Group #29579108 — subgroups

Subgroups — #29579108

Order 15616.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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	#21445936	C₂	1	1	yes	maximal, characteristic
2	#21445672	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839508	C₂	1	4	yes	maximal
4	C₆₁×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92418	C₂²	1	2	yes	—
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×#14519688	—	2	1	no	—
8	#16839517	—	2	1	no	—
9	#16839517	—	2	2	no	—
10	#16839508	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#384	—	2	2	no	—
15	C₆₁×#384	C₄	1	4	yes	—
16	C₄³×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×#92418	—	2	2	no	—
18	C₂²×#92418	—	2	2	no	—
19	C₄⋊#92418	—	2	2	no	—
20	C₄⋊#92418	—	2	4	no	—
21	C₄⋊#92418	D₄	1	4	yes	—
22	#14519688	—	4	2	no	—
23	C₂²×Q₈×C₆₁	—	2	2	no	—
24	C₄⋊Q₈×C₆₁	—	2	4	no	—
25	C₄⋊Q₈×C₆₁	—	2	4	no	—
26	C₄⋊Q₈×C₆₁	—	4	4	no	—
27	C₄⋊Q₈×C₆₁	—	4	2	no	—
28	C₄⋊Q₈×C₆₁	—	4	2	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	D₄	1	2	yes	—
36	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₆₁	—	4	4	no	—
39	C₂×Q₈×C₆₁	—	4	2	no	—
40	C₂×Q₈×C₆₁	—	4	2	no	—
41	C₂×Q₈×C₆₁	—	4	4	no	—
42	C₂²×C₄×C₆₁	—	2	4	no	—
43	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₄⋊C₄×C₆₁	—	2	8	no	—
49	C₄⋊C₄×C₆₁	—	4	8	no	—
50	C₄⋊C₄×C₆₁	—	4	8	no	—
51	C₄⋊C₄×C₆₁	—	4	2	no	—
52	C₄⋊C₄×C₆₁	—	4	2	no	—
53	C₄²×C₆₁	—	2	4	no	—
54	C₄²×C₆₁	—	2	2	no	—
55	C₄²×C₆₁	—	2	4	no	—
56	C₄²×C₆₁	D₈	1	2	yes	—
57	C₄²×C₆₁	SD₁₆	1	2	yes	—
58	C₄²×C₆₁	C₂×D₄	1	2	yes	—
59	C₄²×C₆₁	#41	1	2	yes	—
60	C₂×#92418	—	2	4	no	—
61	C₂×#92418	—	4	2	no	—
62	C₂×#92418	—	4	2	no	—
63	C₂×#92418	—	4	2	no	—
64	C₂×#92418	—	4	4	no	—
65	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	Q₈×C₆₁	—	8	4	no	—
67	Q₈×C₆₁	—	8	4	no	—
68	C₂×C₄×C₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	—	2	2	no	—
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	—	2	2	no	—
73	C₂×C₄×C₆₁	—	4	4	no	—
74	C₂×C₄×C₆₁	—	4	4	no	—
75	C₂×C₄×C₆₁	—	4	4	no	—
76	C₂×C₄×C₆₁	#101	1	4	yes	—
77	C₂×C₄×C₆₁	#102	1	4	yes	—
78	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₆₁	#121	1	2	yes	—
80	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₆₁	#137	1	4	yes	—
86	#92418	—	4	4	no	—
87	#92418	—	8	2	no	—
88	#9705	—	61	1	no	maximal, Sylow
89	C₂²×C₆₁	#390	1	1	yes	characteristic
90	C₂²×C₆₁	C₂×#101	1	2	yes	—
91	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
92	C₂²×C₆₁	#418	1	1	yes	characteristic
93	C₂²×C₆₁	#419	1	2	yes	—
94	C₄×C₆₁	—	2	4	no	—
95	C₄×C₆₁	—	2	4	no	—
96	C₄×C₆₁	—	2	2	no	—
97	C₄×C₆₁	—	8	4	no	—
98	C₄×C₆₁	#459	1	2	yes	—
99	C₄×C₆₁	#460	1	2	yes	—
100	C₄×C₆₁	#491	1	2	yes	—
101	C₄×C₆₁	#497	1	2	yes	—
102	#2868	D₆₁	1	1	yes	characteristic
103	#1831	—	61	1	no	—
104	#1767	—	61	1	no	—
105	C₂×#332	—	61	4	no	—
106	C₂×C₆₁	#1546	1	2	yes	—
107	C₂×C₆₁	#1556	1	2	yes	—
108	C₂×C₆₁	#1960	1	1	yes	characteristic
109	C₂×C₆₁	#1966	1	1	yes	characteristic
110	C₂×C₆₁	#1982	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	—	2	1	no	—
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂×C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
114	C₂×C₄⋊C₈	—	61	2	no	—
115	C₂×C₄⋊C₈	—	61	2	no	—
116	C₂×#97	—	122	1	no	—
117	#384	—	2	2	no	—
118	#384	Dic₆₁	1	4	yes	—
119	C₄³	C₂×D₆₁	1	1	yes	characteristic
120	#336	—	122	2	no	—
121	#336	—	122	1	no	—
122	#332	—	122	8	no	—
123	C₆₁	#9705	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	2	2	no	—
125	C₂²×C₈	—	122	2	no	—
126	C₂²×C₈	—	122	2	no	—
127	C₄⋊Q₈	—	2	4	no	—
128	C₄⋊Q₈	—	2	4	no	—
129	C₄⋊Q₈	—	4	4	no	—
130	C₄⋊Q₈	—	4	2	no	—
131	C₄⋊Q₈	—	4	2	no	—
132	C₂×C₄⋊C₄	—	2	4	no	—
133	C₂×C₄⋊C₄	—	2	4	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
136	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
137	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
138	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
139	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
141	C₄⋊C₈	—	61	4	no	—
142	C₄⋊C₈	—	122	4	no	—
143	C₄⋊C₈	—	122	2	no	—
144	#97	—	244	2	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
151	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
152	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
155	C₂×C₈	—	122	4	no	—
156	C₂×C₈	—	244	4	no	—
157	C₂×C₈	—	244	2	no	—
158	C₂×C₈	—	244	2	no	—
159	C₂×C₈	—	244	2	no	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	4	8	no	—
162	C₄⋊C₄	—	4	8	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	#11758199	1	2	yes	—
169	C₄²	#11758200	1	2	yes	—
170	C₄²	D₄×D₆₁	1	2	yes	—
171	C₄²	#11758224	1	2	yes	—
172	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
173	Q₈	—	8	4	no	—
174	Q₈	—	8	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	#14519696	1	4	yes	—
184	C₂×C₄	#14519698	1	4	yes	—
185	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
186	C₂×C₄	#14519748	1	2	yes	—
187	C₂×C₄	#14519764	1	1	yes	characteristic
188	C₂×C₄	#14519773	1	1	yes	characteristic
189	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
190	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
191	C₂×C₄	#14519803	1	2	yes	—
192	C₂×C₄	#14519808	1	4	yes	—
193	C₈	—	488	2	no	—
194	C₈	—	244	4	no	—
195	C₂²	#16839665	1	1	yes	characteristic
196	C₂²	C₂×#14519696	1	2	yes	—
197	C₂²	C₂×#14519698	1	1	yes	characteristic
198	C₂²	#16839767	1	1	yes	characteristic
199	C₂²	#16839771	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	4	no	—
203	C₄	—	8	4	no	—
204	C₄	#16839954	1	2	yes	—
205	C₄	#16839959	1	2	yes	—
206	C₄	#16840130	1	2	yes	—
207	C₄	#16840154	1	2	yes	—
208	C₂	#21444491	1	2	yes	—
209	C₂	#21444536	1	2	yes	—
210	C₂	#21446640	1	1	yes	characteristic
211	C₂	#21446664	1	1	yes	characteristic
212	C₂	#21446747	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445936

C₂

yes

maximal, characteristic

#21445672

C₂

yes

maximal, characteristic

C₂×#16839508

C₂

yes

maximal

C₆₁×#2868

C₂

yes

maximal, characteristic

C₂×C₄⋊#92418

C₂²

yes

—

C₂×C₄⋊#92418

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₆₁

—

C₂×C₄⋊Q₈×C₆₁

—

C₂×C₄⋊Q₈×C₆₁

C₂²

yes

—

C₆₁×#384

—

C₆₁×#384

C₄

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₂²×#92418

—

C₂²×#92418

—

C₄⋊#92418

—

C₄⋊#92418

—

C₄⋊#92418

D₄

yes

—

#14519688

—

C₂²×Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

D₈

yes

—

C₄²×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

#41

yes

—

—

—

—

—

—

C₂³×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#101

yes

—

C₂×C₄×C₆₁

#102

yes

—

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₁

#137

yes

—

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#459

yes

—

C₄×C₆₁

#460

yes

—

100

C₄×C₆₁

#491

yes

—

101

C₄×C₆₁

#497

yes

—

102

#2868

D₆₁

yes

characteristic

103

#1831

—

104

#1767

—

105

C₂×#332

—

106

C₂×C₆₁

#1546

yes

—

107

C₂×C₆₁

#1556

yes

—

108

C₂×C₆₁

#1960

yes

characteristic

109

C₂×C₆₁

#1966

yes

characteristic

110

C₂×C₆₁

#1982

yes

characteristic

111

C₂×C₄⋊Q₈

—

112

C₂×C₄⋊Q₈

—

113

C₂×C₄⋊Q₈

C₂×D₆₁

yes

—

114

C₂×C₄⋊C₈

—

115

C₂×C₄⋊C₈

—

116

C₂×#97

—

122

—

117

#384

—

118

#384

Dic₆₁

yes

—

119

C₄³

C₂×D₆₁

yes

characteristic

120

#336

—

122

—

121

#336

—

122

—

122

#332

—

122

—

123

C₆₁

#9705

yes

characteristic, complemented, Sylow

124

C₂²×Q₈

—

125

C₂²×C₈

—

122

—

126

C₂²×C₈

—

122

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

C₂×Dic₆₁

yes

—

136

C₂×C₄²

C₂×Dic₆₁

yes

—

137

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

138

C₂×C₄²

C₂²⋊D₆₁

yes

—

139

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

140

C₂×C₄²

C₂²×D₆₁

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

122

—

143

C₄⋊C₈

—

122

—

144

#97

—

244

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂²⋊Dic₆₁

yes

—

151

C₂²×C₄

C₂²⋊Dic₆₁

yes

—

152

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

153

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

154

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

155

C₂×C₈

—

122

—

156

C₂×C₈

—

244

—

157

C₂×C₈

—

244

—

158

C₂×C₈

—

244

—

159

C₂×C₈

—

244

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

#11758199

yes

—

169

C₄²

#11758200

yes

—

170

C₄²

D₄×D₆₁

yes

—

171

C₄²

#11758224

yes

—

172

C₂³

C₂×C₂²⋊Dic₆₁

yes

characteristic

173

Q₈

—

174

Q₈

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

#14519696

yes

—

184

C₂×C₄

#14519698

yes

—

185

C₂×C₄

D₄×Dic₆₁

yes

—

186

C₂×C₄

#14519748

yes

—

187

C₂×C₄

#14519764

yes

characteristic

188

C₂×C₄

#14519773

yes

characteristic

189

C₂×C₄

C₂×#11758199

yes

characteristic

190

C₂×C₄

C₂×#11758200

yes

characteristic

191

C₂×C₄

#14519803

yes

—

192

C₂×C₄

#14519808

yes

—

193

C₈

—

488

—

194

C₈

—

244

—

195

C₂²

#16839665

yes

characteristic

196

C₂²

C₂×#14519696

yes

—

197

C₂²

C₂×#14519698

yes

characteristic

198

C₂²

#16839767

yes

characteristic

199

C₂²

#16839771

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

#16839954

yes

—

205

C₄

#16839959

yes

—

206

C₄

#16840130

yes

—

207

C₄

#16840154

yes

—

208

C₂

#21444491

yes

—

209

C₂

#21444536

yes

—

210

C₂

#21446640

yes

characteristic

211

C₂

#21446664

yes

characteristic

212

C₂

#21446747

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#2868	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#2868	D₆₁
hypocenter	C₆₁	#9705
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₁

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#2868

C₂

socle, prime radical

C₂³×C₆₁

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

N	H ≅ G/N	type
123. C₆₁	#9705	semidirect

H ≅ G/N

type

123. C₆₁

#9705

semidirect

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

subgroup	index
#21445936	2
#21445672	2
C₂×#16839508	2
C₆₁×#2868	2
#9705	61

subgroup

index

C₆₁×#2868

Subgroups — #29579108

Conjugacy classes of subgroups

Complements

Maximal subgroups