Group #33005815 — subgroups

Subgroups — #33005815

Order 18688.

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

normal subgroups only

This group has 302 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	C₂×#17988402	C₂	1	2	yes	maximal
2	#22610802	C₂	1	1	yes	maximal, characteristic
3	#22608031	C₂	1	2	yes	maximal
4	#22608015	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₂	C₂²	1	1	yes	characteristic
7	C₂×#14556531	C₂²	1	2	yes	—
8	C₂×#14556523	C₂²	1	1	yes	characteristic
9	C₂×#14556516	—	2	1	no	—
10	#17988493	—	2	1	no	—
11	#17988402	—	2	4	no	—
12	C₂×#108×C₇₃	—	2	1	no	—
13	C₂×#108×C₇₃	C₂²	1	2	yes	—
14	#423×C₇₃	—	2	1	no	—
15	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₄₆	—	2	1	no	—
17	C₂×Dic₂₉₂	—	2	2	no	—
18	C₂×Dic₂₉₂	—	2	2	no	—
19	C₂×Dic₂₉₂	—	2	1	no	—
20	C₂×Dic₂₉₂	D₄	1	2	yes	—
21	C₂×C₄⋊Dic₇₃	—	2	1	no	—
22	#14556531	—	2	2	no	—
23	#14556531	—	2	2	no	—
24	C₂×#11924003	—	2	2	no	—
25	#14556523	—	2	2	no	—
26	#14556516	—	4	2	no	—
27	#14556498	—	4	2	no	—
28	C₂×M₁₆×C₇₃	—	2	2	no	—
29	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
32	#108×C₇₃	—	2	2	no	—
33	#108×C₇₃	—	2	2	no	—
34	#108×C₇₃	—	4	2	no	—
35	C₄⋊C₈×C₇₃	—	4	2	no	—
36	#97×C₇₃	—	2	2	no	—
37	C₂×Dic₁₄₆	—	2	2	no	—
38	C₂×Dic₁₄₆	—	4	1	no	—
39	C₂×Dic₁₄₆	—	4	1	no	—
40	C₂×Dic₁₄₆	—	4	1	no	—
41	C₂×Dic₁₄₆	—	4	1	no	—
42	C₂×Dic₁₄₆	—	4	1	no	—
43	C₂²×Dic₇₃	—	4	1	no	—
44	C₂²×Dic₇₃	—	4	1	no	—
45	Dic₂₉₂	—	2	4	no	—
46	Dic₂₉₂	—	4	2	no	—
47	C₄⋊Dic₇₃	—	4	1	no	—
48	C₄⋊Dic₇₃	—	4	1	no	—
49	#11924003	—	4	2	no	—
50	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
51	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	M₁₆×C₇₃	—	4	2	no	—
53	M₁₆×C₇₃	—	4	2	no	—
54	C₂×C₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	—	2	1	no	—
56	C₂×C₈×C₇₃	—	4	2	no	—
57	C₂×C₈×C₇₃	D₈	1	2	yes	—
58	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
60	C₂×C₈×C₇₃	#41	1	2	yes	—
61	C₂×C₈×C₇₃	#41	1	2	yes	—
62	C₄²×C₇₃	—	2	1	no	—
63	C₄²×C₇₃	—	2	1	no	—
64	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	4	2	no	—
67	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
73	C₂×Dic₇₃	—	4	2	no	—
74	C₂×Dic₇₃	—	8	1	no	—
75	C₂×Dic₇₃	—	8	1	no	—
76	C₂×Dic₇₃	—	8	2	no	—
77	C₂×Dic₇₃	—	8	1	no	—
78	Dic₁₄₆	—	4	2	no	—
79	Dic₁₄₆	—	4	2	no	—
80	Dic₁₄₆	—	8	1	no	—
81	Dic₁₄₆	—	8	1	no	—
82	C₈×C₇₃	—	2	2	no	—
83	C₈×C₇₃	—	8	2	no	—
84	C₈×C₇₃	#123	1	2	yes	—
85	C₈×C₇₃	#126	1	2	yes	—
86	C₂²×C₇₃	—	2	2	no	—
87	C₂²×C₇₃	#461	1	1	yes	characteristic
88	C₂²×C₇₃	#481	1	1	yes	characteristic
89	C₂²×C₇₃	#486	1	1	yes	characteristic
90	C₄×C₇₃	—	4	2	no	—
91	C₄×C₇₃	#397	1	1	yes	characteristic
92	C₄×C₇₃	#459	1	1	yes	characteristic
93	C₄×C₇₃	#480	1	1	yes	characteristic
94	C₄×C₇₃	#489	1	1	yes	characteristic
95	Dic₇₃	—	8	2	no	—
96	Dic₇₃	—	16	1	no	—
97	#13488	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	—	2	1	no	—
99	C₂×C₇₃	—	2	1	no	—
100	C₂×C₇₃	#2071	1	1	yes	characteristic
101	C₂×C₇₃	#2239	1	2	yes	—
102	C₂×#362	—	73	2	no	—
103	#1971	—	73	1	no	—
104	#1849	D₇₃	1	1	yes	characteristic
105	#1383	—	73	2	no	—
106	#1377	—	73	1	no	—
107	C₇₃	#13488	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₁₆	—	73	1	no	—
109	C₂×M₃₂	—	73	2	no	—
110	C₂×#108	—	2	1	no	—
111	C₂×#108	C₂×D₇₃	1	2	yes	—
112	C₂×#106	—	73	1	no	—
113	#423	—	2	1	no	—
114	C₂×#103	—	146	1	no	—
115	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
116	#391	—	146	1	no	—
117	#362	—	146	4	no	—
118	C₂²×Q₈	—	146	1	no	—
119	C₂×Q₁₆	—	73	2	no	—
120	C₂×Q₁₆	—	146	1	no	—
121	C₂×Q₁₆	—	146	2	no	—
122	C₂×Q₁₆	—	146	2	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	146	1	no	—
127	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
128	M₃₂	—	146	2	no	—
129	M₃₂	—	146	2	no	—
130	C₂×C₁₆	—	146	2	no	—
131	#108	—	2	2	no	—
132	#108	—	2	2	no	—
133	#108	—	4	2	no	—
134	#106	—	146	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	#103	—	292	2	no	—
137	#97	—	2	2	no	—
138	#95	—	292	2	no	—
139	C₂×Q₈	—	292	1	no	—
140	C₂×Q₈	—	292	1	no	—
141	C₂×Q₈	—	292	1	no	—
142	C₂×Q₈	—	292	1	no	—
143	C₂×Q₈	—	292	1	no	—
144	C₂×Q₈	—	146	2	no	—
145	C₂²×C₄	—	292	1	no	—
146	C₂²×C₄	—	292	1	no	—
147	C₂²×C₄	#11924015	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
149	Q₁₆	—	292	2	no	—
150	Q₁₆	—	146	4	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	#11924013	1	2	yes	—
157	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
158	C₂×C₈	#11924027	1	1	yes	characteristic
159	C₂×C₈	#11924028	1	2	yes	—
160	C₂×C₈	#11924029	1	2	yes	—
161	C₄⋊C₄	—	292	1	no	—
162	C₄⋊C₄	—	292	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	292	2	no	—
166	C₂³	C₂×#11924015	1	1	yes	characteristic
167	Q₈	—	292	2	no	—
168	Q₈	—	292	2	no	—
169	Q₈	—	584	1	no	—
170	Q₈	—	584	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	292	2	no	—
174	C₂×C₄	—	584	1	no	—
175	C₂×C₄	—	584	1	no	—
176	C₂×C₄	—	584	2	no	—
177	C₂×C₄	—	584	1	no	—
178	C₂×C₄	#14556562	1	1	yes	characteristic
179	C₂×C₄	#14556574	1	1	yes	characteristic
180	C₂×C₄	#14556580	1	1	yes	characteristic
181	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
182	C₂×C₄	#14556617	1	1	yes	characteristic
183	C₂×C₄	#14556622	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	8	2	no	—
186	C₈	#14556576	1	2	yes	—
187	C₈	#14556586	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	#17988789	1	1	yes	characteristic
190	C₂²	#17988905	1	1	yes	characteristic
191	C₂²	#17988935	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	584	2	no	—
194	C₄	—	1168	1	no	—
195	C₄	#17988513	1	1	yes	characteristic
196	C₄	#17988779	1	1	yes	characteristic
197	C₄	#17988900	1	1	yes	characteristic
198	C₄	#17988949	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#22611335	1	1	yes	characteristic
202	C₂	#22612150	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988402

C₂

yes

maximal

#22610802

C₂

yes

maximal, characteristic

#22608031

C₂

yes

maximal

#22608015

C₂

yes

maximal, characteristic

C₇₃×#1849

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₂

C₂²

yes

characteristic

C₂×#14556531

C₂²

yes

—

C₂×#14556523

C₂²

yes

—

—

—

C₂×#108×C₇₃

—

C₂×#108×C₇₃

C₂²

yes

—

#423×C₇₃

—

C₂×#97×C₇₃

C₂²

yes

characteristic

C₂²×Dic₁₄₆

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

D₄

yes

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

—

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

#108×C₇₃

—

#108×C₇₃

—

#108×C₇₃

—

C₄⋊C₈×C₇₃

—

#97×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

SD₁₆

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

C₂×C₈×C₇₃

yes

C₂×C₈×C₇₃

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×D₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#123

yes

—

C₈×C₇₃

#126

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#13488

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

100

C₂×C₇₃

#2071

yes

characteristic

101

C₂×C₇₃

#2239

yes

—

102

C₂×#362

—

103

#1971

—

104

#1849

D₇₃

yes

characteristic

105

#1383

—

106

#1377

—

107

C₇₃

#13488

yes

characteristic, complemented, Sylow

108

C₂²×Q₁₆

—

109

C₂×M₃₂

—

110

C₂×#108

—

111

C₂×#108

C₂×D₇₃

yes

—

112

C₂×#106

—

113

#423

—

114

C₂×#103

—

146

—

115

C₂×#97

C₂×D₇₃

yes

characteristic

116

#391

—

146

—

117

#362

—

146

—

118

C₂²×Q₈

—

146

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

146

—

121

C₂×Q₁₆

—

146

—

122

C₂×Q₁₆

—

146

—

123

C₂×M₁₆

—

124

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

125

C₂²×C₈

C₂²×D₇₃

yes

characteristic

126

C₂×C₄⋊C₄

—

146

—

127

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

128

M₃₂

—

146

—

129

M₃₂

—

146

—

130

C₂×C₁₆

—

146

—

131

#108

—

132

#108

—

133

#108

—

134

#106

—

146

—

135

C₄⋊C₈

—

136

#103

—

292

—

137

#97

—

138

#95

—

292

—

139

C₂×Q₈

—

292

—

140

C₂×Q₈

—

292

—

141

C₂×Q₈

—

292

—

142

C₂×Q₈

—

292

—

143

C₂×Q₈

—

292

—

144

C₂×Q₈

—

146

—

145

C₂²×C₄

—

292

—

146

C₂²×C₄

—

292

—

147

C₂²×C₄

#11924015

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

149

Q₁₆

—

292

—

150

Q₁₆

—

146

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

#11924013

yes

—

157

C₂×C₈

D₄×D₇₃

yes

characteristic

158

C₂×C₈

#11924027

yes

characteristic

159

C₂×C₈

#11924028

yes

—

160

C₂×C₈

#11924029

yes

—

161

C₄⋊C₄

—

292

—

162

C₄⋊C₄

—

292

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

292

—

166

C₂³

C₂×#11924015

yes

characteristic

167

Q₈

—

292

—

168

Q₈

—

292

—

169

Q₈

—

584

—

170

Q₈

—

584

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

292

—

174

C₂×C₄

—

584

—

175

C₂×C₄

—

584

—

176

C₂×C₄

—

584

—

177

C₂×C₄

—

584

—

178

C₂×C₄

#14556562

yes

characteristic

179

C₂×C₄

#14556574

yes

characteristic

180

C₂×C₄

#14556580

yes

characteristic

181

C₂×C₄

C₂×#11924013

yes

characteristic

182

C₂×C₄

#14556617

yes

characteristic

183

C₂×C₄

#14556622

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

#14556576

yes

—

187

C₈

#14556586

yes

—

188

C₂²

—

189

C₂²

#17988789

yes

characteristic

190

C₂²

#17988905

yes

characteristic

191

C₂²

#17988935

yes

characteristic

192

C₄

—

193

C₄

—

584

—

194

C₄

—

1168

—

195

C₄

#17988513

yes

characteristic

196

C₄

#17988779

yes

characteristic

197

C₄

#17988900

yes

characteristic

198

C₄

#17988949

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#22611335

yes

characteristic

202

C₂

#22612150

yes

—

	subgroup	G/H
center	C₂²	#17988905
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1849	C₂
socle	C₂²×C₇₃	#481
prime radical	C₂³×C₇₃	C₂×SD₁₆
hypercenter	#1849	D₇₃
hypocenter	C₇₃	#13488
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₂²

#17988905

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1849

C₂

socle

C₂²×C₇₃

#481

prime radical

C₂³×C₇₃

C₂×SD₁₆

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

N	H ≅ G/N	type
107. C₇₃	#13488	semidirect

H ≅ G/N

type

107. C₇₃

#13488

semidirect

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

subgroup	index
C₂×#17988402	2
#22610802	2
#22608031	2
#22608015	2
C₇₃×#1849	2
#13488	73

subgroup

index

C₇₃×#1849

Subgroups — #33005815

Conjugacy classes of subgroups

Complements

Maximal subgroups