Group #29595393 — subgroups

Subgroups — #29595393

Order 15616.

308 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 131 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	#21447593	C₂	1	2	yes	maximal
2	M₃₂×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#21445701	C₂	1	2	yes	maximal
4	#21443921	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1385	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	1	yes	characteristic
7	#16839800	C₂²	1	2	yes	—
8	#16839720	—	2	2	no	—
9	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
10	#16839586	C₄	1	8	yes	complemented
11	#16839537	C₄	1	2	yes	complemented
12	C₁₆×Dic₆₁	C₄	1	2	yes	complemented
13	#16839523	—	2	2	no	—
14	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#108×C₆₁	C₂²	1	2	yes	—
16	C₂²×#92418	D₄	1	1	yes	characteristic
17	#14519736	—	2	2	no	—
18	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
19	#14519717	C₂×C₄	1	4	yes	—
20	C₂×#11758185	C₂×C₄	1	2	yes	—
21	#14519711	—	2	2	no	—
22	#14519709	C₂×C₄	1	4	yes	—
23	#14519687	—	4	2	no	—
24	C₈×Dic₆₁	—	4	2	no	—
25	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
28	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
29	C₂×M₁₆×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	M₃₂×C₆₁	—	2	2	no	—
32	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
33	#108×C₆₁	—	2	2	no	—
34	#108×C₆₁	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₆₁	—	2	1	no	—
37	M₁₆×C₆₁	—	2	4	no	—
38	M₁₆×C₆₁	—	4	2	no	—
39	C₂×C₈×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
42	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₆₁	#41	1	2	yes	—
44	C₂²⋊Dic₆₁	—	4	2	no	—
45	C₄⋊Dic₆₁	—	2	4	no	—
46	C₄×Dic₆₁	—	2	1	no	—
47	C₄×Dic₆₁	—	4	2	no	—
48	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
49	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
52	C₂×#92418	SD₁₆	1	2	yes	—
53	C₂×#92418	SD₁₆	1	2	yes	—
54	C₁₆×C₆₁	—	2	2	no	—
55	#11758185	C₄²	1	4	yes	—
56	C₂³×C₆₁	#135	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
61	C₂×Dic₆₁	—	2	1	no	—
62	C₂×Dic₆₁	—	2	1	no	—
63	C₂×Dic₆₁	—	4	2	no	—
64	C₂×Dic₆₁	—	4	4	no	—
65	C₈×C₆₁	—	4	4	no	—
66	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₆₁	#117	1	1	yes	characteristic
68	C₈×C₆₁	C₄×Q₈	1	2	yes	—
69	#92418	#106	1	1	yes	characteristic
70	#92418	#106	1	2	yes	—
71	#92418	#106	1	1	yes	characteristic
72	#12668	—	61	1	no	maximal, Sylow
73	C₂²×C₆₁	—	4	1	no	—
74	C₂²×C₆₁	#427	1	1	yes	characteristic
75	C₂²×C₆₁	C₄×SD₁₆	1	2	yes	—
76	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
78	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
79	Dic₆₁	—	2	2	no	—
80	Dic₆₁	—	4	1	no	—
81	Dic₆₁	—	8	4	no	—
82	#2155	—	61	2	no	—
83	C₄×M₃₂	—	61	1	no	—
84	#1772	—	61	2	no	—
85	#1385	D₆₁	1	1	yes	characteristic
86	#1384	—	61	1	no	—
87	C₂×C₆₁	—	2	1	no	—
88	C₂×C₆₁	—	2	2	no	—
89	C₂×C₆₁	C₄×#106	1	1	yes	characteristic
90	C₂×M₃₂	—	61	1	no	—
91	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
92	C₂×#108	C₂×D₆₁	1	2	yes	—
93	#427	—	61	2	no	—
94	#405	—	122	2	no	—
95	C₂×C₄×C₈	—	61	1	no	—
96	#365	—	61	8	no	—
97	#346	—	61	2	no	—
98	C₄×C₁₆	—	61	2	no	—
99	#339	—	122	2	no	—
100	C₆₁	#12668	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	—	61	1	no	—
103	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
104	#117	—	122	2	no	—
105	C₂×C₄²	—	61	1	no	—
106	M₃₂	—	2	2	no	—
107	M₃₂	—	61	4	no	—
108	C₂×C₁₆	—	61	2	no	—
109	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
110	#108	—	2	2	no	—
111	#108	C₄×D₆₁	1	4	yes	—
112	#107	—	122	2	no	—
113	#106	—	61	4	no	—
114	#97	—	244	2	no	—
115	C₄×C₈	—	61	1	no	—
116	C₄×C₈	—	61	1	no	—
117	C₄×C₈	—	61	1	no	—
118	C₄×C₈	—	61	1	no	—
119	C₄×C₈	—	244	2	no	—
120	C₂²×C₄	—	122	1	no	—
121	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	61	1	no	—
126	C₂×C₈	—	61	1	no	—
127	C₂×C₈	—	61	2	no	—
128	C₂×C₈	—	61	2	no	—
129	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
131	C₂×C₈	#11758224	1	2	yes	—
132	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
133	C₄⋊C₄	—	122	4	no	—
134	C₂²⋊C₄	—	244	2	no	—
135	C₄²	—	61	1	no	—
136	C₄²	—	61	1	no	—
137	C₄²	—	122	1	no	—
138	C₄²	—	244	2	no	—
139	C₁₆	—	2	2	no	—
140	C₁₆	—	61	4	no	—
141	C₂³	#14519800	1	1	yes	characteristic
142	C₂×C₄	—	122	1	no	—
143	C₂×C₄	—	122	1	no	—
144	C₂×C₄	—	244	2	no	—
145	C₂×C₄	—	244	4	no	—
146	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
147	C₂×C₄	#14519737	1	2	yes	—
148	C₂×C₄	SD₁₆×D₆₁	1	2	yes	—
149	C₂×C₄	#14519796	1	1	yes	characteristic
150	C₈	—	4	4	no	—
151	C₈	—	61	1	no	—
152	C₈	—	61	2	no	—
153	C₈	—	61	1	no	—
154	C₈	C₄²×D₆₁	1	1	yes	characteristic
155	C₈	#14519733	1	1	yes	characteristic
156	C₈	#14519742	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#16839799	1	1	yes	characteristic
159	C₂²	#16839847	1	2	yes	—
160	C₄	—	122	2	no	—
161	C₄	—	488	4	no	—
162	C₄	—	244	1	no	—
163	C₄	#16839613	1	1	yes	characteristic
164	C₄	#106×D₆₁	1	1	yes	characteristic
165	C₄	#16839853	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	#21445707	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447593

C₂

yes

maximal

M₃₂×Dic₆₁

C₂

yes

maximal, characteristic

#21445701

C₂

yes

maximal

#21443921

C₂

yes

maximal, characteristic

C₆₁×#1385

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

characteristic

#16839800

C₂²

yes

—

#16839720

—

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

#16839586

C₄

yes

complemented

#16839537

C₄

yes

complemented

C₁₆×Dic₆₁

C₄

yes

complemented

#16839523

—

C₂×M₃₂×C₆₁

C₂²

yes

characteristic

C₂×#108×C₆₁

C₂²

yes

—

C₂²×#92418

D₄

yes

characteristic

#14519736

—

C₂×C₄×Dic₆₁

D₄

yes

characteristic

#14519717

C₂×C₄

yes

—

C₂×#11758185

C₂×C₄

yes

—

#14519711

—

#14519709

C₂×C₄

yes

—

#14519687

—

C₈×Dic₆₁

—

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

Q₈

yes

characteristic

C₈×Dic₆₁

Q₈

yes

characteristic

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

#108×C₆₁

—

#108×C₆₁

C₂×C₄

yes

—

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

C₂²×C₄

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

C₄⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

SD₁₆

yes

—

C₂×#92418

SD₁₆

yes

—

C₁₆×C₆₁

—

#11758185

C₄²

yes

—

C₂³×C₆₁

#135

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

#135

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₄²

yes

characteristic

C₈×C₆₁

#117

yes

characteristic

C₈×C₆₁

C₄×Q₈

yes

—

yes

yes

—

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

#427

yes

characteristic

C₂²×C₆₁

C₄×SD₁₆

yes

—

C₄×C₆₁

C₄×C₄⋊C₄

yes

characteristic

C₄×C₆₁

C₂×#106

yes

characteristic

C₄×C₆₁

C₄×SD₁₆

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#2155

—

C₄×M₃₂

—

#1772

—

#1385

D₆₁

yes

characteristic

#1384

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₄×#106

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₂×D₆₁

yes

characteristic

C₂×#108

C₂×D₆₁

yes

—

#427

—

#405

—

122

—

C₂×C₄×C₈

—

#365

—

#346

—

C₄×C₁₆

—

#339

—

122

—

100

C₆₁

#12668

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂²×D₆₁

yes

characteristic

104

#117

—

122

—

105

C₂×C₄²

—

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₄×D₆₁

yes

—

110

#108

—

111

#108

C₄×D₆₁

yes

—

112

#107

—

122

—

113

#106

—

114

#97

—

244

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

244

—

120

C₂²×C₄

—

122

—

121

C₂²×C₄

D₄×D₆₁

yes

characteristic

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

130

C₂×C₈

C₂×C₄×D₆₁

yes

—

131

C₂×C₈

#11758224

yes

—

132

C₂×C₈

Q₈×D₆₁

yes

characteristic

133

C₄⋊C₄

—

122

—

134

C₂²⋊C₄

—

244

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

122

—

138

C₄²

—

244

—

139

C₁₆

—

140

C₁₆

—

141

C₂³

#14519800

yes

characteristic

142

C₂×C₄

—

122

—

143

C₂×C₄

—

122

—

144

C₂×C₄

—

244

—

145

C₂×C₄

—

244

—

146

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

147

C₂×C₄

#14519737

yes

—

148

C₂×C₄

SD₁₆×D₆₁

yes

—

149

C₂×C₄

#14519796

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

C₄²×D₆₁

yes

characteristic

155

C₈

#14519733

yes

characteristic

156

C₈

#14519742

yes

—

157

C₂²

—

158

C₂²

#16839799

yes

characteristic

159

C₂²

#16839847

yes

—

160

C₄

—

122

—

161

C₄

—

488

—

162

C₄

—

244

—

163

C₄

#16839613

yes

characteristic

164

C₄

#106×D₆₁

yes

characteristic

165

C₄

#16839853

yes

—

166

C₂

—

167

C₂

—

168

C₂

#21445707

yes

characteristic

	subgroup	G/H
center	C₄	#106×D₆₁
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1385	C₂
socle	C₂×C₆₁	C₄×#106
prime radical	C₂³×C₆₁	#135
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1385	D₆₁
hypocenter	C₆₁	#12668
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#106×D₆₁

commutator

C₈×C₆₁

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1385

C₂

C₂×C₆₁

C₂³×C₆₁

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
10. #16839586	C₄	semidirect
11. #16839537	C₄	semidirect
12. C₁₆×Dic₆₁	C₄	semidirect
100. C₆₁	#12668	semidirect

H ≅ G/N

type

10. #16839586

C₄

semidirect

11. #16839537

C₄

semidirect

12. C₁₆×Dic₆₁

C₄

semidirect

100. C₆₁

#12668

semidirect

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

subgroup	index
#21447593	2
M₃₂×Dic₆₁	2
#21445701	2
#21443921	2
C₆₁×#1385	2
#12668	61

subgroup

index

M₃₂×Dic₆₁

C₆₁×#1385

Subgroups — #29595393

Conjugacy classes of subgroups

Complements

Maximal subgroups