Group #29574968 — subgroups

Subgroups — #29574968

Order 15616.

376 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 103 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	#21454925	C₂	1	1	yes	maximal, characteristic
2	#21444484	C₂	1	1	yes	maximal, characteristic
3	#21444292	C₂	1	2	yes	maximal
4	#21443718	C₂	1	2	yes	maximal
5	C₆₁×#3162	C₂	1	1	yes	maximal, characteristic
6	#16839893	—	2	4	no	—
7	#16839823	—	2	2	no	—
8	#16839819	C₂²	1	2	yes	—
9	#16839815	C₂²	1	2	yes	—
10	#16839508	—	2	2	no	—
11	Q₈⋊#92418	—	2	4	no	—
12	C₆₁×#533	C₂²	1	1	yes	characteristic
13	C₆₁×#533	C₂²	1	1	yes	characteristic
14	C₆₁×#501	—	2	2	no	—
15	C₆₁×#501	C₄	1	4	yes	—
16	C₆₁×#427	—	2	2	no	—
17	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
18	C₄⋊#92418	—	2	4	no	—
19	C₄⋊#92418	—	2	2	no	—
20	C₄⋊#92418	D₄	1	4	yes	—
21	C₂²⋊#92418	—	2	2	no	—
22	C₂²⋊#92418	—	2	2	no	—
23	#14519688	—	2	2	no	—
24	#14519688	—	2	2	no	—
25	C₄×#92418	—	2	2	no	—
26	C₄×#92418	—	2	2	no	—
27	C₂²×C₈×C₆₁	—	2	1	no	—
28	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
30	#125×C₆₁	—	2	1	no	—
31	#125×C₆₁	C₂×C₄	1	2	yes	—
32	C₂²⋊Q₈×C₆₁	—	2	2	no	—
33	C₂²⋊Q₈×C₆₁	—	2	2	no	—
34	C₄×Q₈×C₆₁	—	2	2	no	—
35	C₄×Q₈×C₆₁	—	2	2	no	—
36	#117×C₆₁	—	2	1	no	—
37	#117×C₆₁	D₄	1	2	yes	—
38	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	#107×C₆₁	—	2	4	no	—
40	#106×C₆₁	—	2	4	no	—
41	#97×C₆₁	—	2	1	no	—
42	#97×C₆₁	C₂×C₄	1	2	yes	—
43	C₂×Q₈×C₆₁	—	2	2	no	—
44	C₂×Q₈×C₆₁	—	2	2	no	—
45	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	#41	1	2	yes	—
47	C₂×C₈×C₆₁	—	2	4	no	—
48	C₂×C₈×C₆₁	—	2	2	no	—
49	C₄⋊C₄×C₆₁	—	2	4	no	—
50	C₄⋊C₄×C₆₁	—	2	2	no	—
51	C₄⋊C₄×C₆₁	—	2	2	no	—
52	C₄⋊C₄×C₆₁	—	2	2	no	—
53	C₄⋊C₄×C₆₁	—	4	2	no	—
54	C₄⋊C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
55	C₂²⋊C₄×C₆₁	—	2	2	no	—
56	C₂²⋊C₄×C₆₁	—	4	1	no	—
57	C₄²×C₆₁	—	2	2	no	—
58	C₄²×C₆₁	—	4	1	no	—
59	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₆₁	C₂×D₄	1	2	yes	—
62	C₂×#92418	—	2	4	no	—
63	C₂×#92418	—	4	2	no	—
64	C₂³×C₆₁	#123	1	1	yes	characteristic
65	Q₈×C₆₁	—	4	4	no	—
66	Q₈×C₆₁	—	8	2	no	—
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	2	4	no	—
69	C₂×C₄×C₆₁	—	4	2	no	—
70	C₂×C₄×C₆₁	—	4	2	no	—
71	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#121	1	2	yes	—
75	C₈×C₆₁	—	2	4	no	—
76	#92418	—	4	4	no	—
77	#92418	—	8	2	no	—
78	#9006	—	61	1	no	maximal, Sylow
79	C₂²×C₆₁	—	2	2	no	—
80	C₂²×C₆₁	—	2	1	no	—
81	C₂²×C₆₁	#386	1	1	yes	characteristic
82	C₄×C₆₁	—	2	2	no	—
83	C₄×C₆₁	—	2	2	no	—
84	C₄×C₆₁	—	2	1	no	—
85	C₄×C₆₁	—	4	4	no	—
86	C₄×C₆₁	—	8	2	no	—
87	C₄×C₆₁	#456	1	2	yes	—
88	#3162	D₆₁	1	1	yes	characteristic
89	#2987	—	61	1	no	—
90	#1544	—	61	1	no	—
91	#1497	—	61	2	no	—
92	#1302	—	61	2	no	—
93	C₂×C₆₁	—	4	1	no	—
94	C₂×C₆₁	#1881	1	2	yes	—
95	C₂×C₆₁	#1886	1	1	yes	characteristic
96	#533	C₂×D₆₁	1	1	yes	characteristic
97	#533	C₂×D₆₁	1	1	yes	characteristic
98	#501	—	2	2	no	—
99	#501	Dic₆₁	1	4	yes	—
100	#446	—	122	4	no	—
101	#433	—	122	2	no	—
102	#432	—	61	2	no	—
103	#431	—	61	2	no	—
104	#427	—	2	2	no	—
105	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
106	#332	—	122	2	no	—
107	Q₈⋊C₈	—	122	4	no	—
108	C₆₁	#9006	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	2	1	no	—
110	C₄⋊Q₈	C₂×Dic₆₁	1	2	yes	—
111	C₄⋊Q₈	C₂²⋊D₆₁	1	2	yes	—
112	#125	—	2	1	no	—
113	#125	C₂×Dic₆₁	1	2	yes	—
114	C₂²⋊Q₈	—	2	2	no	—
115	C₂²⋊Q₈	—	2	2	no	—
116	C₄×Q₈	—	2	2	no	—
117	C₄×Q₈	—	2	2	no	—
118	#117	—	2	1	no	—
119	#117	C₂²⋊D₆₁	1	2	yes	—
120	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
121	#107	—	2	4	no	—
122	#106	—	2	4	no	—
123	C₄⋊C₈	—	61	4	no	—
124	C₄⋊C₈	—	122	4	no	—
125	C₄⋊C₈	—	122	2	no	—
126	C₂²⋊C₈	—	122	2	no	—
127	C₂²⋊C₈	—	122	2	no	—
128	#97	—	2	1	no	—
129	#97	—	122	2	no	—
130	#97	—	122	2	no	—
131	#97	C₂×Dic₆₁	1	2	yes	—
132	C₄×C₈	—	122	2	no	—
133	C₄×C₈	—	122	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂²×C₄	#11758224	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	122	4	no	—
141	C₂×C₈	—	244	2	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	C₂²⋊Dic₆₁	1	4	yes	—
148	C₂²⋊C₄	—	2	2	no	—
149	C₂²⋊C₄	—	4	1	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	4	1	no	—
152	C₄²	D₄×D₆₁	1	2	yes	—
153	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
154	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
155	C₂³	#14519758	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
163	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
164	C₂×C₄	#14519745	1	1	yes	characteristic
165	C₂×C₄	#14519749	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	488	2	no	—
168	C₈	—	244	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#16839642	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	4	no	—
176	C₄	—	8	2	no	—
177	C₄	#16839944	1	2	yes	—
178	C₂	—	4	1	no	—
179	C₂	#21446183	1	2	yes	—
180	C₂	#21446213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21454925

C₂

yes

maximal, characteristic

#21444484

C₂

yes

maximal, characteristic

#21444292

C₂

yes

maximal

#21443718

C₂

yes

maximal

C₆₁×#3162

C₂

yes

maximal, characteristic

#16839893

—

#16839823

—

#16839819

C₂²

yes

—

#16839815

C₂²

yes

—

#16839508

—

Q₈⋊#92418

—

C₆₁×#533

C₂²

yes

characteristic

C₆₁×#533

C₂²

yes

characteristic

C₆₁×#501

—

C₆₁×#501

C₄

yes

—

C₆₁×#427

—

C₂×#97×C₆₁

C₂²

yes

—

—

D₄

yes

—

—

—

—

—

—

—

C₂²×C₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂×C₄

yes

—

C₄⋊Q₈×C₆₁

D₄

yes

—

#125×C₆₁

—

#125×C₆₁

C₂×C₄

yes

—

C₂²⋊Q₈×C₆₁

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

—

#117×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#107×C₆₁

—

#106×C₆₁

—

#97×C₆₁

—

#97×C₆₁

C₂×C₄

yes

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#123

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#120

yes

characteristic

C₂×C₄×C₆₁

#121

yes

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#386

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#456

yes

—

#3162

D₆₁

yes

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#1881

yes

—

C₂×C₆₁

#1886

yes

characteristic

#533

C₂×D₆₁

yes

characteristic

#533

C₂×D₆₁

yes

characteristic

#501

—

#501

Dic₆₁

yes

—

100

#446

—

122

—

101

#433

—

122

—

102

#432

—

103

#431

—

104

#427

—

105

C₂×#97

C₂×D₆₁

yes

characteristic

106

#332

—

122

—

107

Q₈⋊C₈

—

122

—

108

C₆₁

#9006

yes

characteristic, complemented, Sylow

109

C₂²×C₈

—

110

C₄⋊Q₈

C₂×Dic₆₁

yes

—

111

C₄⋊Q₈

C₂²⋊D₆₁

yes

—

112

#125

—

113

#125

C₂×Dic₆₁

yes

—

114

C₂²⋊Q₈

—

115

C₂²⋊Q₈

—

116

C₄×Q₈

—

117

C₄×Q₈

—

118

#117

—

119

#117

C₂²⋊D₆₁

yes

—

120

C₂×C₄²

C₂²×D₆₁

yes

characteristic

121

#107

—

122

#106

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

122

—

125

C₄⋊C₈

—

122

—

126

C₂²⋊C₈

—

122

—

127

C₂²⋊C₈

—

122

—

128

#97

—

129

#97

—

122

—

130

#97

—

122

—

131

#97

C₂×Dic₆₁

yes

—

132

C₄×C₈

—

122

—

133

C₄×C₈

—

122

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

#11758224

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

122

—

141

C₂×C₈

—

244

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

C₂²⋊Dic₆₁

yes

—

148

C₂²⋊C₄

—

149

C₂²⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

D₄×D₆₁

yes

—

153

C₄²

C₂²×Dic₆₁

yes

characteristic

154

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

155

C₂³

#14519758

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

163

C₂×C₄

D₄×Dic₆₁

yes

—

164

C₂×C₄

#14519745

yes

characteristic

165

C₂×C₄

#14519749

yes

—

166

C₈

—

167

C₈

—

488

—

168

C₈

—

244

—

169

C₂²

—

170

C₂²

—

171

C₂²

#16839642

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#16839944

yes

—

178

C₂

—

179

C₂

#21446183

yes

—

180

C₂

#21446213

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14519745
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#3162	C₂
socle	C₂²×C₆₁	#386
prime radical	C₂³×C₆₁	#123
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#3162	D₆₁
hypocenter	C₆₁	#9006
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

subgroup

G/H

center

C₂×C₄

#14519745

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#3162

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

N	H ≅ G/N	type
108. C₆₁	#9006	semidirect

H ≅ G/N

type

108. C₆₁

#9006

semidirect

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

subgroup	index
#21454925	2
#21444484	2
#21444292	2
#21443718	2
C₆₁×#3162	2
#9006	61

subgroup

index

C₆₁×#3162

Subgroups — #29574968

Conjugacy classes of subgroups

Complements

Maximal subgroups