Group #34018211 — subgroups

Subgroups — #34018211

Order 18688.

612 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 232 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	#22626865	C₂	1	2	yes	maximal
2	#22625508	C₂	1	6	yes	maximal, complemented
3	#22625490	C₂	1	3	yes	maximal, complemented
4	#22625479	C₂	1	3	yes	maximal, complemented
5	C₈×#11924029	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596833	—	2	2	no	—
7	#17989621	—	2	6	no	—
8	#17989606	C₂²	1	6	yes	—
9	#17989591	C₂²	1	6	yes	—
10	#17989451	C₂²	1	1	yes	characteristic
11	#17989451	C₂²	1	3	yes	—
12	#17989451	C₂²	1	6	yes	—
13	#17989448	C₂²	1	6	yes	—
14	C₄×#11924029	C₂²	1	1	yes	characteristic
15	Q₈×#10596831	C₄	1	1	yes	characteristic
16	C₈×D₂₉₂	C₄	1	3	yes	—
17	#17988652	C₄	1	3	yes	—
18	#17988639	C₂²	1	3	yes	—
19	C₄×C₈×D₇₃	C₂²	1	3	yes	—
20	C₈×Q₈×C₇₃	C₄	1	1	yes	characteristic
21	C₂×#11924038	—	2	6	no	—
22	C₄⋊#10596833	—	2	6	no	—
23	C₄⋊#10596833	C₂³	1	2	yes	—
24	C₄⋊#10596833	C₂³	1	6	yes	—
25	#14556648	—	2	6	no	—
26	C₄×#10596833	—	2	6	no	—
27	C₂×#11924029	Q₈	1	1	yes	characteristic
28	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
29	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
30	Q₈×Dic₇₃	D₄	1	1	yes	characteristic
31	C₄×D₂₉₂	C₂×C₄	1	3	yes	—
32	#14556553	D₄	1	3	yes	—
33	#14556549	Q₈	1	3	yes	—
34	C₄²×D₇₃	C₂³	1	3	yes	—
35	C₄⋊#10596831	C₂×C₄	1	3	yes	—
36	#14556503	C₂×C₄	1	6	yes	—
37	C₄×#10596831	C₂×C₄	1	3	yes	—
38	C₈×Dic₇₃	C₂³	1	3	yes	—
39	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₇₃	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₇₃	C₂×C₄	1	3	yes	—
43	#11924038	—	8	6	no	—
44	C₂×#10596833	—	2	6	no	—
45	C₂×#10596833	—	2	2	no	—
46	#11924029	—	2	4	no	—
47	C₂×D₂₉₂	C₄⋊C₄	1	3	yes	—
48	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
49	C₂×C₄×D₇₃	C₂×D₄	1	3	yes	—
50	C₂×C₄×D₇₃	C₂×Q₈	1	3	yes	—
51	C₂×#10596831	C₂²×C₄	1	3	yes	—
52	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
53	C₈×D₇₃	—	2	6	no	—
54	C₈×D₇₃	—	4	3	no	—
55	C₄⋊Dic₇₃	C₄⋊C₄	1	3	yes	—
56	#11924005	C₄⋊C₄	1	6	yes	—
57	C₄×Dic₇₃	C₂×D₄	1	3	yes	—
58	C₄×Dic₇₃	C₂×Q₈	1	3	yes	—
59	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
60	C₂×Q₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
63	C₄⋊C₄×C₇₃	C₄⋊C₄	1	3	yes	—
64	C₄²×C₇₃	C₂²×C₄	1	3	yes	—
65	C₂²×D₇₃	C₂×C₄⋊C₄	1	3	yes	—
66	Q₈×C₇₃	#108	1	4	yes	—
67	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
70	C₂×Dic₇₃	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
72	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
73	D₂₉₂	—	2	6	no	—
74	C₄×D₇₃	—	2	3	no	—
75	C₄×D₇₃	—	2	3	no	—
76	C₄×D₇₃	—	2	6	no	—
77	C₄×D₇₃	—	2	3	no	—
78	C₄×D₇₃	—	2	3	no	—
79	#10596833	—	4	6	no	—
80	#10596833	—	4	2	no	—
81	C₈×C₇₃	—	2	3	no	—
82	C₈×C₇₃	—	2	1	no	—
83	#10596831	—	2	3	no	—
84	#10596831	#131	1	2	yes	—
85	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₂×D₇₃	—	2	3	no	—
87	C₂×D₇₃	—	2	3	no	—
88	C₂×D₇₃	—	2	3	no	—
89	C₄×C₇₃	—	2	3	no	—
90	C₄×C₇₃	C₂×#108	1	6	yes	—
91	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
92	C₄×C₇₃	#568	1	1	yes	characteristic
93	Dic₇₃	—	2	3	no	—
94	Dic₇₃	—	2	1	no	—
95	Dic₇₃	#576	1	1	yes	characteristic
96	Dic₇₃	#578	1	1	yes	characteristic
97	#17837	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	#2906	1	1	yes	characteristic
99	C₂×C₇₃	#2913	1	1	yes	characteristic
100	C₂×C₇₃	C₂²×#108	1	1	yes	characteristic
101	D₇₃	—	4	3	no	—
102	#2991	—	73	2	no	—
103	C₈×#41	—	73	1	no	—
104	#1574	—	73	6	no	—
105	#1567	—	73	3	no	—
106	#1564	—	73	3	no	—
107	C₇₃	#17837	1	1	yes	characteristic, complemented, Sylow
108	C₄×#41	—	73	1	no	—
109	C₈×Q₈	—	73	1	no	—
110	C₈×Q₈	—	146	2	no	—
111	C₈×Q₈	#64408	1	1	yes	characteristic
112	#436	—	146	6	no	—
113	C₈×D₄	—	73	3	no	—
114	C₈×D₄	—	73	3	no	—
115	#431	—	73	3	no	—
116	#431	—	73	6	no	—
117	#423	—	73	6	no	—
118	C₂×C₄×C₈	—	73	3	no	—
119	#335	—	73	1	no	—
120	#335	—	73	3	no	—
121	#335	—	73	6	no	—
122	#334	—	73	6	no	—
123	C₂×#41	—	73	1	no	—
124	C₂×M₁₆	—	146	6	no	—
125	C₂²×C₈	—	73	3	no	—
126	C₂²×C₈	—	73	3	no	—
127	C₄×Q₈	—	73	1	no	—
128	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
129	C₄×D₄	—	73	3	no	—
130	C₄×D₄	—	73	3	no	—
131	#117	—	73	3	no	—
132	C₂×C₄²	—	73	3	no	—
133	C₄⋊C₈	—	73	2	no	—
134	C₄⋊C₈	—	73	6	no	—
135	C₄⋊C₈	—	73	3	no	—
136	C₄⋊C₈	—	146	6	no	—
137	C₄⋊C₈	C₂×#64408	1	3	yes	—
138	C₂²⋊C₈	—	73	6	no	—
139	C₂²⋊C₈	—	146	6	no	—
140	C₄×C₈	—	73	3	no	—
141	C₄×C₈	—	73	1	no	—
142	C₄×C₈	—	73	3	no	—
143	C₄×C₈	—	146	6	no	—
144	C₄×C₈	C₂×#64408	1	3	yes	—
145	#41	—	146	4	no	—
146	C₂×Q₈	C₄⋊#64408	1	1	yes	characteristic
147	C₂×D₄	—	73	3	no	—
148	C₂²×C₄	—	73	3	no	—
149	C₂²×C₄	—	73	3	no	—
150	C₂²×C₄	—	73	3	no	—
151	M₁₆	—	584	6	no	—
152	C₂×C₈	—	292	3	no	—
153	C₂×C₈	—	73	1	no	—
154	C₂×C₈	—	73	3	no	—
155	C₂×C₈	—	146	6	no	—
156	C₂×C₈	—	146	2	no	—
157	C₂×C₈	—	146	6	no	—
158	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
159	C₂×C₈	C₂²×#64408	1	3	yes	—
160	C₄⋊C₄	—	73	3	no	—
161	C₄⋊C₄	C₄⋊#64408	1	3	yes	—
162	C₂²⋊C₄	—	73	6	no	—
163	C₄²	—	73	1	no	—
164	C₄²	—	73	3	no	—
165	C₄²	—	73	3	no	—
166	C₄²	C₂²×#64408	1	3	yes	—
167	C₂³	—	73	3	no	—
168	Q₈	#14556661	1	4	yes	—
169	D₄	—	146	6	no	—
170	C₂×C₄	—	73	1	no	—
171	C₂×C₄	—	73	1	no	—
172	C₂×C₄	—	73	3	no	—
173	C₂×C₄	—	146	3	no	—
174	C₂×C₄	—	146	3	no	—
175	C₂×C₄	—	146	3	no	—
176	C₂×C₄	—	146	3	no	—
177	C₂×C₄	—	146	6	no	—
178	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
179	C₂×C₄	C₂×C₄⋊#64408	1	3	yes	—
180	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
181	C₈	—	2	3	no	—
182	C₈	—	2	1	no	—
183	C₈	—	292	6	no	—
184	C₈	—	292	2	no	—
185	C₈	—	73	2	no	—
186	C₈	—	146	3	no	—
187	C₂²	—	146	3	no	—
188	C₂²	—	146	3	no	—
189	C₂²	—	146	3	no	—
190	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
191	C₄	—	2	3	no	—
192	C₄	—	73	1	no	—
193	C₄	—	73	1	no	—
194	C₄	—	146	3	no	—
195	C₄	—	146	1	no	—
196	C₄	C₂×#14556661	1	6	yes	—
197	C₄	D₇₃⋊#131	1	1	yes	characteristic
198	C₄	#17989667	1	1	yes	characteristic
199	C₂	—	292	3	no	—
200	C₂	#22626579	1	1	yes	characteristic
201	C₂	#22626586	1	1	yes	characteristic
202	C₂	C₂²×#14556661	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626865

C₂

yes

maximal

#22625508

C₂

yes

maximal, complemented

#22625490

C₂

yes

maximal, complemented

#22625479

C₂

yes

maximal, complemented

C₈×#11924029

C₂

yes

maximal, characteristic

Q₈×#10596833

—

#17989621

—

#17989606

C₂²

yes

—

#17989591

C₂²

yes

—

#17989451

C₂²

yes

characteristic

#17989451

C₂²

yes

—

#17989451

C₂²

yes

—

#17989448

C₂²

yes

—

C₄×#11924029

C₂²

yes

characteristic

Q₈×#10596831

C₄

yes

characteristic

C₈×D₂₉₂

C₄

yes

—

#17988652

C₄

yes

—

#17988639

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

—

C₈×Q₈×C₇₃

C₄

yes

—

—

C₂³

yes

—

C₄⋊#10596833

C₂³

yes

—

#14556648

—

C₄×#10596833

—

C₂×#11924029

Q₈

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

Q₈×Dic₇₃

D₄

yes

characteristic

C₄×D₂₉₂

C₂×C₄

yes

—

#14556553

D₄

yes

—

#14556549

Q₈

yes

—

C₄²×D₇₃

C₂³

yes

—

C₄⋊#10596831

C₂×C₄

yes

—

#14556503

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₈×Dic₇₃

C₂³

yes

—

C₈×Dic₇₃

C₂³

yes

characteristic

C₄×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

—

C₂×D₂₉₂

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

—

C₂×C₄×D₇₃

C₂×D₄

yes

—

C₂×C₄×D₇₃

C₂×Q₈

yes

—

C₂×#10596831

C₂²×C₄

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

C₄⋊C₄

yes

—

#11924005

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₂×D₄

yes

—

C₄×Dic₇₃

C₂×Q₈

yes

—

C₄×Dic₇₃

C₂⁴

yes

characteristic

C₂×Q₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

—

C₄⋊C₄×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

—

C₂²×D₇₃

C₂×C₄⋊C₄

yes

—

Q₈×C₇₃

#108

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂³×C₄

yes

characteristic

C₂×Dic₇₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₇₃

C₂²×D₄

yes

characteristic

C₂×Dic₇₃

C₂²×Q₈

yes

characteristic

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

#131

yes

—

C₂²×C₇₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#108

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

100

C₂×C₇₃

C₂²×#108

yes

characteristic

101

D₇₃

—

102

#2991

—

103

C₈×#41

—

104

#1574

—

105

#1567

—

106

#1564

—

107

C₇₃

#17837

yes

characteristic, complemented, Sylow

108

C₄×#41

—

109

C₈×Q₈

—

110

C₈×Q₈

—

146

—

111

C₈×Q₈

#64408

yes

characteristic

112

#436

—

146

—

113

C₈×D₄

—

114

C₈×D₄

—

115

#431

—

116

#431

—

117

#423

—

118

C₂×C₄×C₈

—

119

#335

—

120

#335

—

121

#335

—

122

#334

—

123

C₂×#41

—

124

C₂×M₁₆

—

146

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄×Q₈

—

128

C₄×Q₈

C₂×#64408

yes

characteristic

129

C₄×D₄

—

130

C₄×D₄

—

131

#117

—

132

C₂×C₄²

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

146

—

137

C₄⋊C₈

C₂×#64408

yes

—

138

C₂²⋊C₈

—

139

C₂²⋊C₈

—

146

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

146

—

144

C₄×C₈

C₂×#64408

yes

—

145

#41

—

146

—

146

C₂×Q₈

C₄⋊#64408

yes

characteristic

147

C₂×D₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

M₁₆

—

584

—

152

C₂×C₈

—

292

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

146

—

156

C₂×C₈

—

146

—

157

C₂×C₈

—

146

—

158

C₂×C₈

C₂²×#64408

yes

characteristic

159

C₂×C₈

C₂²×#64408

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

C₄⋊#64408

yes

—

162

C₂²⋊C₄

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

C₂²×#64408

yes

—

167

C₂³

—

168

Q₈

#14556661

yes

—

169

D₄

—

146

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

146

—

174

C₂×C₄

—

146

—

175

C₂×C₄

—

146

—

176

C₂×C₄

—

146

—

177

C₂×C₄

—

146

—

178

C₂×C₄

C₂×C₄⋊#64408

yes

—

179

C₂×C₄

C₂×C₄⋊#64408

yes

—

180

C₂×C₄

C₂³×#64408

yes

characteristic

181

C₈

—

182

C₈

—

183

C₈

—

292

—

184

C₈

—

292

—

185

C₈

—

186

C₈

—

146

—

187

C₂²

—

146

—

188

C₂²

—

146

—

189

C₂²

—

146

—

190

C₂²

C₂²×C₄⋊#64408

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

146

—

195

C₄

—

146

—

196

C₄

C₂×#14556661

yes

—

197

C₄

D₇₃⋊#131

yes

characteristic

198

C₄

#17989667

yes

characteristic

199

C₂

—

292

—

200

C₂

#22626579

yes

characteristic

201

C₂

#22626586

yes

characteristic

202

C₂

C₂²×#14556661

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₈×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂×#11924029	Q₈
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₈×Q₈	#64408
hypocenter	C₇₃	#17837
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×C₄⋊#64408

commutator

C₂×C₄×C₇₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×#64408

Fitting

C₈×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂²×C₄⋊C₄

prime radical

C₂×#11924029

Q₈

cosocle

C₄×Dic₇₃

C₂⁴

C₈×Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
2. #22625508	C₂	semidirect
3. #22625490	C₂	semidirect
4. #22625479	C₂	semidirect
107. C₇₃	#17837	semidirect

H ≅ G/N

type

2. #22625508

C₂

semidirect

3. #22625490

C₂

semidirect

4. #22625479

C₂

semidirect

107. C₇₃

#17837

semidirect

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

subgroup	index
#22626865	2
#22625508	2
#22625490	2
#22625479	2
C₈×#11924029	2
#17837	73

subgroup

index

Subgroups — #34018211

Conjugacy classes of subgroups

Complements

Maximal subgroups