Group #29568331 — subgroups

Subgroups — #29568331

Order 15616.

384 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 119 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	#21445962	C₂	1	2	yes	maximal
2	#21443642	C₂	1	4	yes	maximal
3	C₄×#125×C₆₁	C₂	1	1	yes	maximal, characteristic
4	#16839517	—	2	2	no	—
5	#16839517	—	2	2	no	—
6	#16839517	C₂²	1	4	yes	—
7	C₂×#125×C₆₁	C₄	1	1	yes	characteristic
8	C₆₁×#389	C₂²	1	2	yes	—
9	C₆₁×#387	—	2	2	no	—
10	C₆₁×#382	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
12	C₄×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
13	C₄³×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×#92418	—	2	4	no	—
15	#125×C₆₁	—	2	4	no	—
16	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
17	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
18	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₆₁	—	2	2	no	—
21	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₁	D₄	1	2	yes	—
24	C₂×C₄²×C₆₁	D₄	1	2	yes	—
25	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
26	#95×C₆₁	—	2	2	no	—
27	#95×C₆₁	—	2	2	no	—
28	#95×C₆₁	—	2	2	no	—
29	C₂²×C₄×C₆₁	—	2	2	no	—
30	C₂²×C₄×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
36	C₄⋊C₄×C₆₁	—	2	8	no	—
37	C₄⋊C₄×C₆₁	—	2	2	no	—
38	C₄⋊C₄×C₆₁	—	4	2	no	—
39	C₄⋊C₄×C₆₁	—	4	4	no	—
40	C₄⋊C₄×C₆₁	M₁₆	1	4	yes	—
41	C₄²×C₆₁	—	2	2	no	—
42	C₄²×C₆₁	—	2	2	no	—
43	C₄²×C₆₁	—	2	4	no	—
44	C₄²×C₆₁	—	4	2	no	—
45	C₄²×C₆₁	—	4	1	no	—
46	C₄²×C₆₁	—	4	4	no	—
47	C₂×#92418	—	4	4	no	—
48	C₂×#92418	—	4	4	no	—
49	C₂×#92418	—	4	4	no	—
50	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	—	2	1	no	—
53	C₂×C₄×C₆₁	—	2	1	no	—
54	C₂×C₄×C₆₁	—	2	2	no	—
55	C₂×C₄×C₆₁	—	2	2	no	—
56	C₂×C₄×C₆₁	—	2	2	no	—
57	C₂×C₄×C₆₁	—	2	4	no	—
58	C₂×C₄×C₆₁	—	2	1	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	4	2	no	—
63	C₂×C₄×C₆₁	—	4	2	no	—
64	C₂×C₄×C₆₁	—	4	2	no	—
65	C₂×C₄×C₆₁	—	4	2	no	—
66	C₂×C₄×C₆₁	—	4	2	no	—
67	C₂×C₄×C₆₁	#101	1	2	yes	—
68	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₆₁	#135	1	4	yes	—
70	#92418	—	8	4	no	—
71	#7650	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	#407	1	1	yes	characteristic
73	C₂²×C₆₁	#410	1	1	yes	characteristic
74	C₂²×C₆₁	C₂×#101	1	1	yes	characteristic
75	C₂²×C₆₁	#416	1	2	yes	—
76	C₂²×C₆₁	#421	1	2	yes	—
77	C₄×C₆₁	—	2	2	no	—
78	C₄×C₆₁	—	2	4	no	—
79	C₄×C₆₁	—	2	2	no	—
80	C₄×C₆₁	—	4	1	no	—
81	C₄×C₆₁	—	4	2	no	—
82	C₄×C₆₁	—	4	4	no	—
83	C₄×C₆₁	—	8	2	no	—
84	C₄×#125	D₆₁	1	1	yes	characteristic
85	#1839	—	61	2	no	—
86	#1271	—	61	4	no	—
87	C₂×C₆₁	#1467	1	1	yes	characteristic
88	C₂×C₆₁	#1488	1	2	yes	—
89	C₂×C₆₁	#1521	1	2	yes	—
90	C₂×C₆₁	#1534	1	1	yes	characteristic
91	C₂×C₆₁	#1544	1	1	yes	characteristic
92	C₂×#125	Dic₆₁	1	1	yes	characteristic
93	#389	C₂×D₆₁	1	2	yes	—
94	#387	—	2	2	no	—
95	#382	Dic₆₁	1	1	yes	characteristic
96	C₄×C₄⋊C₄	—	2	2	no	—
97	C₄×C₄⋊C₄	Dic₆₁	1	2	yes	—
98	C₄³	C₂×D₆₁	1	1	yes	characteristic
99	#336	—	61	4	no	—
100	#336	—	122	2	no	—
101	#336	—	122	2	no	—
102	C₆₁	#7650	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₈	—	122	4	no	—
104	#125	—	2	4	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
108	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
109	C₂×C₄²	—	2	2	no	—
110	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
113	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
114	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
115	#95	—	2	2	no	—
116	#95	—	2	2	no	—
117	#95	—	2	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
121	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
122	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
123	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
125	C₂×C₈	—	244	4	no	—
126	C₂×C₈	—	244	4	no	—
127	C₂×C₈	—	244	4	no	—
128	C₄⋊C₄	—	2	8	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	4	no	—
132	C₄⋊C₄	#11758194	1	4	yes	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	4	no	—
136	C₄²	—	4	2	no	—
137	C₄²	—	4	1	no	—
138	C₄²	—	4	4	no	—
139	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	#14519696	1	2	yes	—
157	C₂×C₄	C₂×#11758194	1	2	yes	—
158	C₂×C₄	#14519797	1	4	yes	—
159	C₈	—	488	4	no	—
160	C₂²	#16839726	1	1	yes	characteristic
161	C₂²	#16839736	1	1	yes	characteristic
162	C₂²	C₂×#14519696	1	1	yes	characteristic
163	C₂²	#16839758	1	2	yes	—
164	C₂²	#16839779	1	2	yes	—
165	C₄	—	2	4	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	4	no	—
171	C₄	—	8	2	no	—
172	C₂	#21444163	1	1	yes	characteristic
173	C₂	#21444253	1	2	yes	—
174	C₂	#21444396	1	2	yes	—
175	C₂	#21444446	1	1	yes	characteristic
176	C₂	#21444484	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445962

C₂

yes

maximal

#21443642

C₂

yes

maximal

C₄×#125×C₆₁

C₂

yes

maximal, characteristic

#16839517

—

#16839517

—

#16839517

C₂²

yes

—

C₂×#125×C₆₁

C₄

yes

characteristic

C₆₁×#389

C₂²

yes

—

C₆₁×#387

—

C₆₁×#382

C₄

yes

characteristic

C₄×C₄⋊C₄×C₆₁

—

C₄×C₄⋊C₄×C₆₁

C₄

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₂²×#92418

—

#125×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#95×C₆₁

—

#95×C₆₁

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

M₁₆

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

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

—

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

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

#135

yes

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#421

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×#125

D₆₁

yes

characteristic

#1839

—

#1271

—

C₂×C₆₁

#1467

yes

characteristic

C₂×C₆₁

#1488

yes

—

C₂×C₆₁

#1521

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

Dic₆₁

yes

characteristic

#389

C₂×D₆₁

yes

—

#387

—

#382

Dic₆₁

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₆₁

yes

—

C₄³

C₂×D₆₁

yes

characteristic

#336

—

100

#336

—

122

—

101

#336

—

122

—

102

C₆₁

#7650

yes

characteristic, complemented, Sylow

103

C₂²×C₈

—

122

—

104

#125

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

C₂×Dic₆₁

yes

—

108

C₂×C₄⋊C₄

C₂×Dic₆₁

yes

—

109

C₂×C₄²

—

110

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

111

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

112

C₂×C₄²

C₂²⋊D₆₁

yes

—

113

C₂×C₄²

C₂²⋊D₆₁

yes

—

114

C₂×C₄²

C₂²×D₆₁

yes

characteristic

115

#95

—

116

#95

—

117

#95

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂²⋊Dic₆₁

yes

characteristic

121

C₂²×C₄

C₂²⋊Dic₆₁

yes

—

122

C₂²×C₄

C₂²⋊Dic₆₁

yes

characteristic

123

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

124

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

—

125

C₂×C₈

—

244

—

126

C₂×C₈

—

244

—

127

C₂×C₈

—

244

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

#11758194

yes

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₂³

C₂×C₂²⋊Dic₆₁

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

#14519696

yes

—

157

C₂×C₄

C₂×#11758194

yes

—

158

C₂×C₄

#14519797

yes

—

159

C₈

—

488

—

160

C₂²

#16839726

yes

characteristic

161

C₂²

#16839736

yes

characteristic

162

C₂²

C₂×#14519696

yes

characteristic

163

C₂²

#16839758

yes

—

164

C₂²

#16839779

yes

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₂

#21444163

yes

characteristic

173

C₂

#21444253

yes

—

174

C₂

#21444396

yes

—

175

C₂

#21444446

yes

characteristic

176

C₂

#21444484

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×#125×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#125	D₆₁
hypocenter	C₆₁	#7650
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₁

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₄×#125×C₆₁

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 ◁ C₄×#125

N	H ≅ G/N	type
102. C₆₁	#7650	semidirect

H ≅ G/N

type

102. C₆₁

#7650

semidirect

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

subgroup	index
#21445962	2
#21443642	2
C₄×#125×C₆₁	2
#7650	61

subgroup

index

#21445962

#21443642

C₄×#125×C₆₁

#7650

Subgroups — #29568331

Conjugacy classes of subgroups

Complements

Maximal subgroups