Group #21442003 — subgroups

Subgroups — C₆₁×#1962 #21442003

Order 7808.

360 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 120 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×#532	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#104×C₆₁	C₂	1	4	yes	maximal, complemented
4	C₄³×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#41×C₆₁	C₂²	1	2	yes	—
6	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic, complemented
8	#127×C₆₁	C₄	1	1	yes	characteristic, complemented
9	#124×C₆₁	C₄	1	2	yes	complemented
10	#121×C₆₁	—	2	2	no	—
11	C₄×D₄×C₆₁	—	2	2	no	—
12	C₂×C₄²×C₆₁	—	2	2	no	—
13	C₂×C₄²×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₆₁	—	2	2	no	—
16	#104×C₆₁	—	2	8	no	—
17	#41×C₆₁	—	2	4	no	—
18	#41×C₆₁	—	2	4	no	—
19	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₆₁	—	2	2	no	—
21	C₂×D₄×C₆₁	—	2	2	no	—
22	C₂×D₄×C₆₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₆₁	—	2	2	no	—
24	C₂²×C₄×C₆₁	—	2	2	no	—
25	C₂²×C₄×C₆₁	D₄	1	2	yes	—
26	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₆₁	—	2	4	no	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₂²⋊C₄×C₆₁	—	2	4	no	—
31	C₄²×C₆₁	—	2	8	no	—
32	C₄²×C₆₁	—	2	2	no	—
33	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₆₁	D₄	1	2	yes	—
36	C₄²×C₆₁	D₄	1	4	yes	complemented
37	C₂³×C₆₁	—	2	2	no	—
38	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₆₁	—	2	4	no	—
40	D₄×C₆₁	—	2	4	no	—
41	D₄×C₆₁	—	4	4	no	—
42	D₄×C₆₁	—	4	4	no	—
43	C₂×C₄×C₆₁	—	2	8	no	—
44	C₂×C₄×C₆₁	—	2	4	no	—
45	C₂×C₄×C₆₁	—	2	4	no	—
46	C₂×C₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	—	4	2	no	—
48	C₂×C₄×C₆₁	—	4	2	no	—
49	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#41	1	2	yes	—
55	C₈×C₆₁	—	4	2	no	—
56	C₂²×C₆₁	—	2	2	no	—
57	C₂²×C₆₁	—	4	2	no	—
58	C₂²×C₆₁	—	4	4	no	—
59	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₆₁	#121	1	2	yes	—
61	C₄×C₆₁	—	2	8	no	—
62	C₄×C₆₁	—	2	2	no	—
63	C₄×C₆₁	—	4	2	no	—
64	C₄×C₆₁	#104	1	4	yes	complemented
65	C₄×C₆₁	C₄×D₄	1	2	yes	—
66	C₄×C₆₁	#124	1	1	yes	characteristic
67	C₄×C₆₁	#127	1	1	yes	characteristic
68	#1962	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₆₁	—	2	2	no	—
70	C₂×C₆₁	—	8	2	no	—
71	C₂×C₆₁	#390	1	1	yes	characteristic
72	C₂×C₆₁	C₂×#104	1	2	yes	—
73	#532	C₂×C₆₁	1	1	yes	characteristic
74	#423	C₂×C₆₁	1	1	yes	characteristic, complemented
75	C₂×#104	C₂×C₆₁	1	4	yes	complemented
76	C₄³	C₂×C₆₁	1	1	yes	characteristic, complemented
77	C₆₁	#1962	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×#41	C₂²×C₆₁	1	2	yes	—
79	C₂×M₁₆	C₂²×C₆₁	1	2	yes	—
80	C₄⋊Q₈	C₄×C₆₁	1	1	yes	characteristic, complemented
81	#127	C₄×C₆₁	1	1	yes	characteristic, complemented
82	#124	C₄×C₆₁	1	2	yes	complemented
83	#121	—	2	2	no	—
84	C₄×D₄	—	2	2	no	—
85	C₂×C₄²	—	2	2	no	—
86	C₂×C₄²	C₂²×C₆₁	1	2	yes	—
87	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
88	C₄⋊C₈	—	2	2	no	—
89	#104	—	2	8	no	—
90	#41	—	2	4	no	—
91	#41	—	2	4	no	—
92	C₂×Q₈	C₂×C₄×C₆₁	1	2	yes	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	C₂×C₄×C₆₁	1	2	yes	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	D₄×C₆₁	1	2	yes	—
99	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
100	M₁₆	—	2	4	no	—
101	C₂×C₈	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	2	4	no	—
104	C₄²	—	2	8	no	—
105	C₄²	—	2	2	no	—
106	C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
107	C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
108	C₄²	D₄×C₆₁	1	2	yes	—
109	C₄²	D₄×C₆₁	1	4	yes	complemented
110	C₂³	—	2	2	no	—
111	C₂³	C₂×D₄×C₆₁	1	1	yes	characteristic
112	Q₈	—	2	4	no	—
113	D₄	—	2	4	no	—
114	D₄	—	4	4	no	—
115	D₄	—	4	4	no	—
116	C₂×C₄	—	2	8	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
123	C₂×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
124	C₂×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
125	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
126	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
127	C₂×C₄	#41×C₆₁	1	2	yes	—
128	C₈	—	4	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
133	C₂²	#121×C₆₁	1	2	yes	—
134	C₄	—	2	8	no	—
135	C₄	—	2	2	no	—
136	C₄	—	4	2	no	—
137	C₄	#104×C₆₁	1	4	yes	complemented
138	C₄	C₄×D₄×C₆₁	1	2	yes	—
139	C₄	#124×C₆₁	1	1	yes	characteristic
140	C₄	#127×C₆₁	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	—	8	2	no	—
143	C₂	C₆₁×#390	1	1	yes	characteristic
144	C₂	C₂×#104×C₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#532

C₂

yes

maximal, characteristic

C₆₁×#423

C₂

yes

maximal, characteristic, complemented

C₂×#104×C₆₁

C₂

yes

maximal, complemented

C₄³×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₆₁

C₂²

yes

—

C₂×M₁₆×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₄

yes

characteristic, complemented

#127×C₆₁

C₄

yes

characteristic, complemented

#124×C₆₁

C₄

yes

complemented

#121×C₆₁

—

C₄×D₄×C₆₁

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

#104×C₆₁

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

D₄

yes

complemented

C₂³×C₆₁

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₈×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

#121

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#104

yes

complemented

C₄×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#390

yes

characteristic

C₂×C₆₁

C₂×#104

yes

—

#532

C₂×C₆₁

yes

characteristic

#423

C₂×C₆₁

yes

characteristic, complemented

C₂×#104

C₂×C₆₁

yes

complemented

C₄³

C₂×C₆₁

yes

characteristic, complemented

C₆₁

#1962

yes

characteristic, direct product factor, Sylow

C₂×#41

C₂²×C₆₁

yes

—

C₂×M₁₆

C₂²×C₆₁

yes

—

C₄⋊Q₈

C₄×C₆₁

yes

characteristic, complemented

#127

C₄×C₆₁

yes

characteristic, complemented

#124

C₄×C₆₁

yes

complemented

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₆₁

yes

—

C₂×C₄²

C₂²×C₆₁

yes

characteristic

C₄⋊C₈

—

#104

—

#41

—

#41

—

C₂×Q₈

C₂×C₄×C₆₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₆₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₆₁

yes

—

C₂²×C₄

C₂³×C₆₁

yes

characteristic

100

M₁₆

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂×C₄×C₆₁

yes

characteristic

107

C₄²

C₂×C₄×C₆₁

yes

characteristic

108

C₄²

D₄×C₆₁

yes

—

109

C₄²

D₄×C₆₁

yes

complemented

110

C₂³

—

111

C₂³

C₂×D₄×C₆₁

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂²⋊C₄×C₆₁

yes

—

123

C₂×C₄

C₂²⋊C₄×C₆₁

yes

—

124

C₂×C₄

C₂²×C₄×C₆₁

yes

characteristic

125

C₂×C₄

C₂×D₄×C₆₁

yes

—

126

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

127

C₂×C₄

#41×C₆₁

yes

—

128

C₈

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₂²⋊C₄×C₆₁

yes

characteristic

133

C₂²

#121×C₆₁

yes

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#104×C₆₁

yes

complemented

138

C₄

C₄×D₄×C₆₁

yes

—

139

C₄

#124×C₆₁

yes

characteristic

140

C₄

#127×C₆₁

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

C₆₁×#390

yes

characteristic

144

C₂

C₂×#104×C₆₁

yes

—

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
prime radical	C₆₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₁

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₆₁

Frattini, cosocle

C₂²×C₄

C₂³×C₆₁

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

prime radical

C₆₁×#532

C₂

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

N	H ≅ G/N	type
2. C₆₁×#423	C₂	semidirect
3. C₂×#104×C₆₁	C₂	semidirect
4. C₄³×C₆₁	C₂	semidirect
7. C₄⋊Q₈×C₆₁	C₄	semidirect
8. #127×C₆₁	C₄	semidirect
9. #124×C₆₁	C₄	semidirect
36. C₄²×C₆₁	D₄	semidirect
64. C₄×C₆₁	#104	semidirect
68. #1962	C₆₁	direct (N × H)
74. #423	C₂×C₆₁	semidirect
75. C₂×#104	C₂×C₆₁	semidirect
76. C₄³	C₂×C₆₁	semidirect
77. C₆₁	#1962	direct (N × H)
80. C₄⋊Q₈	C₄×C₆₁	semidirect
81. #127	C₄×C₆₁	semidirect
82. #124	C₄×C₆₁	semidirect
109. C₄²	D₄×C₆₁	semidirect
137. C₄	#104×C₆₁	semidirect

H ≅ G/N

type

2. C₆₁×#423

C₂

semidirect

3. C₂×#104×C₆₁

C₂

semidirect

4. C₄³×C₆₁

C₂

semidirect

7. C₄⋊Q₈×C₆₁

C₄

semidirect

8. #127×C₆₁

C₄

semidirect

9. #124×C₆₁

C₄

semidirect

36. C₄²×C₆₁

D₄

semidirect

64. C₄×C₆₁

#104

semidirect

68. #1962

C₆₁

direct (N × H)

74. #423

C₂×C₆₁

semidirect

75. C₂×#104

C₂×C₆₁

semidirect

76. C₄³

C₂×C₆₁

semidirect

77. C₆₁

#1962

direct (N × H)

80. C₄⋊Q₈

C₄×C₆₁

semidirect

81. #127

C₄×C₆₁

semidirect

82. #124

C₄×C₆₁

semidirect

109. C₄²

D₄×C₆₁

semidirect

137. C₄

#104×C₆₁

semidirect

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

subgroup	index
C₆₁×#532	2
C₆₁×#423	2
C₂×#104×C₆₁	2
C₄³×C₆₁	2
#1962	61

subgroup

index

C₆₁×#532

C₆₁×#423

C₂×#104×C₆₁

C₄³×C₆₁

#1962

Subgroups — C₆₁×#1962 #21442003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#1962 #21442003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#1962 #21442003