Group #29515500 — subgroups

Subgroups — C₆₁×#12067 #29515500

Order 15616.

246 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2177	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1580	C₂	1	2	yes	maximal, complemented
3	C₆₁×#1368	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1333	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆×C₆₁	C₄	1	1	yes	characteristic
7	C₄×D₈×C₆₁	C₂²	1	2	yes	—
8	C₈×D₄×C₆₁	—	2	2	no	—
9	C₆₁×#367	C₄	1	1	yes	characteristic
10	C₆₁×#357	C₄	1	2	yes	—
11	C₆₁×#346	C₂²	1	1	yes	characteristic
12	C₆₁×#346	C₂²	1	1	yes	characteristic
13	C₆₁×#334	C₂²	1	2	yes	—
14	C₆₁×D₄⋊C₈	—	2	2	no	—
15	C₆₁×#322	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₆₁	—	4	2	no	—
18	C₄×D₄×C₆₁	—	2	2	no	—
19	D₁₆×C₆₁	C₈	1	4	yes	complemented
20	C₂×C₁₆×C₆₁	—	2	1	no	—
21	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
23	#107×C₆₁	C₂×C₄	1	2	yes	—
24	C₄⋊C₈×C₆₁	—	2	2	no	—
25	#102×C₆₁	—	2	2	no	—
26	C₂²⋊C₈×C₆₁	—	4	2	no	—
27	C₄×C₈×C₆₁	D₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₆₁	—	2	2	no	—
30	C₂²×C₄×C₆₁	—	4	2	no	—
31	D₈×C₆₁	—	2	2	no	—
32	D₈×C₆₁	C₂×C₈	1	4	yes	complemented
33	C₂×C₈×C₆₁	—	2	2	no	—
34	C₂×C₈×C₆₁	—	4	2	no	—
35	C₂×C₈×C₆₁	—	8	4	no	—
36	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₂²⋊C₄×C₆₁	—	4	2	no	—
40	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₆₁	—	2	1	no	—
42	C₁₆×C₆₁	—	4	1	no	—
43	C₁₆×C₆₁	C₂×C₈	1	2	yes	complemented
44	C₂³×C₆₁	—	4	2	no	—
45	D₄×C₆₁	—	2	4	no	—
46	D₄×C₆₁	—	4	2	no	—
47	C₂×C₄×C₆₁	—	4	2	no	—
48	C₂×C₄×C₆₁	—	8	2	no	—
49	C₂×C₄×C₆₁	—	8	2	no	—
50	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
52	C₈×C₆₁	—	2	1	no	—
53	C₈×C₆₁	—	4	2	no	—
54	C₈×C₆₁	—	8	2	no	—
55	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₆₁	#131	1	1	yes	characteristic
57	#12067	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂²×C₆₁	—	4	4	no	—
59	C₂²×C₆₁	—	8	2	no	—
60	C₂²×C₆₁	—	8	2	no	—
61	C₂²×C₆₁	#442	1	1	yes	characteristic
62	C₄×C₆₁	—	2	2	no	—
63	C₄×C₆₁	—	8	2	no	—
64	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
65	C₄×C₆₁	#444	1	1	yes	characteristic
66	#2177	C₂×C₆₁	1	1	yes	characteristic
67	#1580	C₂×C₆₁	1	2	yes	complemented
68	#1368	C₂×C₆₁	1	1	yes	characteristic, complemented
69	#1333	C₂×C₆₁	1	2	yes	complemented
70	#1313	C₂×C₆₁	1	1	yes	characteristic, complemented
71	C₂×C₆₁	—	8	4	no	—
72	C₂×C₆₁	#1580	1	1	yes	characteristic
73	C₂×C₆₁	AGL(1,ℤ/16)	1	2	yes	—
74	C₂×D₁₆	C₄×C₆₁	1	1	yes	characteristic
75	C₄×D₈	C₂²×C₆₁	1	2	yes	—
76	C₈×D₄	—	2	2	no	—
77	#367	C₄×C₆₁	1	1	yes	characteristic
78	#357	C₄×C₆₁	1	2	yes	—
79	#346	C₂²×C₆₁	1	1	yes	characteristic
80	#346	C₂²×C₆₁	1	1	yes	characteristic
81	#334	C₂²×C₆₁	1	2	yes	—
82	D₄⋊C₈	—	2	2	no	—
83	#322	C₂²×C₆₁	1	1	yes	characteristic
84	C₆₁	#12067	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×D₈	C₂×C₄×C₆₁	1	2	yes	—
86	C₂²×C₈	—	4	2	no	—
87	C₄×D₄	—	2	2	no	—
88	D₁₆	C₈×C₆₁	1	4	yes	complemented
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
91	C₂×C₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
92	#107	C₂×C₄×C₆₁	1	2	yes	—
93	C₄⋊C₈	—	2	2	no	—
94	#102	—	2	2	no	—
95	C₂²⋊C₈	—	4	2	no	—
96	C₄×C₈	D₄×C₆₁	1	2	yes	—
97	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
98	C₂×D₄	—	2	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	D₈	—	2	2	no	—
101	D₈	C₂×C₈×C₆₁	1	4	yes	complemented
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	8	4	no	—
105	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
106	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	4	2	no	—
109	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	C₂×C₈×C₆₁	1	2	yes	complemented
113	C₂³	—	4	2	no	—
114	D₄	—	2	4	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
120	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	4	2	no	—
123	C₈	—	8	2	no	—
124	C₈	C₂²×C₈×C₆₁	1	1	yes	characteristic
125	C₈	#131×C₆₁	1	1	yes	characteristic
126	C₂²	—	4	4	no	—
127	C₂²	—	8	2	no	—
128	C₂²	—	8	2	no	—
129	C₂²	C₆₁×#442	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	8	2	no	—
132	C₄	C₈×D₄×C₆₁	1	1	yes	characteristic
133	C₄	C₆₁×#444	1	1	yes	characteristic
134	C₂	—	8	4	no	—
135	C₂	C₆₁×#1580	1	1	yes	characteristic
136	C₂	C₆₁×AGL(1,ℤ/16)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#2177

C₂

yes

maximal, characteristic

C₆₁×#1580

C₂

yes

maximal, complemented

C₆₁×#1368

C₂

yes

maximal, characteristic, complemented

C₆₁×#1333

C₂

yes

maximal, complemented

C₆₁×#1313

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₆₁

C₄

yes

characteristic

C₄×D₈×C₆₁

C₂²

yes

—

C₈×D₄×C₆₁

—

C₆₁×#367

C₄

yes

characteristic

C₆₁×#357

C₄

yes

—

C₆₁×#346

C₂²

yes

characteristic

C₆₁×#346

C₂²

yes

characteristic

C₆₁×#334

C₂²

yes

—

C₆₁×D₄⋊C₈

—

C₆₁×#322

C₂²

yes

characteristic

C₂×D₈×C₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

—

C₄×D₄×C₆₁

—

D₁₆×C₆₁

C₈

yes

complemented

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

#107×C₆₁

C₂×C₄

yes

—

C₄⋊C₈×C₆₁

—

#102×C₆₁

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×D₄×C₆₁

—

C₂²×C₄×C₆₁

—

D₈×C₆₁

—

D₈×C₆₁

C₂×C₈

yes

complemented

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₁₆×C₆₁

—

C₁₆×C₆₁

—

C₁₆×C₆₁

C₂×C₈

yes

complemented

C₂³×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₈

yes

C₈×C₆₁

yes

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#442

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₈×D₄

yes

C₄×C₆₁

yes

C₂×C₆₁

yes

characteristic

#1580

C₂×C₆₁

yes

complemented

#1368

C₂×C₆₁

yes

characteristic, complemented

#1333

C₂×C₆₁

yes

complemented

#1313

C₂×C₆₁

yes

characteristic, complemented

C₂×C₆₁

—

C₂×C₆₁

#1580

yes

characteristic

C₂×C₆₁

AGL(1,ℤ/16)

yes

—

C₂×D₁₆

C₄×C₆₁

yes

characteristic

C₄×D₈

C₂²×C₆₁

yes

—

C₈×D₄

—

#367

C₄×C₆₁

yes

characteristic

#357

C₄×C₆₁

yes

—

#346

C₂²×C₆₁

yes

characteristic

#346

C₂²×C₆₁

yes

characteristic

#334

C₂²×C₆₁

yes

—

D₄⋊C₈

—

#322

C₂²×C₆₁

yes

characteristic

C₆₁

#12067

yes

characteristic, direct product factor, Sylow

C₂×D₈

C₂×C₄×C₆₁

yes

—

C₂²×C₈

—

C₄×D₄

—

D₁₆

C₈×C₆₁

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₆₁

yes

characteristic

C₂×C₁₆

C₂×C₄×C₆₁

yes

characteristic

#107

C₂×C₄×C₆₁

yes

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

D₄×C₆₁

yes

—

C₄×C₈

C₂³×C₆₁

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

100

D₈

—

101

D₈

C₂×C₈×C₆₁

yes

complemented

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂²×C₄×C₆₁

yes

characteristic

106

C₂×C₈

#41×C₆₁

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

C₂×D₄×C₆₁

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

C₂×C₈×C₆₁

yes

complemented

113

C₂³

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄×D₄×C₆₁

yes

characteristic

120

C₂×C₄

AGL(1,ℤ/8)×C₆₁

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₂²×C₈×C₆₁

yes

characteristic

125

C₈

#131×C₆₁

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₆₁×#442

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

C₈×D₄×C₆₁

yes

characteristic

133

C₄

C₆₁×#444

yes

characteristic

134

C₂

—

135

C₂

C₆₁×#1580

yes

characteristic

136

C₂

C₆₁×AGL(1,ℤ/16)

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₁	#442
commutator	C₈	C₂²×C₈×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
prime radical	C₂×D₁₆×C₆₁	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ C₆₁×#322 ◁ G

subgroup

G/H

center, socle

C₂²×C₆₁

#442

commutator

C₈

C₂²×C₈×C₆₁

Frattini, cosocle

C₄×C₈

C₂³×C₆₁

prime radical

C₂×D₁₆×C₆₁

C₄

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ C₆₁×#322 ◁ G

N	H ≅ G/N	type
2. C₆₁×#1580	C₂	semidirect
3. C₆₁×#1368	C₂	semidirect
4. C₆₁×#1333	C₂	semidirect
5. C₆₁×#1313	C₂	semidirect
19. D₁₆×C₆₁	C₈	semidirect
32. D₈×C₆₁	C₂×C₈	semidirect
43. C₁₆×C₆₁	C₂×C₈	semidirect
57. #12067	C₆₁	direct (N × H)
67. #1580	C₂×C₆₁	semidirect
68. #1368	C₂×C₆₁	semidirect
69. #1333	C₂×C₆₁	semidirect
70. #1313	C₂×C₆₁	semidirect
84. C₆₁	#12067	direct (N × H)
88. D₁₆	C₈×C₆₁	semidirect
101. D₈	C₂×C₈×C₆₁	semidirect
112. C₁₆	C₂×C₈×C₆₁	semidirect

H ≅ G/N

type

2. C₆₁×#1580

C₂

semidirect

3. C₆₁×#1368

C₂

semidirect

4. C₆₁×#1333

C₂

semidirect

5. C₆₁×#1313

C₂

semidirect

19. D₁₆×C₆₁

C₈

semidirect

32. D₈×C₆₁

C₂×C₈

semidirect

43. C₁₆×C₆₁

C₂×C₈

semidirect

57. #12067

C₆₁

direct (N × H)

67. #1580

C₂×C₆₁

semidirect

68. #1368

C₂×C₆₁

semidirect

69. #1333

C₂×C₆₁

semidirect

70. #1313

C₂×C₆₁

semidirect

84. C₆₁

#12067

direct (N × H)

88. D₁₆

C₈×C₆₁

semidirect

101. D₈

C₂×C₈×C₆₁

semidirect

112. C₁₆

C₂×C₈×C₆₁

semidirect

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

subgroup	index
C₆₁×#2177	2
C₆₁×#1580	2
C₆₁×#1368	2
C₆₁×#1333	2
C₆₁×#1313	2
#12067	61

subgroup

index

C₆₁×#2177

C₆₁×#1580

C₆₁×#1368

C₆₁×#1333

C₆₁×#1313

#12067

Subgroups — C₆₁×#12067 #29515500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#12067 #29515500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12067 #29515500