Group #29579449 — subgroups

Subgroups — #29579449

Order 15616.

414 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 159 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	#21445991	C₂	1	1	yes	maximal, characteristic
2	#21444504	C₂	1	2	yes	maximal
3	#21444500	C₂	1	2	yes	maximal
4	#21444102	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#498	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×#14519713	—	2	2	no	—
8	#16839786	C₂²	1	1	yes	characteristic
9	#16839786	C₂²	1	1	yes	characteristic
10	#16839511	—	2	4	no	—
11	#16839508	—	2	4	no	—
12	#16839483	—	2	1	no	—
13	C₈×#92418	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#498	—	2	1	no	—
16	C₆₁×#498	—	2	1	no	—
17	C₆₁×#498	C₄	1	4	yes	—
18	C₂×#106×C₆₁	—	2	2	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#11758194	—	2	1	no	—
21	C₂×#11758194	—	2	1	no	—
22	#14519713	—	4	4	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	D₄	1	4	yes	—
26	C₂²⋊#92418	—	2	2	no	—
27	C₄×#92418	—	2	2	no	—
28	C₂²×Q₈×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	D₄	1	2	yes	—
30	C₄⋊Q₈×C₆₁	—	2	2	no	—
31	C₄⋊Q₈×C₆₁	—	2	2	no	—
32	C₄⋊Q₈×C₆₁	—	2	2	no	—
33	C₄⋊Q₈×C₆₁	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	#106×C₆₁	—	2	4	no	—
38	#106×C₆₁	—	2	4	no	—
39	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₆₁	—	2	4	no	—
43	C₂×Q₈×C₆₁	—	4	2	no	—
44	C₂×Q₈×C₆₁	—	8	4	no	—
45	C₂²×C₄×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	#41	1	2	yes	—
48	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
51	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
52	C₄⋊C₄×C₆₁	—	2	4	no	—
53	C₄⋊C₄×C₆₁	—	2	4	no	—
54	C₄⋊C₄×C₆₁	—	4	2	no	—
55	C₄⋊C₄×C₆₁	—	4	2	no	—
56	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₁	C₂×D₄	1	2	yes	—
59	#11758194	—	4	2	no	—
60	#11758194	—	4	2	no	—
61	C₂×#92418	—	2	4	no	—
62	C₂×#92418	—	4	1	no	—
63	C₂×#92418	—	4	1	no	—
64	C₂³×C₆₁	#124	1	1	yes	characteristic
65	Q₈×C₆₁	—	8	4	no	—
66	Q₈×C₆₁	—	8	4	no	—
67	C₂×C₄×C₆₁	—	2	4	no	—
68	C₂×C₄×C₆₁	—	4	4	no	—
69	C₂×C₄×C₆₁	—	8	4	no	—
70	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₆₁	#121	1	2	yes	—
73	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
75	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
76	C₈×C₆₁	#103	1	4	yes	—
77	C₈×C₆₁	#103	1	4	yes	—
78	#92418	—	4	4	no	—
79	#92418	—	8	1	no	—
80	#92418	—	8	1	no	—
81	#9767	—	61	1	no	maximal, Sylow
82	C₂²×C₆₁	—	2	2	no	—
83	C₂²×C₆₁	#390	1	1	yes	characteristic
84	C₂²×C₆₁	#487	1	2	yes	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	—	8	4	no	—
87	C₄×C₆₁	C₂×#103	1	2	yes	—
88	C₄×C₆₁	#471	1	4	yes	—
89	C₄×C₆₁	#492	1	1	yes	characteristic
90	C₄×C₆₁	#494	1	1	yes	characteristic
91	C₂×#498	D₆₁	1	1	yes	characteristic
92	#1848	—	61	1	no	—
93	#1549	—	61	2	no	—
94	#1548	—	61	2	no	—
95	#1450	—	61	1	no	—
96	C₂×C₆₁	—	2	2	no	—
97	C₂×C₆₁	#1956	1	1	yes	characteristic
98	C₂×C₆₁	#1977	1	2	yes	—
99	C₂×C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
100	#498	—	2	1	no	—
101	#498	—	2	1	no	—
102	#498	Dic₆₁	1	4	yes	—
103	#431	—	61	2	no	—
104	C₂×#108	—	122	2	no	—
105	C₂×#106	—	2	2	no	—
106	#423	—	61	1	no	—
107	#423	—	61	1	no	—
108	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
109	#333	—	122	4	no	—
110	#332	—	122	4	no	—
111	#322	—	122	1	no	—
112	C₈²	—	122	1	no	—
113	C₆₁	#9767	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	2	2	no	—
115	C₂×M₁₆	—	122	1	no	—
116	C₂×M₁₆	—	122	1	no	—
117	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	—	2	2	no	—
120	C₄⋊Q₈	—	2	2	no	—
121	C₄⋊Q₈	C₂×Dic₆₁	1	4	yes	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
125	#108	—	244	4	no	—
126	#106	—	2	4	no	—
127	#106	—	2	4	no	—
128	C₄⋊C₈	—	61	4	no	—
129	C₄⋊C₈	—	122	2	no	—
130	C₄⋊C₈	—	122	2	no	—
131	C₂²⋊C₈	—	122	2	no	—
132	C₄×C₈	—	122	2	no	—
133	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
134	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
135	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	8	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	#11758224	1	2	yes	—
141	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
142	M₁₆	—	244	2	no	—
143	M₁₆	—	244	2	no	—
144	C₂×C₈	—	122	4	no	—
145	C₂×C₈	—	244	1	no	—
146	C₂×C₈	—	244	1	no	—
147	C₂×C₈	#11758201	1	4	yes	—
148	C₂×C₈	#11758202	1	4	yes	—
149	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
150	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄²	D₄×D₆₁	1	2	yes	—
156	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
157	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
158	C₂³	#14519764	1	1	yes	characteristic
159	Q₈	—	8	4	no	—
160	Q₈	—	8	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
165	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
166	C₂×C₄	#14519748	1	2	yes	—
167	C₂×C₄	#14519773	1	1	yes	characteristic
168	C₂×C₄	C₂×#11758201	1	2	yes	—
169	C₂×C₄	C₂×#11758202	1	2	yes	—
170	C₈	—	488	1	no	—
171	C₈	—	488	1	no	—
172	C₈	—	244	4	no	—
173	C₈	#14519701	1	4	yes	—
174	C₈	#14519701	1	4	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	#16839665	1	1	yes	characteristic
177	C₂²	#16840112	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	8	4	no	—
180	C₄	C₂×#14519701	1	2	yes	—
181	C₄	#16840022	1	4	yes	—
182	C₄	#16840136	1	1	yes	characteristic
183	C₄	#16840141	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	#21446622	1	1	yes	characteristic
186	C₂	#21446724	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445991

C₂

yes

maximal, characteristic

#21444504

C₂

yes

maximal

#21444500

C₂

yes

maximal

#21444102

C₂

yes

maximal, characteristic

C₂×C₆₁×#498

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

C₂×#14519713

—

#16839786

C₂²

yes

characteristic

#16839786

C₂²

yes

—

—

—

—

C₂×C₄⋊Q₈×C₆₁

C₂²

yes

—

C₆₁×#498

—

C₆₁×#498

—

C₆₁×#498

C₄

yes

—

C₂×#106×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

—

—

—

—

D₄

yes

—

C₂²⋊#92418

—

C₄×#92418

—

C₂²×Q₈×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#106×C₆₁

—

#106×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

Q₁₆

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

—

—

—

—

—

—

C₂³×C₆₁

#124

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×Q₁₆

yes

—

C₈×C₆₁

#103

yes

—

C₈×C₆₁

#103

yes

—

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#390

yes

characteristic

C₂²×C₆₁

#487

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#103

yes

—

C₄×C₆₁

#471

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

D₆₁

yes

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#1956

yes

characteristic

C₂×C₆₁

#1977

yes

—

C₂×C₄⋊Q₈

C₂×D₆₁

yes

—

100

#498

—

101

#498

—

102

#498

Dic₆₁

yes

—

103

#431

—

104

C₂×#108

—

122

—

105

C₂×#106

—

106

#423

—

107

#423

—

108

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

109

#333

—

122

—

110

#332

—

122

—

111

#322

—

122

—

112

C₈²

—

122

—

113

C₆₁

#9767

yes

characteristic, complemented, Sylow

114

C₂²×Q₈

—

115

C₂×M₁₆

—

122

—

116

C₂×M₁₆

—

122

—

117

C₂²×C₈

C₂²⋊D₆₁

yes

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

C₂×Dic₆₁

yes

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂²×D₆₁

yes

characteristic

125

#108

—

244

—

126

#106

—

127

#106

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

122

—

130

C₄⋊C₈

—

122

—

131

C₂²⋊C₈

—

122

—

132

C₄×C₈

—

122

—

133

C₄×C₈

C₂×Dic₆₁

yes

—

134

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

135

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

#11758224

yes

—

141

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

142

M₁₆

—

244

—

143

M₁₆

—

244

—

144

C₂×C₈

—

122

—

145

C₂×C₈

—

244

—

146

C₂×C₈

—

244

—

147

C₂×C₈

#11758201

yes

—

148

C₂×C₈

#11758202

yes

—

149

C₂×C₈

C₂²⋊Dic₆₁

yes

—

150

C₂×C₈

C₂²⋊Dic₆₁

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

D₄×D₆₁

yes

—

156

C₄²

C₂²×Dic₆₁

yes

characteristic

157

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

158

C₂³

#14519764

yes

characteristic

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

165

C₂×C₄

D₄×Dic₆₁

yes

—

166

C₂×C₄

#14519748

yes

—

167

C₂×C₄

#14519773

yes

characteristic

168

C₂×C₄

C₂×#11758201

yes

—

169

C₂×C₄

C₂×#11758202

yes

—

170

C₈

—

488

—

171

C₈

—

488

—

172

C₈

—

244

—

173

C₈

#14519701

yes

—

174

C₈

#14519701

yes

—

175

C₂²

—

176

C₂²

#16839665

yes

characteristic

177

C₂²

#16840112

yes

—

178

C₄

—

179

C₄

—

180

C₄

C₂×#14519701

yes

—

181

C₄

#16840022

yes

—

182

C₄

#16840136

yes

characteristic

183

C₄

#16840141

yes

characteristic

184

C₂

—

185

C₂

#21446622

yes

characteristic

186

C₂

#21446724

yes

—

	subgroup	G/H
center	C₂²	#16839665
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#498	C₂
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	#124
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#498	D₆₁
hypocenter	C₆₁	#9767
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂²

#16839665

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₆₁×#498

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
113. C₆₁	#9767	semidirect

H ≅ G/N

type

113. C₆₁

#9767

semidirect

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

subgroup	index
#21445991	2
#21444504	2
#21444500	2
#21444102	2
C₂×C₆₁×#498	2
#9767	61

subgroup

index

C₂×C₆₁×#498

Subgroups — #29579449

Conjugacy classes of subgroups

Complements

Maximal subgroups