Group #29571024 — subgroups

Subgroups — #29571024

Order 15616.

384 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 133 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	#21454754	C₂	1	1	yes	maximal, characteristic
2	#21444414	C₂	1	1	yes	maximal, characteristic
3	#21444100	C₂	1	2	yes	maximal
4	#21443710	C₂	1	2	yes	maximal
5	C₆₁×#1544	C₂	1	1	yes	maximal, characteristic
6	#16839893	—	2	2	no	—
7	Q₈×#92418	—	2	2	no	—
8	#16839789	C₄	1	2	yes	—
9	#16839786	C₂²	1	1	yes	characteristic
10	#16839718	C₄	1	2	yes	—
11	C₄×#11758194	C₂²	1	2	yes	—
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	#16839504	C₄	1	4	yes	—
14	#16839484	—	2	2	no	—
15	C₈×#92418	—	2	2	no	—
16	C₆₁×#533	C₂²	1	1	yes	characteristic
17	C₆₁×#431	C₂²	1	2	yes	—
18	C₆₁×#332	C₄	1	4	yes	—
19	C₂×#11758194	—	2	2	no	—
20	C₂×#11758194	C₂×C₄	1	2	yes	—
21	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
22	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	4	no	—
25	#14519688	—	2	2	no	—
26	#14519688	C₂×C₄	1	4	yes	—
27	C₄×#92418	—	2	2	no	—
28	C₄×#92418	—	2	2	no	—
29	C₄×#92418	D₄	1	2	yes	—
30	C₄×#92418	D₄	1	2	yes	—
31	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
32	#125×C₆₁	C₂×C₄	1	2	yes	—
33	C₂²⋊Q₈×C₆₁	—	2	2	no	—
34	C₄×Q₈×C₆₁	—	2	2	no	—
35	#117×C₆₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
38	C₂²⋊C₈×C₆₁	—	2	2	no	—
39	C₄×C₈×C₆₁	—	2	2	no	—
40	C₂×Q₈×C₆₁	—	2	2	no	—
41	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	#41	1	2	yes	—
43	C₂×C₈×C₆₁	—	2	4	no	—
44	C₄⋊C₄×C₆₁	—	2	4	no	—
45	C₄⋊C₄×C₆₁	—	2	2	no	—
46	C₄⋊C₄×C₆₁	C₄²	1	4	yes	—
47	C₂²⋊C₄×C₆₁	—	2	2	no	—
48	C₄²×C₆₁	—	2	2	no	—
49	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₆₁	C₂×D₄	1	2	yes	—
52	#11758194	—	2	8	no	—
53	#11758194	—	4	2	no	—
54	C₂×#92418	—	2	4	no	—
55	C₂×#92418	—	2	2	no	—
56	C₂×#92418	—	2	2	no	—
57	C₂×#92418	—	4	1	no	—
58	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×#92418	C₂²⋊C₄	1	2	yes	—
60	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
61	C₂³×C₆₁	#117	1	1	yes	characteristic
62	Q₈×C₆₁	—	4	4	no	—
63	C₂×C₄×C₆₁	—	2	1	no	—
64	C₂×C₄×C₆₁	—	2	1	no	—
65	C₂×C₄×C₆₁	—	2	4	no	—
66	C₂×C₄×C₆₁	—	2	4	no	—
67	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
71	C₈×C₆₁	—	4	4	no	—
72	#92418	—	2	8	no	—
73	#92418	—	2	1	no	—
74	#92418	—	2	2	no	—
75	#92418	—	4	2	no	—
76	#92418	#101	1	2	yes	—
77	#8242	—	61	1	no	maximal, Sylow
78	C₂²×C₆₁	—	2	1	no	—
79	C₂²×C₆₁	—	2	2	no	—
80	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
81	C₄×C₆₁	—	2	1	no	—
82	C₄×C₆₁	—	2	2	no	—
83	C₄×C₆₁	—	4	4	no	—
84	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
85	C₄×C₆₁	#413	1	1	yes	characteristic
86	C₄×C₆₁	#443	1	4	yes	—
87	#2960	—	61	1	no	—
88	#1544	D₆₁	1	1	yes	characteristic
89	#1526	—	61	1	no	—
90	C₈×M₁₆	—	61	2	no	—
91	#1299	—	61	2	no	—
92	C₂×C₆₁	—	4	1	no	—
93	C₂×C₆₁	C₄×#101	1	1	yes	characteristic
94	C₂×C₆₁	#1764	1	2	yes	—
95	#533	C₂×D₆₁	1	1	yes	characteristic
96	#446	—	122	2	no	—
97	C₈×Q₈	—	122	2	no	—
98	#431	C₂×D₆₁	1	2	yes	—
99	#424	—	61	2	no	—
100	#423	—	61	1	no	—
101	#405	—	61	2	no	—
102	C₄×M₁₆	—	61	2	no	—
103	C₂×C₄×C₈	—	61	1	no	—
104	#332	Dic₆₁	1	4	yes	—
105	#330	—	61	4	no	—
106	#322	—	122	2	no	—
107	C₈²	—	122	2	no	—
108	C₆₁	#8242	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₁₆	—	61	2	no	—
110	C₂×M₁₆	—	122	2	no	—
111	C₂²×C₈	—	61	1	no	—
112	C₂²×C₈	—	61	1	no	—
113	C₄⋊Q₈	C₂×Dic₆₁	1	2	yes	—
114	#125	C₄×D₆₁	1	2	yes	—
115	C₂²⋊Q₈	—	2	2	no	—
116	C₄×Q₈	—	2	2	no	—
117	#117	C₄×D₆₁	1	2	yes	—
118	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
119	C₄⋊C₈	—	122	4	no	—
120	C₄⋊C₈	—	122	2	no	—
121	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
122	C₂²⋊C₈	—	2	2	no	—
123	#97	—	61	4	no	—
124	#97	—	122	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	61	2	no	—
127	C₄×C₈	—	61	2	no	—
128	C₄×C₈	—	122	2	no	—
129	C₄×C₈	—	122	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
132	C₂²×C₄	#11758224	1	2	yes	—
133	M₁₆	—	122	8	no	—
134	M₁₆	—	244	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	61	1	no	—
137	C₂×C₈	—	61	1	no	—
138	C₂×C₈	—	61	2	no	—
139	C₂×C₈	—	122	2	no	—
140	C₂×C₈	—	122	4	no	—
141	C₂×C₈	—	122	2	no	—
142	C₂×C₈	—	244	1	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	C₄×Dic₆₁	1	4	yes	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
149	C₄²	D₄×D₆₁	1	2	yes	—
150	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
151	C₂³	#14519733	1	1	yes	characteristic
152	Q₈	—	4	4	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
159	C₂×C₄	#14519737	1	2	yes	—
160	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
161	C₈	—	4	4	no	—
162	C₈	—	61	2	no	—
163	C₈	—	122	8	no	—
164	C₈	—	122	1	no	—
165	C₈	—	122	2	no	—
166	C₈	—	244	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	C₂²⋊C₄×Dic₆₁	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	4	no	—
173	C₄	#101×D₆₁	1	1	yes	characteristic
174	C₄	#16839745	1	1	yes	characteristic
175	C₄	#16839879	1	4	yes	—
176	C₂	—	4	1	no	—
177	C₂	#101×Dic₆₁	1	1	yes	characteristic
178	C₂	#21445659	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21454754

C₂

yes

maximal, characteristic

#21444414

C₂

yes

maximal, characteristic

#21444100

C₂

yes

maximal

#21443710

C₂

yes

maximal

C₆₁×#1544

C₂

yes

maximal, characteristic

#16839893

—

Q₈×#92418

—

#16839789

C₄

yes

—

#16839786

C₂²

yes

characteristic

#16839718

C₄

yes

—

C₄×#11758194

C₂²

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

#16839504

C₄

yes

—

#16839484

—

C₈×#92418

—

C₆₁×#533

C₂²

yes

characteristic

C₆₁×#431

C₂²

yes

—

C₆₁×#332

C₄

yes

—

C₂×#11758194

—

C₂×#11758194

C₂×C₄

yes

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

—

—

—

C₂×C₄

yes

—

C₄×#92418

—

C₄×#92418

—

C₄×#92418

D₄

yes

—

C₄×#92418

D₄

yes

—

C₄⋊Q₈×C₆₁

C₂×C₄

yes

—

#125×C₆₁

C₂×C₄

yes

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

C₂×C₄

yes

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₄²

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

—

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₂²⋊C₄

yes

—

C₂×#92418

C₂²⋊C₄

yes

characteristic

C₂³×C₆₁

#117

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₈×C₆₁

—

—

—

—

—

yes

—

#8242

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

#2960

—

#1544

D₆₁

yes

characteristic

#1526

—

C₈×M₁₆

—

#1299

—

C₂×C₆₁

—

C₂×C₆₁

C₄×#101

yes

characteristic

C₂×C₆₁

#1764

yes

—

#533

C₂×D₆₁

yes

characteristic

#446

—

122

—

C₈×Q₈

—

122

—

#431

C₂×D₆₁

yes

—

#424

—

100

#423

—

101

#405

—

102

C₄×M₁₆

—

103

C₂×C₄×C₈

—

104

#332

Dic₆₁

yes

—

105

#330

—

106

#322

—

122

—

107

C₈²

—

122

—

108

C₆₁

#8242

yes

characteristic, complemented, Sylow

109

C₂×M₁₆

—

110

C₂×M₁₆

—

122

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₄⋊Q₈

C₂×Dic₆₁

yes

—

114

#125

C₄×D₆₁

yes

—

115

C₂²⋊Q₈

—

116

C₄×Q₈

—

117

#117

C₄×D₆₁

yes

—

118

C₂×C₄²

C₂²×D₆₁

yes

characteristic

119

C₄⋊C₈

—

122

—

120

C₄⋊C₈

—

122

—

121

C₄⋊C₈

C₂×Dic₆₁

yes

—

122

C₂²⋊C₈

—

123

#97

—

124

#97

—

122

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

122

—

129

C₄×C₈

—

122

—

130

C₂×Q₈

—

131

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

132

C₂²×C₄

#11758224

yes

—

133

M₁₆

—

122

—

134

M₁₆

—

244

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

122

—

140

C₂×C₈

—

122

—

141

C₂×C₈

—

122

—

142

C₂×C₈

—

244

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

C₄×Dic₆₁

yes

—

146

C₂²⋊C₄

—

147

C₄²

—

148

C₄²

C₂×C₄×D₆₁

yes

characteristic

149

C₄²

D₄×D₆₁

yes

—

150

C₄²

C₂²×Dic₆₁

yes

characteristic

151

C₂³

#14519733

yes

characteristic

152

Q₈

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂×C₄×Dic₆₁

yes

characteristic

158

C₂×C₄

C₂²⋊C₄×D₆₁

yes

characteristic

159

C₂×C₄

#14519737

yes

—

160

C₂×C₄

D₄×Dic₆₁

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

122

—

164

C₈

—

122

—

165

C₈

—

122

—

166

C₈

—

244

—

167

C₂²

—

168

C₂²

—

169

C₂²

C₂²⋊C₄×Dic₆₁

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#101×D₆₁

yes

characteristic

174

C₄

#16839745

yes

characteristic

175

C₄

#16839879

yes

—

176

C₂

—

177

C₂

#101×Dic₆₁

yes

characteristic

178

C₂

#21445659

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1544	C₂
socle	C₂²×C₆₁	C₄×C₂²⋊C₄
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1544	D₆₁
hypocenter	C₆₁	#8242
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

subgroup

G/H

center

C₂×C₄

C₂²⋊C₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1544

C₂

socle

C₂²×C₆₁

C₄×C₂²⋊C₄

prime radical

C₂³×C₆₁

#117

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

N	H ≅ G/N	type
108. C₆₁	#8242	semidirect

H ≅ G/N

type

108. C₆₁

#8242

semidirect

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

subgroup	index
#21454754	2
#21444414	2
#21444100	2
#21443710	2
C₆₁×#1544	2
#8242	61

subgroup

index

C₆₁×#1544

Subgroups — #29571024

Conjugacy classes of subgroups

Complements

Maximal subgroups