Group #21442006 — subgroups

Subgroups — C₆₁×#1965 #21442006

Order 7808.

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

normal subgroups only

This group has 396 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	C₂×#127×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#100×C₆₁	C₂	1	4	yes	maximal, complemented
4	C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₆₁	C₄	1	4	yes	—
6	C₂²×D₄×C₆₁	C₂²	1	2	yes	—
7	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
8	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
9	#127×C₆₁	—	2	1	no	—
10	#127×C₆₁	—	2	1	no	—
11	#127×C₆₁	—	2	2	no	—
12	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₆₁	—	2	2	no	—
14	#100×C₆₁	—	2	8	no	—
15	#97×C₆₁	—	2	1	no	—
16	C₄×C₈×C₆₁	—	2	1	no	—
17	C₂⁴×C₆₁	C₂×C₄	1	4	yes	—
18	C₂×D₄×C₆₁	—	2	4	no	—
19	C₂×D₄×C₆₁	—	2	4	no	—
20	C₂×D₄×C₆₁	—	2	4	no	—
21	C₂×D₄×C₆₁	—	2	2	no	—
22	C₂×D₄×C₆₁	—	2	2	no	—
23	C₂×D₄×C₆₁	—	2	4	no	—
24	C₂×D₄×C₆₁	—	4	8	no	—
25	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₆₁	—	2	2	no	—
28	M₁₆×C₆₁	—	2	4	no	—
29	M₁₆×C₆₁	D₄	1	4	yes	complemented
30	C₂×C₈×C₆₁	—	2	2	no	—
31	C₂×C₈×C₆₁	—	2	2	no	—
32	C₄²×C₆₁	D₄	1	1	yes	characteristic
33	C₄²×C₆₁	D₄	1	1	yes	characteristic
34	C₄²×C₆₁	D₄	1	2	yes	—
35	C₂³×C₆₁	—	2	4	no	—
36	C₂³×C₆₁	—	2	8	no	—
37	C₂³×C₆₁	—	4	8	no	—
38	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	D₄×C₆₁	—	4	8	no	—
40	D₄×C₆₁	—	4	8	no	—
41	D₄×C₆₁	—	4	4	no	—
42	D₄×C₆₁	—	4	2	no	—
43	D₄×C₆₁	—	4	2	no	—
44	C₂×C₄×C₆₁	—	2	4	no	—
45	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#41	1	2	yes	—
51	C₈×C₆₁	—	2	4	no	—
52	C₈×C₆₁	—	4	2	no	—
53	C₂²×C₆₁	—	2	2	no	—
54	C₂²×C₆₁	—	4	4	no	—
55	C₂²×C₆₁	—	4	8	no	—
56	C₂²×C₆₁	—	8	8	no	—
57	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₆₁	C₄×D₄	1	2	yes	—
59	C₄×C₆₁	—	2	2	no	—
60	C₄×C₆₁	#100	1	4	yes	complemented
61	C₄×C₆₁	#121	1	2	yes	—
62	C₄×C₆₁	#124	1	1	yes	characteristic
63	C₄×C₆₁	#127	1	1	yes	characteristic
64	#1965	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₆₁	—	2	2	no	—
66	C₂×C₆₁	—	8	4	no	—
67	C₂×C₆₁	#390	1	1	yes	characteristic
68	C₂×C₆₁	C₂×#100	1	2	yes	—
69	C₂×#127	C₂×C₆₁	1	1	yes	characteristic
70	#423	C₂×C₆₁	1	1	yes	characteristic, complemented
71	C₂×#100	C₂×C₆₁	1	4	yes	complemented
72	C₄×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
73	C₆₁	#1965	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×D₄	C₄×C₆₁	1	4	yes	—
75	C₂²×D₄	C₂²×C₆₁	1	2	yes	—
76	C₂×M₁₆	C₂²×C₆₁	1	2	yes	—
77	C₂×M₁₆	C₂²×C₆₁	1	2	yes	—
78	#127	—	2	1	no	—
79	#127	—	2	1	no	—
80	#127	—	2	2	no	—
81	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
82	C₄⋊C₈	—	2	2	no	—
83	#100	—	2	8	no	—
84	#97	—	2	1	no	—
85	C₄×C₈	—	2	1	no	—
86	C₂⁴	C₂×C₄×C₆₁	1	4	yes	—
87	C₂×D₄	—	2	4	no	—
88	C₂×D₄	—	2	4	no	—
89	C₂×D₄	—	2	4	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	4	no	—
93	C₂×D₄	—	4	8	no	—
94	C₂²×C₄	C₂×C₄×C₆₁	1	2	yes	—
95	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	2	4	no	—
98	M₁₆	D₄×C₆₁	1	4	yes	complemented
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄²	D₄×C₆₁	1	1	yes	characteristic
102	C₄²	D₄×C₆₁	1	1	yes	characteristic
103	C₄²	D₄×C₆₁	1	2	yes	—
104	C₂³	—	2	4	no	—
105	C₂³	—	2	8	no	—
106	C₂³	—	4	8	no	—
107	C₂³	C₂²×C₄×C₆₁	1	1	yes	characteristic
108	D₄	—	4	8	no	—
109	D₄	—	4	8	no	—
110	D₄	—	4	4	no	—
111	D₄	—	4	2	no	—
112	D₄	—	4	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
115	C₂×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
116	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
117	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
118	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
119	C₂×C₄	#41×C₆₁	1	2	yes	—
120	C₈	—	2	4	no	—
121	C₈	—	4	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	4	4	no	—
124	C₂²	—	4	8	no	—
125	C₂²	—	8	8	no	—
126	C₂²	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
127	C₂²	C₄×D₄×C₆₁	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	#100×C₆₁	1	4	yes	complemented
130	C₄	#121×C₆₁	1	2	yes	—
131	C₄	#124×C₆₁	1	1	yes	characteristic
132	C₄	#127×C₆₁	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	—	8	4	no	—
135	C₂	C₆₁×#390	1	1	yes	characteristic
136	C₂	C₂×#100×C₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#127×C₆₁

C₂

yes

maximal, characteristic

C₆₁×#423

C₂

yes

maximal, characteristic, complemented

C₂×#100×C₆₁

C₂

yes

maximal, complemented

C₄×M₁₆×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₆₁

C₄

yes

—

C₂²×D₄×C₆₁

C₂²

yes

—

C₂×M₁₆×C₆₁

C₂²

yes

—

C₂×M₁₆×C₆₁

C₂²

yes

—

#127×C₆₁

—

#127×C₆₁

—

#127×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

#100×C₆₁

—

#97×C₆₁

—

C₄×C₈×C₆₁

—

C₂⁴×C₆₁

C₂×C₄

yes

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

D₄

yes

complemented

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

—

C₄×C₆₁

#100

yes

complemented

C₄×C₆₁

#121

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×#100

yes

—

C₂×#127

C₂×C₆₁

yes

characteristic

#423

C₂×C₆₁

yes

characteristic, complemented

C₂×#100

C₂×C₆₁

yes

complemented

C₄×M₁₆

C₂×C₆₁

yes

characteristic, complemented

C₆₁

#1965

yes

characteristic, direct product factor, Sylow

C₂²×D₄

C₄×C₆₁

yes

—

C₂²×D₄

C₂²×C₆₁

yes

—

C₂×M₁₆

C₂²×C₆₁

yes

—

C₂×M₁₆

C₂²×C₆₁

yes

—

#127

—

#127

—

#127

—

C₂×C₄²

C₂²×C₆₁

yes

characteristic

C₄⋊C₈

—

#100

—

#97

—

C₄×C₈

—

C₂⁴

C₂×C₄×C₆₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂×C₄×C₆₁

yes

—

C₂²×C₄

C₂³×C₆₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

D₄×C₆₁

yes

complemented

C₂×C₈

—

100

C₂×C₈

—

101

C₄²

D₄×C₆₁

yes

characteristic

102

C₄²

D₄×C₆₁

yes

characteristic

103

C₄²

D₄×C₆₁

yes

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₂²×C₄×C₆₁

yes

characteristic

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²⋊C₄×C₆₁

yes

—

115

C₂×C₄

C₂²⋊C₄×C₆₁

yes

—

116

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

117

C₂×C₄

C₂×D₄×C₆₁

yes

—

118

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

119

C₂×C₄

#41×C₆₁

yes

—

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×C₂²⋊C₄×C₆₁

yes

characteristic

127

C₂²

C₄×D₄×C₆₁

yes

—

128

C₄

—

129

C₄

#100×C₆₁

yes

complemented

130

C₄

#121×C₆₁

yes

—

131

C₄

#124×C₆₁

yes

characteristic

132

C₄

#127×C₆₁

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

C₆₁×#390

yes

characteristic

136

C₂

C₂×#100×C₆₁

yes

—

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

subgroup

G/H

center, socle

C₂²×C₆₁

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₆₁

Frattini, cosocle

C₂²×C₄

C₂³×C₆₁

prime radical

C₂×#127×C₆₁

C₂

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

N	H ≅ G/N	type
2. C₆₁×#423	C₂	semidirect
3. C₂×#100×C₆₁	C₂	semidirect
4. C₄×M₁₆×C₆₁	C₂	semidirect
29. M₁₆×C₆₁	D₄	semidirect
60. C₄×C₆₁	#100	semidirect
64. #1965	C₆₁	direct (N × H)
70. #423	C₂×C₆₁	semidirect
71. C₂×#100	C₂×C₆₁	semidirect
72. C₄×M₁₆	C₂×C₆₁	semidirect
73. C₆₁	#1965	direct (N × H)
98. M₁₆	D₄×C₆₁	semidirect
129. C₄	#100×C₆₁	semidirect

H ≅ G/N

type

2. C₆₁×#423

C₂

semidirect

3. C₂×#100×C₆₁

C₂

semidirect

4. C₄×M₁₆×C₆₁

C₂

semidirect

29. M₁₆×C₆₁

D₄

semidirect

60. C₄×C₆₁

#100

semidirect

64. #1965

C₆₁

direct (N × H)

70. #423

C₂×C₆₁

semidirect

71. C₂×#100

C₂×C₆₁

semidirect

72. C₄×M₁₆

C₂×C₆₁

semidirect

73. C₆₁

#1965

direct (N × H)

98. M₁₆

D₄×C₆₁

semidirect

129. C₄

#100×C₆₁

semidirect

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

subgroup	index
C₂×#127×C₆₁	2
C₆₁×#423	2
C₂×#100×C₆₁	2
C₄×M₁₆×C₆₁	2
#1965	61

subgroup

index

C₂×#127×C₆₁

C₆₁×#423

C₂×#100×C₆₁

C₄×M₁₆×C₆₁

#1965

Subgroups — C₆₁×#1965 #21442006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#1965 #21442006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#1965 #21442006