Group #29573008 — subgroups

Subgroups — #29573008

Order 15616.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#21454749	C₂	1	1	yes	maximal, characteristic
2	#21444580	C₂	1	2	yes	maximal
3	#21444456	C₂	1	2	yes	maximal
4	#21443730	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1307	C₂	1	1	yes	maximal, characteristic
6	#16840319	C₂²	1	1	yes	characteristic
7	#16839889	—	2	2	no	—
8	#16839886	—	2	2	no	—
9	#16839819	C₂²	1	2	yes	—
10	#16839790	C₄	1	2	yes	—
11	#16839720	C₄	1	2	yes	—
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	#16839514	—	2	2	no	—
14	#16839512	—	2	2	no	—
15	#16839503	—	2	4	no	—
16	C₆₁×#432	C₂²	1	2	yes	—
17	C₆₁×#423	C₂²	1	1	yes	characteristic
18	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
19	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
20	#14519777	Q₈	1	2	yes	—
21	#14519769	D₄	1	2	yes	—
22	#14519756	—	2	2	no	—
23	C₄×Dic₁₂₂	—	2	2	no	—
24	#14519736	C₂×C₄	1	2	yes	—
25	C₄⋊#92418	—	2	2	no	—
26	#14519706	—	2	4	no	—
27	C₂²⋊#92418	—	2	2	no	—
28	#14519688	—	2	2	no	—
29	#14519687	—	2	2	no	—
30	C₄×#92418	D₄	1	2	yes	—
31	C₄×#92418	D₄	1	2	yes	—
32	C₈×Dic₆₁	—	2	2	no	—
33	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₁	—	2	2	no	—
36	C₄⋊C₈×C₆₁	—	2	2	no	—
37	C₂²⋊C₈×C₆₁	—	2	2	no	—
38	#97×C₆₁	—	2	2	no	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	#41	1	2	yes	—
41	C₂×Dic₁₂₂	—	2	2	no	—
42	M₁₆×C₆₁	—	4	2	no	—
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	4	2	no	—
45	C₄²×C₆₁	C₂×D₄	1	2	yes	—
46	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
48	C₂²⋊Dic₆₁	—	2	2	no	—
49	C₄⋊Dic₆₁	—	2	2	no	—
50	C₄⋊Dic₆₁	C₄⋊C₄	1	4	yes	—
51	#11758196	—	2	4	no	—
52	C₄×Dic₆₁	—	2	2	no	—
53	C₂×#92418	—	2	2	no	—
54	C₂×#92418	—	4	1	no	—
55	C₂×#92418	—	4	2	no	—
56	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×#92418	C₂²⋊C₄	1	2	yes	—
58	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
59	C₂³×C₆₁	#117	1	1	yes	characteristic
60	C₂×C₄×C₆₁	—	2	1	no	—
61	C₂×C₄×C₆₁	—	2	1	no	—
62	C₂×C₄×C₆₁	—	4	2	no	—
63	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#121	1	2	yes	—
66	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
67	C₂×Dic₆₁	—	2	4	no	—
68	Dic₁₂₂	—	4	4	no	—
69	C₈×C₆₁	—	4	2	no	—
70	C₈×C₆₁	—	8	2	no	—
71	#92418	—	2	1	no	—
72	#92418	—	2	2	no	—
73	#92418	—	8	2	no	—
74	#92418	#101	1	2	yes	—
75	#8631	—	61	1	no	maximal, Sylow
76	C₂²×C₆₁	—	2	1	no	—
77	C₂²×C₆₁	—	2	2	no	—
78	C₂²×C₆₁	#380	1	1	yes	characteristic
79	C₄×C₆₁	—	2	1	no	—
80	C₄×C₆₁	—	2	2	no	—
81	C₄×C₆₁	—	2	2	no	—
82	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
83	C₄×C₆₁	#413	1	1	yes	characteristic
84	Dic₆₁	—	4	4	no	—
85	#2960	—	61	1	no	—
86	#1568	—	61	2	no	—
87	#1537	—	61	2	no	—
88	#1307	—	61	1	no	—
89	#1307	D₆₁	1	1	yes	characteristic
90	C₂×C₆₁	—	4	1	no	—
91	C₂×C₆₁	#1801	1	1	yes	characteristic
92	C₂×C₆₁	#1814	1	2	yes	—
93	#533	—	61	1	no	—
94	#446	—	122	2	no	—
95	C₈×Q₈	—	122	2	no	—
96	#432	—	61	2	no	—
97	#432	C₂×D₆₁	1	2	yes	—
98	#424	—	61	2	no	—
99	#423	C₂×D₆₁	1	1	yes	characteristic
100	#405	—	61	2	no	—
101	C₂×C₄×C₈	—	61	1	no	—
102	#335	—	122	2	no	—
103	#334	—	122	2	no	—
104	#330	—	122	4	no	—
105	C₆₁	#8631	1	1	yes	characteristic, complemented, Sylow
106	C₂×M₁₆	C₄×D₆₁	1	2	yes	—
107	C₂²×C₈	—	61	1	no	—
108	C₂²×C₈	—	61	1	no	—
109	C₄⋊Q₈	—	61	2	no	—
110	#125	—	61	2	no	—
111	C₂²⋊Q₈	—	122	2	no	—
112	C₄×Q₈	—	122	2	no	—
113	#117	—	61	2	no	—
114	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	122	4	no	—
118	C₄⋊C₈	—	122	2	no	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	122	2	no	—
121	#97	—	2	2	no	—
122	#97	—	122	2	no	—
123	#97	—	122	2	no	—
124	C₄×C₈	—	61	2	no	—
125	C₄×C₈	—	61	2	no	—
126	C₄×C₈	—	122	2	no	—
127	C₂×Q₈	—	122	2	no	—
128	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
129	C₂²×C₄	#11758222	1	2	yes	—
130	M₁₆	—	4	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	61	1	no	—
134	C₂×C₈	—	61	1	no	—
135	C₂×C₈	—	61	2	no	—
136	C₂×C₈	—	122	2	no	—
137	C₂×C₈	—	244	1	no	—
138	C₂×C₈	—	244	2	no	—
139	C₄⋊C₄	—	61	4	no	—
140	C₄⋊C₄	—	122	4	no	—
141	C₄⋊C₄	—	122	2	no	—
142	C₂²⋊C₄	—	122	2	no	—
143	C₄²	—	122	2	no	—
144	C₄²	D₄×D₆₁	1	2	yes	—
145	C₄²	D₄×D₆₁	1	1	yes	characteristic
146	C₄²	Q₈×D₆₁	1	1	yes	characteristic
147	C₂³	#14519734	1	1	yes	characteristic
148	Q₈	—	244	4	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	122	4	no	—
153	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
154	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
155	C₂×C₄	#14519750	1	2	yes	—
156	C₂×C₄	#14519755	1	2	yes	—
157	C₈	—	4	2	no	—
158	C₈	—	8	2	no	—
159	C₈	—	61	2	no	—
160	C₈	—	122	1	no	—
161	C₈	—	122	2	no	—
162	C₈	—	488	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	#16839621	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	244	4	no	—
170	C₄	#101×D₆₁	1	1	yes	characteristic
171	C₄	#16839743	1	1	yes	characteristic
172	C₂	—	4	1	no	—
173	C₂	#21445814	1	1	yes	characteristic
174	C₂	#21445872	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21454749

C₂

yes

maximal, characteristic

#21444580

C₂

yes

maximal

#21444456

C₂

yes

maximal

#21443730

C₂

yes

maximal, characteristic

C₆₁×#1307

C₂

yes

maximal, characteristic

#16840319

C₂²

yes

—

—

C₂²

yes

—

#16839790

C₄

yes

—

#16839720

C₄

yes

—

C₂×C₄×#92418

C₂²

yes

—

—

—

C₆₁×#432

C₂²

yes

—

C₆₁×#423

C₂²

yes

characteristic

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

characteristic

#14519777

Q₈

yes

—

#14519769

D₄

yes

—

#14519756

—

C₄×Dic₁₂₂

—

#14519736

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

—

C₄×#92418

D₄

yes

—

C₈×Dic₆₁

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

#97×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂×Dic₁₂₂

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₄⋊C₄

yes

—

#11758196

—

C₄×Dic₆₁

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₂²⋊C₄

yes

—

C₂×#92418

C₂²⋊C₄

yes

characteristic

C₂³×C₆₁

#117

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

—

yes

—

#8631

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#380

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#1801

yes

characteristic

C₂×C₆₁

#1814

yes

—

#533

—

#446

—

122

—

C₈×Q₈

—

122

—

#432

—

#432

C₂×D₆₁

yes

—

#424

—

#423

C₂×D₆₁

yes

characteristic

100

#405

—

101

C₂×C₄×C₈

—

102

#335

—

122

—

103

#334

—

122

—

104

#330

—

122

—

105

C₆₁

#8631

yes

characteristic, complemented, Sylow

106

C₂×M₁₆

C₄×D₆₁

yes

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₄⋊Q₈

—

110

#125

—

111

C₂²⋊Q₈

—

122

—

112

C₄×Q₈

—

122

—

113

#117

—

114

C₂×C₄²

C₂²×D₆₁

yes

characteristic

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

122

—

118

C₄⋊C₈

—

122

—

119

C₂²⋊C₈

—

120

C₂²⋊C₈

—

122

—

121

#97

—

122

#97

—

122

—

123

#97

—

122

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

122

—

127

C₂×Q₈

—

122

—

128

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

129

C₂²×C₄

#11758222

yes

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

122

—

137

C₂×C₈

—

244

—

138

C₂×C₈

—

244

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

122

—

141

C₄⋊C₄

—

122

—

142

C₂²⋊C₄

—

122

—

143

C₄²

—

122

—

144

C₄²

D₄×D₆₁

yes

—

145

C₄²

D₄×D₆₁

yes

characteristic

146

C₄²

Q₈×D₆₁

yes

characteristic

147

C₂³

#14519734

yes

characteristic

148

Q₈

—

244

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

122

—

153

C₂×C₄

C₂²⋊C₄×D₆₁

yes

characteristic

154

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

155

C₂×C₄

#14519750

yes

—

156

C₂×C₄

#14519755

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

122

—

161

C₈

—

122

—

162

C₈

—

488

—

163

C₂²

—

164

C₂²

—

165

C₂²

#16839621

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

244

—

170

C₄

#101×D₆₁

yes

characteristic

171

C₄

#16839743

yes

characteristic

172

C₂

—

173

C₂

#21445814

yes

characteristic

174

C₂

#21445872

yes

—

	subgroup	G/H
center	C₂²	#16839621
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1307	C₂
socle	C₂²×C₆₁	#380
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1307	D₆₁
hypocenter	C₆₁	#8631
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

subgroup

G/H

center

C₂²

#16839621

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1307

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

N	H ≅ G/N	type
105. C₆₁	#8631	semidirect

H ≅ G/N

type

105. C₆₁

#8631

semidirect

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

subgroup	index
#21454749	2
#21444580	2
#21444456	2
#21443730	2
C₆₁×#1307	2
#8631	61

subgroup

index

C₆₁×#1307

Subgroups — #29573008

Conjugacy classes of subgroups

Complements

Maximal subgroups