Group #33005521 — subgroups

Subgroups — #33005521

Order 18688.

418 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22611689	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17988400	C₂	1	2	yes	maximal, complemented
3	#22610842	C₂	1	2	yes	maximal, complemented
4	#22608025	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	1	yes	characteristic
7	C₂×#14556527	C₂²	1	2	yes	—
8	#17988592	—	2	2	no	—
9	C₂×#14556508	—	2	2	no	—
10	#17988542	C₂²	1	1	yes	characteristic
11	#17988405	—	2	2	no	—
12	#17988400	—	2	4	no	—
13	C₂²×D₈×C₇₃	C₂²	1	2	yes	—
14	C₂×D₁₆×C₇₃	—	2	4	no	—
15	C₂×D₁₆×C₇₃	—	2	2	no	—
16	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#11924012	—	2	2	no	—
18	C₂×#11924012	D₄	1	1	yes	characteristic
19	#14556550	D₄	1	1	yes	characteristic
20	#14556531	—	2	2	no	—
21	C₂×#11924003	—	2	1	no	—
22	#14556527	—	2	2	no	—
23	#14556527	—	2	2	no	—
24	#14556512	—	4	4	no	—
25	#14556508	—	4	4	no	—
26	#14556501	—	2	1	no	—
27	C₈×Dic₇₃	—	2	1	no	—
28	C₂²×D₄×C₇₃	—	2	2	no	—
29	C₂×D₈×C₇₃	—	2	2	no	—
30	C₂×D₈×C₇₃	—	2	2	no	—
31	C₂×D₈×C₇₃	—	2	2	no	—
32	C₂×D₈×C₇₃	—	2	2	no	—
33	C₂×D₈×C₇₃	—	2	2	no	—
34	C₂×D₈×C₇₃	D₄	1	4	yes	—
35	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
36	D₁₆×C₇₃	—	2	8	no	—
37	C₂×C₁₆×C₇₃	—	2	2	no	—
38	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
39	#11924012	—	2	2	no	—
40	#11924012	—	4	2	no	—
41	#11924012	—	4	2	no	—
42	C₂×#10596831	—	2	1	no	—
43	C₂×#10596831	—	4	2	no	—
44	C₄⋊Dic₇₃	D₈	1	2	yes	—
45	C₂²⋊Dic₇₃	—	2	1	no	—
46	C₄×Dic₇₃	—	2	1	no	—
47	#11924003	—	4	1	no	—
48	C₂⁴×C₇₃	—	4	2	no	—
49	C₂×D₄×C₇₃	—	2	4	no	—
50	C₂×D₄×C₇₃	—	4	2	no	—
51	C₂×D₄×C₇₃	—	4	2	no	—
52	C₂×D₄×C₇₃	—	4	2	no	—
53	C₂×D₄×C₇₃	—	4	2	no	—
54	C₂×D₄×C₇₃	—	4	2	no	—
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	D₈×C₇₃	—	2	4	no	—
57	D₈×C₇₃	—	2	4	no	—
58	D₈×C₇₃	—	4	2	no	—
59	D₈×C₇₃	—	4	2	no	—
60	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
63	C₂×C₈×C₇₃	#41	1	2	yes	—
64	C₁₆×C₇₃	—	2	2	no	—
65	C₂³×C₇₃	—	4	4	no	—
66	C₂³×C₇₃	—	8	2	no	—
67	C₂³×C₇₃	—	8	2	no	—
68	C₂³×C₇₃	—	16	4	no	—
69	C₂³×C₇₃	#135	1	1	yes	characteristic
70	D₄×C₇₃	—	4	4	no	—
71	D₄×C₇₃	—	4	4	no	—
72	D₄×C₇₃	—	8	2	no	—
73	D₄×C₇₃	—	8	2	no	—
74	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#121	1	2	yes	—
76	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×Dic₇₃	—	2	2	no	—
79	C₈×C₇₃	#120	1	1	yes	characteristic
80	C₈×C₇₃	#123	1	2	yes	—
81	C₈×C₇₃	#124	1	1	yes	characteristic
82	#10596831	—	4	1	no	—
83	#10596831	—	8	2	no	—
84	C₂²×C₇₃	—	2	2	no	—
85	C₂²×C₇₃	—	8	4	no	—
86	C₂²×C₇₃	—	16	4	no	—
87	C₂²×C₇₃	—	16	4	no	—
88	C₂²×C₇₃	—	16	4	no	—
89	C₂²×C₇₃	#463	1	2	yes	—
90	C₂²×C₇₃	#466	1	1	yes	characteristic
91	C₄×C₇₃	#394	1	1	yes	characteristic
92	C₄×C₇₃	#459	1	2	yes	—
93	C₄×C₇₃	#469	1	1	yes	characteristic
94	Dic₇₃	—	4	2	no	—
95	#13439	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	—	2	2	no	—
97	C₂×C₇₃	—	16	4	no	—
98	C₂×C₇₃	#2054	1	1	yes	characteristic
99	C₂×C₇₃	#2221	1	2	yes	—
100	C₂²×D₁₆	D₇₃	1	1	yes	characteristic
101	#2151	—	73	1	no	—
102	C₂×#361	—	73	2	no	—
103	#1978	—	73	2	no	—
104	#1381	—	73	1	no	—
105	C₇₃	#13439	1	1	yes	characteristic, complemented, Sylow
106	C₂²×D₈	C₂×D₇₃	1	2	yes	—
107	C₂×D₁₆	—	2	4	no	—
108	C₂×D₁₆	—	2	2	no	—
109	C₂×M₃₂	—	73	1	no	—
110	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
111	C₂×#108	—	73	2	no	—
112	#418	—	146	2	no	—
113	C₂×#100	—	146	2	no	—
114	#405	—	73	1	no	—
115	C₄⋊C₁₆	—	146	2	no	—
116	#361	—	146	4	no	—
117	C₂²×D₄	—	2	2	no	—
118	C₂×D₈	—	2	2	no	—
119	C₂×D₈	—	2	2	no	—
120	C₂×D₈	—	2	2	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×D₈	C₂²⋊D₇₃	1	4	yes	—
124	C₂×M₁₆	—	73	1	no	—
125	C₂×M₁₆	—	146	2	no	—
126	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
127	#117	—	73	1	no	—
128	D₁₆	—	2	8	no	—
129	M₃₂	—	146	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	146	1	no	—
132	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
133	#108	—	146	2	no	—
134	#108	—	146	2	no	—
135	#102	—	292	4	no	—
136	#100	—	292	4	no	—
137	#97	—	146	1	no	—
138	C₄×C₈	—	146	1	no	—
139	C₂⁴	—	4	2	no	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	4	2	no	—
142	C₂×D₄	—	4	2	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂×D₄	—	4	2	no	—
146	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
147	D₈	—	2	4	no	—
148	D₈	—	2	4	no	—
149	D₈	—	4	2	no	—
150	D₈	—	4	2	no	—
151	M₁₆	—	292	2	no	—
152	M₁₆	—	292	2	no	—
153	M₁₆	—	146	2	no	—
154	C₂×C₈	—	292	2	no	—
155	C₂×C₈	—	146	1	no	—
156	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
158	C₂×C₈	#11924027	1	1	yes	characteristic
159	C₂×C₈	#11924027	1	2	yes	—
160	C₄⋊C₄	—	73	2	no	—
161	C₂²⋊C₄	—	146	1	no	—
162	C₄²	—	146	1	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	292	1	no	—
165	C₂³	—	4	4	no	—
166	C₂³	—	8	2	no	—
167	C₂³	—	8	2	no	—
168	C₂³	—	16	4	no	—
169	C₂³	#14556610	1	1	yes	characteristic
170	D₄	—	4	4	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	2	no	—
173	D₄	—	8	2	no	—
174	C₂×C₄	—	146	2	no	—
175	C₂×C₄	#14556562	1	1	yes	characteristic
176	C₂×C₄	#14556563	1	2	yes	—
177	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
178	C₂×C₄	#14556615	1	2	yes	—
179	C₈	—	292	1	no	—
180	C₈	—	584	2	no	—
181	C₈	#14556560	1	1	yes	characteristic
182	C₈	#14556572	1	2	yes	—
183	C₈	#14556578	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	16	4	no	—
187	C₂²	—	16	4	no	—
188	C₂²	—	16	4	no	—
189	C₂²	#17988800	1	2	yes	—
190	C₂²	#17988820	1	1	yes	characteristic
191	C₄	—	292	2	no	—
192	C₄	#17988501	1	1	yes	characteristic
193	C₄	#17988780	1	2	yes	—
194	C₄	#17988837	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	4	no	—
197	C₂	#22611239	1	1	yes	characteristic
198	C₂	#22612048	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22611689

C₂

yes

maximal, characteristic, complemented

C₂×#17988400

C₂

yes

maximal, complemented

#22610842

C₂

yes

maximal, complemented

#22608025

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556531

C₂²

yes

characteristic

C₂×#14556527

C₂²

yes

—

#17988592

—

C₂×#14556508

—

#17988542

C₂²

yes

characteristic

#17988405

—

#17988400

—

C₂²×D₈×C₇₃

C₂²

yes

—

C₂×D₁₆×C₇₃

—

C₂×D₁₆×C₇₃

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#11924012

—

C₂×#11924012

D₄

yes

characteristic

#14556550

D₄

yes

—

—

—

—

—

—

—

C₈×Dic₇₃

—

C₂²×D₄×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

—

—

—

—

—

C₄⋊Dic₇₃

D₈

yes

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂⁴×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

C₂×C₈×C₇₃

yes

C₂×C₈×C₇₃

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#135

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

—

C₈×C₇₃

#120

yes

characteristic

C₈×C₇₃

#123

yes

—

C₈×C₇₃

yes

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#463

yes

—

C₂²×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₄×C₇₃

#469

yes

characteristic

Dic₇₃

—

#13439

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#2054

yes

characteristic

C₂×C₇₃

#2221

yes

—

100

C₂²×D₁₆

D₇₃

yes

characteristic

101

#2151

—

102

C₂×#361

—

103

#1978

—

104

#1381

—

105

C₇₃

#13439

yes

characteristic, complemented, Sylow

106

C₂²×D₈

C₂×D₇₃

yes

—

107

C₂×D₁₆

—

108

C₂×D₁₆

—

109

C₂×M₃₂

—

110

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

111

C₂×#108

—

112

#418

—

146

—

113

C₂×#100

—

146

—

114

#405

—

115

C₄⋊C₁₆

—

146

—

116

#361

—

146

—

117

C₂²×D₄

—

118

C₂×D₈

—

119

C₂×D₈

—

120

C₂×D₈

—

121

C₂×D₈

—

122

C₂×D₈

—

123

C₂×D₈

C₂²⋊D₇₃

yes

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

146

—

126

C₂²×C₈

C₂²×D₇₃

yes

characteristic

127

#117

—

128

D₁₆

—

129

M₃₂

—

146

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

146

—

132

C₂×C₁₆

C₂²⋊D₇₃

yes

—

133

#108

—

146

—

134

#108

—

146

—

135

#102

—

292

—

136

#100

—

292

—

137

#97

—

146

—

138

C₄×C₈

—

146

—

139

C₂⁴

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂²×C₄

D₄×D₇₃

yes

characteristic

147

D₈

—

148

D₈

—

149

D₈

—

150

D₈

—

151

M₁₆

—

292

—

152

M₁₆

—

292

—

153

M₁₆

—

146

—

154

C₂×C₈

—

292

—

155

C₂×C₈

—

146

—

156

C₂×C₈

C₂×C₂²⋊D₇₃

yes

—

157

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

158

C₂×C₈

#11924027

yes

characteristic

159

C₂×C₈

#11924027

yes

—

160

C₄⋊C₄

—

161

C₂²⋊C₄

—

146

—

162

C₄²

—

146

—

163

C₁₆

—

164

C₁₆

—

292

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

#14556610

yes

characteristic

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

174

C₂×C₄

—

146

—

175

C₂×C₄

#14556562

yes

characteristic

176

C₂×C₄

#14556563

yes

—

177

C₂×C₄

D₈×D₇₃

yes

characteristic

178

C₂×C₄

#14556615

yes

—

179

C₈

—

292

—

180

C₈

—

584

—

181

C₈

#14556560

yes

characteristic

182

C₈

#14556572

yes

—

183

C₈

#14556578

yes

characteristic

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

#17988800

yes

—

190

C₂²

#17988820

yes

characteristic

191

C₄

—

292

—

192

C₄

#17988501

yes

characteristic

193

C₄

#17988780

yes

—

194

C₄

#17988837

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#22611239

yes

characteristic

198

C₂

#22612048

yes

—

	subgroup	G/H
center	C₂²	#17988820
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting, prime radical	C₂²×D₁₆×C₇₃	C₂
socle	C₂²×C₇₃	#466
hypercenter	C₂²×D₁₆	D₇₃
hypocenter	C₇₃	#13439
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

subgroup

G/H

center

C₂²

#17988820

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting, prime radical

C₂²×D₁₆×C₇₃

C₂

socle

C₂²×C₇₃

#466

hypercenter

C₂²×D₁₆

D₇₃

hypocenter

C₇₃

#13439

LCS

G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

N	H ≅ G/N	type
1. #22611689	C₂	semidirect
2. C₂×#17988400	C₂	semidirect
3. #22610842	C₂	semidirect
4. #22608025	C₂	semidirect
105. C₇₃	#13439	semidirect

H ≅ G/N

type

1. #22611689

C₂

semidirect

2. C₂×#17988400

C₂

semidirect

3. #22610842

C₂

semidirect

4. #22608025

C₂

semidirect

105. C₇₃

#13439

semidirect

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

subgroup	index
#22611689	2
C₂×#17988400	2
#22610842	2
#22608025	2
C₂²×D₁₆×C₇₃	2
#13439	73

subgroup

index

C₂²×D₁₆×C₇₃

Subgroups — #33005521

Conjugacy classes of subgroups

Complements

Maximal subgroups