Group #33002335 — subgroups

Subgroups — #33002335

Order 18688.

298 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 123 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×#17988407	C₂	1	2	yes	maximal
2	#22609619	C₂	1	1	yes	maximal, characteristic
3	#22608015	C₂	1	1	yes	maximal, characteristic
4	#22608010	C₂	1	2	yes	maximal
5	C₇₃×#1377	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	C₂×#14556501	C₄	1	1	yes	characteristic
8	C₂×#14556501	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
12	#17988433	C₄	1	1	yes	characteristic
13	#17988407	—	2	4	no	—
14	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
15	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
17	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
20	#14556531	—	2	2	no	—
21	#14556531	—	2	2	no	—
22	C₂×#11924003	—	2	2	no	—
23	#14556501	—	2	2	no	—
24	#14556501	—	2	2	no	—
25	C₄×#10596831	—	2	2	no	—
26	C₈×Dic₇₃	C₈	1	4	yes	—
27	C₈×Dic₇₃	D₄	1	2	yes	—
28	C₈×Dic₇₃	Q₈	1	2	yes	—
29	#14556498	—	2	2	no	—
30	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₇₃	—	2	2	no	—
34	M₃₂×C₇₃	—	2	2	no	—
35	C₂×C₁₆×C₇₃	—	2	2	no	—
36	#97×C₇₃	—	2	2	no	—
37	C₂²×Dic₇₃	—	2	1	no	—
38	C₂²×Dic₇₃	—	2	1	no	—
39	C₂×#10596831	—	2	1	no	—
40	C₂×#10596831	—	2	1	no	—
41	C₂×#10596831	C₄⋊C₄	1	2	yes	—
42	C₂×#10596831	C₂×C₈	1	2	yes	—
43	C₂×#10596831	M₁₆	1	2	yes	—
44	C₂×#10596831	M₁₆	1	2	yes	—
45	C₄×Dic₇₃	—	2	1	no	—
46	C₄×Dic₇₃	—	2	1	no	—
47	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
48	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
49	#11924003	—	4	2	no	—
50	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
51	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	—	2	1	no	—
53	C₂×C₈×C₇₃	—	2	1	no	—
54	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	#41	1	2	yes	—
58	C₂×C₈×C₇₃	#41	1	2	yes	—
59	C₄²×C₇₃	—	2	1	no	—
60	C₄²×C₇₃	—	2	1	no	—
61	C₁₆×C₇₃	—	4	2	no	—
62	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	4	2	no	—
65	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×Dic₇₃	—	2	2	no	—
72	C₂×Dic₇₃	—	2	2	no	—
73	C₂×Dic₇₃	—	4	2	no	—
74	C₂×Dic₇₃	—	4	2	no	—
75	C₈×C₇₃	—	2	2	no	—
76	C₈×C₇₃	#124	1	2	yes	—
77	C₈×C₇₃	#125	1	2	yes	—
78	#10596831	—	2	2	no	—
79	#10596831	C₄⋊C₈	1	4	yes	—
80	C₂²×C₇₃	—	2	2	no	—
81	C₂²×C₇₃	#423	1	1	yes	characteristic
82	C₂²×C₇₃	#431	1	1	yes	characteristic
83	C₂²×C₇₃	#432	1	1	yes	characteristic
84	C₄×C₇₃	—	4	2	no	—
85	C₄×C₇₃	#382	1	1	yes	characteristic
86	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₄×C₇₃	#431	1	1	yes	characteristic
88	C₄×C₇₃	#432	1	1	yes	characteristic
89	Dic₇₃	—	4	2	no	—
90	Dic₇₃	—	4	2	no	—
91	#12881	—	73	1	no	maximal, Sylow
92	C₂×C₇₃	—	2	1	no	—
93	C₂×C₇₃	—	2	1	no	—
94	C₂×C₇₃	#1831	1	1	yes	characteristic
95	C₂×C₇₃	#2166	1	2	yes	—
96	C₂×#364	—	73	2	no	—
97	#1726	—	73	1	no	—
98	#1377	—	73	1	no	—
99	#1377	D₇₃	1	1	yes	characteristic
100	#1375	—	73	2	no	—
101	C₇₃	#12881	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	—	73	2	no	—
103	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
104	C₂×#97	—	73	1	no	—
105	C₂×#97	—	73	1	no	—
106	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
107	C₂×C₄×C₈	—	73	1	no	—
108	C₂×C₄×C₈	—	73	1	no	—
109	C₂×C₄×C₈	—	73	1	no	—
110	#376	—	73	1	no	—
111	#364	—	146	4	no	—
112	C₂²×C₈	—	73	1	no	—
113	C₂²×C₈	—	73	1	no	—
114	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
116	C₂×C₄²	—	73	1	no	—
117	C₂×C₄²	—	73	1	no	—
118	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
119	M₃₂	—	2	2	no	—
120	M₃₂	—	2	2	no	—
121	M₃₂	—	146	2	no	—
122	M₃₂	—	146	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	146	2	no	—
125	#97	—	2	2	no	—
126	#97	—	146	2	no	—
127	#97	—	146	2	no	—
128	C₄×C₈	—	73	4	no	—
129	C₄×C₈	—	73	2	no	—
130	C₄×C₈	—	73	2	no	—
131	C₄×C₈	—	146	2	no	—
132	#95	—	146	2	no	—
133	C₂²×C₄	—	146	1	no	—
134	C₂²×C₄	—	146	1	no	—
135	C₂²×C₄	#11924011	1	2	yes	—
136	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	73	2	no	—
140	C₂×C₈	—	73	2	no	—
141	C₂×C₈	—	73	2	no	—
142	C₂×C₈	—	73	2	no	—
143	C₂×C₈	—	146	1	no	—
144	C₂×C₈	—	146	1	no	—
145	C₂×C₈	C₈×D₇₃	1	2	yes	—
146	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
147	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
148	C₂×C₈	#11924027	1	2	yes	—
149	C₂×C₈	#11924028	1	2	yes	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	73	2	no	—
153	C₄²	—	73	2	no	—
154	C₄²	—	146	1	no	—
155	C₄²	—	146	1	no	—
156	C₁₆	—	4	2	no	—
157	C₁₆	—	292	2	no	—
158	C₂³	C₂×#11924011	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	292	2	no	—
162	C₂×C₄	—	292	2	no	—
163	C₂×C₄	—	146	2	no	—
164	C₂×C₄	—	146	2	no	—
165	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
166	C₂×C₄	#14556547	1	1	yes	characteristic
167	C₂×C₄	#14556548	1	1	yes	characteristic
168	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
169	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
170	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
171	C₈	—	2	2	no	—
172	C₈	—	73	4	no	—
173	C₈	—	146	2	no	—
174	C₈	#14556577	1	2	yes	—
175	C₈	#14556581	1	2	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	#17988609	1	1	yes	characteristic
178	C₂²	#17988638	1	1	yes	characteristic
179	C₂²	#17988641	1	1	yes	characteristic
180	C₄	—	4	2	no	—
181	C₄	—	292	2	no	—
182	C₄	—	292	2	no	—
183	C₄	#17988451	1	1	yes	characteristic
184	C₄	C₄⋊C₈×D₇₃	1	1	yes	characteristic
185	C₄	#17988637	1	1	yes	characteristic
186	C₄	#17988642	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#22610046	1	1	yes	characteristic
190	C₂	#22611744	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988407

C₂

yes

maximal

#22609619

C₂

yes

maximal, characteristic

#22608015

C₂

yes

maximal, characteristic

#22608010

C₂

yes

maximal

C₇₃×#1377

C₂

yes

maximal, characteristic

C₂×#14556531

C₂²

yes

—

C₂×#14556501

C₄

yes

characteristic

C₂×#14556501

C₂²

yes

characteristic

C₂×C₄×#10596831

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988433

C₄

yes

characteristic

#17988407

—

C₂×M₃₂×C₇₃

C₂²

yes

—

C₂×#97×C₇₃

C₂²

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

—

—

—

—

—

C₈×Dic₇₃

C₈

yes

—

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

#14556498

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

#97×C₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

M₁₆

yes

—

C₂×#10596831

M₁₆

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₂×C₈

yes

—

#11924003

—

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#124

yes

—

C₈×C₇₃

#125

yes

—

#10596831

—

#10596831

C₄⋊C₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

#382

yes

characteristic

C₄×C₇₃

C₂×C₄⋊C₈

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#12881

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1831

yes

characteristic

C₂×C₇₃

#2166

yes

—

—

—

—

D₇₃

yes

characteristic

100

#1375

—

101

C₇₃

#12881

yes

characteristic, complemented, Sylow

102

C₂×M₃₂

—

103

C₂×M₃₂

C₂×D₇₃

yes

—

104

C₂×#97

—

105

C₂×#97

—

106

C₂×#97

C₂×D₇₃

yes

characteristic

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

#376

—

111

#364

—

146

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₇₃

yes

characteristic

115

C₂²×C₈

C₂²×D₇₃

yes

characteristic

116

C₂×C₄²

—

117

C₂×C₄²

—

118

C₂×C₄²

C₄×D₇₃

yes

characteristic

119

M₃₂

—

120

M₃₂

—

121

M₃₂

—

146

—

122

M₃₂

—

146

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

146

—

125

#97

—

126

#97

—

146

—

127

#97

—

146

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

146

—

132

#95

—

146

—

133

C₂²×C₄

—

146

—

134

C₂²×C₄

—

146

—

135

C₂²×C₄

#11924011

yes

—

136

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

146

—

144

C₂×C₈

—

146

—

145

C₂×C₈

C₈×D₇₃

yes

—

146

C₂×C₈

D₄×D₇₃

yes

characteristic

147

C₂×C₈

Q₈×D₇₃

yes

characteristic

148

C₂×C₈

#11924027

yes

—

149

C₂×C₈

#11924028

yes

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

146

—

155

C₄²

—

146

—

156

C₁₆

—

157

C₁₆

—

292

—

158

C₂³

C₂×#11924011

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

292

—

162

C₂×C₄

—

292

—

163

C₂×C₄

—

146

—

164

C₂×C₄

—

146

—

165

C₂×C₄

C₄⋊C₄×D₇₃

yes

characteristic

166

C₂×C₄

#14556547

yes

characteristic

167

C₂×C₄

#14556548

yes

characteristic

168

C₂×C₄

C₂×C₈×D₇₃

yes

characteristic

169

C₂×C₄

M₁₆×D₇₃

yes

characteristic

170

C₂×C₄

M₁₆×D₇₃

yes

characteristic

171

C₈

—

172

C₈

—

173

C₈

—

146

—

174

C₈

#14556577

yes

—

175

C₈

#14556581

yes

—

176

C₂²

—

177

C₂²

#17988609

yes

characteristic

178

C₂²

#17988638

yes

characteristic

179

C₂²

#17988641

yes

characteristic

180

C₄

—

181

C₄

—

292

—

182

C₄

—

292

—

183

C₄

#17988451

yes

characteristic

184

C₄

C₄⋊C₈×D₇₃

yes

characteristic

185

C₄

#17988637

yes

characteristic

186

C₄

#17988642

yes

characteristic

187

C₂

—

188

C₂

—

189

C₂

#22610046

yes

characteristic

190

C₂

#22611744

yes

—

	subgroup	G/H
center	C₂×C₄	#14556547
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1377	C₂
socle	C₂²×C₇₃	#432
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1377	D₇₃
hypocenter	C₇₃	#12881
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1377

subgroup

G/H

center

C₂×C₄

#14556547

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1377

C₂

socle

C₂²×C₇₃

#432

prime radical

C₂³×C₇₃

C₂×M₁₆

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1377

N	H ≅ G/N	type
101. C₇₃	#12881	semidirect

H ≅ G/N

type

101. C₇₃

#12881

semidirect

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

subgroup	index
C₂×#17988407	2
#22609619	2
#22608015	2
#22608010	2
C₇₃×#1377	2
#12881	73

subgroup

index

C₇₃×#1377

Subgroups — #33002335

Conjugacy classes of subgroups

Complements

Maximal subgroups