₂×C₄²

—

146

—

118

C₂×C₄²

C₂²×D₇₃

yes

characteristic

119

#108

—

292

—

120

#108

—

292

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

292

—

123

C₄⋊C₈

—

146

—

124

C₄⋊C₈

—

146

—

125

C₂²⋊C₈

—

126

#97

—

127

C₄×C₈

—

128

#95

—

129

#95

—

130

C₂²×C₄

—

146

—

131

C₂²×C₄

—

146

—

132

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

133

C₂²×C₄

Q₈×D₇₃

yes

characteristic

134

C₂²×C₄

#11924027

yes

characteristic

135

M₁₆

—

292

—

136

M₁₆

—

292

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

292

—

143

C₂×C₈

—

292

—

144

C₂×C₈

—

292

—

145

C₂×C₈

—

292

—

146

C₂×C₈

D₅₈₄

yes

—

147

C₂×C₈

#11924017

yes

—

148

C₄⋊C₄

—

292

—

149

C₄²

—

292

—

150

C₄²

—

292

—

151

C₄²

#11924027

yes

—

152

C₄²

#11924028

yes

—

153

C₂³

#14556571

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

292

—

157

C₂×C₄

—

292

—

158

C₂×C₄

—

292

—

159

C₂×C₄

—

292

—

160

C₂×C₄

—

292

—

161

C₂×C₄

—

146

—

162

C₂×C₄

#14556575

yes

characteristic

163

C₂×C₄

#14556579

yes

characteristic

164

C₂×C₄

#14556581

yes

—

165

C₂×C₄

#14556584

yes

—

166

C₂×C₄

C₂×D₅₈₄

yes

characteristic

167

C₂×C₄

C₂×#11924017

yes

characteristic

168

C₂×C₄

#14556624

yes

characteristic

169

C₂×C₄

#14556624

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

584

—

174

C₈

—

584

—

175

C₂²

—

176

C₂²

—

177

C₂²

#17988527

yes

characteristic

178

C₂²

#17988868

yes

characteristic

179

C₂²

#17988873

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

292

—

183

C₄

—

584

—

184

C₄

#17988901

yes

characteristic

185

C₄

#17988918

yes

characteristic

186

C₄

#17988934

yes

characteristic

187

C₄

#17988956

yes

characteristic

188

C₂

—

189

C₂

—

190

C₂

#22611490

yes

characteristic

191

C₂

#22611507

yes

characteristic

192

C₂

#22611512

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556579
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1275	C₂
socle	C₂²×C₇₃	#400
prime radical	C₂³×C₇₃	C₂²⋊Q₈
hypercenter	#1275	D₇₃
hypocenter	C₇₃	#10687
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#14556579

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1275

C₂

socle

C₂²×C₇₃

#400

prime radical

C₂³×C₇₃

C₂²⋊Q₈

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
102. C₇₃	#10687	semidirect

H ≅ G/N

type

102. C₇₃

#10687

semidirect

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

subgroup	index
#22610654	2
#22610106	2
#22607833	2
#22607827	2
C₇₃×#1275	2
#10687	73

subgroup

index

C₇₃×#1275

Subgroups — #32991110

Conjugacy classes of subgroups

Complements

Maximal subgroups