Group #34013282 — subgroups

Subgroups — #34013282

Order 18688.

342 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 116 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	#22626265	C₂	1	2	yes	maximal
2	#22624976	C₂	1	2	yes	maximal
3	#22624975	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556665	C₂²	1	4	yes	—
6	#17989508	—	2	4	no	—
7	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
14	#14556665	—	2	4	no	—
15	#14556665	—	2	4	no	—
16	C₂×#11924031	—	2	4	no	—
17	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
18	C₂²×#10596831	C₂³	1	1	yes	characteristic
19	C₂×C₄×Dic₇₃	—	2	2	no	—
20	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×#10596831	—	2	2	no	—
23	C₈×Dic₇₃	—	2	2	no	—
24	C₈×Dic₇₃	—	2	2	no	—
25	C₈×Dic₇₃	C₈	1	2	yes	—
26	C₈×Dic₇₃	C₈	1	2	yes	—
27	C₈×Dic₇₃	D₄	1	2	yes	—
28	C₈×Dic₇₃	Q₈	1	2	yes	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	—	2	2	no	—
33	#11924031	—	4	4	no	—
34	C₂²×Dic₇₃	—	2	1	no	—
35	C₂²×Dic₇₃	M₁₆	1	2	yes	—
36	C₂×#10596831	—	2	1	no	—
37	C₂×#10596831	—	2	1	no	—
38	C₂×#10596831	C₂×C₈	1	2	yes	—
39	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
40	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
41	C₂×#10596831	#41	1	2	yes	—
42	C₂×#10596831	#41	1	2	yes	—
43	C₄×Dic₇₃	—	2	8	no	—
44	C₄×Dic₇₃	—	2	1	no	—
45	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
46	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
47	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
48	C₄×Dic₇₃	M₁₆	1	2	yes	—
49	C₂²×C₄×C₇₃	—	2	1	no	—
50	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	—	2	1	no	—
52	C₂×C₈×C₇₃	—	2	1	no	—
53	C₂×C₈×C₇₃	—	2	1	no	—
54	C₂×C₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
57	C₄²×C₇₃	—	2	1	no	—
58	C₄²×C₇₃	M₁₆	1	2	yes	—
59	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
60	C₂×C₄×C₇₃	—	2	4	no	—
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×Dic₇₃	—	2	4	no	—
69	C₂×Dic₇₃	—	2	2	no	—
70	C₂×Dic₇₃	—	2	4	no	—
71	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
72	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
73	C₈×C₇₃	—	2	2	no	—
74	C₈×C₇₃	—	2	2	no	—
75	#10596831	—	2	2	no	—
76	#10596831	#124	1	2	yes	—
77	#10596831	#125	1	2	yes	—
78	C₂²×C₇₃	—	2	2	no	—
79	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₂²×C₇₃	#432	1	1	yes	characteristic
81	C₂²×C₇₃	#432	1	1	yes	characteristic
82	C₄×C₇₃	—	2	4	no	—
83	C₄×C₇₃	#382	1	1	yes	characteristic
84	C₄×C₇₃	#423	1	1	yes	characteristic
85	C₄×C₇₃	#431	1	1	yes	characteristic
86	C₄×C₇₃	#431	1	1	yes	characteristic
87	Dic₇₃	—	2	4	no	—
88	Dic₇₃	—	2	2	no	—
89	Dic₇₃	#364	1	4	yes	—
90	#12883	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	—	2	1	no	—
92	C₂×C₇₃	—	2	1	no	—
93	C₂×C₇₃	#1831	1	1	yes	characteristic
94	C₂×C₇₃	C₂×#364	1	2	yes	—
95	#2150	—	73	2	no	—
96	C₄²×C₈	—	73	1	no	—
97	#1376	—	73	2	no	—
98	#1376	—	73	2	no	—
99	C₇₃	#12883	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₃₂	—	73	4	no	—
101	C₂×C₄×C₈	—	73	1	no	—
102	C₂×C₄×C₈	—	73	1	no	—
103	C₂×C₄×C₈	—	73	1	no	—
104	C₂×C₄×C₈	—	73	1	no	—
105	C₂×C₄×C₈	—	73	1	no	—
106	C₂×C₄×C₈	#64408	1	1	yes	characteristic
107	C₄³	—	73	1	no	—
108	C₄⋊C₁₆	—	146	4	no	—
109	C₂²×C₈	—	73	1	no	—
110	C₂²×C₈	—	73	1	no	—
111	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
112	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
113	C₂×C₄²	—	73	1	no	—
114	C₂×C₄²	—	73	1	no	—
115	C₂×C₄²	—	146	2	no	—
116	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
117	M₃₂	—	146	4	no	—
118	M₃₂	—	146	4	no	—
119	C₂×C₁₆	—	146	4	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	73	2	no	—
122	C₄×C₈	—	73	2	no	—
123	C₄×C₈	—	73	2	no	—
124	C₄×C₈	—	73	2	no	—
125	C₄×C₈	—	146	2	no	—
126	C₄×C₈	—	146	2	no	—
127	C₄×C₈	—	146	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	73	2	no	—
130	C₂²×C₄	—	146	1	no	—
131	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	1	no	—
138	C₂×C₈	—	73	2	no	—
139	C₂×C₈	—	73	2	no	—
140	C₂×C₈	—	73	2	no	—
141	C₂×C₈	—	146	1	no	—
142	C₂×C₈	—	146	1	no	—
143	C₂×C₈	C₄⋊#64408	1	2	yes	—
144	C₂×C₈	#11924036	1	2	yes	—
145	C₄²	—	2	1	no	—
146	C₄²	—	73	1	no	—
147	C₄²	—	73	1	no	—
148	C₄²	—	73	2	no	—
149	C₄²	—	73	2	no	—
150	C₄²	—	146	8	no	—
151	C₄²	—	146	1	no	—
152	C₄²	#11924037	1	2	yes	—
153	C₁₆	—	292	4	no	—
154	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	73	2	no	—
158	C₂×C₄	—	73	2	no	—
159	C₂×C₄	—	146	4	no	—
160	C₂×C₄	—	146	4	no	—
161	C₂×C₄	—	146	2	no	—
162	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
163	C₂×C₄	#14556670	1	1	yes	characteristic
164	C₂×C₄	#14556670	1	1	yes	characteristic
165	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
166	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
167	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	2	no	—
170	C₈	—	73	2	no	—
171	C₈	—	73	2	no	—
172	C₈	—	146	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	#17989588	1	1	yes	characteristic
175	C₂²	#17989607	1	1	yes	characteristic
176	C₂²	#17989610	1	1	yes	characteristic
177	C₄	—	2	4	no	—
178	C₄	—	73	4	no	—
179	C₄	—	146	4	no	—
180	C₄	—	146	2	no	—
181	C₄	#17989531	1	1	yes	characteristic
182	C₄	#17989590	1	1	yes	characteristic
183	C₄	#17989604	1	1	yes	characteristic
184	C₄	#17989605	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#22625847	1	1	yes	characteristic
188	C₂	#22626270	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22626265

C₂

yes

maximal

#22624976

C₂

yes

maximal

#22624975

C₂

yes

maximal

C₄×C₈×Dic₇₃

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

—

#17989508

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

C₄²×Dic₇₃

C₄

yes

characteristic

C₂×C₄×C₈×C₇₃

C₄

yes

—

—

—

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

characteristic

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₈×Dic₇₃

C₈

yes

—

C₈×Dic₇₃

C₈

yes

—

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

—

#11924031

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

M₁₆

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₂×#10596831

#41

yes

—

C₂×#10596831

#41

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

M₁₆

yes

—

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄⋊C₈

yes

—

C₂×Dic₇₃

C₄⋊C₈

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

#124

yes

—

#10596831

#125

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#364

yes

—

#12883

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1831

yes

characteristic

C₂×C₇₃

C₂×#364

yes

—

#2150

—

C₄²×C₈

—

#1376

—

#1376

—

C₇₃

#12883

yes

characteristic, complemented, Sylow

100

C₂×M₃₂

—

101

C₂×C₄×C₈

—

102

C₂×C₄×C₈

—

103

C₂×C₄×C₈

—

104

C₂×C₄×C₈

—

105

C₂×C₄×C₈

—

106

C₂×C₄×C₈

#64408

yes

characteristic

107

C₄³

—

108

C₄⋊C₁₆

—

146

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

C₂×#64408

yes

characteristic

112

C₂²×C₈

C₂×#64408

yes

characteristic

113

C₂×C₄²

—

114

C₂×C₄²

—

115

C₂×C₄²

—

146

—

116

C₂×C₄²

C₂×#64408

yes

characteristic

117

M₃₂

—

146

—

118

M₃₂

—

146

—

119

C₂×C₁₆

—

146

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

146

—

126

C₄×C₈

—

146

—

127

C₄×C₈

—

146

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

146

—

131

C₂²×C₄

C₂²×#64408

yes

characteristic

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

146

—

142

C₂×C₈

—

146

—

143

C₂×C₈

C₄⋊#64408

yes

—

144

C₂×C₈

#11924036

yes

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

146

—

151

C₄²

—

146

—

152

C₄²

#11924037

yes

—

153

C₁₆

—

292

—

154

C₂³

D₇₃⋊M₁₆

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

146

—

160

C₂×C₄

—

146

—

161

C₂×C₄

—

146

—

162

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

163

C₂×C₄

#14556670

yes

characteristic

164

C₂×C₄

#14556670

yes

characteristic

165

C₂×C₄

C₂×#11924036

yes

characteristic

166

C₂×C₄

C₂×#11924037

yes

characteristic

167

C₂×C₄

D₇₃⋊M₁₆

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

146

—

173

C₂²

—

174

C₂²

#17989588

yes

characteristic

175

C₂²

#17989607

yes

characteristic

176

C₂²

#17989610

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

146

—

180

C₄

—

146

—

181

C₄

#17989531

yes

characteristic

182

C₄

#17989590

yes

characteristic

183

C₄

#17989604

yes

characteristic

184

C₄

#17989605

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#22625847

yes

characteristic

188

C₂

#22626270

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#12883
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

C₂×C₄×C₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄⋊C₈

prime radical

C₂³×C₇₃

C₂×M₁₆

C₂³

C₂×C₄×C₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
99. C₇₃	#12883	semidirect

H ≅ G/N

type

99. C₇₃

#12883

semidirect

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

subgroup	index
#22626265	2
#22624976	2
#22624975	2
C₄×C₈×Dic₇₃	2
#12883	73

subgroup

index

C₄×C₈×Dic₇₃

Subgroups — #34013282

Conjugacy classes of subgroups

Complements

Maximal subgroups