Group #34011713 — subgroups

Subgroups — #34011713

Order 18688.

286 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 128 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	#22626261	C₂	1	2	yes	maximal
2	#22624954	C₂	1	2	yes	maximal, complemented
3	#22624953	C₂	1	2	yes	maximal, complemented
4	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556664	—	2	2	no	—
6	#17989508	—	2	2	no	—
7	C₄⋊#11924031	C₂²	1	4	yes	—
8	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
9	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
10	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	C₈×#10596831	C₄	1	1	yes	characteristic
12	C₈×#10596831	C₂²	1	1	yes	characteristic
13	C₈×#10596831	C₂²	1	1	yes	characteristic
14	C₈²×C₇₃	C₄	1	1	yes	characteristic
15	#14556664	—	4	4	no	—
16	C₂×#11924031	—	2	4	no	—
17	C₂×C₈×D₇₃	—	2	1	no	—
18	C₂×C₈×D₇₃	C₈	1	1	yes	characteristic
19	C₂×C₈×D₇₃	C₈	1	1	yes	characteristic
20	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
21	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
22	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596831	C₂³	1	1	yes	characteristic
26	C₈×Dic₇₃	—	2	1	no	—
27	C₈×Dic₇₃	C₈	1	1	yes	characteristic
28	C₈×Dic₇₃	C₈	1	1	yes	characteristic
29	C₈×Dic₇₃	D₄	1	1	yes	characteristic
30	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	#11924031	—	4	4	no	—
35	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
38	C₂×#10596831	—	2	1	no	—
39	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
40	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
41	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
42	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
43	C₈×D₇₃	—	2	4	no	—
44	C₈×D₇₃	—	2	2	no	—
45	C₈×D₇₃	—	2	2	no	—
46	C₈×D₇₃	C₁₆	1	4	yes	complemented
47	C₈×D₇₃	SD₁₆	1	4	yes	—
48	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
50	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₇₃	—	2	1	no	—
52	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
61	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
64	C₄×D₇₃	—	2	2	no	—
65	C₄×D₇₃	#106	1	2	yes	—
66	C₄×D₇₃	#108	1	2	yes	—
67	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
68	C₄×D₇₃	M₃₂	1	2	yes	—
69	C₈×C₇₃	—	2	2	no	—
70	C₈×C₇₃	—	2	1	no	—
71	C₈×C₇₃	—	2	1	no	—
72	C₈×C₇₃	#106	1	2	yes	—
73	C₈×C₇₃	C₂×C₁₆	1	2	yes	complemented
74	#10596831	—	2	2	no	—
75	#10596831	—	2	1	no	—
76	#10596831	—	2	1	no	—
77	#10596831	C₂×C₁₆	1	2	yes	complemented
78	#10596831	C₂×SD₁₆	1	2	yes	—
79	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₂×D₇₃	#334	1	2	yes	—
81	C₂×D₇₃	C₄⋊C₁₆	1	2	yes	—
82	C₂×D₇₃	#364	1	2	yes	—
83	C₄×C₇₃	—	2	1	no	—
84	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
85	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
86	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
87	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
88	Dic₇₃	—	2	1	no	—
89	Dic₇₃	#334	1	2	yes	—
90	Dic₇₃	C₄⋊C₁₆	1	2	yes	—
91	C₂×#1367	—	73	1	no	maximal, Sylow
92	C₂×C₇₃	C₂×#334	1	1	yes	characteristic
93	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
94	C₂×C₇₃	C₂×#364	1	1	yes	characteristic
95	D₇₃	#1367	1	4	yes	complemented
96	C₂×C₄⋊C₁₆	—	73	2	no	—
97	C₂×C₈²	—	73	1	no	—
98	#1367	—	73	2	no	—
99	#1367	—	73	2	no	—
100	C₇₃	C₂×#1367	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₁₆	—	146	2	no	—
102	C₂×C₄×C₈	—	73	1	no	—
103	C₂×C₄×C₈	—	73	1	no	—
104	C₂×C₄×C₈	—	73	1	no	—
105	C₄⋊C₁₆	—	73	4	no	—
106	C₄⋊C₁₆	—	146	2	no	—
107	C₈²	—	73	1	no	—
108	C₈²	—	73	1	no	—
109	C₈²	—	73	1	no	—
110	C₈²	#64408	1	1	yes	characteristic
111	C₂²×C₈	—	73	1	no	—
112	C₂²×C₈	—	73	1	no	—
113	C₂²×C₈	—	73	1	no	—
114	C₂²×C₈	—	73	1	no	—
115	C₂²×C₈	—	146	1	no	—
116	C₂×C₄²	—	73	1	no	—
117	C₂×C₁₆	—	292	4	no	—
118	C₂×C₁₆	—	146	4	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	—	73	1	no	—
122	C₄×C₈	—	73	1	no	—
123	C₄×C₈	—	73	1	no	—
124	C₄×C₈	—	73	1	no	—
125	C₄×C₈	—	73	1	no	—
126	C₄×C₈	—	146	1	no	—
127	C₄×C₈	C₂×#64408	1	1	yes	characteristic
128	C₄×C₈	C₂×#64408	1	1	yes	characteristic
129	C₄×C₈	C₂×#64408	1	1	yes	characteristic
130	C₂²×C₄	—	73	1	no	—
131	C₂²×C₄	—	73	1	no	—
132	C₂²×C₄	—	73	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	73	1	no	—
135	C₂×C₈	—	73	1	no	—
136	C₂×C₈	—	73	4	no	—
137	C₂×C₈	—	73	4	no	—
138	C₂×C₈	—	73	1	no	—
139	C₂×C₈	—	73	1	no	—
140	C₂×C₈	—	146	2	no	—
141	C₂×C₈	—	146	2	no	—
142	C₂×C₈	—	146	4	no	—
143	C₂×C₈	—	146	1	no	—
144	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
145	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
146	C₂×C₈	#11924036	1	1	yes	characteristic
147	C₂×C₈	#11924036	1	1	yes	characteristic
148	C₄²	—	73	1	no	—
149	C₄²	—	73	1	no	—
150	C₄²	—	73	1	no	—
151	C₄²	C₂²×#64408	1	1	yes	characteristic
152	C₁₆	—	292	4	no	—
153	C₂³	—	73	1	no	—
154	C₂×C₄	—	73	1	no	—
155	C₂×C₄	—	73	1	no	—
156	C₂×C₄	—	73	2	no	—
157	C₂×C₄	—	73	2	no	—
158	C₂×C₄	—	73	2	no	—
159	C₂×C₄	—	73	2	no	—
160	C₂×C₄	—	73	1	no	—
161	C₂×C₄	—	146	2	no	—
162	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
163	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
164	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	73	2	no	—
169	C₈	—	73	2	no	—
170	C₈	—	146	2	no	—
171	C₈	—	146	1	no	—
172	C₈	—	146	1	no	—
173	C₈	#14556659	1	2	yes	—
174	C₈	#14556664	1	2	yes	complemented
175	C₂²	—	73	2	no	—
176	C₂²	—	73	2	no	—
177	C₂²	—	73	2	no	—
178	C₂²	#17989588	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	73	2	no	—
181	C₄	—	73	2	no	—
182	C₄	—	146	1	no	—
183	C₄	C₂×#14556659	1	1	yes	characteristic
184	C₄	D₇₃⋊#108	1	1	yes	characteristic
185	C₄	C₂×#14556664	1	1	yes	characteristic
186	C₄	#17989643	1	1	yes	characteristic
187	C₂	—	73	4	no	—
188	C₂	#22625473	1	1	yes	characteristic
189	C₂	#22626261	1	1	yes	characteristic
190	C₂	#22626270	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626261

C₂

yes

maximal

#22624954

C₂

yes

maximal, complemented

#22624953

C₂

yes

maximal, complemented

C₈²×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556664

—

#17989508

—

C₄⋊#11924031

C₂²

yes

—

C₄×C₈×D₇₃

C₄

yes

characteristic

C₄×C₈×D₇₃

C₄

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₈×#10596831

C₄

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

C₈²×C₇₃

C₄

yes

characteristic

#14556664

—

C₂×#11924031

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₈

yes

characteristic

C₂×C₈×D₇₃

C₈

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂³

yes

characteristic

C₈×Dic₇₃

—

C₈×Dic₇₃

C₈

yes

characteristic

C₈×Dic₇₃

C₈

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924031

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

C₁₆

yes

complemented

C₈×D₇₃

SD₁₆

yes

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

#106

yes

—

C₄×D₇₃

#108

yes

—

C₄×D₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

M₃₂

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#106

yes

—

C₈×C₇₃

C₂×C₁₆

yes

—

—

—

C₂×C₁₆

yes

complemented

#10596831

C₂×SD₁₆

yes

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂×D₇₃

#334

yes

—

C₂×D₇₃

C₄⋊C₁₆

yes

—

C₂×D₇₃

#364

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

Dic₇₃

—

Dic₇₃

#334

yes

—

Dic₇₃

C₄⋊C₁₆

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₂×C₄⋊C₁₆

yes

C₂×C₇₃

yes

D₇₃

yes

C₂×C₄⋊C₁₆

—

C₂×C₈²

—

#1367

—

#1367

—

100

C₇₃

C₂×#1367

yes

characteristic, complemented, Sylow

101

C₂²×C₁₆

—

146

—

102

C₂×C₄×C₈

—

103

C₂×C₄×C₈

—

104

C₂×C₄×C₈

—

105

C₄⋊C₁₆

—

106

C₄⋊C₁₆

—

146

—

107

C₈²

—

108

C₈²

—

109

C₈²

—

110

C₈²

#64408

yes

characteristic

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

146

—

116

C₂×C₄²

—

117

C₂×C₁₆

—

292

—

118

C₂×C₁₆

—

146

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

146

—

127

C₄×C₈

C₂×#64408

yes

characteristic

128

C₄×C₈

C₂×#64408

yes

characteristic

129

C₄×C₈

C₂×#64408

yes

characteristic

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

146

—

141

C₂×C₈

—

146

—

142

C₂×C₈

—

146

—

143

C₂×C₈

—

146

—

144

C₂×C₈

C₄⋊#64408

yes

characteristic

145

C₂×C₈

C₄⋊#64408

yes

characteristic

146

C₂×C₈

#11924036

yes

characteristic

147

C₂×C₈

#11924036

yes

characteristic

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂²×#64408

yes

characteristic

152

C₁₆

—

292

—

153

C₂³

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

146

—

162

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

163

C₂×C₄

C₂×#11924036

yes

characteristic

164

C₂×C₄

D₇₃⋊M₁₆

yes

characteristic

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₈

—

170

C₈

—

146

—

171

C₈

—

146

—

172

C₈

—

146

—

173

C₈

#14556659

yes

—

174

C₈

#14556664

yes

complemented

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

#17989588

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

146

—

183

C₄

C₂×#14556659

yes

characteristic

184

C₄

D₇₃⋊#108

yes

characteristic

185

C₄

C₂×#14556664

yes

characteristic

186

C₄

#17989643

yes

characteristic

187

C₂

—

188

C₂

#22625473

yes

characteristic

189

C₂

#22626261

yes

characteristic

190

C₂

#22626270

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂²×D₇₃	C₄⋊C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	C₂×#1367
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₈

C₄⋊#64408

commutator

C₄×C₇₃

C₂²×C₁₆

Frattini

C₄²

C₂²×#64408

Fitting

C₈²×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄⋊C₈

prime radical

C₂²×D₇₃

C₄⋊C₈

C₂³

C₈²

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
2. #22624954	C₂	semidirect
3. #22624953	C₂	semidirect
46. C₈×D₇₃	C₁₆	semidirect
73. C₈×C₇₃	C₂×C₁₆	semidirect
77. #10596831	C₂×C₁₆	semidirect
95. D₇₃	#1367	semidirect
100. C₇₃	C₂×#1367	semidirect
174. C₈	#14556664	semidirect

H ≅ G/N

type

2. #22624954

C₂

semidirect

3. #22624953

C₂

semidirect

46. C₈×D₇₃

C₁₆

semidirect

73. C₈×C₇₃

C₂×C₁₆

semidirect

77. #10596831

C₂×C₁₆

semidirect

95. D₇₃

#1367

semidirect

100. C₇₃

C₂×#1367

semidirect

174. C₈

#14556664

semidirect

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

subgroup	index
#22626261	2
#22624954	2
#22624953	2
C₈²×D₇₃	2
C₂×#1367	73

subgroup

index

C₈²×D₇₃

Subgroups — #34011713

Conjugacy classes of subgroups

Complements

Maximal subgroups