Group #29515944 — subgroups

Subgroups — C₆₁×#12511 #29515944

Order 15616.

260 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2241	C₂	1	2	yes	maximal, complemented
2	C₆₁×#2240	C₂	1	2	yes	maximal
3	C₆₁×#1710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1709	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#500	C₂²	1	2	yes	—
7	C₆₁×#494	C₂²	1	1	yes	characteristic
8	C₆₁×#493	C₂²	1	1	yes	characteristic
9	C₆₁×#487	—	2	2	no	—
10	C₆₁×#486	—	2	2	no	—
11	C₆₁×#358	—	2	4	no	—
12	C₆₁×#357	—	2	4	no	—
13	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
14	C₈²×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₆₁	—	2	2	no	—
16	C₂×D₈×C₆₁	—	2	2	no	—
17	C₄⋊Q₈×C₆₁	—	2	1	no	—
18	C₄⋊Q₈×C₆₁	—	2	2	no	—
19	#127×C₆₁	—	2	1	no	—
20	C₂×C₁₆×C₆₁	D₄	1	4	yes	—
21	#107×C₆₁	—	2	2	no	—
22	#107×C₆₁	—	2	2	no	—
23	#103×C₆₁	—	4	4	no	—
24	#102×C₆₁	—	4	4	no	—
25	C₄×C₈×C₆₁	D₄	1	2	yes	—
26	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
27	C₂×Q₈×C₆₁	—	4	1	no	—
28	C₂×Q₈×C₆₁	—	4	2	no	—
29	C₂×D₄×C₆₁	—	4	1	no	—
30	C₂×D₄×C₆₁	—	4	2	no	—
31	Q₁₆×C₆₁	—	4	4	no	—
32	D₈×C₆₁	—	4	4	no	—
33	C₂×C₈×C₆₁	D₈	1	4	yes	—
34	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
35	C₄⋊C₄×C₆₁	—	4	2	no	—
36	C₄⋊C₄×C₆₁	—	4	2	no	—
37	C₄⋊C₄×C₆₁	—	4	2	no	—
38	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₆₁	—	2	4	no	—
40	C₂³×C₆₁	—	8	1	no	—
41	Q₈×C₆₁	—	8	2	no	—
42	Q₈×C₆₁	—	8	4	no	—
43	D₄×C₆₁	—	8	2	no	—
44	D₄×C₆₁	—	8	4	no	—
45	C₂×C₄×C₆₁	—	8	1	no	—
46	C₂×C₄×C₆₁	—	8	2	no	—
47	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
49	C₈×C₆₁	—	2	4	no	—
50	C₈×C₆₁	#135	1	4	yes	—
51	#12511	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₆₁	—	16	1	no	—
53	C₂²×C₆₁	—	16	2	no	—
54	C₂²×C₆₁	#493	1	1	yes	characteristic
55	C₄×C₆₁	—	16	1	no	—
56	C₄×C₆₁	—	16	2	no	—
57	C₄×C₆₁	#495	1	2	yes	—
58	C₄×C₆₁	#508	1	4	yes	—
59	#2241	C₂×C₆₁	1	2	yes	complemented
60	#2240	C₂×C₆₁	1	2	yes	—
61	#1710	C₂×C₆₁	1	1	yes	characteristic, complemented
62	#1709	C₂×C₆₁	1	1	yes	characteristic
63	C₈×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
64	C₂×C₆₁	—	32	1	no	—
65	C₂×C₆₁	#1709	1	1	yes	characteristic
66	C₂×C₆₁	#2249	1	2	yes	—
67	#500	C₂²×C₆₁	1	2	yes	—
68	#494	C₂²×C₆₁	1	1	yes	characteristic
69	#493	C₂²×C₆₁	1	1	yes	characteristic
70	#487	—	2	2	no	—
71	#486	—	2	2	no	—
72	#358	—	2	4	no	—
73	#357	—	2	4	no	—
74	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
75	C₈²	C₂²×C₆₁	1	1	yes	characteristic
76	C₆₁	#12511	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×Q₁₆	—	2	2	no	—
78	C₂×D₈	—	2	2	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	#127	—	2	1	no	—
82	C₂×C₁₆	D₄×C₆₁	1	4	yes	—
83	#107	—	2	2	no	—
84	#107	—	2	2	no	—
85	#103	—	4	4	no	—
86	#102	—	4	4	no	—
87	C₄×C₈	D₄×C₆₁	1	2	yes	—
88	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
89	C₂×Q₈	—	4	1	no	—
90	C₂×Q₈	—	4	2	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	2	no	—
93	Q₁₆	—	4	4	no	—
94	D₈	—	4	4	no	—
95	C₂×C₈	D₈×C₆₁	1	4	yes	—
96	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
101	C₁₆	—	2	4	no	—
102	C₂³	—	8	1	no	—
103	Q₈	—	8	2	no	—
104	Q₈	—	8	4	no	—
105	D₄	—	8	2	no	—
106	D₄	—	8	4	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
110	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#135×C₆₁	1	4	yes	—
113	C₂²	—	16	1	no	—
114	C₂²	—	16	2	no	—
115	C₂²	C₆₁×#493	1	1	yes	characteristic
116	C₄	—	16	1	no	—
117	C₄	—	16	2	no	—
118	C₄	C₆₁×#495	1	2	yes	—
119	C₄	C₆₁×#508	1	4	yes	—
120	C₂	—	32	1	no	—
121	C₂	C₆₁×#1709	1	1	yes	characteristic
122	C₂	C₆₁×#2249	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#2241

C₂

yes

maximal, complemented

C₆₁×#2240

C₂

yes

maximal

C₆₁×#1710

C₂

yes

maximal, characteristic, complemented

C₆₁×#1709

C₂

yes

maximal, characteristic

C₈×C₁₆×C₆₁

C₂

yes

maximal, characteristic, complemented

C₆₁×#500

C₂²

yes

—

C₆₁×#494

C₂²

yes

characteristic

C₆₁×#493

C₂²

yes

characteristic

C₆₁×#487

—

C₆₁×#486

—

C₆₁×#358

—

C₆₁×#357

—

C₄×C₁₆×C₆₁

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Q₁₆×C₆₁

—

C₂×D₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

#127×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

#107×C₆₁

—

#107×C₆₁

—

#103×C₆₁

—

#102×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

Q₁₆×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₁₆×C₆₁

—

C₂³×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₈×C₆₁

—

C₈×C₆₁

#135

yes

—

#12511

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#493

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#495

yes

—

C₄×C₆₁

#508

yes

—

#2241

C₂×C₆₁

yes

complemented

#2240

C₂×C₆₁

yes

—

#1710

C₂×C₆₁

yes

characteristic, complemented

#1709

C₂×C₆₁

yes

characteristic

C₈×C₁₆

C₂×C₆₁

yes

characteristic, complemented

C₂×C₆₁

—

C₂×C₆₁

#1709

yes

characteristic

C₂×C₆₁

#2249

yes

—

#500

C₂²×C₆₁

yes

—

#494

C₂²×C₆₁

yes

characteristic

#493

C₂²×C₆₁

yes

—

—

—

—

C₄×C₁₆

C₂²×C₆₁

yes

—

C₈²

C₂²×C₆₁

yes

characteristic

C₆₁

#12511

yes

characteristic, direct product factor, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₄×C₆₁

yes

—

—

—

—

—

C₄×C₈

D₄×C₆₁

yes

—

C₄×C₈

C₂³×C₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

D₈×C₆₁

yes

—

C₂×C₈

C₂×D₄×C₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

C₂×D₄×C₆₁

yes

characteristic

101

C₁₆

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#127×C₆₁

yes

characteristic

110

C₂×C₄

C₂×D₈×C₆₁

yes

—

111

C₈

—

112

C₈

#135×C₆₁

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₆₁×#493

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

C₆₁×#495

yes

—

119

C₄

C₆₁×#508

yes

—

120

C₂

—

121

C₂

C₆₁×#1709

yes

characteristic

122

C₂

C₆₁×#2249

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₁	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₆₁
prime radical	C₆₁×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ C₈²×C₆₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₁

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₆₁

prime radical

C₆₁×#493

C₂²

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ C₈²×C₆₁ ◁ G

N	H ≅ G/N	type
1. C₆₁×#2241	C₂	semidirect
3. C₆₁×#1710	C₂	semidirect
5. C₈×C₁₆×C₆₁	C₂	semidirect
51. #12511	C₆₁	direct (N × H)
59. #2241	C₂×C₆₁	semidirect
61. #1710	C₂×C₆₁	semidirect
63. C₈×C₁₆	C₂×C₆₁	semidirect
76. C₆₁	#12511	direct (N × H)

H ≅ G/N

type

1. C₆₁×#2241

C₂

semidirect

3. C₆₁×#1710

C₂

semidirect

5. C₈×C₁₆×C₆₁

C₂

semidirect

51. #12511

C₆₁

direct (N × H)

59. #2241

C₂×C₆₁

semidirect

61. #1710

C₂×C₆₁

semidirect

63. C₈×C₁₆

C₂×C₆₁

semidirect

76. C₆₁

#12511

direct (N × H)

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

subgroup	index
C₆₁×#2241	2
C₆₁×#2240	2
C₆₁×#1710	2
C₆₁×#1709	2
C₈×C₁₆×C₆₁	2
#12511	61

subgroup

index

C₆₁×#2241

C₆₁×#2240

C₆₁×#1710

C₆₁×#1709

C₈×C₁₆×C₆₁

#12511

Subgroups — C₆₁×#12511 #29515944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#12511 #29515944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12511 #29515944