Group #34012456 — subgroups

Subgroups — #34012456

Order 18688.

382 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 124 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×#17989645	C₂	1	1	yes	maximal, characteristic
2	#22626167	C₂	1	1	yes	maximal, characteristic
3	#22626154	C₂	1	2	yes	maximal, complemented
4	#22624986	C₂	1	2	yes	maximal, complemented
5	#22609734	C₂	1	1	yes	maximal, characteristic
6	#17989645	—	2	4	no	—
7	C₂×#14556665	C₄	1	2	yes	complemented
8	C₂×#14556665	C₂²	1	2	yes	—
9	C₂×#14556665	C₂²	1	1	yes	characteristic
10	C₂²×#11924031	C₄	1	2	yes	complemented
11	C₂²×#11924031	C₂²	1	1	yes	characteristic
12	#17989483	C₄	1	4	yes	complemented
13	#17989480	—	2	4	no	—
14	C₂×#14556598	C₂²	1	1	yes	characteristic
15	#17988561	C₄	1	1	yes	characteristic
16	#17988545	C₂²	1	2	yes	—
17	#17988359	C₄	1	2	yes	—
18	#413×C₇₃	C₄	1	1	yes	characteristic
19	#14556665	—	2	4	no	—
20	#14556665	—	2	2	no	—
21	#14556665	C₂×C₄	1	4	yes	—
22	#14556665	D₄	1	4	yes	—
23	C₂×#11924031	—	2	4	no	—
24	C₂×#11924031	—	2	2	no	—
25	C₂×#11924031	—	2	2	no	—
26	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
27	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
28	C₂×#11924031	C₂×C₄	1	2	yes	—
29	#14556598	—	2	2	no	—
30	#14556598	—	2	2	no	—
31	C₂×#11924012	C₂×C₄	1	2	yes	—
32	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
33	C₂²×#10596831	C₂×C₄	1	2	yes	—
34	C₂²×#10596831	C₂³	1	1	yes	characteristic
35	#14556550	C₂×C₄	1	2	yes	—
36	#14556506	—	2	2	no	—
37	#14556501	—	2	2	no	—
38	C₈×Dic₇₃	—	2	2	no	—
39	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
41	#101×C₇₃	—	2	1	no	—
42	#100×C₇₃	—	2	1	no	—
43	#11924031	—	2	1	no	—
44	#11924031	—	2	1	no	—
45	#11924031	—	2	4	no	—
46	#11924031	C₂²⋊C₄	1	4	yes	—
47	#11924012	—	2	2	no	—
48	#11924012	—	2	4	no	—
49	C₂×#10596831	—	2	2	no	—
50	C₂×#10596831	—	2	2	no	—
51	C₂×#10596831	C₄²	1	2	yes	—
52	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
53	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
54	C₂×#10596831	C₂×D₄	1	2	yes	—
55	C₂×#10596831	#41	1	2	yes	—
56	C₄⋊Dic₇₃	—	2	2	no	—
57	C₂²⋊Dic₇₃	—	2	2	no	—
58	C₄×Dic₇₃	—	2	2	no	—
59	#41×C₇₃	—	2	2	no	—
60	#41×C₇₃	—	2	2	no	—
61	C₂×Q₈×C₇₃	M₁₆	1	1	yes	characteristic
62	C₂×D₄×C₇₃	M₁₆	1	1	yes	characteristic
63	C₂×D₄×C₇₃	M₁₆	1	2	yes	—
64	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
65	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	M₁₆×C₇₃	—	2	2	no	—
67	M₁₆×C₇₃	—	2	2	no	—
68	C₂×C₈×C₇₃	—	2	2	no	—
69	C₂³×C₇₃	#97	1	2	yes	—
70	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₇₃	—	2	2	no	—
72	D₄×C₇₃	—	2	4	no	—
73	D₄×C₇₃	—	2	2	no	—
74	C₂×C₄×C₇₃	—	2	2	no	—
75	C₂×C₄×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	#97	1	2	yes	—
77	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#131	1	2	yes	—
81	C₂×Dic₇₃	—	4	2	no	—
82	C₈×C₇₃	—	4	2	no	—
83	#10596831	—	2	2	no	—
84	#10596831	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	#10596831	#117	1	1	yes	characteristic
86	#10596831	C₄×D₄	1	2	yes	—
87	C₂²×C₇₃	—	2	2	no	—
88	C₂²×C₇₃	—	4	1	no	—
89	C₂²×C₇₃	—	4	2	no	—
90	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
91	C₂²×C₇₃	#435	1	2	yes	—
92	C₄×C₇₃	—	2	2	no	—
93	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₇₃	#405	1	1	yes	characteristic
95	C₄×C₇₃	#435	1	2	yes	—
96	Dic₇₃	—	8	2	no	—
97	#12617	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	—	2	2	no	—
99	C₂×C₇₃	—	2	1	no	—
100	C₂×C₇₃	—	4	2	no	—
101	C₂×C₇₃	#1752	1	1	yes	characteristic
102	C₂×#504	—	73	1	no	—
103	#2115	—	73	1	no	—
104	#2110	—	73	2	no	—
105	#1757	—	73	1	no	—
106	#1379	—	73	2	no	—
107	C₇₃	#12617	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	—	73	1	no	—
109	#504	—	146	4	no	—
110	C₂×M₃₂	—	73	1	no	—
111	C₂×M₃₂	—	73	2	no	—
112	C₂×M₃₂	—	73	2	no	—
113	C₂²×C₁₆	—	73	2	no	—
114	C₂²×C₁₆	—	73	1	no	—
115	#413	#64408	1	1	yes	characteristic
116	#410	—	73	1	no	—
117	#405	—	73	2	no	—
118	#349	—	73	4	no	—
119	#347	—	146	4	no	—
120	#344	—	73	2	no	—
121	C₂×#41	C₂×#64408	1	1	yes	characteristic
122	#131	—	146	2	no	—
123	#131	—	146	2	no	—
124	C₂×M₁₆	—	73	2	no	—
125	C₂×M₁₆	—	73	1	no	—
126	C₂×M₁₆	C₂×#64408	1	2	yes	—
127	C₂²×C₈	—	73	1	no	—
128	C₂²×C₈	—	73	2	no	—
129	#117	—	73	2	no	—
130	M₃₂	—	73	4	no	—
131	M₃₂	—	73	4	no	—
132	M₃₂	—	146	4	no	—
133	M₃₂	—	146	2	no	—
134	C₂×C₁₆	—	73	2	no	—
135	C₂×C₁₆	—	73	1	no	—
136	C₂×C₁₆	—	73	1	no	—
137	C₂×C₁₆	—	146	4	no	—
138	C₂×C₁₆	—	146	2	no	—
139	C₂×C₁₆	—	146	2	no	—
140	#101	—	2	1	no	—
141	#100	—	2	1	no	—
142	#99	—	146	2	no	—
143	#97	—	146	2	no	—
144	C₄×C₈	—	146	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	#11924037	1	1	yes	characteristic
148	C₂×D₄	#11924037	1	1	yes	characteristic
149	C₂×D₄	#11924038	1	2	yes	—
150	C₂²×C₄	C₄×#64408	1	2	yes	—
151	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	2	no	—
154	M₁₆	—	146	4	no	—
155	M₁₆	—	146	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	73	1	no	—
158	C₂×C₈	—	73	1	no	—
159	C₂×C₈	—	73	2	no	—
160	C₂×C₈	—	73	2	no	—
161	C₂×C₈	—	73	2	no	—
162	C₂×C₈	—	146	2	no	—
163	C₂×C₈	—	146	2	no	—
164	C₄⋊C₄	—	146	2	no	—
165	C₂²⋊C₄	—	146	2	no	—
166	C₄²	—	146	2	no	—
167	C₁₆	—	73	4	no	—
168	C₁₆	—	146	1	no	—
169	C₁₆	—	146	1	no	—
170	C₁₆	—	146	4	no	—
171	C₂³	#14556646	1	2	yes	—
172	C₂³	C₂×#11924038	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	292	2	no	—
179	C₂×C₄	#14556646	1	2	yes	—
180	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
181	C₂×C₄	D₄×#64408	1	2	yes	—
182	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
183	C₂×C₄	#14556678	1	2	yes	—
184	C₈	—	4	2	no	—
185	C₈	—	73	1	no	—
186	C₈	—	73	2	no	—
187	C₈	—	73	1	no	—
188	C₈	—	146	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×#14556646	1	1	yes	characteristic
193	C₂²	#17989616	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	584	2	no	—
196	C₄	C₂²⋊C₄×#64408	1	1	yes	characteristic
197	C₄	#17989551	1	1	yes	characteristic
198	C₄	#17989618	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	#22625668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989645

C₂

yes

maximal, characteristic

#22626167

C₂

yes

maximal, characteristic

#22626154

C₂

yes

maximal, complemented

#22624986

C₂

yes

maximal, complemented

#22609734

C₂

yes

maximal, characteristic

#17989645

—

C₂×#14556665

C₄

yes

complemented

C₂×#14556665

C₂²

yes

—

C₂×#14556665

C₂²

yes

characteristic

C₂²×#11924031

C₄

yes

complemented

C₂²×#11924031

C₂²

yes

characteristic

#17989483

C₄

yes

complemented

#17989480

—

C₂×#14556598

C₂²

yes

characteristic

#17988561

C₄

yes

characteristic

#17988545

C₂²

yes

—

#17988359

C₄

yes

—

#413×C₇₃

C₄

yes

—

—

C₂×C₄

yes

—

#14556665

D₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

—

#14556598

—

#14556598

—

C₂×#11924012

C₂×C₄

yes

—

C₂×#11924012

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

—

C₂²×#10596831

C₂³

yes

characteristic

#14556550

C₂×C₄

yes

—

#14556506

—

#14556501

—

C₈×Dic₇₃

—

C₂×#41×C₇₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₃

C₂×C₄

yes

—

#101×C₇₃

—

#100×C₇₃

—

—

—

—

C₂²⋊C₄

yes

—

—

—

—

—

C₄²

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂×D₄

yes

—

C₂×#10596831

#41

yes

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

M₁₆

yes

characteristic

C₂×D₄×C₇₃

M₁₆

yes

characteristic

C₂×D₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₄²

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

#97

yes

—

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#97

yes

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

#131

yes

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

C₂×C₂²⋊C₄

yes

yes

C₄×D₄

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#97

yes

characteristic

C₂²×C₇₃

#435

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₄×C₂²⋊C₄

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12617

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

100

C₂×C₇₃

—

101

C₂×C₇₃

#1752

yes

characteristic

102

C₂×#504

—

103

#2115

—

104

#2110

—

105

#1757

—

106

#1379

—

107

C₇₃

#12617

yes

characteristic, complemented, Sylow

108

C₂×#131

—

109

#504

—

146

—

110

C₂×M₃₂

—

111

C₂×M₃₂

—

112

C₂×M₃₂

—

113

C₂²×C₁₆

—

114

C₂²×C₁₆

—

115

yes

116

—

117

#405

—

118

#349

—

119

#347

—

146

—

120

#344

—

121

C₂×#41

C₂×#64408

yes

characteristic

122

#131

—

146

—

123

#131

—

146

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂×M₁₆

C₂×#64408

yes

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

#117

—

130

M₃₂

—

131

M₃₂

—

132

M₃₂

—

146

—

133

M₃₂

—

146

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

146

—

138

C₂×C₁₆

—

146

—

139

C₂×C₁₆

—

146

—

140

#101

—

141

#100

—

142

#99

—

146

—

143

#97

—

146

—

144

C₄×C₈

—

146

—

145

#41

—

146

#41

—

147

C₂×Q₈

#11924037

yes

characteristic

148

C₂×D₄

#11924037

yes

characteristic

149

C₂×D₄

#11924038

yes

—

150

C₂²×C₄

C₄×#64408

yes

—

151

C₂²×C₄

C₂²×#64408

yes

characteristic

152

M₁₆

—

153

M₁₆

—

154

M₁₆

—

146

—

155

M₁₆

—

146

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

146

—

163

C₂×C₈

—

146

—

164

C₄⋊C₄

—

146

—

165

C₂²⋊C₄

—

146

—

166

C₄²

—

146

—

167

C₁₆

—

168

C₁₆

—

146

—

169

C₁₆

—

146

—

170

C₁₆

—

146

—

171

C₂³

#14556646

yes

—

172

C₂³

C₂×#11924038

yes

characteristic

173

Q₈

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

292

—

179

C₂×C₄

#14556646

yes

—

180

C₂×C₄

C₂×C₄×#64408

yes

characteristic

181

C₂×C₄

D₄×#64408

yes

—

182

C₂×C₄

C₂×#11924037

yes

characteristic

183

C₂×C₄

#14556678

yes

—

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

188

C₈

—

146

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

C₂×#14556646

yes

characteristic

193

C₂²

#17989616

yes

—

194

C₄

—

195

C₄

—

584

—

196

C₄

C₂²⋊C₄×#64408

yes

characteristic

197

C₄

#17989551

yes

characteristic

198

C₄

#17989618

yes

—

199

C₂

—

200

C₂

—

201

C₂

—

202

C₂

#22625668

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#413×C₇₃	C₄
socle	C₂×C₇₃	#1752
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₂²×#10596831	C₂³
hypercenter	#413	#64408
hypocenter	C₇₃	#12617
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

C₂²⋊C₄×#64408

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#413×C₇₃

C₄

socle

C₂×C₇₃

#1752

prime radical

C₂×#41×C₇₃

C₂×C₄

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. #22626154	C₂	semidirect
4. #22624986	C₂	semidirect
7. C₂×#14556665	C₄	semidirect
10. C₂²×#11924031	C₄	semidirect
12. #17989483	C₄	semidirect
107. C₇₃	#12617	semidirect

H ≅ G/N

type

3. #22626154

C₂

semidirect

4. #22624986

C₂

semidirect

7. C₂×#14556665

C₄

semidirect

10. C₂²×#11924031

C₄

semidirect

12. #17989483

C₄

semidirect

107. C₇₃

#12617

semidirect

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

subgroup	index
C₂×#17989645	2
#22626167	2
#22626154	2
#22624986	2
#22609734	2
#12617	73

subgroup

index

Subgroups — #34012456

Conjugacy classes of subgroups

Complements

Maximal subgroups