Group #29596197 — subgroups

Subgroups — #29596197

Order 15616.

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

normal subgroups only

This group has 374 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	C₂²×#14519716	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839591	C₂	1	2	yes	maximal, complemented
3	#21445854	C₂	1	2	yes	maximal
4	#21443913	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
7	C₂×#14519716	—	2	4	no	—
8	C₂×#14519716	C₄	1	4	yes	—
9	C₂²×#11758185	C₂²	1	1	yes	characteristic
10	C₂×#14519713	C₂²	1	2	yes	—
11	#16839725	—	2	2	no	—
12	#16839591	—	2	4	no	—
13	#16839526	—	2	2	no	—
14	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#108×C₆₁	C₂²	1	2	yes	—
16	C₂²×C₄×D₆₁	D₄	1	1	yes	characteristic
17	C₂×#11758194	—	2	2	no	—
18	C₂²×#92418	D₄	1	1	yes	characteristic
19	C₂×C₈×D₆₁	—	2	2	no	—
20	C₂×C₈×D₆₁	—	2	2	no	—
21	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
22	C₂×C₈×D₆₁	Q₈	1	2	yes	—
23	#14519716	—	2	8	no	—
24	C₂×#11758185	C₂×C₄	1	2	yes	—
25	C₂×#11758185	D₄	1	4	yes	—
26	#14519713	—	2	2	no	—
27	#14519713	—	2	2	no	—
28	#14519689	—	4	4	no	—
29	C₂×M₁₆×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₆₁	—	2	2	no	—
32	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
33	#108×C₆₁	—	2	2	no	—
34	#108×C₆₁	—	2	2	no	—
35	C₂³×D₆₁	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₆₁	Q₁₆	1	1	yes	characteristic
38	C₂×C₄×D₆₁	—	2	2	no	—
39	C₂×C₄×D₆₁	—	2	2	no	—
40	C₂×C₄×D₆₁	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×D₆₁	D₈	1	2	yes	—
42	M₁₆×C₆₁	—	4	2	no	—
43	M₁₆×C₆₁	—	4	2	no	—
44	C₂×C₈×C₆₁	—	4	2	no	—
45	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₆₁	#41	1	2	yes	—
49	#11758194	—	4	2	no	—
50	#11758194	—	4	2	no	—
51	C₂×#92418	—	2	2	no	—
52	C₂×#92418	—	4	2	no	—
53	C₂×#92418	C₄⋊C₄	1	2	yes	—
54	C₈×D₆₁	—	2	2	no	—
55	C₈×D₆₁	—	2	2	no	—
56	C₈×D₆₁	—	2	4	no	—
57	C₁₆×C₆₁	—	2	2	no	—
58	#11758185	C₂²⋊C₄	1	4	yes	—
59	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×D₆₁	—	2	2	no	—
61	C₂²×D₆₁	—	4	2	no	—
62	C₂²×D₆₁	#107	1	2	yes	—
63	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#121	1	2	yes	—
65	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#135	1	2	yes	—
67	C₂×Dic₆₁	—	2	2	no	—
68	C₂×Dic₆₁	#107	1	2	yes	—
69	C₄×D₆₁	—	2	2	no	—
70	C₄×D₆₁	—	2	2	no	—
71	C₄×D₆₁	—	2	4	no	—
72	C₈×C₆₁	—	8	2	no	—
73	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₈×C₆₁	#117	1	1	yes	characteristic
75	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
76	#92418	—	2	2	no	—
77	#92418	—	8	2	no	—
78	#12838	—	61	1	no	maximal, Sylow
79	C₂²×C₆₁	—	2	2	no	—
80	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
81	C₂²×C₆₁	#467	1	2	yes	—
82	C₂×D₆₁	—	2	2	no	—
83	C₂×D₆₁	—	2	4	no	—
84	C₂×D₆₁	—	4	4	no	—
85	C₄×C₆₁	#380	1	1	yes	characteristic
86	C₄×C₆₁	#427	1	1	yes	characteristic
87	C₄×C₆₁	#466	1	2	yes	—
88	Dic₆₁	—	2	2	no	—
89	C₂²×M₃₂	—	61	1	no	—
90	C₂×#368	—	61	2	no	—
91	#1810	—	61	2	no	—
92	#1381	—	61	1	no	—
93	#1381	D₆₁	1	1	yes	characteristic, complemented
94	C₂×C₆₁	—	2	2	no	—
95	C₂×C₆₁	#1808	1	1	yes	characteristic
96	C₂×C₆₁	#2161	1	2	yes	—
97	D₆₁	—	4	2	no	—
98	C₂³×C₈	—	61	1	no	—
99	C₂×M₃₂	—	61	4	no	—
100	C₂×M₃₂	—	122	4	no	—
101	C₂²×C₁₆	—	61	1	no	—
102	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
103	C₂×#108	—	61	2	no	—
104	C₂×#108	C₂×D₆₁	1	2	yes	—
105	#407	—	122	2	no	—
106	#368	—	122	4	no	—
107	#341	—	122	2	no	—
108	C₆₁	#12838	1	1	yes	characteristic, complemented, Sylow
109	C₂³×C₄	—	61	1	no	—
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	122	2	no	—
112	C₂²×C₈	—	61	1	no	—
113	C₂²×C₈	—	61	2	no	—
114	C₂²×C₈	—	61	2	no	—
115	C₂²×C₈	—	122	2	no	—
116	C₂²×C₈	—	122	2	no	—
117	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
118	M₃₂	—	122	8	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	61	2	no	—
121	C₂×C₁₆	—	61	4	no	—
122	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
123	#108	—	2	2	no	—
124	#108	—	2	2	no	—
125	#108	—	122	2	no	—
126	#108	—	122	2	no	—
127	C₂²⋊C₈	—	244	4	no	—
128	C₂⁴	—	61	1	no	—
129	C₂²×C₄	—	61	1	no	—
130	C₂²×C₄	—	61	2	no	—
131	C₂²×C₄	—	61	2	no	—
132	C₂²×C₄	—	122	2	no	—
133	C₂²×C₄	—	122	2	no	—
134	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
135	M₁₆	—	4	2	no	—
136	M₁₆	—	4	2	no	—
137	M₁₆	—	244	2	no	—
138	M₁₆	—	244	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	61	2	no	—
141	C₂×C₈	—	122	2	no	—
142	C₂×C₈	—	122	2	no	—
143	C₂×C₈	—	122	2	no	—
144	C₂×C₈	—	122	4	no	—
145	C₂×C₈	—	244	2	no	—
146	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
147	C₂×C₈	#11758222	1	2	yes	—
148	C₂×C₈	D₄×D₆₁	1	2	yes	—
149	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
150	C₁₆	—	2	2	no	—
151	C₁₆	—	61	4	no	—
152	C₂³	—	61	2	no	—
153	C₂³	—	122	2	no	—
154	C₂³	—	244	2	no	—
155	C₂³	Q₁₆×D₆₁	1	1	yes	characteristic
156	C₂×C₄	—	61	2	no	—
157	C₂×C₄	—	122	2	no	—
158	C₂×C₄	—	122	4	no	—
159	C₂×C₄	—	122	2	no	—
160	C₂×C₄	—	122	2	no	—
161	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
162	C₂×C₄	#14519750	1	2	yes	—
163	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
164	C₂×C₄	#14519798	1	2	yes	—
165	C₈	—	8	2	no	—
166	C₈	—	122	2	no	—
167	C₈	—	488	2	no	—
168	C₈	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
169	C₈	#14519734	1	1	yes	characteristic
170	C₈	#14519755	1	2	yes	—
171	C₂²	—	2	2	no	—
172	C₂²	—	122	4	no	—
173	C₂²	—	122	2	no	—
174	C₂²	—	244	4	no	—
175	C₂²	#107×D₆₁	1	1	yes	characteristic
176	C₂²	#16840002	1	2	yes	—
177	C₄	—	122	2	no	—
178	C₄	#16839621	1	1	yes	characteristic
179	C₄	#16839797	1	1	yes	characteristic
180	C₄	#16839997	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	—	244	2	no	—
183	C₂	#21445847	1	1	yes	characteristic
184	C₂	#21447612	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14519716

C₂

yes

maximal, characteristic

C₂×#16839591

C₂

yes

maximal, complemented

#21445854

C₂

yes

maximal

#21443913

C₂

yes

maximal, characteristic, complemented

C₆₁×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₆₁

C₂²

yes

characteristic

C₂×#14519716

—

C₂×#14519716

C₄

yes

—

C₂²×#11758185

C₂²

yes

characteristic

C₂×#14519713

C₂²

yes

—

#16839725

—

#16839591

—

#16839526

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×#108×C₆₁

C₂²

yes

—

C₂²×C₄×D₆₁

D₄

yes

characteristic

C₂×#11758194

—

C₂²×#92418

D₄

yes

characteristic

C₂×C₈×D₆₁

—

C₂×C₈×D₆₁

—

C₂×C₈×D₆₁

C₂×C₄

yes

—

C₂×C₈×D₆₁

Q₈

yes

—

#14519716

—

C₂×#11758185

C₂×C₄

yes

—

C₂×#11758185

D₄

yes

—

#14519713

—

#14519713

—

#14519689

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

#108×C₆₁

—

#108×C₆₁

—

C₂³×D₆₁

Q₁₆

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

Q₁₆

yes

characteristic

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

C₄⋊C₄

yes

—

C₂×C₄×D₆₁

D₈

yes

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₈×D₆₁

—

C₈×D₆₁

—

C₈×D₆₁

—

C₁₆×C₆₁

—

#11758185

C₂²⋊C₄

yes

—

C₂³×C₆₁

C₂×Q₁₆

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

#107

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#135

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#107

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₁

#117

yes

characteristic

C₈×C₆₁

C₂²⋊Q₈

yes

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#107

yes

characteristic

C₂²×C₆₁

#467

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

C₂²×M₃₂

—

—

—

—

D₆₁

yes

characteristic, complemented

C₂×C₆₁

—

C₂×C₆₁

#1808

yes

characteristic

C₂×C₆₁

#2161

yes

—

D₆₁

—

C₂³×C₈

—

C₂×M₃₂

—

100

C₂×M₃₂

—

122

—

101

C₂²×C₁₆

—

102

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

103

C₂×#108

—

104

C₂×#108

C₂×D₆₁

yes

—

105

#407

—

122

—

106

#368

—

122

—

107

#341

—

122

—

108

C₆₁

#12838

yes

characteristic, complemented, Sylow

109

C₂³×C₄

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

122

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

122

—

116

C₂²×C₈

—

122

—

117

C₂²×C₈

C₂²×D₆₁

yes

characteristic

118

M₃₂

—

122

—

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

C₄×D₆₁

yes

—

123

#108

—

124

#108

—

125

#108

—

122

—

126

#108

—

122

—

127

C₂²⋊C₈

—

244

—

128

C₂⁴

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

122

—

133

C₂²×C₄

—

122

—

134

C₂²×C₄

D₄×D₆₁

yes

characteristic

135

M₁₆

—

136

M₁₆

—

137

M₁₆

—

244

—

138

M₁₆

—

244

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

122

—

142

C₂×C₈

—

122

—

143

C₂×C₈

—

122

—

144

C₂×C₈

—

122

—

145

C₂×C₈

—

244

—

146

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

147

C₂×C₈

#11758222

yes

—

148

C₂×C₈

D₄×D₆₁

yes

—

149

C₂×C₈

Q₈×D₆₁

yes

characteristic

150

C₁₆

—

151

C₁₆

—

152

C₂³

—

153

C₂³

—

122

—

154

C₂³

—

244

—

155

C₂³

Q₁₆×D₆₁

yes

characteristic

156

C₂×C₄

—

157

C₂×C₄

—

122

—

158

C₂×C₄

—

122

—

159

C₂×C₄

—

122

—

160

C₂×C₄

—

122

—

161

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

162

C₂×C₄

#14519750

yes

—

163

C₂×C₄

D₈×D₆₁

yes

characteristic

164

C₂×C₄

#14519798

yes

—

165

C₈

—

166

C₈

—

122

—

167

C₈

—

488

—

168

C₈

C₂²⋊C₄×D₆₁

yes

characteristic

169

C₈

#14519734

yes

characteristic

170

C₈

#14519755

yes

—

171

C₂²

—

172

C₂²

—

122

—

173

C₂²

—

122

—

174

C₂²

—

244

—

175

C₂²

#107×D₆₁

yes

characteristic

176

C₂²

#16840002

yes

—

177

C₄

—

122

—

178

C₄

#16839621

yes

characteristic

179

C₄

#16839797

yes

characteristic

180

C₄

#16839997

yes

—

181

C₂

—

182

C₂

—

244

—

183

C₂

#21445847

yes

characteristic

184

C₂

#21447612

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₁
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1381	C₂
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂³×D₆₁	Q₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1381	D₆₁
hypocenter	C₆₁	#12838
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₆₁

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1381

C₂

socle

C₂²×C₆₁

C₂×#107

prime radical

C₂³×D₆₁

Q₁₆

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
2. C₂×#16839591	C₂	semidirect
4. #21443913	C₂	semidirect
5. C₆₁×#1381	C₂	semidirect
93. #1381	D₆₁	semidirect
108. C₆₁	#12838	semidirect

H ≅ G/N

type

2. C₂×#16839591

C₂

semidirect

4. #21443913

C₂

semidirect

5. C₆₁×#1381

C₂

semidirect

93. #1381

D₆₁

semidirect

108. C₆₁

#12838

semidirect

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

subgroup	index
C₂²×#14519716	2
C₂×#16839591	2
#21445854	2
#21443913	2
C₆₁×#1381	2
#12838	61

subgroup

index

C₆₁×#1381

Subgroups — #29596197

Conjugacy classes of subgroups

Complements

Maximal subgroups