Group #29599658 — subgroups

Subgroups — #29599658

Order 15616.

356 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#16839584	C₂	1	2	yes	maximal
2	#21446567	C₂	1	1	yes	maximal, characteristic
3	#21443926	C₂	1	2	yes	maximal
4	#21443901	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	2	yes	—
7	C₂×#14519713	C₂²	1	1	yes	characteristic
8	#16839800	C₂²	1	1	yes	characteristic
9	#16839788	—	2	1	no	—
10	#16839584	—	2	4	no	—
11	#16839522	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
13	C₆₁×#498	—	2	4	no	—
14	C₂×#106×C₆₁	—	2	1	no	—
15	C₂×#106×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#11758194	—	2	1	no	—
18	#14519736	—	2	1	no	—
19	#14519717	—	2	2	no	—
20	#14519717	Q₈	1	4	yes	complemented
21	C₂×#11758185	—	2	2	no	—
22	#14519713	—	2	2	no	—
23	#14519711	Q₈	1	2	yes	complemented
24	#14519709	—	2	1	no	—
25	#14519706	—	4	2	no	—
26	C₂²×Q₈×C₆₁	—	4	1	no	—
27	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₆₁	—	2	4	no	—
30	C₄⋊Q₈×C₆₁	—	4	2	no	—
31	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
33	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
34	#106×C₆₁	—	2	4	no	—
35	#106×C₆₁	—	2	2	no	—
36	#106×C₆₁	D₄	1	4	yes	—
37	C₄×C₈×C₆₁	—	2	2	no	—
38	C₂×Q₈×C₆₁	—	4	4	no	—
39	C₂×Q₈×C₆₁	—	8	4	no	—
40	C₂²×C₄×C₆₁	—	2	1	no	—
41	C₂²×C₄×C₆₁	—	4	2	no	—
42	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	—	2	1	no	—
44	C₂×C₈×C₆₁	—	2	1	no	—
45	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
46	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₆₁	—	2	4	no	—
51	C₄⋊C₄×C₆₁	—	4	4	no	—
52	C₄⋊C₄×C₆₁	—	4	2	no	—
53	C₄²×C₆₁	—	2	1	no	—
54	C₄²×C₆₁	SD₁₆	1	2	yes	—
55	C₂²⋊Dic₆₁	—	4	1	no	—
56	C₄⋊Dic₆₁	—	2	2	no	—
57	C₄×Dic₆₁	—	4	1	no	—
58	#11758194	—	4	1	no	—
59	#11758194	—	4	1	no	—
60	C₂×#92418	—	4	1	no	—
61	#11758185	—	2	4	no	—
62	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	Q₈×C₆₁	—	8	8	no	—
64	C₂×C₄×C₆₁	—	2	4	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	4	4	no	—
67	C₂×C₄×C₆₁	—	8	4	no	—
68	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
74	C₂×Dic₆₁	—	4	2	no	—
75	C₈×C₆₁	—	2	2	no	—
76	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
78	#92418	—	8	1	no	—
79	#13457	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	—	2	2	no	—
81	C₂²×C₆₁	#450	1	1	yes	characteristic
82	C₂²×C₆₁	#476	1	1	yes	characteristic
83	C₂²×C₆₁	#501	1	1	yes	characteristic
84	C₄×C₆₁	—	2	4	no	—
85	C₄×C₆₁	—	8	4	no	—
86	C₄×C₆₁	#395	1	1	yes	characteristic
87	C₄×C₆₁	#448	1	1	yes	characteristic
88	C₄×C₆₁	#475	1	1	yes	characteristic
89	C₄×C₆₁	#498	1	1	yes	characteristic
90	Dic₆₁	—	8	2	no	—
91	C₂×#498	D₆₁	1	1	yes	characteristic
92	C₂×#365	—	61	2	no	—
93	#1945	—	61	1	no	—
94	#1386	—	61	2	no	—
95	#1376	—	61	1	no	—
96	C₂×C₆₁	—	2	1	no	—
97	C₂×C₆₁	—	2	1	no	—
98	C₂×C₆₁	#2058	1	1	yes	characteristic
99	C₂×C₆₁	#2230	1	2	yes	—
100	C₂×C₄⋊Q₈	—	2	1	no	—
101	C₂×M₃₂	—	61	2	no	—
102	#498	—	2	4	no	—
103	C₂×#108	—	61	1	no	—
104	#427	—	61	1	no	—
105	C₂×#106	—	2	1	no	—
106	C₂×#106	C₂×D₆₁	1	2	yes	—
107	#424	—	122	1	no	—
108	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
109	#365	—	122	4	no	—
110	#339	—	122	1	no	—
111	C₆₁	#13457	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	4	1	no	—
113	C₂×M₁₆	—	122	1	no	—
114	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	—	4	2	no	—
118	#117	—	122	1	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
122	M₃₂	—	61	4	no	—
123	M₃₂	—	122	2	no	—
124	C₂×C₁₆	—	122	2	no	—
125	#108	—	122	2	no	—
126	#107	—	61	2	no	—
127	#106	—	2	4	no	—
128	#106	—	2	2	no	—
129	#106	—	122	1	no	—
130	#106	C₂²⋊D₆₁	1	4	yes	—
131	C₄⋊C₈	—	244	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×Q₈	—	8	4	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
138	M₁₆	—	244	1	no	—
139	M₁₆	—	244	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	244	1	no	—
143	C₂×C₈	#11758201	1	2	yes	—
144	C₂×C₈	Q₈×D₆₁	1	2	yes	—
145	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
146	C₂×C₈	#11758226	1	1	yes	characteristic
147	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	122	2	no	—
152	C₂²⋊C₄	—	244	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	244	1	no	—
155	C₄²	#11758200	1	2	yes	—
156	C₁₆	—	122	4	no	—
157	C₂³	#14519803	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	244	2	no	—
164	C₂×C₄	#14519746	1	1	yes	characteristic
165	C₂×C₄	#14519753	1	1	yes	characteristic
166	C₂×C₄	#14519775	1	1	yes	characteristic
167	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
168	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
169	C₂×C₄	#14519808	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	488	1	no	—
172	C₈	#14519753	1	2	yes	—
173	C₈	#14519775	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	#16839913	1	1	yes	characteristic
176	C₂²	#16840049	1	1	yes	characteristic
177	C₂²	#16840168	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	4	no	—
180	C₄	—	488	2	no	—
181	C₄	#16839681	1	1	yes	characteristic
182	C₄	#16839901	1	1	yes	characteristic
183	C₄	#16840044	1	1	yes	characteristic
184	C₄	#16840158	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#21447149	1	1	yes	characteristic
188	C₂	#21447987	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16839584

C₂

yes

maximal

#21446567

C₂

yes

maximal, characteristic

#21443926

C₂

yes

maximal

#21443901

C₂

yes

maximal, characteristic

C₂×C₆₁×#498

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

—

C₂×#14519713

C₂²

yes

characteristic

#16839800

C₂²

yes

—

—

—

C₂×C₄⋊Q₈×C₆₁

—

C₆₁×#498

—

C₂×#106×C₆₁

—

C₂×#106×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

—

—

Q₈

yes

—

—

Q₈

yes

complemented

#14519709

—

#14519706

—

C₂²×Q₈×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

#106×C₆₁

—

#106×C₆₁

—

#106×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

SD₁₆

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

—

—

—

—

C₂³×C₆₁

AGL(1,ℤ/8)

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#120

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

#137

yes

characteristic

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊Q₈

yes

—

C₈×C₆₁

C₄⋊Q₈

yes

—

#92418

—

#13457

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

C₂×#498

D₆₁

yes

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#2058

yes

characteristic

C₂×C₆₁

#2230

yes

—

100

C₂×C₄⋊Q₈

—

101

C₂×M₃₂

—

102

#498

—

103

C₂×#108

—

104

#427

—

105

C₂×#106

—

106

C₂×#106

C₂×D₆₁

yes

—

107

#424

—

122

—

108

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

109

#365

—

122

—

110

#339

—

122

—

111

C₆₁

#13457

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂×M₁₆

—

122

—

114

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

115

C₂²×C₈

C₂²×D₆₁

yes

characteristic

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

#117

—

122

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

122

M₃₂

—

123

M₃₂

—

122

—

124

C₂×C₁₆

—

122

—

125

#108

—

122

—

126

#107

—

127

#106

—

128

#106

—

129

#106

—

122

—

130

#106

C₂²⋊D₆₁

yes

—

131

C₄⋊C₈

—

244

—

132

C₄×C₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

138

M₁₆

—

244

—

139

M₁₆

—

244

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

244

—

143

C₂×C₈

#11758201

yes

—

144

C₂×C₈

Q₈×D₆₁

yes

—

145

C₂×C₈

Q₈×D₆₁

yes

characteristic

146

C₂×C₈

#11758226

yes

characteristic

147

C₂×C₈

C₂×C₂²⋊D₆₁

yes

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

122

—

152

C₂²⋊C₄

—

244

—

153

C₄²

—

154

C₄²

—

244

—

155

C₄²

#11758200

yes

—

156

C₁₆

—

122

—

157

C₂³

#14519803

yes

characteristic

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

244

—

164

C₂×C₄

#14519746

yes

characteristic

165

C₂×C₄

#14519753

yes

characteristic

166

C₂×C₄

#14519775

yes

characteristic

167

C₂×C₄

C₂×#11758201

yes

characteristic

168

C₂×C₄

C₂×#11758200

yes

characteristic

169

C₂×C₄

#14519808

yes

characteristic

170

C₈

—

171

C₈

—

488

—

172

C₈

#14519753

yes

—

173

C₈

#14519775

yes

—

174

C₂²

—

175

C₂²

#16839913

yes

characteristic

176

C₂²

#16840049

yes

characteristic

177

C₂²

#16840168

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

488

—

181

C₄

#16839681

yes

characteristic

182

C₄

#16839901

yes

characteristic

183

C₄

#16840044

yes

characteristic

184

C₄

#16840158

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#21447149

yes

characteristic

188

C₂

#21447987

yes

—

	subgroup	G/H
center	C₂²	#16840049
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#498	C₂
socle	C₂²×C₆₁	#476
prime radical	C₂³×C₆₁	AGL(1,ℤ/8)
hypercenter	C₂×#498	D₆₁
hypocenter	C₆₁	#13457
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

subgroup

G/H

center

C₂²

#16840049

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₂×C₆₁×#498

C₂

C₂²×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

N	H ≅ G/N	type
20. #14519717	Q₈	semidirect
23. #14519711	Q₈	semidirect
111. C₆₁	#13457	semidirect

H ≅ G/N

type

20. #14519717

Q₈

semidirect

23. #14519711

Q₈

semidirect

111. C₆₁

#13457

semidirect

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

subgroup	index
C₂×#16839584	2
#21446567	2
#21443926	2
#21443901	2
C₂×C₆₁×#498	2
#13457	61

subgroup

index

C₂×C₆₁×#498

Subgroups — #29599658

Conjugacy classes of subgroups

Complements

Maximal subgroups