Group #29579477 — subgroups

Subgroups — #29579477

Order 15616.

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

normal subgroups only

This group has 478 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	#21445991	C₂	1	1	yes	maximal, characteristic
2	#21444491	C₂	1	4	yes	maximal
3	#21444102	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₁×#494	C₂	1	1	yes	maximal, characteristic
5	#16839815	C₂²	1	2	yes	—
6	C₂×#14519713	—	2	2	no	—
7	#16839786	C₂²	1	2	yes	—
8	#16839508	—	2	8	no	—
9	#16839483	—	2	1	no	—
10	C₈×#92418	—	2	1	no	—
11	C₂²×Q₁₆×C₆₁	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#494	—	2	1	no	—
14	C₆₁×#494	—	2	1	no	—
15	C₆₁×#494	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#11758194	—	2	2	no	—
18	#14519713	—	4	4	no	—
19	C₄⋊#92418	—	2	4	no	—
20	C₄⋊#92418	D₄	1	4	yes	—
21	C₂²⋊#92418	—	2	2	no	—
22	C₄×#92418	—	2	2	no	—
23	C₂²×Q₈×C₆₁	—	2	2	no	—
24	C₂×Q₁₆×C₆₁	—	2	4	no	—
25	C₂×Q₁₆×C₆₁	—	2	4	no	—
26	C₂×Q₁₆×C₆₁	—	4	4	no	—
27	C₂×Q₁₆×C₆₁	—	4	4	no	—
28	C₂²×C₈×C₆₁	D₄	1	2	yes	—
29	C₄⋊Q₈×C₆₁	—	2	4	no	—
30	C₄⋊Q₈×C₆₁	—	2	2	no	—
31	C₄⋊Q₈×C₆₁	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
33	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
36	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₆₁	—	2	4	no	—
38	C₂×Q₈×C₆₁	—	4	2	no	—
39	C₂×Q₈×C₆₁	—	4	4	no	—
40	C₂×Q₈×C₆₁	—	4	4	no	—
41	C₂²×C₄×C₆₁	—	4	2	no	—
42	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	#41	1	2	yes	—
44	Q₁₆×C₆₁	—	4	8	no	—
45	Q₁₆×C₆₁	—	4	8	no	—
46	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₁	D₈	1	4	yes	—
49	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
50	C₄⋊C₄×C₆₁	—	2	8	no	—
51	C₄⋊C₄×C₆₁	—	4	4	no	—
52	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₆₁	C₂×D₄	1	2	yes	—
55	#11758194	—	4	4	no	—
56	C₂×#92418	—	2	4	no	—
57	C₂×#92418	—	4	2	no	—
58	C₂³×C₆₁	#124	1	1	yes	characteristic
59	Q₈×C₆₁	—	4	8	no	—
60	Q₈×C₆₁	—	8	2	no	—
61	Q₈×C₆₁	—	8	2	no	—
62	C₂×C₄×C₆₁	—	2	4	no	—
63	C₂×C₄×C₆₁	—	4	4	no	—
64	C₂×C₄×C₆₁	—	8	4	no	—
65	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₆₁	#121	1	2	yes	—
68	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
71	C₈×C₆₁	#102	1	4	yes	—
72	C₈×C₆₁	#102	1	4	yes	—
73	#92418	—	4	4	no	—
74	#92418	—	8	2	no	—
75	#9772	—	61	1	no	maximal, Sylow
76	C₂²×C₆₁	—	2	2	no	—
77	C₂²×C₆₁	#390	1	1	yes	characteristic
78	C₂²×C₆₁	#486	1	2	yes	—
79	C₄×C₆₁	—	2	2	no	—
80	C₄×C₆₁	—	8	4	no	—
81	C₄×C₆₁	C₂×#102	1	2	yes	—
82	C₄×C₆₁	#473	1	4	yes	—
83	C₄×C₆₁	#492	1	1	yes	characteristic
84	C₄×C₆₁	#493	1	1	yes	characteristic
85	C₂×#494	D₆₁	1	1	yes	characteristic
86	#1848	—	61	1	no	—
87	#1546	—	61	4	no	—
88	#1450	—	61	1	no	—
89	C₂×C₆₁	—	2	2	no	—
90	C₂×C₆₁	#1957	1	1	yes	characteristic
91	C₂×C₆₁	#1979	1	2	yes	—
92	C₂²×Q₁₆	—	2	2	no	—
93	C₂×C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
94	#494	—	2	1	no	—
95	#494	—	2	1	no	—
96	#494	Dic₆₁	1	4	yes	—
97	#431	—	61	2	no	—
98	C₂×#108	—	122	2	no	—
99	#423	—	61	2	no	—
100	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
101	#332	—	122	8	no	—
102	#322	—	122	1	no	—
103	C₈²	—	122	1	no	—
104	C₆₁	#9772	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	2	2	no	—
106	C₂×Q₁₆	—	2	4	no	—
107	C₂×Q₁₆	—	2	4	no	—
108	C₂×Q₁₆	—	4	4	no	—
109	C₂×Q₁₆	—	4	4	no	—
110	C₂×M₁₆	—	122	2	no	—
111	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	C₂×Dic₆₁	1	4	yes	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
117	#108	—	244	4	no	—
118	C₄⋊C₈	—	61	4	no	—
119	C₄⋊C₈	—	122	4	no	—
120	C₂²⋊C₈	—	122	2	no	—
121	C₄×C₈	—	122	2	no	—
122	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
123	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	#11758224	1	2	yes	—
131	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
132	Q₁₆	—	4	8	no	—
133	Q₁₆	—	4	8	no	—
134	M₁₆	—	244	4	no	—
135	C₂×C₈	—	122	4	no	—
136	C₂×C₈	—	244	2	no	—
137	C₂×C₈	#11758199	1	4	yes	—
138	C₂×C₈	#11758200	1	4	yes	—
139	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
140	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	D₄×D₆₁	1	2	yes	—
144	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
145	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
146	C₂³	#14519764	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	Q₈	—	8	2	no	—
149	Q₈	—	8	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
154	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
155	C₂×C₄	#14519748	1	2	yes	—
156	C₂×C₄	#14519773	1	1	yes	characteristic
157	C₂×C₄	C₂×#11758199	1	2	yes	—
158	C₂×C₄	C₂×#11758200	1	2	yes	—
159	C₈	—	488	2	no	—
160	C₈	—	244	4	no	—
161	C₈	#14519698	1	4	yes	—
162	C₈	#14519698	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	#16839665	1	1	yes	characteristic
165	C₂²	#16840108	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	8	4	no	—
168	C₄	C₂×#14519698	1	2	yes	—
169	C₄	#16840034	1	4	yes	—
170	C₄	#16840134	1	1	yes	characteristic
171	C₄	#16840138	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	#21446626	1	1	yes	characteristic
174	C₂	#21446735	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445991

C₂

yes

maximal, characteristic

#21444491

C₂

yes

maximal

#21444102

C₂

yes

maximal, characteristic

C₂×C₆₁×#494

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

C₂×#14519713

—

#16839786

C₂²

yes

—

#16839508

—

#16839483

—

C₈×#92418

—

C₂²×Q₁₆×C₆₁

—

C₂×C₄⋊Q₈×C₆₁

C₂²

yes

—

C₆₁×#494

—

C₆₁×#494

—

C₆₁×#494

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

—

—

D₄

yes

—

C₂²⋊#92418

—

C₄×#92418

—

C₂²×Q₈×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

—

#11758194

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#124

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×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

#121

yes

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₈×C₆₁

#102

yes

—

C₈×C₆₁

#102

yes

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#390

yes

characteristic

C₂²×C₆₁

#486

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#102

yes

—

C₄×C₆₁

#473

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

D₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#1957

yes

characteristic

C₂×C₆₁

#1979

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₆₁

yes

—

#494

—

#494

—

#494

Dic₆₁

yes

—

#431

—

C₂×#108

—

122

—

#423

—

100

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

101

#332

—

122

—

102

#322

—

122

—

103

C₈²

—

122

—

104

C₆₁

#9772

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂×Q₁₆

—

110

C₂×M₁₆

—

122

—

111

C₂²×C₈

C₂²⋊D₆₁

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×Dic₆₁

yes

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²×D₆₁

yes

characteristic

117

#108

—

244

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

122

—

120

C₂²⋊C₈

—

122

—

121

C₄×C₈

—

122

—

122

C₄×C₈

C₂×Dic₆₁

yes

—

123

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

124

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

#11758224

yes

—

131

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

132

Q₁₆

—

133

Q₁₆

—

134

M₁₆

—

244

—

135

C₂×C₈

—

122

—

136

C₂×C₈

—

244

—

137

C₂×C₈

#11758199

yes

—

138

C₂×C₈

#11758200

yes

—

139

C₂×C₈

C₂²⋊Dic₆₁

yes

—

140

C₂×C₈

C₂²⋊Dic₆₁

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

D₄×D₆₁

yes

—

144

C₄²

C₂²×Dic₆₁

yes

characteristic

145

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

146

C₂³

#14519764

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

154

C₂×C₄

D₄×Dic₆₁

yes

—

155

C₂×C₄

#14519748

yes

—

156

C₂×C₄

#14519773

yes

characteristic

157

C₂×C₄

C₂×#11758199

yes

—

158

C₂×C₄

C₂×#11758200

yes

—

159

C₈

—

488

—

160

C₈

—

244

—

161

C₈

#14519698

yes

—

162

C₈

#14519698

yes

—

163

C₂²

—

164

C₂²

#16839665

yes

characteristic

165

C₂²

#16840108

yes

—

166

C₄

—

167

C₄

—

168

C₄

C₂×#14519698

yes

—

169

C₄

#16840034

yes

—

170

C₄

#16840134

yes

characteristic

171

C₄

#16840138

yes

characteristic

172

C₂

—

173

C₂

#21446626

yes

characteristic

174

C₂

#21446735

yes

—

	subgroup	G/H
center	C₂²	#16839665
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#494	C₂
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	#124
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#494	D₆₁
hypocenter	C₆₁	#9772
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#16839665

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₆₁×#494

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
104. C₆₁	#9772	semidirect

H ≅ G/N

type

104. C₆₁

#9772

semidirect

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

subgroup	index
#21445991	2
#21444491	2
#21444102	2
C₂×C₆₁×#494	2
#9772	61

subgroup

index

C₂×C₆₁×#494

Subgroups — #29579477

Conjugacy classes of subgroups

Complements

Maximal subgroups