Group #29599908 — subgroups

Subgroups — #29599908

Order 15616.

276 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 87 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	#21447614	C₂	1	1	yes	maximal, characteristic, complemented
2	#21445873	C₂	1	2	yes	maximal
3	#21443921	C₂	1	1	yes	maximal, characteristic, complemented
4	#21443915	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2113	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519713	C₂²	1	2	yes	—
7	#16839800	C₂²	1	2	yes	—
8	#16839779	—	2	2	no	—
9	#16839592	—	2	2	no	—
10	#16839534	—	2	2	no	—
11	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×C₂²⋊C₁₆	—	2	2	no	—
15	C₂×#11758194	—	2	2	no	—
16	#14519736	—	2	2	no	—
17	#14519713	—	2	2	no	—
18	#14519713	—	2	2	no	—
19	#14519711	—	2	2	no	—
20	#14519709	—	2	2	no	—
21	#14519704	—	4	4	no	—
22	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
23	#131×C₆₁	—	2	2	no	—
24	#131×C₆₁	—	2	2	no	—
25	C₂×M₁₆×C₆₁	—	2	1	no	—
26	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	—	2	1	no	—
28	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	M₃₂×C₆₁	—	2	2	no	—
30	C₂×C₁₆×C₆₁	—	2	1	no	—
31	C₂×C₁₆×C₆₁	—	2	2	no	—
32	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
33	#41×C₆₁	—	2	2	no	—
34	#41×C₆₁	—	2	2	no	—
35	C₂×Q₈×C₆₁	Q₁₆	1	1	yes	characteristic
36	C₂×D₄×C₆₁	—	2	1	no	—
37	C₂×D₄×C₆₁	Q₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	—	2	1	no	—
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	M₁₆×C₆₁	—	2	2	no	—
41	M₁₆×C₆₁	—	4	2	no	—
42	C₂×C₈×C₆₁	—	2	1	no	—
43	C₂×C₈×C₆₁	—	4	1	no	—
44	C₂×C₈×C₆₁	—	4	1	no	—
45	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	#41	1	2	yes	—
47	C₂×C₈×C₆₁	#41	1	2	yes	—
48	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
49	C₂²⋊Dic₆₁	—	4	2	no	—
50	C₄⋊Dic₆₁	—	4	2	no	—
51	C₄×Dic₆₁	—	4	2	no	—
52	#11758194	—	4	2	no	—
53	#11758194	—	4	2	no	—
54	C₂×#92418	—	4	2	no	—
55	C₁₆×C₆₁	—	2	2	no	—
56	C₁₆×C₆₁	—	4	1	no	—
57	C₂³×C₆₁	—	2	1	no	—
58	C₂³×C₆₁	#135	1	1	yes	characteristic
59	Q₈×C₆₁	—	2	2	no	—
60	D₄×C₆₁	—	2	2	no	—
61	D₄×C₆₁	—	4	2	no	—
62	C₂×C₄×C₆₁	—	2	1	no	—
63	C₂×C₄×C₆₁	—	4	1	no	—
64	C₂×C₄×C₆₁	—	4	1	no	—
65	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#123	1	2	yes	—
67	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×Dic₆₁	—	8	2	no	—
70	C₈×C₆₁	—	4	1	no	—
71	C₈×C₆₁	#123	1	1	yes	characteristic
72	C₈×C₆₁	#124	1	1	yes	characteristic
73	C₈×C₆₁	#126	1	2	yes	—
74	#92418	—	8	2	no	—
75	#13499	—	61	1	no	maximal, Sylow
76	C₂²×C₆₁	—	4	1	no	—
77	C₂²×C₆₁	—	4	2	no	—
78	C₂²×C₆₁	—	4	1	no	—
79	C₂²×C₆₁	#467	1	1	yes	characteristic
80	C₂²×C₆₁	#482	1	2	yes	—
81	C₄×C₆₁	—	4	1	no	—
82	C₄×C₆₁	#397	1	1	yes	characteristic
83	C₄×C₆₁	#469	1	1	yes	characteristic
84	C₄×C₆₁	#484	1	2	yes	—
85	Dic₆₁	—	16	2	no	—
86	#2161	—	61	1	no	—
87	#2113	D₆₁	1	1	yes	characteristic
88	#1814	—	61	2	no	—
89	#1384	—	61	1	no	—
90	#1382	—	61	2	no	—
91	C₂×C₆₁	—	2	2	no	—
92	C₂×C₆₁	—	2	1	no	—
93	C₂×C₆₁	—	8	1	no	—
94	C₂×C₆₁	#2076	1	1	yes	characteristic
95	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
96	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
97	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
98	C₂×#108	—	61	2	no	—
99	#427	—	61	2	no	—
100	#421	—	122	2	no	—
101	#368	—	122	2	no	—
102	C₂²⋊C₁₆	—	2	2	no	—
103	#344	—	122	2	no	—
104	C₆₁	#13499	1	1	yes	characteristic, complemented, Sylow
105	C₂×#41	D₂₄₄	1	1	yes	characteristic
106	#131	—	2	2	no	—
107	#131	—	2	2	no	—
108	C₂×M₁₆	—	2	1	no	—
109	C₂×M₁₆	—	122	2	no	—
110	C₂×M₁₆	D₂₄₄	1	1	yes	characteristic
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
113	#117	—	122	2	no	—
114	M₃₂	—	2	2	no	—
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	2	1	no	—
117	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
118	#108	—	122	2	no	—
119	#108	—	122	2	no	—
120	#107	—	122	2	no	—
121	#106	—	122	2	no	—
122	#104	—	244	4	no	—
123	#41	—	2	2	no	—
124	#41	—	2	2	no	—
125	C₂×Q₈	Dic₂₄₄	1	1	yes	characteristic
126	C₂×D₄	—	2	1	no	—
127	C₂×D₄	Dic₂₄₄	1	1	yes	characteristic
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
130	M₁₆	—	2	2	no	—
131	M₁₆	—	4	2	no	—
132	M₁₆	—	244	2	no	—
133	M₁₆	—	244	2	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	244	2	no	—
138	C₂×C₈	#11758222	1	1	yes	characteristic
139	C₂×C₈	#11758224	1	2	yes	—
140	C₂×C₈	#11758224	1	2	yes	—
141	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
142	C₄⋊C₄	—	244	2	no	—
143	C₂²⋊C₄	—	244	2	no	—
144	C₄²	—	244	2	no	—
145	C₁₆	—	2	2	no	—
146	C₁₆	—	4	1	no	—
147	C₂³	—	2	1	no	—
148	C₂³	#14519799	1	1	yes	characteristic
149	Q₈	—	2	2	no	—
150	D₄	—	2	2	no	—
151	D₄	—	4	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	488	2	no	—
156	C₂×C₄	#14519751	1	1	yes	characteristic
157	C₂×C₄	#14519761	1	2	yes	—
158	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
159	C₂×C₄	#14519805	1	2	yes	—
160	C₈	—	4	1	no	—
161	C₈	—	488	2	no	—
162	C₈	#14519758	1	1	yes	characteristic
163	C₈	#14519764	1	1	yes	characteristic
164	C₈	#14519770	1	2	yes	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	1	no	—
168	C₂²	#16840003	1	1	yes	characteristic
169	C₂²	#16840087	1	2	yes	—
170	C₄	—	4	1	no	—
171	C₄	—	976	2	no	—
172	C₄	#16839687	1	1	yes	characteristic
173	C₄	#16840015	1	1	yes	characteristic
174	C₄	#16840098	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	#21447247	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447614

C₂

yes

maximal, characteristic, complemented

#21445873

C₂

yes

maximal

#21443921

C₂

yes

maximal, characteristic, complemented

#21443915

C₂

yes

maximal, complemented

C₆₁×#2113

C₂

yes

maximal, characteristic

C₂×#14519713

C₂²

yes

—

#16839800

C₂²

yes

—

#16839779

—

#16839592

—

#16839534

—

C₂×#131×C₆₁

C₂²

yes

characteristic

C₂×M₃₂×C₆₁

C₂²

yes

characteristic

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₆₁×C₂²⋊C₁₆

—

—

—

—

—

—

—

—

C₂×#41×C₆₁

D₄

yes

characteristic

#131×C₆₁

—

#131×C₆₁

—

C₂×M₁₆×C₆₁

—

C₂×M₁₆×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

Q₁₆

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

Q₁₆

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

#11758194

—

#11758194

—

C₂×#92418

—

C₁₆×C₆₁

—

C₁₆×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

#135

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

C₈×C₆₁

yes

—

#92418

—

#13499

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#467

yes

characteristic

C₂²×C₆₁

#482

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

#2161

—

#2113

D₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#2076

yes

characteristic

C₂×#131

C₂×D₆₁

yes

characteristic

C₂×M₃₂

C₂×D₆₁

yes

characteristic

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

C₂×#108

—

#427

—

100

#421

—

122

—

101

#368

—

122

—

102

C₂²⋊C₁₆

—

103

#344

—

122

—

104

C₆₁

#13499

yes

characteristic, complemented, Sylow

105

C₂×#41

D₂₄₄

yes

characteristic

106

#131

—

107

#131

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

122

—

110

C₂×M₁₆

D₂₄₄

yes

characteristic

111

C₂²×C₈

—

112

C₂²×C₈

C₂²×D₆₁

yes

characteristic

113

#117

—

122

—

114

M₃₂

—

115

C₂×C₁₆

—

116

C₂×C₁₆

—

117

C₂×C₁₆

C₂²⋊D₆₁

yes

—

118

#108

—

122

—

119

#108

—

122

—

120

#107

—

122

—

121

#106

—

122

—

122

#104

—

244

—

123

#41

—

124

#41

—

125

C₂×Q₈

Dic₂₄₄

yes

characteristic

126

C₂×D₄

—

127

C₂×D₄

Dic₂₄₄

yes

characteristic

128

C₂²×C₄

—

129

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

130

M₁₆

—

131

M₁₆

—

132

M₁₆

—

244

—

133

M₁₆

—

244

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

244

—

138

C₂×C₈

#11758222

yes

characteristic

139

C₂×C₈

#11758224

yes

—

140

C₂×C₈

#11758224

yes

—

141

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

142

C₄⋊C₄

—

244

—

143

C₂²⋊C₄

—

244

—

144

C₄²

—

244

—

145

C₁₆

—

146

C₁₆

—

147

C₂³

—

148

C₂³

#14519799

yes

characteristic

149

Q₈

—

150

D₄

—

151

D₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

488

—

156

C₂×C₄

#14519751

yes

characteristic

157

C₂×C₄

#14519761

yes

—

158

C₂×C₄

C₂×Dic₂₄₄

yes

characteristic

159

C₂×C₄

#14519805

yes

—

160

C₈

—

161

C₈

—

488

—

162

C₈

#14519758

yes

characteristic

163

C₈

#14519764

yes

characteristic

164

C₈

#14519770

yes

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#16840003

yes

characteristic

169

C₂²

#16840087

yes

—

170

C₄

—

171

C₄

—

976

—

172

C₄

#16839687

yes

characteristic

173

C₄

#16840015

yes

characteristic

174

C₄

#16840098

yes

—

175

C₂

—

176

C₂

—

177

C₂

—

178

C₂

#21447247

yes

characteristic

	subgroup	G/H
center	C₄	#16840015
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2113	C₂
socle	C₂×C₆₁	#2076
prime radical	C₂×#41×C₆₁	D₄
hypercenter	#2113	D₆₁
hypocenter	C₆₁	#13499
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

subgroup

G/H

center

C₄

#16840015

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#2113

C₂

socle

C₂×C₆₁

#2076

prime radical

C₂×#41×C₆₁

D₄

D₆₁

C₆₁

G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

N	H ≅ G/N	type
1. #21447614	C₂	semidirect
3. #21443921	C₂	semidirect
4. #21443915	C₂	semidirect
104. C₆₁	#13499	semidirect

H ≅ G/N

type

1. #21447614

C₂

semidirect

3. #21443921

C₂

semidirect

4. #21443915

C₂

semidirect

104. C₆₁

#13499

semidirect

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

subgroup	index
#21447614	2
#21445873	2
#21443921	2
#21443915	2
C₆₁×#2113	2
#13499	61

subgroup

index

C₆₁×#2113

Subgroups — #29599908

Conjugacy classes of subgroups

Complements

Maximal subgroups