Group #29513294 — subgroups

Subgroups — C₆₁×#9861 #29513294

Order 15616.

406 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 174 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×#3160	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1848	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1493	C₂	1	4	yes	maximal
4	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#533	C₂²	1	2	yes	—
6	C₆₁×#500	C₄	1	2	yes	—
7	C₆₁×#499	—	2	2	no	—
8	C₆₁×#498	C₄	1	2	yes	—
9	C₂×#108×C₆₁	—	2	2	no	—
10	C₆₁×#427	—	2	2	no	—
11	C₆₁×#423	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×Q₈⋊C₈	—	2	8	no	—
15	C₆₁×#322	—	2	2	no	—
16	C₂×M₁₆×C₆₁	—	2	2	no	—
17	C₂²×C₈×C₆₁	—	2	2	no	—
18	C₂²×C₈×C₆₁	D₄	1	2	yes	—
19	C₄⋊Q₈×C₆₁	C₂×C₄	1	4	yes	—
20	#125×C₆₁	—	2	2	no	—
21	C₂²⋊Q₈×C₆₁	—	2	4	no	—
22	C₄×Q₈×C₆₁	—	2	4	no	—
23	#117×C₆₁	—	2	2	no	—
24	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
25	#108×C₆₁	—	4	4	no	—
26	#107×C₆₁	—	2	4	no	—
27	#106×C₆₁	—	2	4	no	—
28	C₄⋊C₈×C₆₁	—	2	4	no	—
29	C₄×C₈×C₆₁	—	2	2	no	—
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	D₄	1	4	yes	—
33	C₄×C₈×C₆₁	D₄	1	2	yes	—
34	C₂×Q₈×C₆₁	—	2	4	no	—
35	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	#41	1	2	yes	—
37	M₁₆×C₆₁	—	4	4	no	—
38	C₂×C₈×C₆₁	—	2	4	no	—
39	C₂×C₈×C₆₁	—	4	4	no	—
40	C₂×C₈×C₆₁	—	4	2	no	—
41	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
44	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
45	C₄⋊C₄×C₆₁	—	2	4	no	—
46	C₄⋊C₄×C₆₁	—	2	4	no	—
47	C₄⋊C₄×C₆₁	—	4	4	no	—
48	C₂²⋊C₄×C₆₁	—	4	2	no	—
49	C₄²×C₆₁	—	4	2	no	—
50	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₆₁	C₂×D₄	1	2	yes	—
53	C₂³×C₆₁	#124	1	1	yes	characteristic
54	Q₈×C₆₁	—	4	8	no	—
55	C₂×C₄×C₆₁	—	2	4	no	—
56	C₂×C₄×C₆₁	—	4	4	no	—
57	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₆₁	#121	1	2	yes	—
60	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
63	C₈×C₆₁	—	4	4	no	—
64	C₈×C₆₁	—	8	2	no	—
65	C₈×C₆₁	#103	1	4	yes	—
66	C₈×C₆₁	#103	1	4	yes	—
67	#9861	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₆₁	—	2	2	no	—
69	C₂²×C₆₁	#390	1	1	yes	characteristic
70	C₂²×C₆₁	#487	1	2	yes	—
71	C₄×C₆₁	—	2	2	no	—
72	C₄×C₆₁	—	8	4	no	—
73	C₄×C₆₁	C₂×#103	1	2	yes	—
74	C₄×C₆₁	#444	1	4	yes	—
75	C₄×C₆₁	#492	1	1	yes	characteristic
76	C₄×C₆₁	#494	1	1	yes	characteristic
77	#3160	C₂×C₆₁	1	1	yes	characteristic
78	#1848	C₂×C₆₁	1	1	yes	characteristic
79	#1493	C₂×C₆₁	1	4	yes	—
80	C₂×#322	C₂×C₆₁	1	1	yes	characteristic
81	C₂×C₆₁	—	2	2	no	—
82	C₂×C₆₁	#1956	1	1	yes	characteristic
83	C₂×C₆₁	#1975	1	2	yes	—
84	#533	C₂²×C₆₁	1	2	yes	—
85	#500	C₄×C₆₁	1	2	yes	—
86	#499	—	2	2	no	—
87	#498	C₄×C₆₁	1	2	yes	—
88	C₂×#108	—	2	2	no	—
89	#427	—	2	2	no	—
90	#423	C₂²×C₆₁	1	2	yes	—
91	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
92	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
93	Q₈⋊C₈	—	2	8	no	—
94	#322	—	2	2	no	—
95	C₆₁	#9861	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×M₁₆	—	2	2	no	—
97	C₂²×C₈	—	2	2	no	—
98	C₂²×C₈	D₄×C₆₁	1	2	yes	—
99	C₄⋊Q₈	C₂×C₄×C₆₁	1	4	yes	—
100	#125	—	2	2	no	—
101	C₂²⋊Q₈	—	2	4	no	—
102	C₄×Q₈	—	2	4	no	—
103	#117	—	2	2	no	—
104	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
105	#108	—	4	4	no	—
106	#107	—	2	4	no	—
107	#106	—	2	4	no	—
108	C₄⋊C₈	—	2	4	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
111	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
112	C₄×C₈	D₄×C₆₁	1	4	yes	—
113	C₄×C₈	D₄×C₆₁	1	2	yes	—
114	C₂×Q₈	—	2	4	no	—
115	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
116	C₂²×C₄	#41×C₆₁	1	2	yes	—
117	M₁₆	—	4	4	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	4	4	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
122	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
123	C₂×C₈	SD₁₆×C₆₁	1	4	yes	—
124	C₂×C₈	Q₁₆×C₆₁	1	4	yes	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	—	4	2	no	—
130	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
131	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
132	C₄²	C₂×D₄×C₆₁	1	2	yes	—
133	C₂³	#124×C₆₁	1	1	yes	characteristic
134	Q₈	—	4	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
138	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
139	C₂×C₄	#121×C₆₁	1	2	yes	—
140	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
141	C₂×C₄	C₂×SD₁₆×C₆₁	1	2	yes	—
142	C₂×C₄	C₂×Q₁₆×C₆₁	1	2	yes	—
143	C₈	—	4	4	no	—
144	C₈	—	8	2	no	—
145	C₈	#103×C₆₁	1	4	yes	—
146	C₈	#103×C₆₁	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₆₁×#390	1	1	yes	characteristic
149	C₂²	C₆₁×#487	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	8	4	no	—
152	C₄	C₂×#103×C₆₁	1	2	yes	—
153	C₄	C₆₁×#444	1	4	yes	—
154	C₄	C₆₁×#492	1	1	yes	characteristic
155	C₄	C₆₁×#494	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₆₁×#1956	1	1	yes	characteristic
158	C₂	C₆₁×#1975	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#3160

C₂

yes

maximal, characteristic

C₆₁×#1848

C₂

yes

maximal, characteristic

C₆₁×#1493

C₂

yes

maximal

C₂×C₆₁×#322

C₂

yes

maximal, characteristic

C₆₁×#533

C₂²

yes

—

C₆₁×#500

C₄

yes

—

C₆₁×#499

—

C₆₁×#498

C₄

yes

—

C₂×#108×C₆₁

—

C₆₁×#427

—

C₆₁×#423

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×Q₈⋊C₈

—

C₆₁×#322

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

—

C₄⋊Q₈×C₆₁

C₂×C₄

yes

—

#125×C₆₁

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#108×C₆₁

—

#107×C₆₁

—

#106×C₆₁

—

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

Q₁₆

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

—

C₂³×C₆₁

#124

yes

characteristic

Q₈×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×SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×Q₁₆

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#103

yes

—

C₈×C₆₁

#103

yes

—

#9861

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#390

yes

characteristic

C₂²×C₆₁

#487

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#103

yes

—

C₄×C₆₁

#444

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₂×C₆₁

yes

characteristic

#1848

C₂×C₆₁

yes

characteristic

#1493

C₂×C₆₁

yes

—

C₂×#322

C₂×C₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#1956

yes

characteristic

C₂×C₆₁

#1975

yes

—

#533

C₂²×C₆₁

yes

—

#500

C₄×C₆₁

yes

—

#499

—

#498

C₄×C₆₁

yes

—

C₂×#108

—

#427

—

#423

C₂²×C₆₁

yes

—

C₂×C₄×C₈

C₂²×C₆₁

yes

—

C₂×C₄×C₈

C₂²×C₆₁

yes

characteristic

Q₈⋊C₈

—

#322

—

C₆₁

#9861

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄×C₆₁

yes

—

C₄⋊Q₈

C₂×C₄×C₆₁

yes

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

—

104

C₂×C₄²

C₂³×C₆₁

yes

characteristic

105

#108

—

106

#107

—

107

#106

—

108

C₄⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₂×C₄×C₆₁

yes

characteristic

111

C₄×C₈

C₂×C₄×C₆₁

yes

characteristic

112

C₄×C₈

D₄×C₆₁

yes

—

113

C₄×C₈

D₄×C₆₁

yes

—

114

C₂×Q₈

—

115

C₂²×C₄

C₂×D₄×C₆₁

yes

characteristic

116

C₂²×C₄

#41×C₆₁

yes

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₂²⋊C₄×C₆₁

yes

—

122

C₂×C₈

C₂²⋊C₄×C₆₁

yes

—

123

C₂×C₈

SD₁₆×C₆₁

yes

—

124

C₂×C₈

Q₁₆×C₆₁

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

C₂²×C₄×C₆₁

yes

characteristic

131

C₄²

C₂×D₄×C₆₁

yes

characteristic

132

C₄²

C₂×D₄×C₆₁

yes

—

133

C₂³

#124×C₆₁

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂×C₂²⋊C₄×C₆₁

yes

characteristic

138

C₂×C₄

C₄×D₄×C₆₁

yes

—

139

C₂×C₄

#121×C₆₁

yes

—

140

C₂×C₄

#127×C₆₁

yes

characteristic

141

C₂×C₄

C₂×SD₁₆×C₆₁

yes

—

142

C₂×C₄

C₂×Q₁₆×C₆₁

yes

—

143

C₈

—

144

C₈

—

145

C₈

#103×C₆₁

yes

—

146

C₈

#103×C₆₁

yes

—

147

C₂²

—

148

C₂²

C₆₁×#390

yes

characteristic

149

C₂²

C₆₁×#487

yes

—

150

C₄

—

151

C₄

—

152

C₄

C₂×#103×C₆₁

yes

—

153

C₄

C₆₁×#444

yes

—

154

C₄

C₆₁×#492

yes

characteristic

155

C₄

C₆₁×#494

yes

characteristic

156

C₂

—

157

C₂

C₆₁×#1956

yes

characteristic

158

C₂

C₆₁×#1975

yes

—

	subgroup	G/H
center	C₂×C₄×C₆₁	#127
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₁

#127

commutator

C₄²

C₂²×C₄×C₆₁

Frattini, cosocle

C₂×C₄²

C₂³×C₆₁

C₂²×C₆₁

C₂³×C₆₁

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

N	H ≅ G/N	type
67. #9861	C₆₁	direct (N × H)
95. C₆₁	#9861	direct (N × H)

H ≅ G/N

type

67. #9861

C₆₁

direct (N × H)

95. C₆₁

#9861

direct (N × H)

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

subgroup	index
C₆₁×#3160	2
C₆₁×#1848	2
C₆₁×#1493	2
C₂×C₆₁×#322	2
#9861	61

subgroup

index

C₆₁×#3160

C₆₁×#1848

C₆₁×#1493

C₂×C₆₁×#322

#9861

Subgroups — C₆₁×#9861 #29513294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#9861 #29513294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#9861 #29513294