Group #21442013 — subgroups

Subgroups — C₆₁×#1972 #21442013

Order 7808.

376 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 128 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	C₂²×D₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#106×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#102×C₆₁	C₂	1	2	yes	maximal, complemented
4	C₂×#97×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#390	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄×C₆₁	C₂²	1	2	yes	—
7	C₂×D₈×C₆₁	—	2	2	no	—
8	C₂×D₈×C₆₁	—	2	2	no	—
9	C₂×D₈×C₆₁	C₄	1	4	yes	complemented
10	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
14	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	#106×C₆₁	—	2	2	no	—
16	#102×C₆₁	—	2	4	no	—
17	#97×C₆₁	—	2	2	no	—
18	C₂⁴×C₆₁	—	2	2	no	—
19	C₂×D₄×C₆₁	—	2	2	no	—
20	C₂×D₄×C₆₁	—	2	2	no	—
21	C₂×D₄×C₆₁	—	2	2	no	—
22	C₂×D₄×C₆₁	—	2	2	no	—
23	C₂×D₄×C₆₁	—	2	2	no	—
24	C₂×D₄×C₆₁	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₆₁	—	2	2	no	—
26	C₂²×C₄×C₆₁	D₄	1	2	yes	—
27	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
28	D₈×C₆₁	—	2	8	no	—
29	C₂×C₈×C₆₁	—	2	1	no	—
30	C₂×C₈×C₆₁	—	2	1	no	—
31	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₆₁	D₄	1	2	yes	—
33	C₂×C₈×C₆₁	D₄	1	2	yes	—
34	C₂×C₈×C₆₁	D₄	1	2	yes	—
35	C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₂²⋊C₄×C₆₁	—	4	8	no	—
38	C₄²×C₆₁	—	2	1	no	—
39	C₄²×C₆₁	—	2	1	no	—
40	C₂³×C₆₁	—	2	4	no	—
41	C₂³×C₆₁	—	4	2	no	—
42	C₂³×C₆₁	—	4	2	no	—
43	C₂³×C₆₁	—	4	4	no	—
44	C₂³×C₆₁	—	4	4	no	—
45	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
46	D₄×C₆₁	—	2	4	no	—
47	D₄×C₆₁	—	2	4	no	—
48	D₄×C₆₁	—	4	2	no	—
49	D₄×C₆₁	—	4	2	no	—
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	—	2	4	no	—
52	C₂×C₄×C₆₁	—	4	2	no	—
53	C₂×C₄×C₆₁	—	4	4	no	—
54	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
59	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
60	C₈×C₆₁	—	2	2	no	—
61	C₈×C₆₁	C₂²⋊C₄	1	4	yes	complemented
62	C₂²×C₆₁	—	4	4	no	—
63	C₂²×C₆₁	—	4	8	no	—
64	C₂²×C₆₁	—	8	4	no	—
65	C₂²×C₆₁	—	8	2	no	—
66	C₂²×C₆₁	—	8	2	no	—
67	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
68	C₂²×C₆₁	#121	1	1	yes	characteristic
69	C₂²×C₆₁	#127	1	1	yes	characteristic
70	C₂²×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
71	C₄×C₆₁	—	4	2	no	—
72	C₄×C₆₁	—	8	2	no	—
73	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
75	C₄×C₆₁	#121	1	1	yes	characteristic
76	C₄×C₆₁	#124	1	1	yes	characteristic
77	#1972	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×C₆₁	—	8	4	no	—
79	C₂×C₆₁	#390	1	1	yes	characteristic
80	C₂×C₆₁	#442	1	2	yes	—
81	C₂×C₆₁	#469	1	2	yes	—
82	C₂×C₆₁	#496	1	2	yes	—
83	C₂²×D₈	C₂×C₆₁	1	1	yes	characteristic
84	C₂×#106	C₂×C₆₁	1	1	yes	characteristic, complemented
85	C₂×#102	C₂×C₆₁	1	2	yes	complemented
86	C₂×#97	C₂×C₆₁	1	1	yes	characteristic, complemented
87	#390	C₂×C₆₁	1	2	yes	complemented
88	C₆₁	#1972	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×D₄	C₂²×C₆₁	1	2	yes	—
90	C₂×D₈	—	2	2	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	C₄×C₆₁	1	4	yes	complemented
93	C₂²×C₈	C₂²×C₆₁	1	1	yes	characteristic
94	C₂²×C₈	C₂²×C₆₁	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
96	C₂×C₂²⋊C₄	—	2	4	no	—
97	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
98	#106	—	2	2	no	—
99	#102	—	2	4	no	—
100	#97	—	2	2	no	—
101	C₂⁴	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	C₂×C₄×C₆₁	1	4	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	D₄×C₆₁	1	2	yes	—
110	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
111	D₈	—	2	8	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	C₂×C₄×C₆₁	1	2	yes	—
115	C₂×C₈	D₄×C₆₁	1	2	yes	—
116	C₂×C₈	D₄×C₆₁	1	2	yes	—
117	C₂×C₈	D₄×C₆₁	1	2	yes	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	4	8	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	1	no	—
123	C₂³	—	2	4	no	—
124	C₂³	—	4	2	no	—
125	C₂³	—	4	2	no	—
126	C₂³	—	4	4	no	—
127	C₂³	—	4	4	no	—
128	C₂³	C₂×D₄×C₆₁	1	1	yes	characteristic
129	D₄	—	2	4	no	—
130	D₄	—	2	4	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
138	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
139	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
140	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
141	C₂×C₄	#41×C₆₁	1	1	yes	characteristic
142	C₂×C₄	#41×C₆₁	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	C₂²⋊C₄×C₆₁	1	4	yes	complemented
145	C₂²	—	4	4	no	—
146	C₂²	—	4	8	no	—
147	C₂²	—	8	4	no	—
148	C₂²	—	8	2	no	—
149	C₂²	—	8	2	no	—
150	C₂²	C₄×D₄×C₆₁	1	1	yes	characteristic
151	C₂²	#121×C₆₁	1	1	yes	characteristic
152	C₂²	#127×C₆₁	1	1	yes	characteristic
153	C₂²	AGL(1,ℤ/8)×C₆₁	1	4	yes	—
154	C₄	—	4	2	no	—
155	C₄	—	8	2	no	—
156	C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
157	C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
158	C₄	#121×C₆₁	1	1	yes	characteristic
159	C₄	#124×C₆₁	1	1	yes	characteristic
160	C₂	—	8	4	no	—
161	C₂	C₆₁×#390	1	1	yes	characteristic
162	C₂	C₆₁×#442	1	2	yes	—
163	C₂	C₆₁×#469	1	2	yes	—
164	C₂	C₆₁×#496	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#106×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#102×C₆₁

C₂

yes

maximal, complemented

C₂×#97×C₆₁

C₂

yes

maximal, characteristic, complemented

C₆₁×#390

C₂

yes

maximal, complemented

C₂²×D₄×C₆₁

C₂²

yes

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

C₄

yes

complemented

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#106×C₆₁

—

#102×C₆₁

—

#97×C₆₁

—

C₂⁴×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

D₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊C₄

yes

complemented

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×D₄

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

C₄×D₄

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₁

—

C₂×C₆₁

#390

yes

characteristic

C₂×C₆₁

#442

yes

—

C₂×C₆₁

#469

yes

—

C₂×C₆₁

#496

yes

—

C₂²×D₈

C₂×C₆₁

yes

characteristic

C₂×#106

C₂×C₆₁

yes

characteristic, complemented

C₂×#102

C₂×C₆₁

yes

complemented

C₂×#97

C₂×C₆₁

yes

characteristic, complemented

#390

C₂×C₆₁

yes

complemented

C₆₁

#1972

yes

characteristic, direct product factor, Sylow

C₂²×D₄

C₂²×C₆₁

yes

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₄×C₆₁

yes

complemented

C₂²×C₈

C₂²×C₆₁

yes

characteristic

C₂²×C₈

C₂²×C₆₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₆₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₆₁

yes

characteristic

#106

—

#102

—

100

#97

—

101

C₂⁴

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂×C₄×C₆₁

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

D₄×C₆₁

yes

—

110

C₂²×C₄

C₂³×C₆₁

yes

characteristic

111

D₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂×C₄×C₆₁

yes

—

115

C₂×C₈

D₄×C₆₁

yes

—

116

C₂×C₈

D₄×C₆₁

yes

—

117

C₂×C₈

D₄×C₆₁

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

C₂×D₄×C₆₁

yes

characteristic

129

D₄

—

130

D₄

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂²×C₄×C₆₁

yes

characteristic

138

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

139

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

140

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

141

C₂×C₄

#41×C₆₁

yes

characteristic

142

C₂×C₄

#41×C₆₁

yes

characteristic

143

C₈

—

144

C₈

C₂²⋊C₄×C₆₁

yes

complemented

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₄×D₄×C₆₁

yes

characteristic

151

C₂²

#121×C₆₁

yes

characteristic

152

C₂²

#127×C₆₁

yes

characteristic

153

C₂²

AGL(1,ℤ/8)×C₆₁

yes

—

154

C₄

—

155

C₄

—

156

C₄

C₂×C₂²⋊C₄×C₆₁

yes

characteristic

157

C₄

C₄×D₄×C₆₁

yes

characteristic

158

C₄

#121×C₆₁

yes

characteristic

159

C₄

#124×C₆₁

yes

characteristic

160

C₂

—

161

C₂

C₆₁×#390

yes

characteristic

162

C₂

C₆₁×#442

yes

—

163

C₂

C₆₁×#469

yes

—

164

C₂

C₆₁×#496

yes

—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂³×C₆₁
prime radical	C₂²×D₈×C₆₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₆₁

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₆₁

Frattini, cosocle

C₂²×C₄

C₂³×C₆₁

prime radical

C₂²×D₈×C₆₁

C₂

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

N	H ≅ G/N	type
2. C₂×#106×C₆₁	C₂	semidirect
3. C₂×#102×C₆₁	C₂	semidirect
4. C₂×#97×C₆₁	C₂	semidirect
5. C₆₁×#390	C₂	semidirect
9. C₂×D₈×C₆₁	C₄	semidirect
61. C₈×C₆₁	C₂²⋊C₄	semidirect
77. #1972	C₆₁	direct (N × H)
84. C₂×#106	C₂×C₆₁	semidirect
85. C₂×#102	C₂×C₆₁	semidirect
86. C₂×#97	C₂×C₆₁	semidirect
87. #390	C₂×C₆₁	semidirect
88. C₆₁	#1972	direct (N × H)
92. C₂×D₈	C₄×C₆₁	semidirect
144. C₈	C₂²⋊C₄×C₆₁	semidirect

H ≅ G/N

type

2. C₂×#106×C₆₁

C₂

semidirect

3. C₂×#102×C₆₁

C₂

semidirect

4. C₂×#97×C₆₁

C₂

semidirect

5. C₆₁×#390

C₂

semidirect

9. C₂×D₈×C₆₁

C₄

semidirect

61. C₈×C₆₁

C₂²⋊C₄

semidirect

77. #1972

C₆₁

direct (N × H)

84. C₂×#106

C₂×C₆₁

semidirect

85. C₂×#102

C₂×C₆₁

semidirect

86. C₂×#97

C₂×C₆₁

semidirect

87. #390

C₂×C₆₁

semidirect

88. C₆₁

#1972

direct (N × H)

92. C₂×D₈

C₄×C₆₁

semidirect

144. C₈

C₂²⋊C₄×C₆₁

semidirect

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

subgroup	index
C₂²×D₈×C₆₁	2
C₂×#106×C₆₁	2
C₂×#102×C₆₁	2
C₂×#97×C₆₁	2
C₆₁×#390	2
#1972	61

subgroup

index

C₂²×D₈×C₆₁

C₂×#106×C₆₁

C₂×#102×C₆₁

C₂×#97×C₆₁

C₆₁×#390

#1972

Subgroups — C₆₁×#1972 #21442013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#1972 #21442013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#1972 #21442013