Group #33004093 — subgroups

Subgroups — #33004093

Order 18688.

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

normal subgroups only

This group has 294 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	#22611717	C₂	1	2	yes	maximal
2	C₂×#17988408	C₂	1	1	yes	maximal, characteristic
3	#22609810	C₂	1	1	yes	maximal, characteristic
4	#22608039	C₂	1	2	yes	maximal
5	C₇₃×#1846	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	#17988414	—	2	2	no	—
11	#17988412	—	2	2	no	—
12	#17988408	C₄	1	4	yes	complemented
13	#17988347	—	2	1	no	—
14	C₂×#107×C₇₃	C₂²	1	2	yes	—
15	C₂×#106×C₇₃	—	2	1	no	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#382×C₇₃	—	2	1	no	—
18	C₂×#11924012	—	2	1	no	—
19	#14556550	—	2	1	no	—
20	#14556531	—	2	2	no	—
21	#14556531	C₂×C₄	1	4	yes	—
22	C₂×#11924003	—	2	2	no	—
23	#14556527	—	2	1	no	—
24	#14556527	Q₈	1	2	yes	—
25	#14556525	D₄	1	2	yes	—
26	#14556523	—	2	1	no	—
27	#14556501	—	4	1	no	—
28	C₈×Dic₇₃	—	4	1	no	—
29	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
32	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
33	#107×C₇₃	—	2	2	no	—
34	#107×C₇₃	—	2	2	no	—
35	#106×C₇₃	—	4	1	no	—
36	#106×C₇₃	—	4	1	no	—
37	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₃	Q₈	1	2	yes	—
39	#95×C₇₃	—	4	2	no	—
40	#11924012	—	2	2	no	—
41	#11924012	—	4	1	no	—
42	C₂×#10596831	—	4	1	no	—
43	C₄⋊Dic₇₃	—	2	2	no	—
44	C₂²⋊Dic₇₃	—	4	1	no	—
45	C₄×Dic₇₃	—	4	1	no	—
46	#11924003	—	2	4	no	—
47	C₂²×C₄×C₇₃	—	2	1	no	—
48	C₂²×C₄×C₇₃	—	4	2	no	—
49	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
53	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
54	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₇₃	—	4	2	no	—
61	C₄⋊C₄×C₇₃	—	4	2	no	—
62	C₄²×C₇₃	—	2	1	no	—
63	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
64	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	4	2	no	—
67	C₂×C₄×C₇₃	—	8	4	no	—
68	C₂×C₄×C₇₃	—	8	2	no	—
69	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
75	C₂×Dic₇₃	—	4	2	no	—
76	C₈×C₇₃	#107	1	2	yes	—
77	C₈×C₇₃	#107	1	2	yes	—
78	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
79	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₇₃	#125	1	1	yes	characteristic
81	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
82	#10596831	—	4	2	no	—
83	C₂²×C₇₃	—	2	2	no	—
84	C₂²×C₇₃	#428	1	1	yes	characteristic
85	C₂²×C₇₃	#459	1	1	yes	characteristic
86	C₂²×C₇₃	#477	1	1	yes	characteristic
87	C₄×C₇₃	—	4	2	no	—
88	C₄×C₇₃	—	16	2	no	—
89	C₄×C₇₃	#389	1	1	yes	characteristic
90	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
91	C₄×C₇₃	#462	1	1	yes	characteristic
92	C₄×C₇₃	#474	1	1	yes	characteristic
93	Dic₇₃	—	8	2	no	—
94	#13180	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	—	2	1	no	—
96	C₂×C₇₃	—	2	1	no	—
97	C₂×C₇₃	#1947	1	1	yes	characteristic
98	C₂×C₇₃	#2212	1	2	yes	—
99	#2159	—	73	2	no	—
100	C₂×#364	—	73	1	no	—
101	#1846	D₇₃	1	1	yes	characteristic
102	#1772	—	73	1	no	—
103	#1387	—	73	2	no	—
104	C₇₃	#13180	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₃₂	—	73	2	no	—
106	C₂×#108	—	73	1	no	—
107	#427	—	73	1	no	—
108	C₂×#107	C₂×D₇₃	1	2	yes	—
109	C₂×#106	—	2	1	no	—
110	#405	—	146	1	no	—
111	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
112	#382	—	2	1	no	—
113	#367	—	146	2	no	—
114	#366	—	146	2	no	—
115	#364	—	73	4	no	—
116	#339	—	146	1	no	—
117	C₂×M₁₆	—	146	1	no	—
118	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
120	#117	—	146	1	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
123	M₃₂	—	73	4	no	—
124	M₃₂	—	146	2	no	—
125	C₂×C₁₆	—	146	2	no	—
126	#108	—	73	2	no	—
127	#108	—	146	1	no	—
128	#107	—	2	2	no	—
129	#107	—	2	2	no	—
130	#107	—	73	2	no	—
131	#106	—	4	1	no	—
132	#106	—	4	1	no	—
133	#106	—	146	1	no	—
134	#97	—	292	1	no	—
135	C₄×C₈	—	292	1	no	—
136	C₄×C₈	Dic₁₄₆	1	2	yes	—
137	C₄×C₈	C₄×D₇₃	1	2	yes	—
138	#95	—	4	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
142	M₁₆	—	292	1	no	—
143	M₁₆	—	146	2	no	—
144	C₂×C₈	—	292	1	no	—
145	C₂×C₈	#11924007	1	2	yes	—
146	C₂×C₈	#11924013	1	1	yes	characteristic
147	C₂×C₈	#11924013	1	1	yes	characteristic
148	C₂×C₈	#11924018	1	2	yes	—
149	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
150	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
153	C₂×C₈	#11924027	1	1	yes	characteristic
154	C₂×C₈	#11924029	1	1	yes	characteristic
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	146	2	no	—
158	C₂²⋊C₄	—	292	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	292	1	no	—
161	C₄²	#11924007	1	2	yes	—
162	C₁₆	—	146	4	no	—
163	C₂³	#14556617	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	292	2	no	—
169	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
170	C₂×C₄	#14556562	1	1	yes	characteristic
171	C₂×C₄	#14556567	1	1	yes	characteristic
172	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
173	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
174	C₂×C₄	#14556622	1	1	yes	characteristic
175	C₈	—	292	2	no	—
176	C₈	#14556524	1	2	yes	—
177	C₈	#14556524	1	2	yes	—
178	C₈	#14556553	1	1	yes	characteristic
179	C₈	#14556556	1	1	yes	characteristic
180	C₈	#14556582	1	1	yes	characteristic
181	C₈	#14556590	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	#17988627	1	1	yes	characteristic
184	C₂²	#17988779	1	1	yes	characteristic
185	C₂²	#17988879	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	16	2	no	—
188	C₄	—	584	2	no	—
189	C₄	#17988483	1	1	yes	characteristic
190	C₄	C₂×#14556524	1	1	yes	characteristic
191	C₄	#17988794	1	1	yes	characteristic
192	C₄	#17988864	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#22610687	1	1	yes	characteristic
196	C₂	#22611997	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22611717

C₂

yes

maximal

C₂×#17988408

C₂

yes

maximal, characteristic

#22609810

C₂

yes

maximal, characteristic

#22608039

C₂

yes

maximal

C₇₃×#1846

C₂

yes

maximal, characteristic

C₂×#14556531

C₂²

yes

—

C₂×#14556527

C₂²

yes

characteristic

#17988625

C₂²

yes

—

—

—

C₄

yes

complemented

#17988347

—

C₂×#107×C₇₃

C₂²

yes

—

C₂×#106×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#382×C₇₃

—

—

—

—

C₂×C₄

yes

—

C₂×#11924003

—

#14556527

—

#14556527

Q₈

yes

—

#14556525

D₄

yes

—

#14556523

—

#14556501

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

#107×C₇₃

—

#107×C₇₃

—

#106×C₇₃

—

#106×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

#95×C₇₃

—

#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

D₈

yes

characteristic

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

Q₁₆

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

—

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×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

#137

yes

characteristic

C₂×Dic₇₃

—

C₈×C₇₃

#107

yes

—

C₈×C₇₃

#107

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

—

#13180

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1947

yes

characteristic

C₂×C₇₃

#2212

yes

—

#2159

—

100

C₂×#364

—

101

#1846

D₇₃

yes

characteristic

102

#1772

—

103

#1387

—

104

C₇₃

#13180

yes

characteristic, complemented, Sylow

105

C₂×M₃₂

—

106

C₂×#108

—

107

#427

—

108

C₂×#107

C₂×D₇₃

yes

—

109

C₂×#106

—

110

#405

—

146

—

111

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

112

#382

—

113

#367

—

146

—

114

#366

—

146

—

115

#364

—

116

#339

—

146

—

117

C₂×M₁₆

—

146

—

118

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

119

C₂²×C₈

C₂²×D₇₃

yes

characteristic

120

#117

—

146

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

123

M₃₂

—

124

M₃₂

—

146

—

125

C₂×C₁₆

—

146

—

126

#108

—

127

#108

—

146

—

128

#107

—

129

#107

—

130

#107

—

131

#106

—

132

#106

—

133

#106

—

146

—

134

#97

—

292

—

135

C₄×C₈

—

292

—

136

C₄×C₈

Dic₁₄₆

yes

—

137

C₄×C₈

C₄×D₇₃

yes

—

138

#95

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

142

M₁₆

—

292

—

143

M₁₆

—

146

—

144

C₂×C₈

—

292

—

145

C₂×C₈

#11924007

yes

—

146

C₂×C₈

#11924013

yes

characteristic

147

C₂×C₈

#11924013

yes

characteristic

148

C₂×C₈

#11924018

yes

—

149

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

150

C₂×C₈

D₄×D₇₃

yes

characteristic

151

C₂×C₈

Q₈×D₇₃

yes

characteristic

152

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

153

C₂×C₈

#11924027

yes

characteristic

154

C₂×C₈

#11924029

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

146

—

158

C₂²⋊C₄

—

292

—

159

C₄²

—

160

C₄²

—

292

—

161

C₄²

#11924007

yes

—

162

C₁₆

—

146

—

163

C₂³

#14556617

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

292

—

169

C₂×C₄

C₂×#11924007

yes

characteristic

170

C₂×C₄

#14556562

yes

characteristic

171

C₂×C₄

#14556567

yes

characteristic

172

C₂×C₄

C₂×#11924013

yes

characteristic

173

C₂×C₄

C₂×#11924018

yes

characteristic

174

C₂×C₄

#14556622

yes

characteristic

175

C₈

—

292

—

176

C₈

#14556524

yes

—

177

C₈

#14556524

yes

—

178

C₈

#14556553

yes

characteristic

179

C₈

#14556556

yes

characteristic

180

C₈

#14556582

yes

characteristic

181

C₈

#14556590

yes

characteristic

182

C₂²

—

183

C₂²

#17988627

yes

characteristic

184

C₂²

#17988779

yes

characteristic

185

C₂²

#17988879

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

584

—

189

C₄

#17988483

yes

characteristic

190

C₄

C₂×#14556524

yes

characteristic

191

C₄

#17988794

yes

characteristic

192

C₄

#17988864

yes

characteristic

193

C₂

—

194

C₂

—

195

C₂

#22610687

yes

characteristic

196

C₂

#22611997

yes

—

	subgroup	G/H
center	C₂²	#17988779
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1846	C₂
socle	C₂²×C₇₃	#459
prime radical	C₂³×C₇₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1846	D₇₃
hypocenter	C₇₃	#13180
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂²

#17988779

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1846

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 ◁ #1846

N	H ≅ G/N	type
12. #17988408	C₄	semidirect
104. C₇₃	#13180	semidirect

H ≅ G/N

type

12. #17988408

C₄

semidirect

104. C₇₃

#13180

semidirect

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

subgroup	index
#22611717	2
C₂×#17988408	2
#22609810	2
#22608039	2
C₇₃×#1846	2
#13180	73

subgroup

index

C₇₃×#1846

Subgroups — #33004093

Conjugacy classes of subgroups

Complements

Maximal subgroups