Group #29600924 — subgroups

Subgroups — #29600924

Order 15616.

158 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 57 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	#21448096	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#21444020	C₂	1	1	yes	maximal, characteristic
4	#21444019	C₂	1	1	yes	maximal, characteristic
5	#21444014	C₂	1	1	yes	maximal, characteristic
6	#21444000	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1416	C₂	1	1	yes	maximal, characteristic
8	#16840163	—	2	1	no	—
9	Q₁₆×Dic₆₁	—	2	1	no	—
10	C₂×#14519683	C₂²	1	1	yes	characteristic
11	#16839589	—	2	1	no	—
12	#16839588	C₂²	1	1	yes	characteristic
13	#16839588	C₂²	1	1	yes	characteristic
14	#16839566	—	2	1	no	—
15	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#366	C₂²	1	1	yes	characteristic
19	#14519768	—	4	1	no	—
20	Q₈×Dic₆₁	—	4	1	no	—
21	C₂×#11758185	D₄	1	1	yes	characteristic
22	#14519711	—	2	1	no	—
23	#14519711	—	2	1	no	—
24	#14519708	—	4	1	no	—
25	#14519701	—	4	1	no	—
26	C₈×Dic₆₁	D₄	1	1	yes	characteristic
27	#14519683	—	2	1	no	—
28	C₂×Q₁₆×C₆₁	—	2	1	no	—
29	Q₃₂×C₆₁	—	2	1	no	—
30	Q₃₂×C₆₁	Q₈	1	2	yes	—
31	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
32	#107×C₆₁	—	2	1	no	—
33	C₃₂×C₆₁	—	2	1	no	—
34	C₂×Q₈×C₆₁	—	4	1	no	—
35	Q₁₆×C₆₁	—	2	2	no	—
36	Q₁₆×C₆₁	—	4	1	no	—
37	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₆₁	—	4	1	no	—
39	C₄⋊Dic₆₁	—	4	1	no	—
40	C₄⋊Dic₆₁	—	4	1	no	—
41	C₄⋊Dic₆₁	—	8	1	no	—
42	C₄⋊Dic₆₁	—	8	1	no	—
43	#11758196	—	8	1	no	—
44	C₄×Dic₆₁	—	8	1	no	—
45	C₄×Dic₆₁	D₈	1	1	yes	characteristic
46	C₂×#92418	D₈	1	1	yes	characteristic
47	C₁₆×C₆₁	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₆₁	#41	1	1	yes	characteristic
49	#11758185	—	2	1	no	—
50	Q₈×C₆₁	—	4	2	no	—
51	Q₈×C₆₁	—	8	1	no	—
52	C₂×C₄×C₆₁	—	8	1	no	—
53	C₂×C₄×C₆₁	—	8	1	no	—
54	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₆₁	—	8	1	no	—
56	C₂×Dic₆₁	—	8	1	no	—
57	C₂×Dic₆₁	D₁₆	1	1	yes	characteristic
58	C₂×Dic₆₁	D₁₆	1	1	yes	characteristic
59	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
60	C₈×C₆₁	#137	1	1	yes	characteristic
61	#92418	—	2	1	no	—
62	#13708	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₆₁	—	8	2	no	—
65	C₄×C₆₁	—	16	1	no	—
66	C₄×C₆₁	#474	1	1	yes	characteristic
67	C₄×C₆₁	#510	1	1	yes	characteristic
68	Dic₆₁	—	2	1	no	—
69	Dic₆₁	—	2	1	no	—
70	Dic₆₁	—	16	1	no	—
71	Dic₆₁	—	16	1	no	—
72	#2253	—	61	1	no	—
73	C₄×Q₃₂	—	61	1	no	—
74	#1424	—	61	1	no	—
75	#1423	—	61	1	no	—
76	C₄⋊C₃₂	—	61	1	no	—
77	#1416	—	61	1	no	—
78	#1416	D₆₁	1	1	yes	characteristic
79	C₂×C₆₁	#2224	1	1	yes	characteristic
80	C₂×C₆₁	#2262	1	1	yes	characteristic
81	C₂×C₆₁	#2264	1	1	yes	characteristic
82	C₂×Q₃₂	C₂×D₆₁	1	1	yes	characteristic
83	#499	—	122	1	no	—
84	C₄×Q₁₆	—	122	1	no	—
85	C₂×C₃₂	—	61	1	no	—
86	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
87	#367	—	122	1	no	—
88	#366	—	61	1	no	—
89	#366	—	61	1	no	—
90	#366	C₂×D₆₁	1	1	yes	characteristic
91	#358	—	122	1	no	—
92	C₄×C₁₆	—	61	1	no	—
93	C₆₁	#13708	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	2	1	no	—
95	#125	—	244	1	no	—
96	C₄×Q₈	—	244	1	no	—
97	Q₃₂	—	2	1	no	—
98	Q₃₂	Dic₁₂₂	1	2	yes	—
99	C₂×C₁₆	—	61	1	no	—
100	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	122	1	no	—
103	#107	—	122	1	no	—
104	#106	—	244	1	no	—
105	#103	—	244	1	no	—
106	C₄×C₈	—	61	1	no	—
107	C₃₂	—	2	1	no	—
108	C₃₂	—	122	1	no	—
109	C₂×Q₈	—	4	1	no	—
110	Q₁₆	—	2	2	no	—
111	Q₁₆	—	4	1	no	—
112	C₂×C₈	—	61	1	no	—
113	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	488	1	no	—
116	C₄⋊C₄	—	488	1	no	—
117	C₄⋊C₄	—	488	1	no	—
118	C₄⋊C₄	—	244	1	no	—
119	C₄⋊C₄	—	244	1	no	—
120	C₄²	—	61	1	no	—
121	C₄²	—	488	1	no	—
122	C₁₆	—	122	1	no	—
123	C₁₆	C₂×Dic₁₂₂	1	1	yes	characteristic
124	C₁₆	#11758224	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	61	1	no	—
130	C₂×C₄	—	61	1	no	—
131	C₂×C₄	—	488	1	no	—
132	C₂×C₄	—	488	1	no	—
133	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
134	C₈	—	122	1	no	—
135	C₈	#14519754	1	1	yes	characteristic
136	C₈	#14519806	1	1	yes	characteristic
137	C₂²	D₁₆×D₆₁	1	1	yes	characteristic
138	C₄	—	8	2	no	—
139	C₄	—	16	1	no	—
140	C₄	—	122	1	no	—
141	C₄	—	122	1	no	—
142	C₄	—	976	1	no	—
143	C₄	—	976	1	no	—
144	C₄	#16840040	1	1	yes	characteristic
145	C₄	#16840201	1	1	yes	characteristic
146	C₂	#21447954	1	1	yes	characteristic
147	C₂	#21448129	1	1	yes	characteristic
148	C₂	#21448135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448096

C₂

yes

maximal, characteristic

Q₃₂×Dic₆₁

C₂

yes

maximal, characteristic

#21444020

C₂

yes

maximal, characteristic

#21444019

C₂

yes

maximal, characteristic

#21444014

C₂

yes

maximal, characteristic

#21444000

C₂

yes

maximal, characteristic

C₆₁×#1416

C₂

yes

maximal, characteristic

#16840163

—

Q₁₆×Dic₆₁

—

C₂×#14519683

C₂²

yes

characteristic

#16839589

—

#16839588

C₂²

yes

characteristic

#16839588

C₂²

yes

characteristic

#16839566

—

C₁₆×Dic₆₁

C₂²

yes

characteristic

C₂×Q₃₂×C₆₁

C₂²

yes

characteristic

C₂×C₃₂×C₆₁

C₂²

yes

characteristic

C₆₁×#366

C₂²

yes

characteristic

#14519768

—

Q₈×Dic₆₁

—

C₂×#11758185

D₄

yes

—

—

—

—

C₈×Dic₆₁

D₄

yes

characteristic

#14519683

—

C₂×Q₁₆×C₆₁

—

Q₃₂×C₆₁

—

Q₃₂×C₆₁

Q₈

yes

—

C₂×C₁₆×C₆₁

C₂³

yes

characteristic

#107×C₆₁

—

C₃₂×C₆₁

—

C₂×Q₈×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₈

yes

characteristic

C₂×#92418

D₈

yes

characteristic

C₁₆×C₆₁

C₂×Q₈

yes

C₁₆×C₆₁

yes

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

D₁₆

yes

characteristic

C₂×Dic₆₁

D₁₆

yes

characteristic

C₈×C₆₁

C₂²⋊Q₈

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

C₂×D₁₆

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#2253

—

C₄×Q₃₂

—

#1424

—

#1423

—

C₄⋊C₃₂

—

#1416

—

#1416

D₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×Q₃₂

C₂×D₆₁

yes

characteristic

#499

—

122

—

C₄×Q₁₆

—

122

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₆₁

yes

characteristic

#367

—

122

—

#366

—

#366

—

#366

C₂×D₆₁

yes

characteristic

#358

—

122

—

C₄×C₁₆

—

C₆₁

#13708

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

#125

—

244

—

C₄×Q₈

—

244

—

Q₃₂

—

Q₃₂

Dic₁₂₂

yes

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂²×D₆₁

yes

characteristic

101

#107

—

102

#107

—

122

—

103

#107

—

122

—

104

#106

—

244

—

105

#103

—

244

—

106

C₄×C₈

—

107

C₃₂

—

108

C₃₂

—

122

—

109

C₂×Q₈

—

110

Q₁₆

—

111

Q₁₆

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₆₁

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

488

—

116

C₄⋊C₄

—

488

—

117

C₄⋊C₄

—

488

—

118

C₄⋊C₄

—

244

—

119

C₄⋊C₄

—

244

—

120

C₄²

—

121

C₄²

—

488

—

122

C₁₆

—

122

—

123

C₁₆

C₂×Dic₁₂₂

yes

characteristic

124

C₁₆

#11758224

yes

characteristic

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

488

—

132

C₂×C₄

—

488

—

133

C₂×C₄

D₈×D₆₁

yes

characteristic

134

C₈

—

122

—

135

C₈

#14519754

yes

characteristic

136

C₈

#14519806

yes

characteristic

137

C₂²

D₁₆×D₆₁

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

122

—

141

C₄

—

122

—

142

C₄

—

976

—

143

C₄

—

976

—

144

C₄

#16840040

yes

characteristic

145

C₄

#16840201

yes

characteristic

146

C₂

#21447954

yes

characteristic

147

C₂

#21448129

yes

characteristic

148

C₂

#21448135

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₆₁
commutator, cosocle	C₂×C₁₆×C₆₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₆₁×#1416	C₂
socle, prime radical	C₂²×C₆₁	C₂×D₁₆
hypercenter	#1416	D₆₁
hypocenter	C₆₁	#13708
LCS	G ▷ C₂×C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₆₁

commutator, cosocle

C₂×C₁₆×C₆₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₆₁

Fitting

C₆₁×#1416

C₂

socle, prime radical

C₂²×C₆₁

C₂×D₁₆

D₆₁

C₆₁

G ▷ C₂×C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
93. C₆₁	#13708	semidirect

H ≅ G/N

type

93. C₆₁

#13708

semidirect

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

subgroup	index
#21448096	2
Q₃₂×Dic₆₁	2
#21444020	2
#21444019	2
#21444014	2
#21444000	2
C₆₁×#1416	2
#13708	61

subgroup

index

Q₃₂×Dic₆₁

C₆₁×#1416

Subgroups — #29600924

Conjugacy classes of subgroups

Complements

Maximal subgroups