Group #29514930 — subgroups

Subgroups — C₆₁×#11497 #29514930

Order 15616.

288 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 130 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	C₆₁×#2960	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2163	C₂	1	2	yes	maximal
3	C₆₁×#1323	C₂	1	4	yes	maximal
4	C₆₁×#533	C₄	1	1	yes	characteristic
5	C₆₁×#446	—	2	2	no	—
6	C₈×Q₈×C₆₁	—	2	2	no	—
7	C₆₁×#424	C₂²	1	2	yes	—
8	C₆₁×#423	C₄	1	1	yes	characteristic
9	C₆₁×#405	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×C₄⋊C₁₆	—	2	2	no	—
12	C₆₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
13	C₆₁×#346	—	2	2	no	—
14	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
17	C₄⋊Q₈×C₆₁	C₈	1	2	yes	—
18	#125×C₆₁	C₈	1	2	yes	—
19	C₂²⋊Q₈×C₆₁	—	2	2	no	—
20	C₄×Q₈×C₆₁	—	2	2	no	—
21	#117×C₆₁	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₆₁	—	2	4	no	—
24	C₄⋊C₈×C₆₁	—	2	4	no	—
25	C₄⋊C₈×C₆₁	—	2	2	no	—
26	#97×C₆₁	—	2	2	no	—
27	C₄×C₈×C₆₁	—	2	2	no	—
28	C₄×C₈×C₆₁	D₄	1	2	yes	—
29	C₄×C₈×C₆₁	D₄	1	2	yes	—
30	C₂×Q₈×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
32	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₆₁	—	4	2	no	—
34	C₂×C₈×C₆₁	—	2	2	no	—
35	C₂×C₈×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	—	4	1	no	—
37	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
40	C₄⋊C₄×C₆₁	—	2	4	no	—
41	C₄⋊C₄×C₆₁	—	2	2	no	—
42	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
43	C₂²⋊C₄×C₆₁	—	2	2	no	—
44	C₄²×C₆₁	—	2	2	no	—
45	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
46	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
48	C₁₆×C₆₁	—	4	4	no	—
49	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
50	Q₈×C₆₁	—	4	4	no	—
51	C₂×C₄×C₆₁	—	2	1	no	—
52	C₂×C₄×C₆₁	—	2	1	no	—
53	C₂×C₄×C₆₁	—	2	4	no	—
54	C₂×C₄×C₆₁	—	4	2	no	—
55	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₆₁	—	2	2	no	—
60	C₈×C₆₁	—	2	1	no	—
61	C₈×C₆₁	—	4	2	no	—
62	C₈×C₆₁	#101	1	2	yes	—
63	#11497	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂²×C₆₁	—	2	2	no	—
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	—	2	2	no	—
69	C₄×C₆₁	—	2	1	no	—
70	C₄×C₆₁	—	4	4	no	—
71	C₄×C₆₁	#410	1	1	yes	characteristic
72	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
73	#2960	C₂×C₆₁	1	1	yes	characteristic
74	#2163	C₂×C₆₁	1	2	yes	—
75	#1323	C₂×C₆₁	1	4	yes	—
76	C₂×C₆₁	—	4	1	no	—
77	C₂×C₆₁	#1461	1	1	yes	characteristic
78	C₂×C₆₁	#2117	1	2	yes	—
79	#533	C₄×C₆₁	1	1	yes	characteristic
80	#446	—	2	2	no	—
81	C₈×Q₈	—	2	2	no	—
82	#424	C₂²×C₆₁	1	2	yes	—
83	#423	C₄×C₆₁	1	1	yes	characteristic
84	#405	C₄×C₆₁	1	2	yes	—
85	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
86	C₄⋊C₁₆	—	2	2	no	—
87	C₂²⋊C₁₆	C₂²×C₆₁	1	4	yes	—
88	#346	—	2	2	no	—
89	C₆₁	#11497	1	1	yes	characteristic, direct product factor, Sylow
90	C₂×M₁₆	C₂×C₄×C₆₁	1	2	yes	—
91	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
92	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
93	C₄⋊Q₈	C₈×C₆₁	1	2	yes	—
94	#125	C₈×C₆₁	1	2	yes	—
95	C₂²⋊Q₈	—	2	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	#117	C₂×C₄×C₆₁	1	2	yes	—
98	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
99	C₂×C₁₆	—	2	4	no	—
100	C₄⋊C₈	—	2	4	no	—
101	C₄⋊C₈	—	2	2	no	—
102	#97	—	2	2	no	—
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	D₄×C₆₁	1	2	yes	—
105	C₄×C₈	D₄×C₆₁	1	2	yes	—
106	C₂×Q₈	—	2	2	no	—
107	C₂²×C₄	M₁₆×C₆₁	1	2	yes	—
108	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
109	M₁₆	—	4	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
114	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
115	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂×C₈×C₆₁	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
122	C₄²	C₂×C₈×C₆₁	1	1	yes	characteristic
123	C₄²	C₂×C₈×C₆₁	1	1	yes	characteristic
124	C₁₆	—	4	4	no	—
125	C₂³	C₂×M₁₆×C₆₁	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
132	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
134	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	1	no	—
137	C₈	—	4	2	no	—
138	C₈	#101×C₆₁	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	4	no	—
146	C₄	C₆₁×#410	1	1	yes	characteristic
147	C₄	C₂×#101×C₆₁	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	C₆₁×#1461	1	1	yes	characteristic
150	C₂	C₆₁×#2117	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#2960

C₂

yes

maximal, characteristic

C₆₁×#2163

C₂

yes

maximal

C₆₁×#1323

C₂

yes

maximal

C₆₁×#533

C₄

yes

characteristic

C₆₁×#446

—

C₈×Q₈×C₆₁

—

C₆₁×#424

C₂²

yes

—

C₆₁×#423

C₄

yes

characteristic

C₆₁×#405

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

—

C₆₁×C₂²⋊C₁₆

C₂²

yes

—

C₆₁×#346

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊Q₈×C₆₁

C₈

yes

—

#125×C₆₁

C₈

yes

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

#97×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₈

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₁₆×C₆₁

—

C₂³×C₆₁

C₂×M₁₆

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#101

yes

—

#11497

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₂×C₆₁

yes

characteristic

#2163

C₂×C₆₁

yes

—

#1323

C₂×C₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

#1461

yes

characteristic

C₂×C₆₁

#2117

yes

—

#533

C₄×C₆₁

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₂²×C₆₁

yes

—

#423

C₄×C₆₁

yes

characteristic

#405

C₄×C₆₁

yes

—

C₂×C₄×C₈

C₂²×C₆₁

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

C₂²×C₆₁

yes

—

#346

—

C₆₁

#11497

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

C₂×C₄×C₆₁

yes

—

C₂²×C₈

C₂×C₄×C₆₁

yes

characteristic

C₂²×C₈

C₂³×C₆₁

yes

characteristic

C₄⋊Q₈

C₈×C₆₁

yes

—

#125

C₈×C₆₁

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×C₄×C₆₁

yes

—

C₂×C₄²

C₂×C₄×C₆₁

yes

characteristic

C₂×C₁₆

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#97

—

103

C₄×C₈

—

104

C₄×C₈

D₄×C₆₁

yes

—

105

C₄×C₈

D₄×C₆₁

yes

—

106

C₂×Q₈

—

107

C₂²×C₄

M₁₆×C₆₁

yes

—

108

C₂²×C₄

C₂²×C₄×C₆₁

yes

characteristic

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²⋊C₄×C₆₁

yes

characteristic

114

C₂×C₈

C₂²⋊C₄×C₆₁

yes

characteristic

115

C₂×C₈

C₂×D₄×C₆₁

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂×C₈×C₆₁

yes

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂²⋊C₄×C₆₁

yes

—

122

C₄²

C₂×C₈×C₆₁

yes

characteristic

123

C₄²

C₂×C₈×C₆₁

yes

characteristic

124

C₁₆

—

125

C₂³

C₂×M₁₆×C₆₁

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊C₈×C₆₁

yes

—

132

C₂×C₄

C₂²⋊C₈×C₆₁

yes

—

133

C₂×C₄

C₂×C₂²⋊C₄×C₆₁

yes

characteristic

134

C₂×C₄

C₂²×C₈×C₆₁

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#101×C₆₁

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊C₈×C₆₁

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

C₆₁×#410

yes

characteristic

147

C₄

C₂×#101×C₆₁

yes

characteristic

148

C₂

—

149

C₂

C₆₁×#1461

yes

characteristic

150

C₂

C₆₁×#2117

yes

—

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₈×C₆₁

Frattini, cosocle

C₂²×C₈

C₂³×C₆₁

socle

C₂²×C₆₁

C₂×C₂²⋊C₈

prime radical

C₂³×C₆₁

C₂×M₁₆

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

N	H ≅ G/N	type
63. #11497	C₆₁	direct (N × H)
89. C₆₁	#11497	direct (N × H)

H ≅ G/N

type

63. #11497

C₆₁

direct (N × H)

89. C₆₁

#11497

direct (N × H)

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

subgroup	index
C₆₁×#2960	2
C₆₁×#2163	2
C₆₁×#1323	2
#11497	61

subgroup

index

C₆₁×#2960

C₆₁×#2163

C₆₁×#1323

#11497

Subgroups — C₆₁×#11497 #29514930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#11497 #29514930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#11497 #29514930