Group #29598061 — subgroups

Subgroups — #29598061

Order 15616.

286 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 115 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	#21447603	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#21446565	C₂	1	2	yes	maximal
4	#21443916	C₂	1	2	yes	maximal
5	C₂×C₆₁×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758185	C₂²	1	1	yes	characteristic
7	#16839800	C₂²	1	2	yes	—
8	#16839790	—	2	2	no	—
9	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
10	#16839590	C₄	1	2	yes	complemented
11	#16839588	C₄	1	2	yes	complemented
12	C₁₆×Dic₆₁	—	2	2	no	—
13	#16839523	—	2	2	no	—
14	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#108×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#368	—	2	2	no	—
17	C₂²×#92418	D₄	1	1	yes	characteristic
18	#14519736	—	2	2	no	—
19	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
20	C₂×#11758185	—	2	1	no	—
21	C₂×#11758185	—	2	1	no	—
22	C₂×#11758185	Q₈	1	2	yes	—
23	#14519711	C₂×C₄	1	4	yes	—
24	#14519709	—	2	2	no	—
25	#14519706	—	4	4	no	—
26	C₈×Dic₆₁	—	2	1	no	—
27	C₈×Dic₆₁	—	2	1	no	—
28	C₂×M₁₆×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
33	C₂×C₁₆×C₆₁	Q₈	1	2	yes	—
34	#108×C₆₁	—	2	2	no	—
35	#108×C₆₁	—	2	2	no	—
36	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₆₁	—	2	1	no	—
38	M₁₆×C₆₁	—	4	2	no	—
39	M₁₆×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	—	4	2	no	—
41	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
46	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
47	C₂²⋊Dic₆₁	—	4	2	no	—
48	C₄⋊Dic₆₁	—	2	4	no	—
49	C₄×Dic₆₁	—	2	1	no	—
50	C₄×Dic₆₁	—	4	2	no	—
51	C₄×Dic₆₁	Q₁₆	1	2	yes	—
52	C₂×#92418	—	2	1	no	—
53	C₂×#92418	—	2	1	no	—
54	C₂×#92418	D₈	1	2	yes	—
55	C₁₆×C₆₁	C₄⋊C₄	1	2	yes	—
56	C₁₆×C₆₁	C₄⋊C₄	1	2	yes	—
57	#11758185	—	2	2	no	—
58	C₂³×C₆₁	#135	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
65	C₂×Dic₆₁	—	2	4	no	—
66	C₂×Dic₆₁	—	2	2	no	—
67	C₂×Dic₆₁	—	4	4	no	—
68	C₈×C₆₁	—	8	2	no	—
69	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
71	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
72	C₈×C₆₁	#125	1	1	yes	characteristic
73	#92418	—	2	2	no	—
74	#13169	—	61	1	no	maximal, Sylow
75	C₂²×C₆₁	—	2	2	no	—
76	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
77	C₂²×C₆₁	#467	1	1	yes	characteristic
78	C₂²×C₆₁	#499	1	1	yes	characteristic
79	C₄×C₆₁	#389	1	1	yes	characteristic
80	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
81	C₄×C₆₁	#466	1	1	yes	characteristic
82	C₄×C₆₁	#500	1	1	yes	characteristic
83	Dic₆₁	—	2	4	no	—
84	Dic₆₁	—	8	4	no	—
85	C₂×#368	D₆₁	1	1	yes	characteristic
86	#2158	—	61	1	no	—
87	C₂×C₄×C₁₆	—	61	1	no	—
88	#1945	—	61	2	no	—
89	#1382	—	61	2	no	—
90	C₂×C₆₁	—	2	1	no	—
91	C₂×C₆₁	—	2	1	no	—
92	C₂×C₆₁	#1942	1	1	yes	characteristic
93	C₂×C₆₁	#2178	1	2	yes	—
94	C₂²×C₁₆	—	61	1	no	—
95	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
96	C₂×#108	C₂×D₆₁	1	2	yes	—
97	#427	—	61	2	no	—
98	#424	—	122	2	no	—
99	C₂×C₄×C₈	—	61	1	no	—
100	#368	—	2	2	no	—
101	#367	—	61	2	no	—
102	#366	—	61	2	no	—
103	C₄×C₁₆	—	122	2	no	—
104	#339	—	122	2	no	—
105	C₆₁	#13169	1	1	yes	characteristic, complemented, Sylow
106	C₂×M₁₆	—	2	2	no	—
107	C₂²×C₈	—	61	1	no	—
108	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
109	#117	—	122	2	no	—
110	C₂×C₄²	—	61	1	no	—
111	C₂×C₁₆	—	61	2	no	—
112	C₂×C₁₆	—	122	1	no	—
113	C₂×C₁₆	—	122	1	no	—
114	C₂×C₁₆	Dic₁₂₂	1	2	yes	—
115	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
116	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
117	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
118	#108	—	2	2	no	—
119	#108	—	2	2	no	—
120	#107	—	61	4	no	—
121	#106	—	122	2	no	—
122	C₄⋊C₈	—	244	4	no	—
123	C₄×C₈	—	122	1	no	—
124	C₄×C₈	—	122	1	no	—
125	C₂²×C₄	—	122	1	no	—
126	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
127	M₁₆	—	4	2	no	—
128	M₁₆	—	4	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	61	2	no	—
131	C₂×C₈	—	122	1	no	—
132	C₂×C₈	—	122	1	no	—
133	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
134	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
135	C₂×C₈	#11758224	1	1	yes	characteristic
136	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
137	C₂×C₈	#11758226	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
139	C₄⋊C₄	—	122	4	no	—
140	C₂²⋊C₄	—	244	2	no	—
141	C₄²	—	61	2	no	—
142	C₄²	—	122	1	no	—
143	C₄²	—	244	2	no	—
144	C₁₆	—	122	2	no	—
145	C₁₆	#11758196	1	2	yes	—
146	C₁₆	#11758196	1	2	yes	—
147	C₂³	#14519800	1	1	yes	characteristic
148	C₂×C₄	—	122	4	no	—
149	C₂×C₄	—	122	2	no	—
150	C₂×C₄	—	244	4	no	—
151	C₂×C₄	#14519739	1	1	yes	characteristic
152	C₂×C₄	#14519748	1	1	yes	characteristic
153	C₂×C₄	#14519776	1	1	yes	characteristic
154	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
155	C₂×C₄	Q₁₆×D₆₁	1	1	yes	characteristic
156	C₂×C₄	#14519796	1	1	yes	characteristic
157	C₈	—	8	2	no	—
158	C₈	—	122	2	no	—
159	C₈	C₂×#11758196	1	1	yes	characteristic
160	C₈	#14519742	1	1	yes	characteristic
161	C₈	#14519753	1	1	yes	characteristic
162	C₈	#14519768	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#16839844	1	1	yes	characteristic
165	C₂²	#16840000	1	1	yes	characteristic
166	C₂²	#16840163	1	1	yes	characteristic
167	C₄	—	122	4	no	—
168	C₄	—	488	4	no	—
169	C₄	#16839658	1	1	yes	characteristic
170	C₄	#16839856	1	1	yes	characteristic
171	C₄	#16839995	1	1	yes	characteristic
172	C₄	#16840166	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#21446550	1	1	yes	characteristic
176	C₂	#21447699	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447603

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₆₁

C₂

yes

maximal, characteristic

#21446565

C₂

yes

maximal

#21443916

C₂

yes

maximal

C₂×C₆₁×#368

C₂

yes

maximal, characteristic

C₂²×#11758185

C₂²

yes

characteristic

#16839800

C₂²

yes

—

#16839790

—

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

#16839590

C₄

yes

complemented

#16839588

C₄

yes

complemented

C₁₆×Dic₆₁

—

#16839523

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×#108×C₆₁

C₂²

yes

—

C₆₁×#368

—

C₂²×#92418

D₄

yes

characteristic

#14519736

—

C₂×C₄×Dic₆₁

D₄

yes

—

—

Q₈

yes

—

#14519711

C₂×C₄

yes

—

#14519709

—

#14519706

—

C₈×Dic₆₁

—

C₈×Dic₆₁

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

D₄

yes

—

C₂×C₁₆×C₆₁

Q₈

yes

—

#108×C₆₁

—

#108×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

C₂×C₈×C₆₁

yes

C₂×C₈×C₆₁

yes

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

Q₁₆

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

D₈

yes

—

C₁₆×C₆₁

C₄⋊C₄

yes

—

C₁₆×C₆₁

C₄⋊C₄

yes

—

#11758185

—

C₂³×C₆₁

#135

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₁

#135

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₆₁

C₄×Q₈

yes

characteristic

C₈×C₆₁

C₂²⋊Q₈

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×D₈

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₄×Q₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

C₂×#368

D₆₁

yes

characteristic

#2158

—

C₂×C₄×C₁₆

—

#1945

—

#1382

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1942

yes

characteristic

C₂×C₆₁

#2178

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

C₂×#108

C₂×D₆₁

yes

—

#427

—

#424

—

122

—

C₂×C₄×C₈

—

100

#368

—

101

#367

—

102

#366

—

103

C₄×C₁₆

—

122

—

104

#339

—

122

—

105

C₆₁

#13169

yes

characteristic, complemented, Sylow

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂²×D₆₁

yes

characteristic

109

#117

—

122

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

122

—

113

C₂×C₁₆

—

122

—

114

C₂×C₁₆

Dic₁₂₂

yes

—

115

C₂×C₁₆

C₄×D₆₁

yes

characteristic

116

C₂×C₁₆

C₄×D₆₁

yes

characteristic

117

C₂×C₁₆

C₂²⋊D₆₁

yes

—

118

#108

—

119

#108

—

120

#107

—

121

#106

—

122

—

122

C₄⋊C₈

—

244

—

123

C₄×C₈

—

122

—

124

C₄×C₈

—

122

—

125

C₂²×C₄

—

122

—

126

C₂²×C₄

D₄×D₆₁

yes

characteristic

127

M₁₆

—

128

M₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

122

—

132

C₂×C₈

—

122

—

133

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

134

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

135

C₂×C₈

#11758224

yes

characteristic

136

C₂×C₈

Q₈×D₆₁

yes

characteristic

137

C₂×C₈

#11758226

yes

characteristic

138

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

139

C₄⋊C₄

—

122

—

140

C₂²⋊C₄

—

244

—

141

C₄²

—

142

C₄²

—

122

—

143

C₄²

—

244

—

144

C₁₆

—

122

—

145

C₁₆

#11758196

yes

—

146

C₁₆

#11758196

yes

—

147

C₂³

#14519800

yes

characteristic

148

C₂×C₄

—

122

—

149

C₂×C₄

—

122

—

150

C₂×C₄

—

244

—

151

C₂×C₄

#14519739

yes

characteristic

152

C₂×C₄

#14519748

yes

characteristic

153

C₂×C₄

#14519776

yes

characteristic

154

C₂×C₄

D₈×D₆₁

yes

characteristic

155

C₂×C₄

Q₁₆×D₆₁

yes

characteristic

156

C₂×C₄

#14519796

yes

characteristic

157

C₈

—

158

C₈

—

122

—

159

C₈

C₂×#11758196

yes

characteristic

160

C₈

#14519742

yes

characteristic

161

C₈

#14519753

yes

characteristic

162

C₈

#14519768

yes

characteristic

163

C₂²

—

164

C₂²

#16839844

yes

characteristic

165

C₂²

#16840000

yes

characteristic

166

C₂²

#16840163

yes

characteristic

167

C₄

—

122

—

168

C₄

—

488

—

169

C₄

#16839658

yes

characteristic

170

C₄

#16839856

yes

characteristic

171

C₄

#16839995

yes

characteristic

172

C₄

#16840166

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#21446550

yes

characteristic

176

C₂

#21447699

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₁
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#368	C₂
socle	C₂²×C₆₁	C₄×D₈
prime radical	C₂³×C₆₁	#135
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#368	D₆₁
hypocenter	C₆₁	#13169
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₆₁

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₂×C₆₁×#368

C₂

socle

C₂²×C₆₁

C₄×D₈

prime radical

C₂³×C₆₁

#135

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
10. #16839590	C₄	semidirect
11. #16839588	C₄	semidirect
105. C₆₁	#13169	semidirect

H ≅ G/N

type

10. #16839590

C₄

semidirect

11. #16839588

C₄

semidirect

105. C₆₁

#13169

semidirect

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

subgroup	index
#21447603	2
C₂×C₁₆×Dic₆₁	2
#21446565	2
#21443916	2
C₂×C₆₁×#368	2
#13169	61

subgroup

index

C₂×C₁₆×Dic₆₁

C₂×C₆₁×#368

Subgroups — #29598061

Conjugacy classes of subgroups

Complements

Maximal subgroups