Group #29573664 — subgroups

Subgroups — #29573664

Order 15616.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#21445572	C₂	1	1	yes	maximal, characteristic
2	#21444565	C₂	1	2	yes	maximal
3	#21443664	C₂	1	1	yes	maximal, characteristic
4	#21443647	C₂	1	2	yes	maximal
5	C₆₁×#1281	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	#16839608	C₂²	1	1	yes	characteristic
9	#16839516	C₄	1	2	yes	—
10	#16839516	C₄	1	2	yes	—
11	#16839514	—	2	2	no	—
12	#16839512	—	2	2	no	—
13	C₆₁×#431	C₂²	1	2	yes	—
14	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×#92418	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
18	C₄⋊#92418	—	2	2	no	—
19	C₂²⋊#92418	—	2	2	no	—
20	C₄×#92418	—	2	2	no	—
21	C₄×#92418	D₄	1	1	yes	characteristic
22	C₄×#92418	D₄	1	1	yes	characteristic
23	C₄×#92418	Q₈	1	2	yes	—
24	#14519684	—	2	2	no	—
25	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₆₁	—	2	2	no	—
28	C₂²⋊C₈×C₆₁	—	2	2	no	—
29	#97×C₆₁	—	2	2	no	—
30	C₄×C₈×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	#41	1	2	yes	—
33	C₂²×Dic₆₁	—	2	1	no	—
34	C₂²×Dic₆₁	—	2	1	no	—
35	C₂×C₈×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	—	2	2	no	—
37	C₂×C₈×C₆₁	—	2	2	no	—
38	C₂×C₈×C₆₁	—	4	2	no	—
39	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₆₁	#41	1	2	yes	—
42	C₄×Dic₆₁	—	2	2	no	—
43	C₄×Dic₆₁	—	2	2	no	—
44	C₂×#92418	—	2	2	no	—
45	C₂×#92418	—	2	2	no	—
46	C₂×#92418	—	4	2	no	—
47	C₂×#92418	C₄⋊C₄	1	2	yes	—
48	C₂×#92418	C₄⋊C₄	1	2	yes	—
49	C₂³×C₆₁	#117	1	1	yes	characteristic
50	C₂×C₄×C₆₁	—	4	2	no	—
51	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#124	1	2	yes	—
54	C₂×C₄×C₆₁	#125	1	2	yes	—
55	C₂×C₄×C₆₁	#131	1	2	yes	—
56	C₂×C₄×C₆₁	#131	1	2	yes	—
57	C₂×Dic₆₁	—	4	2	no	—
58	C₂×Dic₆₁	—	4	2	no	—
59	C₂×Dic₆₁	—	4	1	no	—
60	C₂×Dic₆₁	—	4	1	no	—
61	C₈×C₆₁	—	4	2	no	—
62	C₈×C₆₁	—	8	2	no	—
63	#92418	—	2	2	no	—
64	#92418	—	2	2	no	—
65	#92418	—	8	2	no	—
66	#8782	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	—	2	2	no	—
68	C₂²×C₆₁	#382	1	1	yes	characteristic
69	C₂²×C₆₁	#433	1	1	yes	characteristic
70	C₂²×C₆₁	#433	1	1	yes	characteristic
71	C₄×C₆₁	—	2	2	no	—
72	C₄×C₆₁	—	2	2	no	—
73	C₄×C₆₁	#424	1	2	yes	—
74	C₄×C₆₁	#433	1	2	yes	—
75	Dic₆₁	—	8	1	no	—
76	Dic₆₁	—	8	1	no	—
77	#1746	—	61	1	no	—
78	#1564	—	61	2	no	—
79	#1281	—	61	1	no	—
80	#1281	D₆₁	1	1	yes	characteristic
81	#1274	—	61	2	no	—
82	C₂×C₆₁	—	2	1	no	—
83	C₂×C₆₁	—	2	1	no	—
84	C₂×C₆₁	#1832	1	1	yes	characteristic
85	C₂×C₆₁	#1833	1	2	yes	—
86	#431	—	61	2	no	—
87	#431	C₂×D₆₁	1	2	yes	—
88	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
89	C₂×C₄×C₈	—	61	1	no	—
90	#376	—	61	1	no	—
91	#336	—	61	2	no	—
92	#336	—	61	2	no	—
93	#335	—	122	2	no	—
94	#334	—	122	2	no	—
95	C₆₁	#8782	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	61	2	no	—
97	C₂²×C₈	C₄×D₆₁	1	2	yes	—
98	C₂×C₄²	—	61	1	no	—
99	C₂×C₄²	—	61	1	no	—
100	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	122	2	no	—
103	C₂²⋊C₈	—	2	2	no	—
104	C₂²⋊C₈	—	122	2	no	—
105	#97	—	2	2	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	61	1	no	—
108	C₄×C₈	—	61	1	no	—
109	C₄×C₈	—	61	2	no	—
110	C₄×C₈	—	122	2	no	—
111	#95	—	122	2	no	—
112	C₂²×C₄	—	122	1	no	—
113	C₂²×C₄	—	122	1	no	—
114	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
115	C₂²×C₄	#11758222	1	2	yes	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	61	2	no	—
121	C₂×C₈	—	61	2	no	—
122	C₂×C₈	—	122	2	no	—
123	C₂×C₈	—	122	2	no	—
124	C₂×C₈	—	244	2	no	—
125	C₄²	—	122	2	no	—
126	C₄²	—	122	2	no	—
127	C₄²	D₄×D₆₁	1	1	yes	characteristic
128	C₄²	#11758224	1	2	yes	—
129	C₄²	Q₈×D₆₁	1	1	yes	characteristic
130	C₂³	#14519734	1	1	yes	characteristic
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	244	2	no	—
133	C₂×C₄	—	244	2	no	—
134	C₂×C₄	—	244	1	no	—
135	C₂×C₄	—	244	1	no	—
136	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
137	C₂×C₄	#14519733	1	1	yes	characteristic
138	C₂×C₄	#14519763	1	2	yes	—
139	C₂×C₄	#14519767	1	2	yes	—
140	C₂×C₄	#14519783	1	2	yes	—
141	C₂×C₄	#14519785	1	2	yes	—
142	C₈	—	4	2	no	—
143	C₈	—	8	2	no	—
144	C₈	—	122	2	no	—
145	C₈	—	122	2	no	—
146	C₈	—	488	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	#16839626	1	1	yes	characteristic
149	C₂²	#16839821	1	1	yes	characteristic
150	C₂²	#16839824	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	488	1	no	—
154	C₄	—	488	1	no	—
155	C₄	#16839787	1	2	yes	—
156	C₄	#16839820	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	#21445937	1	1	yes	characteristic
160	C₂	#21445940	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445572

C₂

yes

maximal, characteristic

#21444565

C₂

yes

maximal

#21443664

C₂

yes

maximal, characteristic

#21443647

C₂

yes

maximal

C₆₁×#1281

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

#16839608

C₂²

yes

characteristic

#16839516

C₄

yes

—

#16839516

C₄

yes

—

#16839514

—

#16839512

—

C₆₁×#431

C₂²

yes

—

C₂×#97×C₆₁

C₂²

yes

characteristic

C₂²×#92418

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

—

—

D₄

yes

characteristic

C₄×#92418

D₄

yes

characteristic

C₄×#92418

Q₈

yes

—

#14519684

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

#97×C₆₁

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

—

—

—

C₄⋊C₄

yes

—

C₂×#92418

C₄⋊C₄

yes

—

C₂³×C₆₁

#117

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#125

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#424

yes

—

C₄×C₆₁

#433

yes

—

Dic₆₁

—

Dic₆₁

—

—

—

—

D₆₁

yes

characteristic

#1274

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1832

yes

characteristic

C₂×C₆₁

#1833

yes

—

#431

—

#431

C₂×D₆₁

yes

—

C₂×#97

C₂×D₆₁

yes

characteristic

C₂×C₄×C₈

—

—

—

—

—

122

—

#334

—

122

—

C₆₁

#8782

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

C₂²×D₆₁

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

122

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

—

122

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

122

—

111

#95

—

122

—

112

C₂²×C₄

—

122

—

113

C₂²×C₄

—

122

—

114

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

115

C₂²×C₄

#11758222

yes

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

122

—

123

C₂×C₈

—

122

—

124

C₂×C₈

—

244

—

125

C₄²

—

122

—

126

C₄²

—

122

—

127

C₄²

D₄×D₆₁

yes

characteristic

128

C₄²

#11758224

yes

—

129

C₄²

Q₈×D₆₁

yes

characteristic

130

C₂³

#14519734

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

244

—

133

C₂×C₄

—

244

—

134

C₂×C₄

—

244

—

135

C₂×C₄

—

244

—

136

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

137

C₂×C₄

#14519733

yes

characteristic

138

C₂×C₄

#14519763

yes

—

139

C₂×C₄

#14519767

yes

—

140

C₂×C₄

#14519783

yes

—

141

C₂×C₄

#14519785

yes

—

142

C₈

—

143

C₈

—

144

C₈

—

122

—

145

C₈

—

122

—

146

C₈

—

488

—

147

C₂²

—

148

C₂²

#16839626

yes

characteristic

149

C₂²

#16839821

yes

characteristic

150

C₂²

#16839824

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

488

—

154

C₄

—

488

—

155

C₄

#16839787

yes

—

156

C₄

#16839820

yes

—

157

C₂

—

158

C₂

—

159

C₂

#21445937

yes

characteristic

160

C₂

#21445940

yes

—

	subgroup	G/H
center	C₂×C₄	#14519733
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1281	C₂
socle	C₂²×C₆₁	#382
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1281	D₆₁
hypocenter	C₆₁	#8782
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂×C₄

#14519733

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1281

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
95. C₆₁	#8782	semidirect

H ≅ G/N

type

95. C₆₁

#8782

semidirect

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

subgroup	index
#21445572	2
#21444565	2
#21443664	2
#21443647	2
C₆₁×#1281	2
#8782	61

subgroup

index

C₆₁×#1281

Subgroups — #29573664

Conjugacy classes of subgroups

Complements

Maximal subgroups