Group #29598545 — subgroups

Subgroups — #29598545

Order 15616.

342 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 159 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	#21447575	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic
3	#21443908	C₂	1	4	yes	maximal
4	C₆₁×#2960	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519717	C₂²	1	2	yes	—
6	C₂²×#11758185	C₂²	1	2	yes	—
7	C₄⋊#11758185	—	2	2	no	—
8	C₄×#11758185	—	2	2	no	—
9	C₆₁×#533	C₄	1	1	yes	characteristic
10	C₆₁×#446	—	2	2	no	—
11	C₈×Q₈×C₆₁	—	2	2	no	—
12	C₆₁×#424	C₄	1	2	yes	—
13	C₆₁×#423	C₄	1	1	yes	characteristic
14	C₆₁×#405	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	#14519717	—	2	4	no	—
17	C₂×#11758185	—	2	2	no	—
18	C₂×#11758185	—	2	2	no	—
19	C₂×#11758185	—	2	2	no	—
20	C₂×#11758185	Q₈	1	4	yes	complemented
21	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
24	C₄⋊Q₈×C₆₁	C₈	1	2	yes	—
25	#125×C₆₁	C₈	1	2	yes	—
26	C₂²⋊Q₈×C₆₁	—	2	2	no	—
27	C₄×Q₈×C₆₁	—	2	2	no	—
28	#117×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄⋊C₈×C₆₁	—	2	4	no	—
31	C₄⋊C₈×C₆₁	—	2	2	no	—
32	#97×C₆₁	—	2	2	no	—
33	C₄×C₈×C₆₁	—	2	2	no	—
34	C₄×C₈×C₆₁	D₄	1	2	yes	—
35	C₄×C₈×C₆₁	D₄	1	2	yes	—
36	C₂×Q₈×C₆₁	—	2	2	no	—
37	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	M₁₆×C₆₁	—	2	4	no	—
40	C₂×C₈×C₆₁	—	2	2	no	—
41	C₂×C₈×C₆₁	—	2	2	no	—
42	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₆₁	#41	1	2	yes	—
47	C₄⋊C₄×C₆₁	—	2	4	no	—
48	C₄⋊C₄×C₆₁	—	2	2	no	—
49	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
50	C₂²⋊C₄×C₆₁	—	2	2	no	—
51	C₄²×C₆₁	—	2	2	no	—
52	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
55	#11758185	—	2	4	no	—
56	#11758185	—	4	2	no	—
57	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
58	Q₈×C₆₁	—	4	4	no	—
59	C₂×C₄×C₆₁	—	2	4	no	—
60	C₂×C₄×C₆₁	—	2	4	no	—
61	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#131	1	2	yes	—
67	C₈×C₆₁	—	2	2	no	—
68	C₈×C₆₁	—	4	2	no	—
69	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
70	C₈×C₆₁	#124	1	1	yes	characteristic
71	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
72	#13266	—	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	—	4	4	no	—
78	C₄×C₆₁	#350	1	4	yes	—
79	C₄×C₆₁	#391	1	1	yes	characteristic
80	C₄×C₆₁	#408	1	1	yes	characteristic
81	C₄×C₆₁	C₈×Q₈	1	2	yes	—
82	#2960	D₆₁	1	1	yes	characteristic
83	#2150	—	61	1	no	—
84	C₂×C₄×C₁₆	—	61	1	no	—
85	#1379	—	61	4	no	—
86	C₂×C₆₁	—	2	2	no	—
87	C₂×C₆₁	#1984	1	1	yes	characteristic
88	C₂×C₆₁	C₂×#350	1	2	yes	—
89	#533	Dic₆₁	1	1	yes	characteristic
90	C₂×M₃₂	—	61	2	no	—
91	C₂²×C₁₆	—	61	2	no	—
92	#446	—	2	2	no	—
93	C₈×Q₈	—	2	2	no	—
94	#424	Dic₆₁	1	2	yes	—
95	#423	Dic₆₁	1	1	yes	characteristic
96	#405	C₂×D₆₁	1	2	yes	—
97	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
98	C₄⋊C₁₆	—	122	2	no	—
99	C₄×C₁₆	—	122	2	no	—
100	C₆₁	#13266	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
102	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
104	C₄⋊Q₈	#92418	1	2	yes	—
105	#125	#92418	1	2	yes	—
106	C₂²⋊Q₈	—	2	2	no	—
107	C₄×Q₈	—	2	2	no	—
108	#117	C₂×Dic₆₁	1	2	yes	—
109	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
110	M₃₂	—	122	4	no	—
111	C₂×C₁₆	—	61	4	no	—
112	C₂×C₁₆	—	122	2	no	—
113	C₂×C₁₆	—	122	2	no	—
114	C₂×C₁₆	—	122	2	no	—
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	#97	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
120	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
121	C₂×Q₈	—	2	2	no	—
122	C₂²×C₄	#11758194	1	2	yes	—
123	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
128	C₂×C₈	Q₈×D₆₁	1	2	yes	—
129	C₂×C₈	#11758226	1	2	yes	—
130	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
131	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	C₂×#92418	1	4	yes	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	C₂×#92418	1	1	yes	characteristic
138	C₄²	C₂×#92418	1	1	yes	characteristic
139	C₄²	C₂²⋊Dic₆₁	1	2	yes	—
140	C₁₆	—	122	4	no	—
141	C₁₆	—	244	2	no	—
142	C₂³	C₂×#11758194	1	1	yes	characteristic
143	Q₈	—	4	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	C₂²⋊#92418	1	2	yes	—
147	C₂×C₄	C₂²⋊#92418	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
149	C₂×C₄	Q₈×Dic₆₁	1	2	yes	—
150	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
151	C₂×C₄	#14519784	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	4	2	no	—
154	C₈	#14519753	1	2	yes	—
155	C₈	#14519766	1	1	yes	characteristic
156	C₈	#14519775	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×C₂²⋊#92418	1	1	yes	characteristic
159	C₂²	#16839893	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#16839546	1	4	yes	—
163	C₄	#16839669	1	1	yes	characteristic
164	C₄	#16839729	1	1	yes	characteristic
165	C₄	Q₈×#92418	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#21446756	1	1	yes	characteristic
168	C₂	C₂×#16839546	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447575

C₂

yes

maximal, characteristic

C₂×C₄×#11758185

C₂

yes

maximal, characteristic

#21443908

C₂

yes

maximal

C₆₁×#2960

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

—

C₂²×#11758185

C₂²

yes

—

C₄⋊#11758185

—

C₄×#11758185

—

C₆₁×#533

C₄

yes

characteristic

C₆₁×#446

—

C₈×Q₈×C₆₁

—

C₆₁×#424

C₄

yes

—

C₆₁×#423

C₄

yes

characteristic

C₆₁×#405

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

—

—

—

Q₈

yes

complemented

C₂×M₁₆×C₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊Q₈×C₆₁

C₈

yes

—

#125×C₆₁

C₈

yes

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

#97×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

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

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₈

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

#11758185

—

#11758185

—

C₂³×C₆₁

C₂×M₁₆

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

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

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊Q₈

yes

—

C₈×C₆₁

#124

yes

characteristic

C₈×C₆₁

C₄⋊Q₈

yes

characteristic

#13266

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₆₁

#446

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#350

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₈×Q₈

yes

—

#2960

D₆₁

yes

characteristic

#2150

—

C₂×C₄×C₁₆

—

#1379

—

C₂×C₆₁

—

C₂×C₆₁

#1984

yes

characteristic

C₂×C₆₁

C₂×#350

yes

—

#533

Dic₆₁

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#446

—

C₈×Q₈

—

#424

Dic₆₁

yes

—

#423

Dic₆₁

yes

characteristic

#405

C₂×D₆₁

yes

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

122

—

C₄×C₁₆

—

122

—

100

C₆₁

#13266

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

C₂×Dic₆₁

yes

—

102

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

103

C₂²×C₈

C₂²×D₆₁

yes

characteristic

104

C₄⋊Q₈

#92418

yes

—

105

#125

#92418

yes

—

106

C₂²⋊Q₈

—

107

C₄×Q₈

—

108

#117

C₂×Dic₆₁

yes

—

109

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

110

M₃₂

—

122

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

122

—

113

C₂×C₁₆

—

122

—

114

C₂×C₁₆

—

122

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

#97

—

118

C₄×C₈

—

119

C₄×C₈

C₂²⋊D₆₁

yes

—

120

C₄×C₈

C₂²⋊D₆₁

yes

—

121

C₂×Q₈

—

122

C₂²×C₄

#11758194

yes

—

123

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂²⋊Dic₆₁

yes

—

128

C₂×C₈

Q₈×D₆₁

yes

—

129

C₂×C₈

#11758226

yes

—

130

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

131

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

C₂×#92418

yes

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

C₂×#92418

yes

characteristic

138

C₄²

C₂×#92418

yes

characteristic

139

C₄²

C₂²⋊Dic₆₁

yes

—

140

C₁₆

—

122

—

141

C₁₆

—

244

—

142

C₂³

C₂×#11758194

yes

characteristic

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊#92418

yes

—

147

C₂×C₄

C₂²⋊#92418

yes

—

148

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

149

C₂×C₄

Q₈×Dic₆₁

yes

—

150

C₂×C₄

C₂²×#92418

yes

characteristic

151

C₂×C₄

#14519784

yes

—

152

C₈

—

153

C₈

—

154

C₈

#14519753

yes

—

155

C₈

#14519766

yes

characteristic

156

C₈

#14519775

yes

characteristic

157

C₂²

—

158

C₂²

C₂×C₂²⋊#92418

yes

characteristic

159

C₂²

#16839893

yes

—

160

C₄

—

161

C₄

—

162

C₄

#16839546

yes

—

163

C₄

#16839669

yes

characteristic

164

C₄

#16839729

yes

characteristic

165

C₄

Q₈×#92418

yes

—

166

C₂

—

167

C₂

#21446756

yes

characteristic

168

C₂

C₂×#16839546

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2960	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#2960	D₆₁
hypocenter	C₆₁	#13266
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#2960

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 ◁ #2960

N	H ≅ G/N	type
20. C₂×#11758185	Q₈	semidirect
100. C₆₁	#13266	semidirect

H ≅ G/N

type

20. C₂×#11758185

Q₈

semidirect

100. C₆₁

#13266

semidirect

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

subgroup	index
#21447575	2
C₂×C₄×#11758185	2
#21443908	2
C₆₁×#2960	2
#13266	61

subgroup

index

C₆₁×#2960

Subgroups — #29598545

Conjugacy classes of subgroups

Complements

Maximal subgroups