Group #29599666 — subgroups

Subgroups — #29599666

Order 15616.

324 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 103 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×#16840164	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839586	C₂	1	2	yes	maximal
3	#21443924	C₂	1	2	yes	maximal
4	#21443901	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1945	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519769	—	2	1	no	—
7	C₂×#14519717	C₂²	1	2	yes	—
8	#16840164	—	2	4	no	—
9	C₂×#14519711	—	2	1	no	—
10	C₂×#14519709	C₂²	1	2	yes	—
11	#16839586	—	2	4	no	—
12	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#427	C₂²	1	1	yes	characteristic
14	C₆₁×#424	—	2	1	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#339	—	2	1	no	—
17	#14519769	—	2	4	no	—
18	#14519769	—	4	2	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×#11758196	—	4	2	no	—
21	#14519717	—	2	2	no	—
22	#14519717	Q₈	1	4	yes	—
23	C₂×#11758185	—	2	2	no	—
24	#14519711	—	2	4	no	—
25	#14519709	—	2	2	no	—
26	#14519709	D₄	1	4	yes	—
27	C₂×M₁₆×C₆₁	—	2	1	no	—
28	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	#117×C₆₁	—	2	1	no	—
31	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
32	#108×C₆₁	—	2	2	no	—
33	#107×C₆₁	Q₈	1	2	yes	—
34	#106×C₆₁	—	2	1	no	—
35	C₄⋊C₈×C₆₁	—	4	2	no	—
36	C₄×C₈×C₆₁	—	2	2	no	—
37	C₂²×C₄×C₆₁	—	2	1	no	—
38	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₆₁	—	4	2	no	—
40	M₁₆×C₆₁	—	4	1	no	—
41	M₁₆×C₆₁	—	4	1	no	—
42	C₂×C₈×C₆₁	—	2	1	no	—
43	C₂×C₈×C₆₁	—	2	1	no	—
44	C₂×C₈×C₆₁	—	4	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	2	no	—
51	C₂²⋊C₄×C₆₁	—	4	1	no	—
52	C₄²×C₆₁	—	2	1	no	—
53	C₄²×C₆₁	—	4	1	no	—
54	C₄²×C₆₁	SD₁₆	1	2	yes	—
55	C₄⋊Dic₆₁	—	2	4	no	—
56	C₄⋊Dic₆₁	—	4	2	no	—
57	#11758196	—	4	8	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	2	no	—
63	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
69	C₂×Dic₆₁	—	4	4	no	—
70	C₂×Dic₆₁	—	8	4	no	—
71	C₈×C₆₁	—	2	2	no	—
72	C₈×C₆₁	—	8	1	no	—
73	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
74	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
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	2	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	4	no	—
87	C₂×#499	—	61	1	no	—
88	C₂×#365	—	61	2	no	—
89	#1945	D₆₁	1	1	yes	characteristic
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	—	122	1	no	—
97	C₂×M₃₂	—	61	2	no	—
98	#499	—	122	4	no	—
99	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
100	#427	C₂×D₆₁	1	1	yes	characteristic
101	C₂×#107	—	122	1	no	—
102	C₂×#106	—	61	2	no	—
103	#424	—	2	1	no	—
104	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
105	#365	—	122	4	no	—
106	#339	—	2	1	no	—
107	C₆₁	#13458	1	1	yes	characteristic, complemented, Sylow
108	C₂×M₁₆	—	2	1	no	—
109	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
111	#125	—	122	4	no	—
112	#125	—	244	2	no	—
113	#117	—	2	1	no	—
114	C₂×C₄⋊C₄	—	122	2	no	—
115	C₂×C₄⋊C₄	—	244	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	—	2	2	no	—
121	#107	—	122	4	no	—
122	#107	Dic₁₂₂	1	2	yes	—
123	#106	—	2	1	no	—
124	#106	—	61	4	no	—
125	#106	—	122	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	244	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
131	M₁₆	—	4	1	no	—
132	M₁₆	—	4	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	#11758200	1	2	yes	—
137	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
138	C₂×C₈	#11758222	1	1	yes	characteristic
139	C₂×C₈	D₄×D₆₁	1	2	yes	—
140	C₂×C₈	Q₈×D₆₁	1	2	yes	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	122	4	no	—
143	C₄⋊C₄	—	244	8	no	—
144	C₄⋊C₄	—	244	2	no	—
145	C₂²⋊C₄	—	4	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	4	1	no	—
148	C₄²	#11758201	1	2	yes	—
149	C₁₆	—	122	4	no	—
150	C₂³	#14519802	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	488	4	no	—
155	C₂×C₄	—	244	4	no	—
156	C₂×C₄	#14519745	1	1	yes	characteristic
157	C₂×C₄	#14519756	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	#14519807	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	8	1	no	—
164	C₈	#14519755	1	2	yes	—
165	C₈	#14519776	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#16839912	1	1	yes	characteristic
168	C₂²	#16840052	1	1	yes	characteristic
169	C₂²	#16840172	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	8	2	no	—
172	C₄	—	488	4	no	—
173	C₄	#16839682	1	1	yes	characteristic
174	C₄	#16839900	1	1	yes	characteristic
175	C₄	#16840047	1	1	yes	characteristic
176	C₄	#16840158	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#21447152	1	1	yes	characteristic
180	C₂	#21447995	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840164

C₂

yes

maximal, characteristic

C₂×#16839586

C₂

yes

maximal

#21443924

C₂

yes

maximal

#21443901

C₂

yes

maximal, characteristic

C₆₁×#1945

C₂

yes

maximal, characteristic

C₂×#14519769

—

C₂×#14519717

C₂²

yes

—

#16840164

—

C₂×#14519711

—

C₂×#14519709

C₂²

yes

—

#16839586

—

C₂×#108×C₆₁

C₂²

yes

characteristic

C₆₁×#427

C₂²

yes

characteristic

C₆₁×#424

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#339

—

#14519769

—

#14519769

—

C₂×C₄⋊Dic₆₁

—

C₂×#11758196

—

#14519717

—

#14519717

Q₈

yes

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#117×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

#108×C₆₁

—

#107×C₆₁

Q₈

yes

—

#106×C₆₁

—

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

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

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

SD₁₆

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

#11758185

—

C₂³×C₆₁

AGL(1,ℤ/8)

yes

characteristic

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

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊Q₈

yes

—

C₈×C₆₁

C₄⋊Q₈

yes

—

#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

—

C₂×#365

—

#1945

D₆₁

yes

characteristic

#1386

—

#1376

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#2058

yes

characteristic

C₂×C₆₁

#2231

yes

—

C₂×#125

—

122

—

C₂×M₃₂

—

#499

—

122

—

C₂×#108

C₂×D₆₁

yes

characteristic

100

#427

C₂×D₆₁

yes

characteristic

101

C₂×#107

—

122

—

102

C₂×#106

—

103

#424

—

104

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

105

#365

—

122

—

106

#339

—

107

C₆₁

#13458

yes

characteristic, complemented, Sylow

108

C₂×M₁₆

—

109

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

110

C₂²×C₈

C₂²×D₆₁

yes

characteristic

111

#125

—

122

—

112

#125

—

244

—

113

#117

—

114

C₂×C₄⋊C₄

—

122

—

115

C₂×C₄⋊C₄

—

244

—

116

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

117

M₃₂

—

118

M₃₂

—

122

—

119

C₂×C₁₆

—

122

—

120

#108

—

121

#107

—

122

—

122

#107

Dic₁₂₂

yes

—

123

#106

—

124

#106

—

125

#106

—

122

—

126

C₄⋊C₈

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

244

—

130

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

131

M₁₆

—

132

M₁₆

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

#11758200

yes

—

137

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

138

C₂×C₈

#11758222

yes

characteristic

139

C₂×C₈

D₄×D₆₁

yes

—

140

C₂×C₈

Q₈×D₆₁

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

122

—

143

C₄⋊C₄

—

244

—

144

C₄⋊C₄

—

244

—

145

C₂²⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

#11758201

yes

—

149

C₁₆

—

122

—

150

C₂³

#14519802

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

488

—

155

C₂×C₄

—

244

—

156

C₂×C₄

#14519745

yes

characteristic

157

C₂×C₄

#14519756

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

#14519807

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

#14519755

yes

—

165

C₈

#14519776

yes

—

166

C₂²

—

167

C₂²

#16839912

yes

characteristic

168

C₂²

#16840052

yes

characteristic

169

C₂²

#16840172

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

488

—

173

C₄

#16839682

yes

characteristic

174

C₄

#16839900

yes

characteristic

175

C₄

#16840047

yes

characteristic

176

C₄

#16840158

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#21447152

yes

characteristic

180

C₂

#21447995

yes

—

	subgroup	G/H
center	C₂²	#16840052
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1945	C₂
socle	C₂²×C₆₁	#476
prime radical	C₂³×C₆₁	AGL(1,ℤ/8)
hypercenter	#1945	D₆₁
hypocenter	C₆₁	#13458
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

subgroup

G/H

center

C₂²

#16840052

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1945

C₂

C₂²×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

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×#16840164	2
C₂×#16839586	2
#21443924	2
#21443901	2
C₆₁×#1945	2
#13458	61

subgroup

index

C₆₁×#1945

Subgroups — #29599666

Conjugacy classes of subgroups

Complements

Maximal subgroups