Group #29578632 — subgroups

Subgroups — #29578632

Order 15616.

294 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₄×#14519713	C₂	1	1	yes	maximal, characteristic
2	#21444569	C₂	1	2	yes	maximal
3	#21444111	C₂	1	1	yes	maximal, characteristic
4	#21443669	C₂	1	2	yes	maximal
5	C₂×C₆₁×#334	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	1	yes	characteristic
7	#16839815	C₂²	1	1	yes	characteristic
8	C₂×#14519713	C₄	1	2	yes	—
9	C₄×#11758194	C₂²	1	2	yes	—
10	#16839527	C₄	1	1	yes	characteristic
11	#16839527	C₄	1	1	yes	characteristic
12	#16839514	—	2	2	no	—
13	#16839512	—	2	2	no	—
14	#16839484	—	2	2	no	—
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#334	—	2	2	no	—
18	C₂×#11758194	C₂×C₄	1	2	yes	—
19	C₂×#11758194	C₂×C₄	1	2	yes	—
20	#14519713	—	2	4	no	—
21	C₄⋊#92418	—	2	1	no	—
22	C₄⋊#92418	D₄	1	2	yes	—
23	C₂²⋊#92418	—	2	1	no	—
24	C₂²⋊#92418	Q₈	1	2	yes	—
25	#14519688	—	2	2	no	—
26	C₄×#92418	—	2	2	no	—
27	C₄×#92418	—	2	1	no	—
28	C₄×#92418	—	2	1	no	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₁	—	2	2	no	—
34	C₄⋊C₈×C₆₁	—	2	2	no	—
35	C₄×C₈×C₆₁	D₄	1	2	yes	—
36	C₄×C₈×C₆₁	Q₈	1	2	yes	—
37	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
40	C₂×C₈×C₆₁	—	2	2	no	—
41	C₂×C₈×C₆₁	—	2	2	no	—
42	C₂×C₈×C₆₁	—	4	2	no	—
43	C₂×C₈×C₆₁	—	8	2	no	—
44	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
46	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₆₁	#41	1	1	yes	characteristic
50	#11758194	—	2	2	no	—
51	#11758194	—	2	2	no	—
52	#11758194	—	2	2	no	—
53	#11758194	M₁₆	1	4	yes	—
54	C₂×#92418	—	2	2	no	—
55	C₂×#92418	—	2	2	no	—
56	C₂×#92418	—	2	2	no	—
57	C₂×#92418	—	2	2	no	—
58	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	4	1	no	—
61	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₆₁	#131	1	2	yes	—
69	C₈×C₆₁	—	2	2	no	—
70	C₈×C₆₁	—	2	2	no	—
71	C₈×C₆₁	—	8	2	no	—
72	#92418	—	2	4	no	—
73	#92418	—	4	2	no	—
74	#92418	—	4	2	no	—
75	#92418	—	4	2	no	—
76	#9609	—	61	1	no	maximal, Sylow
77	C₂²×C₆₁	—	2	1	no	—
78	C₂²×C₆₁	—	2	1	no	—
79	C₂²×C₆₁	#389	1	1	yes	characteristic
80	C₂²×C₆₁	#446	1	2	yes	—
81	C₄×C₆₁	—	2	1	no	—
82	C₄×C₆₁	—	2	1	no	—
83	C₄×C₆₁	—	2	1	no	—
84	C₄×C₆₁	#423	1	1	yes	characteristic
85	C₄×C₆₁	#424	1	1	yes	characteristic
86	C₄×C₆₁	#435	1	2	yes	—
87	C₄×C₆₁	#471	1	2	yes	—
88	C₄×#108	—	61	1	no	—
89	#1565	—	61	2	no	—
90	C₂×#334	D₆₁	1	1	yes	characteristic
91	#1452	—	61	1	no	—
92	#1283	—	61	2	no	—
93	C₂×C₆₁	—	2	2	no	—
94	C₂×C₆₁	#1940	1	1	yes	characteristic
95	C₂×C₆₁	#1951	1	1	yes	characteristic
96	C₂×C₆₁	#1953	1	1	yes	characteristic
97	#431	—	61	1	no	—
98	#431	—	61	1	no	—
99	C₂×#108	—	61	2	no	—
100	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
101	C₄×M₁₆	—	61	2	no	—
102	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
103	#341	—	61	1	no	—
104	#341	—	61	1	no	—
105	#335	—	122	2	no	—
106	#334	—	2	2	no	—
107	#334	—	122	2	no	—
108	#322	—	122	2	no	—
109	C₆₁	#9609	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	—	61	2	no	—
111	C₂×M₁₆	—	61	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
116	#108	—	122	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	61	2	no	—
120	C₄⋊C₈	—	122	1	no	—
121	C₂²⋊C₈	—	61	2	no	—
122	C₂²⋊C₈	—	122	1	no	—
123	#97	—	122	2	no	—
124	C₄×C₈	—	122	2	no	—
125	C₄×C₈	—	122	1	no	—
126	C₄×C₈	—	122	1	no	—
127	C₄×C₈	Dic₁₂₂	1	2	yes	—
128	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
129	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₂²×C₄	#11758224	1	1	yes	characteristic
131	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
132	M₁₆	—	61	4	no	—
133	M₁₆	—	122	2	no	—
134	M₁₆	—	122	2	no	—
135	M₁₆	—	122	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	—	122	2	no	—
141	C₂×C₈	—	122	2	no	—
142	C₂×C₈	—	122	2	no	—
143	C₂×C₈	—	122	2	no	—
144	C₂×C₈	#11758196	1	2	yes	—
145	C₂×C₈	#11758196	1	2	yes	—
146	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
147	C₄²	D₄×D₆₁	1	1	yes	characteristic
148	C₄²	#11758226	1	1	yes	characteristic
149	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
150	C₂³	#14519742	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
154	C₂×C₄	#14519739	1	1	yes	characteristic
155	C₂×C₄	#14519748	1	1	yes	characteristic
156	C₂×C₄	#14519753	1	1	yes	characteristic
157	C₂×C₄	#14519768	1	1	yes	characteristic
158	C₂×C₄	#14519776	1	1	yes	characteristic
159	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
160	C₂×C₄	#14519783	1	2	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	8	2	no	—
164	C₈	—	122	4	no	—
165	C₈	—	244	2	no	—
166	C₈	—	244	2	no	—
167	C₈	—	244	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#16839658	1	1	yes	characteristic
171	C₂²	#16839891	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#16839785	1	1	yes	characteristic
176	C₄	#16839788	1	1	yes	characteristic
177	C₄	#16839832	1	2	yes	—
178	C₄	#16840022	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#21446540	1	1	yes	characteristic
181	C₂	#21446597	1	1	yes	characteristic
182	C₂	#21446607	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14519713

C₂

yes

maximal, characteristic

#21444569

C₂

yes

maximal

#21444111

C₂

yes

maximal, characteristic

#21443669

C₂

yes

maximal

C₂×C₆₁×#334

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

characteristic

#16839815

C₂²

yes

characteristic

C₂×#14519713

C₄

yes

—

C₄×#11758194

C₂²

yes

—

#16839527

C₄

yes

characteristic

#16839527

C₄

yes

—

—

—

C₂×C₄⋊C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#334

—

C₂×#11758194

C₂×C₄

yes

—

C₂×#11758194

C₂×C₄

yes

—

#14519713

—

C₄⋊#92418

—

C₄⋊#92418

D₄

yes

—

C₂²⋊#92418

—

C₂²⋊#92418

Q₈

yes

—

—

—

—

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

C₄²×C₆₁

yes

—

—

—

M₁₆

yes

—

—

—

—

—

C₂³×C₆₁

C₄×Q₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

#125

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#389

yes

characteristic

C₂²×C₆₁

#446

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×C₆₁

#471

yes

—

C₄×#108

—

#1565

—

C₂×#334

D₆₁

yes

characteristic

#1452

—

#1283

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

100

C₂×C₄⋊C₈

C₂×D₆₁

yes

—

101

C₄×M₁₆

—

102

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

103

#341

—

104

#341

—

105

#335

—

122

—

106

#334

—

107

#334

—

122

—

108

#322

—

122

—

109

C₆₁

#9609

yes

characteristic, complemented, Sylow

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₂²×C₈

C₄×D₆₁

yes

characteristic

114

C₂²×C₈

C₄×D₆₁

yes

characteristic

115

C₂×C₄²

C₂²×D₆₁

yes

characteristic

116

#108

—

122

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

122

—

121

C₂²⋊C₈

—

122

C₂²⋊C₈

—

122

—

123

#97

—

122

—

124

C₄×C₈

—

122

—

125

C₄×C₈

—

122

—

126

C₄×C₈

—

122

—

127

C₄×C₈

Dic₁₂₂

yes

—

128

C₄×C₈

C₂²⋊D₆₁

yes

—

129

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

130

C₂²×C₄

#11758224

yes

characteristic

131

C₂²×C₄

Q₈×D₆₁

yes

characteristic

132

M₁₆

—

133

M₁₆

—

122

—

134

M₁₆

—

122

—

135

M₁₆

—

122

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

122

—

141

C₂×C₈

—

122

—

142

C₂×C₈

—

122

—

143

C₂×C₈

—

122

—

144

C₂×C₈

#11758196

yes

—

145

C₂×C₈

#11758196

yes

—

146

C₄²

C₂×Dic₁₂₂

yes

characteristic

147

C₄²

D₄×D₆₁

yes

characteristic

148

C₄²

#11758226

yes

characteristic

149

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

150

C₂³

#14519742

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×#11758196

yes

characteristic

154

C₂×C₄

#14519739

yes

characteristic

155

C₂×C₄

#14519748

yes

characteristic

156

C₂×C₄

#14519753

yes

characteristic

157

C₂×C₄

#14519768

yes

characteristic

158

C₂×C₄

#14519776

yes

characteristic

159

C₂×C₄

M₁₆×D₆₁

yes

—

160

C₂×C₄

#14519783

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

122

—

165

C₈

—

244

—

166

C₈

—

244

—

167

C₈

—

244

—

168

C₂²

—

169

C₂²

—

170

C₂²

#16839658

yes

characteristic

171

C₂²

#16839891

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#16839785

yes

characteristic

176

C₄

#16839788

yes

characteristic

177

C₄

#16839832

yes

—

178

C₄

#16840022

yes

—

179

C₂

—

180

C₂

#21446540

yes

characteristic

181

C₂

#21446597

yes

characteristic

182

C₂

#21446607

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14519748
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#334	C₂
socle	C₂²×C₆₁	#389
prime radical	C₂³×C₆₁	C₄×Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#334	D₆₁
hypocenter	C₆₁	#9609
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

subgroup

G/H

center

C₂×C₄

#14519748

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₆₁×#334

C₂

socle

C₂²×C₆₁

#389

prime radical

C₂³×C₆₁

C₄×Q₈

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

N	H ≅ G/N	type
109. C₆₁	#9609	semidirect

H ≅ G/N

type

109. C₆₁

#9609

semidirect

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

subgroup	index
C₄×#14519713	2
#21444569	2
#21444111	2
#21443669	2
C₂×C₆₁×#334	2
#9609	61

subgroup

index

C₂×C₆₁×#334

Subgroups — #29578632

Conjugacy classes of subgroups

Complements

Maximal subgroups