Group #33005376 — subgroups

Subgroups — #33005376

Order 18688.

436 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 99 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×#17988967	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988402	C₂	1	2	yes	maximal
3	#22611650	C₂	1	2	yes	maximal
4	#22608013	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1945	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₂	—	2	1	no	—
7	C₂²×Dic₂₉₂	C₂²	1	2	yes	—
8	C₂×#14556591	—	2	1	no	—
9	C₂×#14556531	C₂²	1	2	yes	—
10	#17988967	—	2	4	no	—
11	#17988402	—	2	4	no	—
12	#17988390	—	2	4	no	—
13	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
14	#427×C₇₃	C₂²	1	1	yes	characteristic
15	#424×C₇₃	—	2	1	no	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#339×C₇₃	—	2	1	no	—
18	C₂²×Dic₁₄₆	—	2	2	no	—
19	C₂×Dic₂₉₂	—	2	4	no	—
20	C₂×Dic₂₉₂	—	2	2	no	—
21	C₂×Dic₂₉₂	—	2	2	no	—
22	C₂×Dic₂₉₂	—	2	2	no	—
23	C₂×Dic₂₉₂	—	2	2	no	—
24	C₂×Dic₂₉₂	—	2	2	no	—
25	C₂×Dic₂₉₂	—	4	2	no	—
26	C₂×Dic₂₉₂	—	4	2	no	—
27	C₂×Dic₂₉₂	D₄	1	4	yes	—
28	#14556591	—	2	4	no	—
29	#14556591	—	4	2	no	—
30	C₂×C₄⋊Dic₇₃	—	4	2	no	—
31	#14556531	—	2	2	no	—
32	#14556531	—	2	2	no	—
33	C₂×#11924003	—	2	2	no	—
34	C₂×M₁₆×C₇₃	—	2	1	no	—
35	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
37	#117×C₇₃	—	2	1	no	—
38	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
39	#108×C₇₃	—	2	2	no	—
40	#107×C₇₃	D₄	1	2	yes	—
41	#106×C₇₃	—	2	1	no	—
42	C₄⋊C₈×C₇₃	—	4	2	no	—
43	C₄×C₈×C₇₃	—	2	2	no	—
44	C₂×Dic₁₄₆	—	2	4	no	—
45	C₂×Dic₁₄₆	—	4	2	no	—
46	C₂×Dic₁₄₆	—	4	2	no	—
47	C₂×Dic₁₄₆	—	4	2	no	—
48	C₂×Dic₁₄₆	—	4	2	no	—
49	C₂×Dic₁₄₆	—	4	2	no	—
50	C₂²×Dic₇₃	—	4	2	no	—
51	Dic₂₉₂	—	2	4	no	—
52	Dic₂₉₂	—	2	4	no	—
53	Dic₂₉₂	—	4	8	no	—
54	Dic₂₉₂	—	4	2	no	—
55	Dic₂₉₂	—	4	2	no	—
56	C₄⋊Dic₇₃	—	4	8	no	—
57	#11924003	—	4	2	no	—
58	C₂²×C₄×C₇₃	—	2	1	no	—
59	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	M₁₆×C₇₃	—	4	1	no	—
61	M₁₆×C₇₃	—	4	1	no	—
62	C₂×C₈×C₇₃	—	2	1	no	—
63	C₂×C₈×C₇₃	—	2	1	no	—
64	C₂×C₈×C₇₃	—	4	1	no	—
65	C₂×C₈×C₇₃	D₈	1	2	yes	—
66	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
69	C₂×C₈×C₇₃	#41	1	2	yes	—
70	C₄⋊C₄×C₇₃	—	2	2	no	—
71	C₂²⋊C₄×C₇₃	—	4	1	no	—
72	C₄²×C₇₃	—	2	1	no	—
73	C₄²×C₇₃	—	4	1	no	—
74	C₄²×C₇₃	SD₁₆	1	2	yes	—
75	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×C₄×C₇₃	—	2	4	no	—
77	C₂×C₄×C₇₃	—	2	2	no	—
78	C₂×C₄×C₇₃	—	4	2	no	—
79	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×Dic₇₃	—	4	4	no	—
86	C₂×Dic₇₃	—	8	2	no	—
87	C₂×Dic₇₃	—	8	2	no	—
88	Dic₁₄₆	—	4	4	no	—
89	Dic₁₄₆	—	4	4	no	—
90	Dic₁₄₆	—	8	2	no	—
91	Dic₁₄₆	—	8	2	no	—
92	C₈×C₇₃	—	2	2	no	—
93	C₈×C₇₃	—	8	1	no	—
94	C₈×C₇₃	#121	1	2	yes	—
95	C₈×C₇₃	#123	1	2	yes	—
96	C₂²×C₇₃	—	2	2	no	—
97	C₂²×C₇₃	#450	1	1	yes	characteristic
98	C₂²×C₇₃	#459	1	1	yes	characteristic
99	C₂²×C₇₃	#490	1	1	yes	characteristic
100	C₄×C₇₃	—	2	4	no	—
101	C₄×C₇₃	—	8	2	no	—
102	C₄×C₇₃	#394	1	1	yes	characteristic
103	C₄×C₇₃	#447	1	1	yes	characteristic
104	C₄×C₇₃	#460	1	1	yes	characteristic
105	C₄×C₇₃	#486	1	1	yes	characteristic
106	Dic₇₃	—	8	4	no	—
107	#13415	—	73	1	no	maximal, Sylow
108	C₂×C₇₃	—	2	1	no	—
109	C₂×C₇₃	—	2	1	no	—
110	C₂×C₇₃	#2041	1	1	yes	characteristic
111	C₂×C₇₃	#2214	1	2	yes	—
112	C₂×#494	—	73	1	no	—
113	C₂×#362	—	73	2	no	—
114	#2141	—	73	2	no	—
115	#1945	D₇₃	1	1	yes	characteristic
116	#1376	—	73	1	no	—
117	C₇₃	#13415	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	—	73	2	no	—
119	C₂²×Q₁₆	—	146	1	no	—
120	C₂×C₄⋊Q₈	—	146	1	no	—
121	C₂×M₃₂	—	73	2	no	—
122	#494	—	146	4	no	—
123	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
124	#427	C₂×D₇₃	1	1	yes	characteristic
125	#424	—	2	1	no	—
126	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
127	#362	—	146	4	no	—
128	#358	—	146	4	no	—
129	#339	—	2	1	no	—
130	C₂²×Q₈	—	146	2	no	—
131	C₂×Q₁₆	—	292	2	no	—
132	C₂×Q₁₆	—	292	2	no	—
133	C₂×Q₁₆	—	73	4	no	—
134	C₂×Q₁₆	—	146	4	no	—
135	C₂×Q₁₆	—	146	2	no	—
136	C₂×Q₁₆	—	146	2	no	—
137	C₂×Q₁₆	—	146	2	no	—
138	C₂×Q₁₆	—	146	2	no	—
139	C₂×Q₁₆	—	146	2	no	—
140	C₂×M₁₆	—	2	1	no	—
141	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
143	C₄⋊Q₈	—	292	2	no	—
144	C₄⋊Q₈	—	146	4	no	—
145	#117	—	2	1	no	—
146	C₂×C₄⋊C₄	—	292	2	no	—
147	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
148	M₃₂	—	146	2	no	—
149	M₃₂	—	146	2	no	—
150	C₂×C₁₆	—	146	2	no	—
151	#108	—	2	2	no	—
152	#107	D₂₉₂	1	2	yes	—
153	#106	—	2	1	no	—
154	C₄⋊C₈	—	4	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂×Q₈	—	292	2	no	—
157	C₂×Q₈	—	292	2	no	—
158	C₂×Q₈	—	292	2	no	—
159	C₂×Q₈	—	292	2	no	—
160	C₂×Q₈	—	292	2	no	—
161	C₂×Q₈	—	146	4	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	292	2	no	—
164	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
165	Q₁₆	—	292	8	no	—
166	Q₁₆	—	292	2	no	—
167	Q₁₆	—	292	2	no	—
168	Q₁₆	—	146	4	no	—
169	Q₁₆	—	146	4	no	—
170	M₁₆	—	4	1	no	—
171	M₁₆	—	4	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	#11924013	1	2	yes	—
176	C₂×C₈	D₄×D₇₃	1	2	yes	—
177	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
178	C₂×C₈	#11924028	1	1	yes	characteristic
179	C₂×C₈	#11924029	1	2	yes	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	292	8	no	—
182	C₂²⋊C₄	—	4	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	4	1	no	—
185	C₄²	#11924015	1	2	yes	—
186	C₁₆	—	292	2	no	—
187	C₂³	#14556616	1	1	yes	characteristic
188	Q₈	—	292	4	no	—
189	Q₈	—	292	4	no	—
190	Q₈	—	584	2	no	—
191	Q₈	—	584	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	292	4	no	—
196	C₂×C₄	—	584	2	no	—
197	C₂×C₄	—	584	2	no	—
198	C₂×C₄	#14556559	1	1	yes	characteristic
199	C₂×C₄	#14556565	1	1	yes	characteristic
200	C₂×C₄	#14556580	1	1	yes	characteristic
201	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
202	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
203	C₂×C₄	#14556616	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	8	1	no	—
206	C₈	#14556566	1	2	yes	—
207	C₈	#14556576	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	#17988737	1	1	yes	characteristic
210	C₂²	#17988782	1	1	yes	characteristic
211	C₂²	#17988953	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	8	2	no	—
214	C₄	—	584	4	no	—
215	C₄	#17988504	1	1	yes	characteristic
216	C₄	#17988720	1	1	yes	characteristic
217	C₄	#17988787	1	1	yes	characteristic
218	C₄	#17988935	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#22611165	1	1	yes	characteristic
222	C₂	#22612012	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988967

C₂

yes

maximal, characteristic

C₂×#17988402

C₂

yes

maximal

#22611650

C₂

yes

maximal

#22608013

C₂

yes

maximal, characteristic

C₇₃×#1945

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₂

—

C₂²×Dic₂₉₂

C₂²

yes

—

C₂×#14556591

—

C₂×#14556531

C₂²

yes

—

#17988967

—

#17988402

—

#17988390

—

C₂×#108×C₇₃

C₂²

yes

characteristic

#427×C₇₃

C₂²

yes

characteristic

#424×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#339×C₇₃

—

C₂²×Dic₁₄₆

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

D₄

yes

—

#14556591

—

#14556591

—

C₂×C₄⋊Dic₇₃

—

#14556531

—

#14556531

—

C₂×#11924003

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

#117×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

#108×C₇₃

—

#107×C₇₃

D₄

yes

—

#106×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

—

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

C₂×D₄

yes

C₂×C₈×C₇₃

yes

C₂×C₈×C₇₃

yes

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

—

C₂³×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#121

yes

—

C₈×C₇₃

#123

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

100

C₄×C₇₃

—

101

C₄×C₇₃

—

102

C₄×C₇₃

#394

yes

characteristic

103

C₄×C₇₃

#447

yes

characteristic

104

C₄×C₇₃

#460

yes

characteristic

105

C₄×C₇₃

#486

yes

characteristic

106

Dic₇₃

—

107

#13415

—

maximal, Sylow

108

C₂×C₇₃

—

109

C₂×C₇₃

—

110

C₂×C₇₃

#2041

yes

characteristic

111

C₂×C₇₃

#2214

yes

—

112

C₂×#494

—

113

C₂×#362

—

114

#2141

—

115

#1945

D₇₃

yes

characteristic

116

#1376

—

117

C₇₃

#13415

yes

characteristic, complemented, Sylow

118

C₂²×Q₁₆

—

119

C₂²×Q₁₆

—

146

—

120

C₂×C₄⋊Q₈

—

146

—

121

C₂×M₃₂

—

122

#494

—

146

—

123

C₂×#108

C₂×D₇₃

yes

characteristic

124

#427

C₂×D₇₃

yes

characteristic

125

#424

—

126

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

127

#362

—

146

—

128

#358

—

146

—

129

#339

—

130

C₂²×Q₈

—

146

—

131

C₂×Q₁₆

—

292

—

132

C₂×Q₁₆

—

292

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

146

—

135

C₂×Q₁₆

—

146

—

136

C₂×Q₁₆

—

146

—

137

C₂×Q₁₆

—

146

—

138

C₂×Q₁₆

—

146

—

139

C₂×Q₁₆

—

146

—

140

C₂×M₁₆

—

141

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

142

C₂²×C₈

C₂²×D₇₃

yes

characteristic

143

C₄⋊Q₈

—

292

—

144

C₄⋊Q₈

—

146

—

145

#117

—

146

C₂×C₄⋊C₄

—

292

—

147

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

148

M₃₂

—

146

—

149

M₃₂

—

146

—

150

C₂×C₁₆

—

146

—

151

#108

—

152

#107

D₂₉₂

yes

—

153

#106

—

154

C₄⋊C₈

—

155

C₄×C₈

—

156

C₂×Q₈

—

292

—

157

C₂×Q₈

—

292

—

158

C₂×Q₈

—

292

—

159

C₂×Q₈

—

292

—

160

C₂×Q₈

—

292

—

161

C₂×Q₈

—

146

—

162

C₂²×C₄

—

163

C₂²×C₄

—

292

—

164

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

165

Q₁₆

—

292

—

166

Q₁₆

—

292

—

167

Q₁₆

—

292

—

168

Q₁₆

—

146

—

169

Q₁₆

—

146

—

170

M₁₆

—

171

M₁₆

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

#11924013

yes

—

176

C₂×C₈

D₄×D₇₃

yes

—

177

C₂×C₈

C₂×D₂₉₂

yes

characteristic

178

C₂×C₈

#11924028

yes

characteristic

179

C₂×C₈

#11924029

yes

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

292

—

182

C₂²⋊C₄

—

183

C₄²

—

184

C₄²

—

185

C₄²

#11924015

yes

—

186

C₁₆

—

292

—

187

C₂³

#14556616

yes

characteristic

188

Q₈

—

292

—

189

Q₈

—

292

—

190

Q₈

—

584

—

191

Q₈

—

584

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

292

—

196

C₂×C₄

—

584

—

197

C₂×C₄

—

584

—

198

C₂×C₄

#14556559

yes

characteristic

199

C₂×C₄

#14556565

yes

characteristic

200

C₂×C₄

#14556580

yes

characteristic

201

C₂×C₄

C₂×#11924013

yes

characteristic

202

C₂×C₄

C₂×#11924015

yes

characteristic

203

C₂×C₄

#14556616

yes

characteristic

204

C₈

—

205

C₈

—

206

C₈

#14556566

yes

—

207

C₈

#14556576

yes

—

208

C₂²

—

209

C₂²

#17988737

yes

characteristic

210

C₂²

#17988782

yes

characteristic

211

C₂²

#17988953

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

584

—

215

C₄

#17988504

yes

characteristic

216

C₄

#17988720

yes

characteristic

217

C₄

#17988787

yes

characteristic

218

C₄

#17988935

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#22611165

yes

characteristic

222

C₂

#22612012

yes

—

	subgroup	G/H
center	C₂²	#17988782
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1945	C₂
socle	C₂²×C₇₃	#459
prime radical	C₂³×C₇₃	AGL(1,ℤ/8)
hypercenter	#1945	D₇₃
hypocenter	C₇₃	#13415
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

subgroup

G/H

center

C₂²

#17988782

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1945

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

N	H ≅ G/N	type
117. C₇₃	#13415	semidirect

H ≅ G/N

type

117. C₇₃

#13415

semidirect

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

subgroup	index
C₂×#17988967	2
C₂×#17988402	2
#22611650	2
#22608013	2
C₇₃×#1945	2
#13415	73

subgroup

index

C₇₃×#1945

Subgroups — #33005376

Conjugacy classes of subgroups

Complements

Maximal subgroups