Group #34013354 — subgroups

Subgroups — #34013354

Order 18688.

370 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 112 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	#22626898	C₂	1	1	yes	maximal, characteristic
2	#22626274	C₂	1	2	yes	maximal, complemented
3	#22626161	C₂	1	1	yes	maximal, characteristic
4	#22624985	C₂	1	2	yes	maximal, complemented
5	#22609733	C₂	1	1	yes	maximal, characteristic
6	#17989645	—	2	4	no	—
7	C₂×#14556665	—	2	2	no	—
8	C₂×#14556665	C₂²	1	1	yes	characteristic
9	C₂×#14556665	C₂²	1	2	yes	—
10	C₂²×#11924031	C₂²	1	1	yes	characteristic
11	#17989510	—	2	4	no	—
12	C₂²⋊#11924031	—	2	2	no	—
13	C₂×#14556598	C₂²	1	1	yes	characteristic
14	#17988571	C₄	1	1	yes	characteristic
15	#17988543	C₂²	1	2	yes	—
16	#17988357	C₄	1	2	yes	—
17	#410×C₇₃	C₄	1	1	yes	characteristic
18	#14556665	—	2	4	no	—
19	#14556665	—	2	2	no	—
20	#14556665	—	2	2	no	—
21	#14556665	D₄	1	4	yes	—
22	C₂×#11924031	—	2	4	no	—
23	C₂×#11924031	—	2	2	no	—
24	C₂×#11924031	—	2	1	no	—
25	C₂×#11924031	—	2	1	no	—
26	C₂×#11924031	—	2	2	no	—
27	#14556598	—	2	2	no	—
28	#14556598	—	2	2	no	—
29	C₂×#11924012	C₂×C₄	1	2	yes	—
30	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
31	C₂×#11924012	Q₈	1	2	yes	—
32	C₂²×#10596831	D₄	1	2	yes	—
33	C₂²×#10596831	C₂³	1	1	yes	characteristic
34	#14556510	C₈	1	2	yes	—
35	#14556508	C₈	1	2	yes	—
36	#14556502	—	2	2	no	—
37	C₄×#10596831	—	2	2	no	—
38	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
39	#117×C₇₃	C₂×C₄	1	2	yes	—
40	#99×C₇₃	—	2	2	no	—
41	#11924031	—	2	4	no	—
42	#11924031	—	4	1	no	—
43	#11924031	—	4	1	no	—
44	#11924012	—	2	2	no	—
45	#11924012	—	2	2	no	—
46	#11924012	—	2	4	no	—
47	#11924012	C₂×C₈	1	4	yes	—
48	C₂×#10596831	—	2	2	no	—
49	C₂×#10596831	—	2	2	no	—
50	C₂×#10596831	—	2	2	no	—
51	C₂×#10596831	C₄⋊C₄	1	2	yes	—
52	C₂×#10596831	C₂×D₄	1	2	yes	—
53	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
54	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
55	C₂×#10596831	#41	1	2	yes	—
56	#41×C₇₃	—	2	2	no	—
57	#41×C₇₃	—	2	2	no	—
58	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
59	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
60	C₂×D₄×C₇₃	M₁₆	1	2	yes	—
61	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₄⋊C₄×C₇₃	—	2	2	no	—
64	C₂²⋊C₄×C₇₃	—	2	2	no	—
65	C₄²×C₇₃	—	2	2	no	—
66	C₂³×C₇₃	C₄⋊C₈	1	2	yes	—
67	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
68	Q₈×C₇₃	—	2	2	no	—
69	D₄×C₇₃	—	2	4	no	—
70	D₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	—	4	2	no	—
74	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#131	1	2	yes	—
79	#10596831	—	2	2	no	—
80	#10596831	—	2	4	no	—
81	#10596831	#120	1	1	yes	characteristic
82	#10596831	C₂²⋊Q₈	1	2	yes	—
83	#10596831	#123	1	1	yes	characteristic
84	C₂²×C₇₃	—	2	2	no	—
85	C₂²×C₇₃	—	4	1	no	—
86	C₂²×C₇₃	—	4	2	no	—
87	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₇₃	#435	1	2	yes	—
89	C₄×C₇₃	—	2	2	no	—
90	C₄×C₇₃	—	8	2	no	—
91	C₄×C₇₃	#385	1	1	yes	characteristic
92	C₄×C₇₃	#424	1	1	yes	characteristic
93	C₄×C₇₃	C₈×D₄	1	2	yes	—
94	#12916	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	—	2	2	no	—
96	C₂×C₇₃	—	2	1	no	—
97	C₂×C₇₃	—	4	2	no	—
98	C₂×C₇₃	#1850	1	1	yes	characteristic
99	#3407	—	73	1	no	—
100	#2153	—	73	2	no	—
101	#2113	—	73	1	no	—
102	#1757	—	73	1	no	—
103	#1379	—	73	2	no	—
104	C₇₃	#12916	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	—	73	1	no	—
106	#504	—	146	4	no	—
107	C₂×M₃₂	—	73	2	no	—
108	C₂×M₃₂	—	73	1	no	—
109	C₂×M₃₂	—	146	2	no	—
110	C₂²×C₁₆	—	73	1	no	—
111	#413	—	73	1	no	—
112	#410	#64408	1	1	yes	characteristic
113	#405	—	73	2	no	—
114	#364	—	146	4	no	—
115	C₂²⋊C₁₆	—	146	2	no	—
116	#344	—	73	2	no	—
117	C₂×#41	C₂×#64408	1	1	yes	characteristic
118	#131	—	146	2	no	—
119	#131	—	146	2	no	—
120	C₂×M₁₆	—	73	2	no	—
121	C₂×M₁₆	—	73	1	no	—
122	C₂×M₁₆	—	73	2	no	—
123	C₂²×C₈	—	73	1	no	—
124	C₂²×C₈	—	73	2	no	—
125	#117	C₂×#64408	1	2	yes	—
126	M₃₂	—	73	4	no	—
127	M₃₂	—	146	2	no	—
128	M₃₂	—	146	2	no	—
129	M₃₂	—	146	4	no	—
130	C₂×C₁₆	—	146	1	no	—
131	C₂×C₁₆	—	146	1	no	—
132	C₂×C₁₆	—	146	2	no	—
133	C₂×C₁₆	—	146	2	no	—
134	C₂×C₁₆	—	146	4	no	—
135	#101	—	73	2	no	—
136	#100	—	73	2	no	—
137	#99	—	2	2	no	—
138	#97	—	146	2	no	—
139	C₄×C₈	—	146	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	#11924036	1	1	yes	characteristic
143	C₂×D₄	#11924036	1	1	yes	characteristic
144	C₂×D₄	#11924037	1	2	yes	—
145	C₂²×C₄	C₄⋊#64408	1	2	yes	—
146	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
147	M₁₆	—	73	4	no	—
148	M₁₆	—	146	4	no	—
149	M₁₆	—	146	2	no	—
150	M₁₆	—	146	2	no	—
151	C₂×C₈	—	73	1	no	—
152	C₂×C₈	—	73	1	no	—
153	C₂×C₈	—	73	2	no	—
154	C₂×C₈	—	73	2	no	—
155	C₂×C₈	—	73	2	no	—
156	C₂×C₈	—	146	2	no	—
157	C₂×C₈	—	146	2	no	—
158	C₂×C₈	—	146	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₁₆	—	292	1	no	—
163	C₁₆	—	292	1	no	—
164	C₁₆	—	146	4	no	—
165	C₂³	#14556658	1	2	yes	—
166	C₂³	C₂×#11924037	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	#14556658	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
175	C₂×C₄	D₄×#64408	1	2	yes	—
176	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
177	C₂×C₄	#14556678	1	2	yes	—
178	C₈	—	73	1	no	—
179	C₈	—	73	1	no	—
180	C₈	—	73	2	no	—
181	C₈	—	146	2	no	—
182	C₈	—	146	4	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×#14556658	1	1	yes	characteristic
187	C₂²	#17989618	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	8	2	no	—
190	C₄	#17989534	1	1	yes	characteristic
191	C₄	#17989593	1	1	yes	characteristic
192	C₄	#17989614	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#22625875	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626898

C₂

yes

maximal, characteristic

#22626274

C₂

yes

maximal, complemented

#22626161

C₂

yes

maximal, characteristic

#22624985

C₂

yes

maximal, complemented

#22609733

C₂

yes

maximal, characteristic

#17989645

—

C₂×#14556665

—

C₂×#14556665

C₂²

yes

characteristic

C₂×#14556665

C₂²

yes

—

C₂²×#11924031

C₂²

yes

—

—

C₂²

yes

characteristic

#17988571

C₄

yes

characteristic

#17988543

C₂²

yes

—

#17988357

C₄

yes

—

#410×C₇₃

C₄

yes

—

—

—

D₄

yes

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11924012

C₂×C₄

yes

characteristic

C₂×#11924012

Q₈

yes

—

C₂²×#10596831

D₄

yes

—

C₂²×#10596831

C₂³

yes

characteristic

#14556510

C₈

yes

—

#14556508

C₈

yes

—

#14556502

—

C₄×#10596831

—

C₂×#41×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

C₂×C₄

yes

—

#99×C₇₃

—

—

—

—

—

—

—

C₂×C₈

yes

—

—

—

—

C₄⋊C₄

yes

—

C₂×#10596831

C₂×D₄

yes

—

C₂×#10596831

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₂×#10596831

#41

yes

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₈

yes

characteristic

C₂×D₄×C₇₃

C₂×C₈

yes

characteristic

C₂×D₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₄⋊C₈

yes

—

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

#131

yes

—

—

—

yes

C₂²⋊Q₈

yes

—

#10596831

#123

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₃

#435

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₈×D₄

yes

—

#12916

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1850

yes

characteristic

#3407

—

100

#2153

—

101

#2113

—

102

#1757

—

103

#1379

—

104

C₇₃

#12916

yes

characteristic, complemented, Sylow

105

C₂×#131

—

106

#504

—

146

—

107

C₂×M₃₂

—

108

C₂×M₃₂

—

109

C₂×M₃₂

—

146

—

110

C₂²×C₁₆

—

111

#413

—

112

yes

113

—

114

#364

—

146

—

115

C₂²⋊C₁₆

—

146

—

116

#344

—

117

C₂×#41

C₂×#64408

yes

characteristic

118

#131

—

146

—

119

#131

—

146

—

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

#117

C₂×#64408

yes

—

126

M₃₂

—

127

M₃₂

—

146

—

128

M₃₂

—

146

—

129

M₃₂

—

146

—

130

C₂×C₁₆

—

146

—

131

C₂×C₁₆

—

146

—

132

C₂×C₁₆

—

146

—

133

C₂×C₁₆

—

146

—

134

C₂×C₁₆

—

146

—

135

#101

—

136

#100

—

137

#99

—

138

#97

—

146

—

139

C₄×C₈

—

146

—

140

#41

—

141

#41

—

142

C₂×Q₈

#11924036

yes

characteristic

143

C₂×D₄

#11924036

yes

characteristic

144

C₂×D₄

#11924037

yes

—

145

C₂²×C₄

C₄⋊#64408

yes

—

146

C₂²×C₄

C₂²×#64408

yes

characteristic

147

M₁₆

—

148

M₁₆

—

146

—

149

M₁₆

—

146

—

150

M₁₆

—

146

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

146

—

157

C₂×C₈

—

146

—

158

C₂×C₈

—

146

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

—

162

C₁₆

—

292

—

163

C₁₆

—

292

—

164

C₁₆

—

146

—

165

C₂³

#14556658

yes

—

166

C₂³

C₂×#11924037

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#14556658

yes

—

174

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

175

C₂×C₄

D₄×#64408

yes

—

176

C₂×C₄

C₂×#11924036

yes

characteristic

177

C₂×C₄

#14556678

yes

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

146

—

182

C₈

—

146

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×#14556658

yes

characteristic

187

C₂²

#17989618

yes

—

188

C₄

—

189

C₄

—

190

C₄

#17989534

yes

characteristic

191

C₄

#17989593

yes

characteristic

192

C₄

#17989614

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#22625875

yes

characteristic

	subgroup	G/H
center	C₄	#17989534
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#410×C₇₃	C₄
socle	C₂×C₇₃	#1850
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₂²×#10596831	C₂³
hypercenter	#410	#64408
hypocenter	C₇₃	#12916
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#17989534

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#410×C₇₃

C₄

socle

C₂×C₇₃

#1850

prime radical

C₂×#41×C₇₃

C₂×C₄

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #22626274	C₂	semidirect
4. #22624985	C₂	semidirect
104. C₇₃	#12916	semidirect

H ≅ G/N

type

2. #22626274

C₂

semidirect

4. #22624985

C₂

semidirect

104. C₇₃

#12916

semidirect

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

subgroup	index
#22626898	2
#22626274	2
#22626161	2
#22624985	2
#22609733	2
#12916	73

subgroup

index

Subgroups — #34013354

Conjugacy classes of subgroups

Complements

Maximal subgroups