Group #29599663 — subgroups

Subgroups — #29599663

Order 15616.

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

normal subgroups only

This group has 324 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×#499	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×#125×C₆₁	—	2	1	no	—
13	C₆₁×#499	—	2	4	no	—
14	C₂×#107×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	—
21	C₂×#11758185	—	2	2	no	—
22	#14519713	—	2	2	no	—
23	#14519711	Q₈	1	2	yes	—
24	#14519709	—	2	1	no	—
25	#14519706	—	4	2	no	—
26	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
28	#125×C₆₁	—	2	4	no	—
29	#125×C₆₁	—	4	2	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
32	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
33	#107×C₆₁	—	2	4	no	—
34	#106×C₆₁	—	2	2	no	—
35	#106×C₆₁	D₄	1	4	yes	—
36	C₄×C₈×C₆₁	—	2	2	no	—
37	C₂²×C₄×C₆₁	—	2	1	no	—
38	C₂²×C₄×C₆₁	—	4	2	no	—
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	—	2	1	no	—
41	C₂×C₈×C₆₁	—	2	1	no	—
42	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
43	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
47	C₄⋊C₄×C₆₁	—	2	4	no	—
48	C₄⋊C₄×C₆₁	—	4	8	no	—
49	C₄⋊C₄×C₆₁	—	4	2	no	—
50	C₄²×C₆₁	—	2	1	no	—
51	C₄²×C₆₁	SD₁₆	1	2	yes	—
52	C₂²⋊Dic₆₁	—	4	1	no	—
53	C₄⋊Dic₆₁	—	2	2	no	—
54	C₄×Dic₆₁	—	4	1	no	—
55	#11758194	—	4	1	no	—
56	#11758194	—	4	1	no	—
57	C₂×#92418	—	4	1	no	—
58	#11758185	—	2	4	no	—
59	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	4	2	no	—
62	C₂×C₄×C₆₁	—	4	4	no	—
63	C₂×C₄×C₆₁	—	8	4	no	—
64	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
70	C₂×Dic₆₁	—	4	2	no	—
71	C₈×C₆₁	—	2	2	no	—
72	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
73	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
74	#92418	—	8	1	no	—
75	#13458	—	61	1	no	maximal, Sylow
76	C₂²×C₆₁	—	2	2	no	—
77	C₂²×C₆₁	#450	1	1	yes	characteristic
78	C₂²×C₆₁	#476	1	1	yes	characteristic
79	C₂²×C₆₁	#501	1	1	yes	characteristic
80	C₄×C₆₁	—	4	2	no	—
81	C₄×C₆₁	—	8	4	no	—
82	C₄×C₆₁	#395	1	1	yes	characteristic
83	C₄×C₆₁	#448	1	1	yes	characteristic
84	C₄×C₆₁	#475	1	1	yes	characteristic
85	C₄×C₆₁	#498	1	1	yes	characteristic
86	Dic₆₁	—	8	2	no	—
87	C₂×#499	D₆₁	1	1	yes	characteristic
88	C₂×#365	—	61	2	no	—
89	#1945	—	61	1	no	—
90	#1386	—	61	2	no	—
91	#1376	—	61	1	no	—
92	C₂×C₆₁	—	2	1	no	—
93	C₂×C₆₁	—	2	1	no	—
94	C₂×C₆₁	#2058	1	1	yes	characteristic
95	C₂×C₆₁	#2231	1	2	yes	—
96	C₂×#125	—	2	1	no	—
97	C₂×M₃₂	—	61	2	no	—
98	#499	—	2	4	no	—
99	C₂×#108	—	61	1	no	—
100	#427	—	61	1	no	—
101	C₂×#107	—	2	1	no	—
102	C₂×#106	C₂×D₆₁	1	2	yes	—
103	#424	—	122	1	no	—
104	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
105	#365	—	122	4	no	—
106	#339	—	122	1	no	—
107	C₆₁	#13458	1	1	yes	characteristic, complemented, Sylow
108	C₂×M₁₆	—	122	1	no	—
109	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
111	#125	—	2	4	no	—
112	#125	—	4	2	no	—
113	#117	—	122	1	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	4	2	no	—
116	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
117	M₃₂	—	61	4	no	—
118	M₃₂	—	122	2	no	—
119	C₂×C₁₆	—	122	2	no	—
120	#108	—	122	2	no	—
121	#107	—	2	4	no	—
122	#107	—	61	2	no	—
123	#106	—	2	2	no	—
124	#106	—	122	1	no	—
125	#106	C₂²⋊D₆₁	1	4	yes	—
126	C₄⋊C₈	—	244	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
131	M₁₆	—	244	1	no	—
132	M₁₆	—	244	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	244	1	no	—
136	C₂×C₈	#11758201	1	2	yes	—
137	C₂×C₈	Q₈×D₆₁	1	2	yes	—
138	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
139	C₂×C₈	#11758226	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	122	2	no	—
145	C₂²⋊C₄	—	244	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	244	1	no	—
148	C₄²	#11758200	1	2	yes	—
149	C₁₆	—	122	4	no	—
150	C₂³	#14519803	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	244	2	no	—
156	C₂×C₄	#14519746	1	1	yes	characteristic
157	C₂×C₄	#14519753	1	1	yes	characteristic
158	C₂×C₄	#14519775	1	1	yes	characteristic
159	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
160	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
161	C₂×C₄	#14519808	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	488	1	no	—
164	C₈	#14519753	1	2	yes	—
165	C₈	#14519775	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#16839913	1	1	yes	characteristic
168	C₂²	#16840049	1	1	yes	characteristic
169	C₂²	#16840168	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	8	4	no	—
172	C₄	—	488	2	no	—
173	C₄	#16839681	1	1	yes	characteristic
174	C₄	#16839901	1	1	yes	characteristic
175	C₄	#16840044	1	1	yes	characteristic
176	C₄	#16840158	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#21447149	1	1	yes	characteristic
180	C₂	#21447992	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×#499

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

—

C₂×#14519713

C₂²

yes

characteristic

#16839800

C₂²

yes

—

—

—

C₂×#125×C₆₁

—

C₆₁×#499

—

C₂×#107×C₆₁

—

C₂×#106×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

—

—

Q₈

yes

—

C₂×#11758185

—

#14519713

—

#14519711

Q₈

yes

—

#14519709

—

#14519706

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#125×C₆₁

—

#125×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

#107×C₆₁

—

#106×C₆₁

—

#106×C₆₁

D₄

yes

—

C₄×C₈×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

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

—

#13458

—

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×#499

D₆₁

yes

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#2058

yes

characteristic

C₂×C₆₁

#2231

yes

—

C₂×#125

—

C₂×M₃₂

—

#499

—

C₂×#108

—

100

#427

—

101

C₂×#107

—

102

C₂×#106

C₂×D₆₁

yes

—

103

#424

—

122

—

104

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

105

#365

—

122

—

106

#339

—

122

—

107

C₆₁

#13458

yes

characteristic, complemented, Sylow

108

C₂×M₁₆

—

122

—

109

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

110

C₂²×C₈

C₂²×D₆₁

yes

characteristic

111

#125

—

112

#125

—

113

#117

—

122

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

117

M₃₂

—

118

M₃₂

—

122

—

119

C₂×C₁₆

—

122

—

120

#108

—

122

—

121

#107

—

122

#107

—

123

#106

—

124

#106

—

122

—

125

#106

C₂²⋊D₆₁

yes

—

126

C₄⋊C₈

—

244

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

131

M₁₆

—

244

—

132

M₁₆

—

244

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

244

—

136

C₂×C₈

#11758201

yes

—

137

C₂×C₈

Q₈×D₆₁

yes

—

138

C₂×C₈

Q₈×D₆₁

yes

characteristic

139

C₂×C₈

#11758226

yes

characteristic

140

C₂×C₈

C₂×C₂²⋊D₆₁

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

122

—

145

C₂²⋊C₄

—

244

—

146

C₄²

—

147

C₄²

—

244

—

148

C₄²

#11758200

yes

—

149

C₁₆

—

122

—

150

C₂³

#14519803

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

244

—

156

C₂×C₄

#14519746

yes

characteristic

157

C₂×C₄

#14519753

yes

characteristic

158

C₂×C₄

#14519775

yes

characteristic

159

C₂×C₄

C₂×#11758201

yes

characteristic

160

C₂×C₄

C₂×#11758200

yes

characteristic

161

C₂×C₄

#14519808

yes

characteristic

162

C₈

—

163

C₈

—

488

—

164

C₈

#14519753

yes

—

165

C₈

#14519775

yes

—

166

C₂²

—

167

C₂²

#16839913

yes

characteristic

168

C₂²

#16840049

yes

characteristic

169

C₂²

#16840168

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

488

—

173

C₄

#16839681

yes

characteristic

174

C₄

#16839901

yes

characteristic

175

C₄

#16840044

yes

characteristic

176

C₄

#16840158

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#21447149

yes

characteristic

180

C₂

#21447992

yes

—

	subgroup	G/H
center	C₂²	#16840049
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#499	C₂
socle	C₂²×C₆₁	#476
prime radical	C₂³×C₆₁	AGL(1,ℤ/8)
hypercenter	C₂×#499	D₆₁
hypocenter	C₆₁	#13458
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

#16840049

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₂×C₆₁×#499

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×#499

N	H ≅ G/N	type
107. C₆₁	#13458	semidirect

H ≅ G/N

type

107. C₆₁

#13458

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×#499	2
#13458	61

subgroup

index

C₂×C₆₁×#499

Subgroups — #29599663

Conjugacy classes of subgroups

Complements

Maximal subgroups