Group #29598548 — subgroups

Subgroups — #29598548

Order 15616.

334 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 139 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	#21454753	C₂	1	1	yes	maximal, characteristic
2	#21447575	C₂	1	1	yes	maximal, characteristic
3	#21443907	C₂	1	4	yes	maximal
4	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic
5	#16840319	C₄	1	1	yes	characteristic
6	C₂×#14519717	C₂²	1	2	yes	—
7	#16839892	—	2	2	no	—
8	C₈×Dic₁₂₂	—	2	2	no	—
9	#16839788	C₄	1	2	yes	—
10	#16839786	C₄	1	1	yes	characteristic
11	#16839717	C₂²	1	2	yes	—
12	C₄⋊#11758185	—	2	2	no	—
13	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#346	—	2	2	no	—
16	C₂×#11758194	C₂×C₄	1	2	yes	—
17	#14519777	C₈	1	2	yes	—
18	#14519769	C₈	1	2	yes	—
19	#14519756	—	2	2	no	—
20	C₄×Dic₁₂₂	—	2	2	no	—
21	#14519736	C₂×C₄	1	2	yes	—
22	#14519717	—	2	4	no	—
23	C₂×#11758185	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	#14519706	—	2	4	no	—
26	#14519687	—	2	2	no	—
27	C₈×Dic₆₁	—	2	2	no	—
28	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₆₁	—	2	2	no	—
32	C₂×C₁₆×C₆₁	—	2	2	no	—
33	C₂×C₁₆×C₆₁	Q₈	1	4	yes	complemented
34	C₄×C₈×C₆₁	D₄	1	2	yes	—
35	C₄×C₈×C₆₁	D₄	1	2	yes	—
36	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
37	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₁₂₂	—	2	2	no	—
39	C₂×C₈×C₆₁	—	2	2	no	—
40	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₆₁	#41	1	2	yes	—
45	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
46	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
48	C₂²⋊Dic₆₁	—	2	2	no	—
49	C₄⋊Dic₆₁	—	2	2	no	—
50	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
51	#11758196	—	2	4	no	—
52	C₄×Dic₆₁	—	2	2	no	—
53	#11758194	—	2	4	no	—
54	C₂×#92418	—	2	2	no	—
55	C₁₆×C₆₁	—	2	4	no	—
56	#11758185	—	4	2	no	—
57	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₆₁	—	4	2	no	—
59	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#131	1	2	yes	—
65	C₂×Dic₆₁	—	2	4	no	—
66	Dic₁₂₂	—	4	4	no	—
67	C₈×C₆₁	—	2	2	no	—
68	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
69	C₈×C₆₁	#124	1	1	yes	characteristic
70	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
71	#92418	—	4	2	no	—
72	#13267	—	61	1	no	maximal, Sylow
73	C₂²×C₆₁	—	2	2	no	—
74	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₂²×C₆₁	#446	1	2	yes	—
76	C₄×C₆₁	—	2	2	no	—
77	C₄×C₆₁	—	2	2	no	—
78	C₄×C₆₁	#391	1	1	yes	characteristic
79	C₄×C₆₁	#408	1	1	yes	characteristic
80	C₄×C₆₁	C₈×Q₈	1	2	yes	—
81	Dic₆₁	—	4	4	no	—
82	#2960	—	61	1	no	—
83	#2150	—	61	1	no	—
84	C₂×#346	D₆₁	1	1	yes	characteristic
85	#1379	—	61	4	no	—
86	C₂×C₆₁	—	2	2	no	—
87	C₂×C₆₁	#1984	1	1	yes	characteristic
88	C₂×C₆₁	#2117	1	2	yes	—
89	#533	—	61	1	no	—
90	C₂×M₃₂	—	61	2	no	—
91	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
92	#446	—	122	2	no	—
93	C₈×Q₈	—	122	2	no	—
94	#424	—	61	2	no	—
95	#423	—	61	1	no	—
96	#405	—	61	2	no	—
97	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
98	C₄⋊C₁₆	—	122	2	no	—
99	#346	—	2	2	no	—
100	C₆₁	#13267	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	61	2	no	—
102	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
104	C₄⋊Q₈	—	61	2	no	—
105	#125	—	61	2	no	—
106	C₂²⋊Q₈	—	122	2	no	—
107	C₄×Q₈	—	122	2	no	—
108	#117	—	61	2	no	—
109	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
110	M₃₂	—	122	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	122	2	no	—
114	C₂×C₁₆	Dic₁₂₂	1	4	yes	complemented
115	C₄⋊C₈	—	122	4	no	—
116	C₄⋊C₈	—	122	2	no	—
117	#97	—	122	2	no	—
118	C₄×C₈	—	122	2	no	—
119	C₄×C₈	D₂₄₄	1	2	yes	—
120	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
121	C₂×Q₈	—	122	2	no	—
122	C₂²×C₄	#11758189	1	2	yes	—
123	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
124	M₁₆	—	122	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	122	2	no	—
127	C₂×C₈	#11758198	1	2	yes	—
128	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
129	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
130	C₂×C₈	#11758222	1	2	yes	—
131	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
132	C₄⋊C₄	—	61	4	no	—
133	C₄⋊C₄	—	122	4	no	—
134	C₄⋊C₄	—	122	2	no	—
135	C₂²⋊C₄	—	122	2	no	—
136	C₄²	—	122	2	no	—
137	C₄²	C₈×D₆₁	1	1	yes	characteristic
138	C₄²	C₈×D₆₁	1	1	yes	characteristic
139	C₄²	#11758198	1	2	yes	—
140	C₁₆	—	2	4	no	—
141	C₁₆	—	244	2	no	—
142	C₂³	C₂×#11758189	1	1	yes	characteristic
143	Q₈	—	244	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	122	4	no	—
146	C₂×C₄	#14519689	1	2	yes	—
147	C₂×C₄	#14519689	1	2	yes	—
148	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
149	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
150	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
151	C₂×C₄	#14519783	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	244	2	no	—
154	C₈	#14519756	1	2	yes	—
155	C₈	#14519765	1	1	yes	characteristic
156	C₈	#14519777	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×#14519689	1	1	yes	characteristic
159	C₂²	#16839892	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	244	4	no	—
163	C₄	#16839668	1	1	yes	characteristic
164	C₄	#16839728	1	1	yes	characteristic
165	C₄	C₈×Dic₁₂₂	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#21446755	1	1	yes	characteristic
168	C₂	#21447446	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21454753

C₂

yes

maximal, characteristic

#21447575

C₂

yes

maximal, characteristic

#21443907

C₂

yes

maximal

C₂×C₆₁×#346

C₂

yes

maximal, characteristic

#16840319

C₄

yes

characteristic

C₂×#14519717

C₂²

yes

—

#16839892

—

C₈×Dic₁₂₂

—

#16839788

C₄

yes

—

#16839786

C₄

yes

characteristic

#16839717

C₂²

yes

—

C₄⋊#11758185

—

C₂²×C₁₆×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#346

—

C₂×#11758194

C₂×C₄

yes

—

#14519777

C₈

yes

—

#14519769

C₈

yes

—

#14519756

—

C₄×Dic₁₂₂

—

#14519736

C₂×C₄

yes

—

—

—

—

—

—

C₈×Dic₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

Q₈

yes

complemented

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄²×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₈

yes

—

#11758196

—

C₄×Dic₆₁

—

#11758194

—

C₂×#92418

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×Q₈

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

#131

yes

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊Q₈

yes

—

C₈×C₆₁

#124

yes

characteristic

C₈×C₆₁

C₄⋊Q₈

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₆₁

#446

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₈×Q₈

yes

—

Dic₆₁

—

#2960

—

#2150

—

C₂×#346

D₆₁

yes

characteristic

#1379

—

C₂×C₆₁

—

C₂×C₆₁

#1984

yes

characteristic

C₂×C₆₁

#2117

yes

—

#533

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₆₁

yes

—

#446

—

122

—

C₈×Q₈

—

122

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

122

—

#346

—

100

C₆₁

#13267

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

102

C₂²×C₈

C₄×D₆₁

yes

characteristic

103

C₂²×C₈

C₂²×D₆₁

yes

characteristic

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

122

—

107

C₄×Q₈

—

122

—

108

#117

—

109

C₂×C₄²

C₄×D₆₁

yes

characteristic

110

M₃₂

—

122

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

122

—

114

C₂×C₁₆

Dic₁₂₂

yes

complemented

115

C₄⋊C₈

—

122

—

116

C₄⋊C₈

—

122

—

117

#97

—

122

—

118

C₄×C₈

—

122

—

119

C₄×C₈

D₂₄₄

yes

—

120

C₄×C₈

C₂²⋊D₆₁

yes

—

121

C₂×Q₈

—

122

—

122

C₂²×C₄

#11758189

yes

—

123

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

124

M₁₆

—

122

—

125

C₂×C₈

—

126

C₂×C₈

—

122

—

127

C₂×C₈

#11758198

yes

—

128

C₂×C₈

C₂×Dic₁₂₂

yes

—

129

C₂×C₈

C₂×D₂₄₄

yes

characteristic

130

C₂×C₈

#11758222

yes

—

131

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

122

—

134

C₄⋊C₄

—

122

—

135

C₂²⋊C₄

—

122

—

136

C₄²

—

122

—

137

C₄²

C₈×D₆₁

yes

characteristic

138

C₄²

C₈×D₆₁

yes

characteristic

139

C₄²

#11758198

yes

—

140

C₁₆

—

141

C₁₆

—

244

—

142

C₂³

C₂×#11758189

yes

characteristic

143

Q₈

—

244

—

144

C₂×C₄

—

145

C₂×C₄

—

122

—

146

C₂×C₄

#14519689

yes

—

147

C₂×C₄

#14519689

yes

—

148

C₂×C₄

C₂×#11758198

yes

characteristic

149

C₂×C₄

C₄×Dic₁₂₂

yes

—

150

C₂×C₄

C₂×C₈×D₆₁

yes

characteristic

151

C₂×C₄

#14519783

yes

—

152

C₈

—

153

C₈

—

244

—

154

C₈

#14519756

yes

—

155

C₈

#14519765

yes

characteristic

156

C₈

#14519777

yes

characteristic

157

C₂²

—

158

C₂²

C₂×#14519689

yes

characteristic

159

C₂²

#16839892

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

244

—

163

C₄

#16839668

yes

characteristic

164

C₄

#16839728

yes

characteristic

165

C₄

C₈×Dic₁₂₂

yes

—

166

C₂

—

167

C₂

#21446755

yes

characteristic

168

C₂

#21447446

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#346	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#346	D₆₁
hypocenter	C₆₁	#13267
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₈

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₂×C₆₁×#346

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₈

prime radical

C₂³×C₆₁

C₂×M₁₆

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
33. C₂×C₁₆×C₆₁	Q₈	semidirect
100. C₆₁	#13267	semidirect
114. C₂×C₁₆	Dic₁₂₂	semidirect

H ≅ G/N

type

33. C₂×C₁₆×C₆₁

Q₈

semidirect

100. C₆₁

#13267

semidirect

114. C₂×C₁₆

Dic₁₂₂

semidirect

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

subgroup	index
#21454753	2
#21447575	2
#21443907	2
C₂×C₆₁×#346	2
#13267	61

subgroup

index

C₂×C₆₁×#346

Subgroups — #29598548

Conjugacy classes of subgroups

Complements

Maximal subgroups