Group #29571180 — subgroups

Subgroups — #29571180

Order 15616.

390 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 195 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	C₄×#14519687	C₂	1	1	yes	maximal, characteristic
2	#21444565	C₂	1	2	yes	maximal
3	#21444102	C₂	1	1	yes	maximal, characteristic
4	#21443651	C₂	1	2	yes	maximal
5	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×#14519687	C₄	1	2	yes	—
8	C₂×#14519687	C₄	1	2	yes	—
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
11	#16839514	C₄	1	4	yes	complemented
12	#16839512	C₄	1	4	yes	complemented
13	#16839483	—	2	1	no	—
14	C₈×#92418	—	2	1	no	—
15	C₆₁×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#322	—	2	1	no	—
18	C₈²×C₆₁	—	2	1	no	—
19	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₁	—	2	2	no	—
22	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
23	C₄⋊#92418	C₂×C₄	1	4	yes	—
24	C₂²⋊#92418	—	2	2	no	—
25	#14519687	—	2	4	no	—
26	#14519687	C₈	1	8	yes	complemented
27	C₄×#92418	—	2	2	no	—
28	C₄×#92418	C₂×C₄	1	2	yes	—
29	C₄×#92418	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
33	C₂²⋊C₈×C₆₁	—	2	2	no	—
34	C₄×C₈×C₆₁	—	2	2	no	—
35	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
36	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
37	C₄×C₈×C₆₁	Q₈	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	#41	1	2	yes	—
40	C₂²×Dic₆₁	—	2	2	no	—
41	C₂×C₈×C₆₁	—	2	4	no	—
42	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
46	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
47	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₄×Dic₆₁	—	2	2	no	—
50	C₄×Dic₆₁	—	4	4	no	—
51	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
52	C₂×#92418	—	2	2	no	—
53	C₂×#92418	—	2	4	no	—
54	C₂×#92418	C₄²	1	1	yes	characteristic
55	C₂×#92418	C₄²	1	1	yes	characteristic
56	C₂×#92418	C₄²	1	2	yes	—
57	C₂×#92418	C₂×C₈	1	4	yes	—
58	C₂³×C₆₁	#117	1	1	yes	characteristic
59	C₂×C₄×C₆₁	—	2	4	no	—
60	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
66	C₂×Dic₆₁	—	2	4	no	—
67	C₂×Dic₆₁	—	4	4	no	—
68	C₈×C₆₁	—	4	4	no	—
69	C₈×C₆₁	C₄⋊C₈	1	4	yes	—
70	C₈×C₆₁	C₄⋊C₈	1	4	yes	—
71	#92418	—	2	1	no	—
72	#92418	—	2	1	no	—
73	#92418	—	4	4	no	—
74	#92418	C₄×C₈	1	4	yes	—
75	#8268	—	61	1	no	maximal, Sylow
76	C₂²×C₆₁	—	2	2	no	—
77	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₆₁	#431	1	2	yes	—
79	C₄×C₆₁	—	2	2	no	—
80	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
81	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
82	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
83	C₄×C₆₁	#444	1	4	yes	—
84	Dic₆₁	—	4	4	no	—
85	C₄×#97	—	61	1	no	—
86	#1564	—	61	2	no	—
87	#1450	—	61	1	no	—
88	#1450	D₆₁	1	1	yes	characteristic
89	#1276	—	61	2	no	—
90	C₂×C₆₁	—	2	2	no	—
91	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₆₁	#1773	1	2	yes	—
93	#431	—	61	2	no	—
94	#431	C₂×D₆₁	1	2	yes	—
95	C₂×#97	—	61	2	no	—
96	C₂×#97	—	61	2	no	—
97	C₂×C₄×C₈	—	61	1	no	—
98	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
99	C₄³	—	61	1	no	—
100	#335	—	61	4	no	—
101	#334	—	61	4	no	—
102	#322	—	2	1	no	—
103	#322	—	122	1	no	—
104	C₈²	—	2	1	no	—
105	C₈²	—	122	1	no	—
106	C₆₁	#8268	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	—	61	1	no	—
108	C₂²×C₈	—	61	1	no	—
109	C₂²×C₈	C₄×D₆₁	1	2	yes	—
110	C₂×C₄²	—	61	2	no	—
111	C₂×C₄²	—	122	2	no	—
112	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
113	C₄⋊C₈	—	61	4	no	—
114	C₄⋊C₈	C₄×D₆₁	1	4	yes	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₂²⋊C₈	—	122	2	no	—
117	#97	—	61	8	no	—
118	#97	—	122	4	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	61	2	no	—
121	C₄×C₈	—	61	2	no	—
122	C₄×C₈	—	122	2	no	—
123	C₄×C₈	Dic₁₂₂	1	2	yes	—
124	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
126	C₂²×C₄	—	122	2	no	—
127	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
128	C₂²×C₄	#11758222	1	2	yes	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	61	2	no	—
131	C₂×C₈	—	61	4	no	—
132	C₂×C₈	—	61	1	no	—
133	C₂×C₈	—	61	1	no	—
134	C₂×C₈	—	122	2	no	—
135	C₂×C₈	—	122	4	no	—
136	C₂×C₈	C₈×D₆₁	1	4	yes	—
137	C₂×C₈	#11758189	1	4	yes	—
138	C₂×C₈	#11758196	1	2	yes	—
139	C₂×C₈	#11758196	1	2	yes	—
140	C₄²	—	61	4	no	—
141	C₄²	—	122	2	no	—
142	C₄²	—	244	4	no	—
143	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
144	C₄²	C₂×C₄×D₆₁	1	2	yes	—
145	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
146	C₂³	#14519734	1	1	yes	characteristic
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	122	4	no	—
149	C₂×C₄	—	244	4	no	—
150	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
151	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
152	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
153	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
154	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
155	C₂×C₄	C₂×#11758189	1	2	yes	—
156	C₈	—	4	4	no	—
157	C₈	—	61	4	no	—
158	C₈	—	122	1	no	—
159	C₈	—	122	1	no	—
160	C₈	—	244	4	no	—
161	C₈	#14519706	1	4	yes	—
162	C₈	#14519706	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	C₄×#11758196	1	1	yes	characteristic
165	C₂²	#16839813	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	244	4	no	—
168	C₄	C₄×C₈×D₆₁	1	1	yes	characteristic
169	C₄	C₄×#11758189	1	1	yes	characteristic
170	C₄	C₂×#14519706	1	2	yes	—
171	C₄	#16839882	1	4	yes	—
172	C₂	—	2	2	no	—
173	C₂	C₄×#14519706	1	1	yes	characteristic
174	C₂	#21445704	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14519687

C₂

yes

maximal, characteristic

#21444565

C₂

yes

maximal

#21444102

C₂

yes

maximal, characteristic

#21443651

C₂

yes

maximal

C₆₁×#1450

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

C₂×#14519687

C₄

yes

—

C₂×#14519687

C₄

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₄²×Dic₆₁

C₂²

yes

characteristic

#16839514

C₄

yes

complemented

#16839512

C₄

yes

complemented

#16839483

—

C₈×#92418

—

C₆₁×#431

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#322

—

C₈²×C₆₁

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₄⋊#92418

C₂×C₄

yes

—

C₂²⋊#92418

—

#14519687

—

#14519687

C₈

yes

complemented

C₄×#92418

—

C₄×#92418

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

C₂×C₄

yes

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

M₁₆

yes

—

C₄²×C₆₁

C₂²×C₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×C₈

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₄²

yes

characteristic

C₂×#92418

C₄²

yes

characteristic

C₂×#92418

C₄²

yes

—

C₂×#92418

C₂×C₈

yes

—

C₂³×C₆₁

#117

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

C₄×Q₈

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₄⋊C₈

yes

—

C₈×C₆₁

C₄⋊C₈

yes

—

—

—

—

C₄×C₈

yes

—

#8268

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

#431

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄×C₈

yes

characteristic

C₄×C₆₁

C₄×M₁₆

yes

characteristic

C₄×C₆₁

C₂×C₄⋊C₈

yes

—

C₄×C₆₁

#444

yes

—

Dic₆₁

—

C₄×#97

—

#1564

—

#1450

—

#1450

D₆₁

yes

characteristic

#1276

—

C₂×C₆₁

—

C₂×C₆₁

C₄×C₄⋊C₈

yes

characteristic

C₂×C₆₁

#1773

yes

—

#431

—

#431

C₂×D₆₁

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄³

—

100

#335

—

101

#334

—

102

#322

—

103

#322

—

122

—

104

C₈²

—

105

C₈²

—

122

—

106

C₆₁

#8268

yes

characteristic, complemented, Sylow

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂²×C₈

C₄×D₆₁

yes

—

110

C₂×C₄²

—

111

C₂×C₄²

—

122

—

112

C₂×C₄²

C₂²×D₆₁

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

C₄×D₆₁

yes

—

115

C₂²⋊C₈

—

116

C₂²⋊C₈

—

122

—

117

#97

—

118

#97

—

122

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

122

—

123

C₄×C₈

Dic₁₂₂

yes

—

124

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

125

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

126

C₂²×C₄

—

122

—

127

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

128

C₂²×C₄

#11758222

yes

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

122

—

135

C₂×C₈

—

122

—

136

C₂×C₈

C₈×D₆₁

yes

—

137

C₂×C₈

#11758189

yes

—

138

C₂×C₈

#11758196

yes

—

139

C₂×C₈

#11758196

yes

—

140

C₄²

—

141

C₄²

—

122

—

142

C₄²

—

244

—

143

C₄²

C₂×Dic₁₂₂

yes

characteristic

144

C₄²

C₂×C₄×D₆₁

yes

—

145

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

146

C₂³

#14519734

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

122

—

149

C₂×C₄

—

244

—

150

C₂×C₄

C₄²×D₆₁

yes

characteristic

151

C₂×C₄

C₂×#11758196

yes

characteristic

152

C₂×C₄

C₄×C₂²⋊D₆₁

yes

—

153

C₂×C₄

C₄×Dic₁₂₂

yes

—

154

C₂×C₄

C₂×C₈×D₆₁

yes

—

155

C₂×C₄

C₂×#11758189

yes

—

156

C₈

—

157

C₈

—

158

C₈

—

122

—

159

C₈

—

122

—

160

C₈

—

244

—

161

C₈

#14519706

yes

—

162

C₈

#14519706

yes

—

163

C₂²

—

164

C₂²

C₄×#11758196

yes

characteristic

165

C₂²

#16839813

yes

—

166

C₄

—

167

C₄

—

244

—

168

C₄

C₄×C₈×D₆₁

yes

characteristic

169

C₄

C₄×#11758189

yes

characteristic

170

C₄

C₂×#14519706

yes

—

171

C₄

#16839882

yes

—

172

C₂

—

173

C₂

C₄×#14519706

yes

characteristic

174

C₂

#21445704

yes

—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1450	C₂
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#8268
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1450

subgroup

G/H

center

C₄²

C₂×C₂²⋊D₆₁

commutator

C₄×C₆₁

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1450

C₂

socle

C₂²×C₆₁

C₄×C₄⋊C₄

prime radical

C₂³×C₆₁

#117

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1450

N	H ≅ G/N	type
11. #16839514	C₄	semidirect
12. #16839512	C₄	semidirect
26. #14519687	C₈	semidirect
106. C₆₁	#8268	semidirect

H ≅ G/N

type

11. #16839514

C₄

semidirect

12. #16839512

C₄

semidirect

26. #14519687

C₈

semidirect

106. C₆₁

#8268

semidirect

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

subgroup	index
C₄×#14519687	2
#21444565	2
#21444102	2
#21443651	2
C₆₁×#1450	2
#8268	61

subgroup

index

C₆₁×#1450

Subgroups — #29571180

Conjugacy classes of subgroups

Complements

Maximal subgroups