Group #33004123 — subgroups

Subgroups — #33004123

Order 18688.

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

normal subgroups only

This group has 310 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	C₂×#17988409	C₂	1	2	yes	maximal
2	C₂×#17988408	C₂	1	1	yes	maximal, characteristic
3	#22609810	C₂	1	1	yes	maximal, characteristic
4	#22608038	C₂	1	2	yes	maximal
5	C₇₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	C₂×#14556527	C₂²	1	1	yes	characteristic
8	#17988625	C₂²	1	1	yes	characteristic
9	#17988542	—	2	1	no	—
10	#17988409	—	2	4	no	—
11	#17988408	C₄	1	4	yes	complemented
12	#17988347	—	2	1	no	—
13	C₂×#106×C₇₃	—	2	1	no	—
14	C₂×#106×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#384×C₇₃	—	2	1	no	—
17	C₂×#11924012	—	2	1	no	—
18	#14556550	—	2	1	no	—
19	#14556531	—	2	2	no	—
20	#14556531	C₂×C₄	1	4	yes	—
21	C₂×#11924003	—	2	2	no	—
22	#14556527	—	2	1	no	—
23	#14556527	D₄	1	2	yes	—
24	#14556525	Q₈	1	2	yes	—
25	#14556523	—	2	1	no	—
26	#14556501	—	4	1	no	—
27	C₈×Dic₇₃	—	4	1	no	—
28	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₇₃	—	4	2	no	—
32	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
33	#106×C₇₃	—	2	2	no	—
34	#106×C₇₃	—	2	2	no	—
35	#106×C₇₃	—	4	2	no	—
36	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₇₃	Q₈	1	2	yes	—
38	#11924012	—	2	2	no	—
39	#11924012	—	4	1	no	—
40	C₂×#10596831	—	4	1	no	—
41	C₄⋊Dic₇₃	—	2	2	no	—
42	C₂²⋊Dic₇₃	—	4	1	no	—
43	C₄×Dic₇₃	—	4	1	no	—
44	#11924003	—	2	4	no	—
45	C₂²×C₄×C₇₃	—	2	1	no	—
46	C₂²×C₄×C₇₃	—	4	2	no	—
47	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
52	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
57	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
58	C₄⋊C₄×C₇₃	—	4	2	no	—
59	C₄⋊C₄×C₇₃	—	4	2	no	—
60	C₄⋊C₄×C₇₃	—	8	4	no	—
61	C₄²×C₇₃	—	2	1	no	—
62	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
63	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×C₄×C₇₃	—	2	4	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	8	4	no	—
67	C₂×C₄×C₇₃	—	8	2	no	—
68	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
74	C₂×Dic₇₃	—	4	2	no	—
75	C₈×C₇₃	#106	1	2	yes	—
76	C₈×C₇₃	#106	1	2	yes	—
77	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
78	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₇₃	#125	1	1	yes	characteristic
80	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
81	#10596831	—	4	2	no	—
82	C₂²×C₇₃	—	2	2	no	—
83	C₂²×C₇₃	#428	1	1	yes	characteristic
84	C₂²×C₇₃	#460	1	1	yes	characteristic
85	C₂²×C₇₃	#476	1	1	yes	characteristic
86	C₄×C₇₃	—	2	4	no	—
87	C₄×C₇₃	—	16	2	no	—
88	C₄×C₇₃	#389	1	1	yes	characteristic
89	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
90	C₄×C₇₃	#461	1	1	yes	characteristic
91	C₄×C₇₃	#475	1	1	yes	characteristic
92	Dic₇₃	—	8	2	no	—
93	#13186	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	—	2	1	no	—
95	C₂×C₇₃	—	2	1	no	—
96	C₂×C₇₃	#1948	1	1	yes	characteristic
97	C₂×C₇₃	#2230	1	2	yes	—
98	C₂×#365	—	73	2	no	—
99	C₂×#364	—	73	1	no	—
100	#1844	D₇₃	1	1	yes	characteristic
101	#1772	—	73	1	no	—
102	#1386	—	73	2	no	—
103	C₇₃	#13186	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₃₂	—	73	2	no	—
105	C₂×#108	—	73	1	no	—
106	#427	—	73	1	no	—
107	C₂×#106	—	2	1	no	—
108	C₂×#106	C₂×D₇₃	1	2	yes	—
109	#405	—	146	1	no	—
110	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
111	#384	—	2	1	no	—
112	#365	—	146	4	no	—
113	#364	—	73	4	no	—
114	#339	—	146	1	no	—
115	C₂×M₁₆	—	146	1	no	—
116	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
118	#117	—	146	1	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	2	no	—
121	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
122	M₃₂	—	73	4	no	—
123	M₃₂	—	146	2	no	—
124	C₂×C₁₆	—	146	2	no	—
125	#108	—	73	2	no	—
126	#108	—	146	1	no	—
127	#107	—	73	2	no	—
128	#106	—	2	2	no	—
129	#106	—	2	2	no	—
130	#106	—	4	2	no	—
131	#106	—	146	1	no	—
132	#97	—	292	1	no	—
133	C₄×C₈	—	292	1	no	—
134	C₄×C₈	Dic₁₄₆	1	2	yes	—
135	C₄×C₈	C₄×D₇₃	1	2	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
139	M₁₆	—	292	1	no	—
140	M₁₆	—	146	2	no	—
141	C₂×C₈	—	292	1	no	—
142	C₂×C₈	#11924007	1	2	yes	—
143	C₂×C₈	#11924015	1	1	yes	characteristic
144	C₂×C₈	#11924015	1	1	yes	characteristic
145	C₂×C₈	#11924016	1	2	yes	—
146	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
147	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
148	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
149	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
150	C₂×C₈	#11924027	1	1	yes	characteristic
151	C₂×C₈	#11924029	1	1	yes	characteristic
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	8	4	no	—
155	C₄⋊C₄	—	146	2	no	—
156	C₂²⋊C₄	—	292	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	292	1	no	—
159	C₄²	#11924007	1	2	yes	—
160	C₁₆	—	146	4	no	—
161	C₂³	#14556617	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	8	4	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	292	2	no	—
167	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
168	C₂×C₄	#14556562	1	1	yes	characteristic
169	C₂×C₄	#14556567	1	1	yes	characteristic
170	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
171	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
172	C₂×C₄	#14556622	1	1	yes	characteristic
173	C₈	—	292	2	no	—
174	C₈	#14556522	1	2	yes	—
175	C₈	#14556522	1	2	yes	—
176	C₈	#14556553	1	1	yes	characteristic
177	C₈	#14556556	1	1	yes	characteristic
178	C₈	#14556582	1	1	yes	characteristic
179	C₈	#14556590	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	#17988627	1	1	yes	characteristic
182	C₂²	#17988784	1	1	yes	characteristic
183	C₂²	#17988874	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	16	2	no	—
186	C₄	—	584	2	no	—
187	C₄	#17988483	1	1	yes	characteristic
188	C₄	C₂×#14556522	1	1	yes	characteristic
189	C₄	#17988789	1	1	yes	characteristic
190	C₄	#17988869	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#22610692	1	1	yes	characteristic
194	C₂	#22612099	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988409

C₂

yes

maximal

C₂×#17988408

C₂

yes

maximal, characteristic

#22609810

C₂

yes

maximal, characteristic

#22608038

C₂

yes

maximal

C₇₃×#1844

C₂

yes

maximal, characteristic

C₂×#14556531

C₂²

yes

—

C₂×#14556527

C₂²

yes

characteristic

#17988625

C₂²

yes

—

—

C₄

yes

complemented

#17988347

—

C₂×#106×C₇₃

—

C₂×#106×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#384×C₇₃

—

—

—

—

C₂×C₄

yes

—

C₂×#11924003

—

#14556527

—

#14556527

D₄

yes

—

#14556525

Q₈

yes

—

#14556523

—

#14556501

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

#106×C₇₃

—

#106×C₇₃

—

#106×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

characteristic

C₂×C₈×C₇₃

SD₁₆

yes

characteristic

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

C₂×C₈×C₇₃

yes

C₂×C₈×C₇₃

yes

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₄⋊C₄

yes

—

C₂³×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

#137

yes

characteristic

C₂×Dic₇₃

—

C₈×C₇₃

#106

yes

—

C₈×C₇₃

#106

yes

—

C₈×C₇₃

C₄×D₄

yes

characteristic

C₈×C₇₃

C₄×Q₈

yes

characteristic

C₈×C₇₃

#125

yes

characteristic

C₈×C₇₃

C₄⋊Q₈

yes

characteristic

#10596831

—

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

—

#13186

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1948

yes

characteristic

C₂×C₇₃

#2230

yes

—

C₂×#365

—

C₂×#364

—

100

#1844

D₇₃

yes

characteristic

101

#1772

—

102

#1386

—

103

C₇₃

#13186

yes

characteristic, complemented, Sylow

104

C₂×M₃₂

—

105

C₂×#108

—

106

#427

—

107

C₂×#106

—

108

C₂×#106

C₂×D₇₃

yes

—

109

#405

—

146

—

110

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

111

#384

—

112

#365

—

146

—

113

#364

—

114

#339

—

146

—

115

C₂×M₁₆

—

146

—

116

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

117

C₂²×C₈

C₂²×D₇₃

yes

characteristic

118

#117

—

146

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

122

M₃₂

—

123

M₃₂

—

146

—

124

C₂×C₁₆

—

146

—

125

#108

—

126

#108

—

146

—

127

#107

—

128

#106

—

129

#106

—

130

#106

—

131

#106

—

146

—

132

#97

—

292

—

133

C₄×C₈

—

292

—

134

C₄×C₈

Dic₁₄₆

yes

—

135

C₄×C₈

C₄×D₇₃

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

139

M₁₆

—

292

—

140

M₁₆

—

146

—

141

C₂×C₈

—

292

—

142

C₂×C₈

#11924007

yes

—

143

C₂×C₈

#11924015

yes

characteristic

144

C₂×C₈

#11924015

yes

characteristic

145

C₂×C₈

#11924016

yes

—

146

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

147

C₂×C₈

D₄×D₇₃

yes

characteristic

148

C₂×C₈

Q₈×D₇₃

yes

characteristic

149

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

150

C₂×C₈

#11924027

yes

characteristic

151

C₂×C₈

#11924029

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

146

—

156

C₂²⋊C₄

—

292

—

157

C₄²

—

158

C₄²

—

292

—

159

C₄²

#11924007

yes

—

160

C₁₆

—

146

—

161

C₂³

#14556617

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

292

—

167

C₂×C₄

C₂×#11924007

yes

characteristic

168

C₂×C₄

#14556562

yes

characteristic

169

C₂×C₄

#14556567

yes

characteristic

170

C₂×C₄

C₂×#11924015

yes

characteristic

171

C₂×C₄

C₂×#11924016

yes

characteristic

172

C₂×C₄

#14556622

yes

characteristic

173

C₈

—

292

—

174

C₈

#14556522

yes

—

175

C₈

#14556522

yes

—

176

C₈

#14556553

yes

characteristic

177

C₈

#14556556

yes

characteristic

178

C₈

#14556582

yes

characteristic

179

C₈

#14556590

yes

characteristic

180

C₂²

—

181

C₂²

#17988627

yes

characteristic

182

C₂²

#17988784

yes

characteristic

183

C₂²

#17988874

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

—

584

—

187

C₄

#17988483

yes

characteristic

188

C₄

C₂×#14556522

yes

characteristic

189

C₄

#17988789

yes

characteristic

190

C₄

#17988869

yes

characteristic

191

C₂

—

192

C₂

—

193

C₂

#22610692

yes

characteristic

194

C₂

#22612099

yes

—

	subgroup	G/H
center	C₂²	#17988874
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1844	C₂
socle	C₂²×C₇₃	#476
prime radical	C₂³×C₇₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1844	D₇₃
hypocenter	C₇₃	#13186
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

subgroup

G/H

center

C₂²

#17988874

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1844

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

N	H ≅ G/N	type
11. #17988408	C₄	semidirect
103. C₇₃	#13186	semidirect

H ≅ G/N

type

11. #17988408

C₄

semidirect

103. C₇₃

#13186

semidirect

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

subgroup	index
C₂×#17988409	2
C₂×#17988408	2
#22609810	2
#22608038	2
C₇₃×#1844	2
#13186	73

subgroup

index

C₇₃×#1844

Subgroups — #33004123

Conjugacy classes of subgroups

Complements

Maximal subgroups