Group #21441671 — subgroups

Subgroups — C₆₁×#1630 #21441671

Order 7808.

218 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 68 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×#559	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#492	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#477	C₂	1	2	yes	maximal, complemented
4	C₆₁×#331	C₂	1	1	yes	maximal, characteristic
5	C₆₁×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆×C₆₁	—	2	2	no	—
7	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
8	#127×C₆₁	C₂²	1	1	yes	characteristic
9	#124×C₆₁	—	2	2	no	—
10	#123×C₆₁	—	2	2	no	—
11	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
12	C₄×D₄×C₆₁	—	2	2	no	—
13	#107×C₆₁	—	2	2	no	—
14	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	#103×C₆₁	—	2	2	no	—
16	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₆₁	—	2	1	no	—
18	C₂×Q₈×C₆₁	D₄	1	2	yes	—
19	C₂×D₄×C₆₁	—	2	2	no	—
20	C₂×D₄×C₆₁	D₄	1	2	yes	—
21	C₂²×C₄×C₆₁	—	2	2	no	—
22	SD₁₆×C₆₁	—	4	4	no	—
23	C₂×C₈×C₆₁	—	2	1	no	—
24	C₂×C₈×C₆₁	—	2	2	no	—
25	C₄⋊C₄×C₆₁	—	2	2	no	—
26	C₄⋊C₄×C₆₁	—	2	2	no	—
27	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
28	C₂²⋊C₄×C₆₁	—	2	4	no	—
29	C₂²⋊C₄×C₆₁	—	2	2	no	—
30	C₄²×C₆₁	—	2	2	no	—
31	C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	C₂³×C₆₁	—	2	2	no	—
33	Q₈×C₆₁	—	2	2	no	—
34	Q₈×C₆₁	—	2	2	no	—
35	Q₈×C₆₁	—	4	2	no	—
36	D₄×C₆₁	—	4	2	no	—
37	D₄×C₆₁	—	4	2	no	—
38	D₄×C₆₁	—	4	2	no	—
39	C₂×C₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	—	2	2	no	—
41	C₂×C₄×C₆₁	—	4	1	no	—
42	C₂×C₄×C₆₁	—	8	2	no	—
43	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
45	C₈×C₆₁	—	2	2	no	—
46	C₈×C₆₁	—	4	2	no	—
47	C₂²×C₆₁	—	4	4	no	—
48	C₂²×C₆₁	—	4	2	no	—
49	C₂²×C₆₁	#120	1	1	yes	characteristic
50	C₄×C₆₁	—	2	1	no	—
51	C₄×C₆₁	—	4	2	no	—
52	C₄×C₆₁	—	4	2	no	—
53	C₄×C₆₁	—	8	1	no	—
54	C₄×C₆₁	#135	1	2	yes	—
55	C₄×C₆₁	#137	1	2	yes	—
56	#1630	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×C₆₁	—	8	2	no	—
58	C₂×C₆₁	#452	1	2	yes	—
59	C₂×C₆₁	#453	1	1	yes	characteristic
60	#559	C₂×C₆₁	1	1	yes	characteristic
61	#492	C₂×C₆₁	1	1	yes	characteristic
62	#477	C₂×C₆₁	1	2	yes	complemented
63	#331	C₂×C₆₁	1	1	yes	characteristic
64	Q₈⋊C₈	C₂×C₆₁	1	2	yes	complemented
65	C₆₁	#1630	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×SD₁₆	—	2	2	no	—
67	C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
68	#127	C₂²×C₆₁	1	1	yes	characteristic
69	#124	—	2	2	no	—
70	#123	—	2	2	no	—
71	C₄×Q₈	C₂²×C₆₁	1	2	yes	—
72	C₄×D₄	—	2	2	no	—
73	#107	—	2	2	no	—
74	C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
75	#103	—	2	2	no	—
76	C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	D₄×C₆₁	1	2	yes	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	D₄×C₆₁	1	2	yes	—
81	C₂²×C₄	—	2	2	no	—
82	SD₁₆	—	4	4	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	D₄×C₆₁	1	2	yes	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₂²⋊C₄	—	2	2	no	—
90	C₄²	—	2	2	no	—
91	C₄²	C₂³×C₆₁	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	2	2	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
104	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
105	C₈	—	2	2	no	—
106	C₈	—	4	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#120×C₆₁	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	4	2	no	—
112	C₄	—	4	2	no	—
113	C₄	—	8	1	no	—
114	C₄	#135×C₆₁	1	2	yes	—
115	C₄	#137×C₆₁	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	C₆₁×#452	1	2	yes	—
118	C₂	C₆₁×#453	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#559

C₂

yes

maximal, characteristic

C₆₁×#492

C₂

yes

maximal, characteristic

C₆₁×#477

C₂

yes

maximal, complemented

C₆₁×#331

C₂

yes

maximal, characteristic

C₆₁×Q₈⋊C₈

C₂

yes

maximal, complemented

C₂×SD₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#127×C₆₁

C₂²

yes

characteristic

#124×C₆₁

—

#123×C₆₁

—

C₄×Q₈×C₆₁

C₂²

yes

—

C₄×D₄×C₆₁

—

#107×C₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

—

#103×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

D₄

yes

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

—

SD₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₂²⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

C₂³×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#120

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#135

yes

—

C₄×C₆₁

#137

yes

—

#1630

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₁

—

C₂×C₆₁

#452

yes

—

C₂×C₆₁

#453

yes

characteristic

#559

C₂×C₆₁

yes

characteristic

#492

C₂×C₆₁

yes

characteristic

#477

C₂×C₆₁

yes

complemented

#331

C₂×C₆₁

yes

characteristic

Q₈⋊C₈

C₂×C₆₁

yes

complemented

C₆₁

#1630

yes

characteristic, direct product factor, Sylow

C₂×SD₁₆

—

C₄⋊Q₈

C₂²×C₆₁

yes

characteristic

#127

C₂²×C₆₁

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂²×C₆₁

yes

—

C₄×D₄

—

#107

—

C₄⋊C₈

C₂²×C₆₁

yes

—

#103

—

C₄×C₈

C₂²×C₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₆₁

yes

—

C₂×D₄

—

C₂×D₄

D₄×C₆₁

yes

—

C₂²×C₄

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₆₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂³×C₆₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

104

C₂×C₄

C₂×D₄×C₆₁

yes

—

105

C₈

—

106

C₈

—

107

C₂²

—

108

C₂²

—

109

C₂²

#120×C₆₁

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#135×C₆₁

yes

—

115

C₄

#137×C₆₁

yes

—

116

C₂

—

117

C₂

C₆₁×#452

yes

—

118

C₂

C₆₁×#453

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₆₁
prime radical	#127×C₆₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₁

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₆₁

prime radical

#127×C₆₁

C₂²

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ G

N	H ≅ G/N	type
3. C₆₁×#477	C₂	semidirect
5. C₆₁×Q₈⋊C₈	C₂	semidirect
56. #1630	C₆₁	direct (N × H)
62. #477	C₂×C₆₁	semidirect
64. Q₈⋊C₈	C₂×C₆₁	semidirect
65. C₆₁	#1630	direct (N × H)

H ≅ G/N

type

3. C₆₁×#477

C₂

semidirect

5. C₆₁×Q₈⋊C₈

C₂

semidirect

56. #1630

C₆₁

direct (N × H)

62. #477

C₂×C₆₁

semidirect

64. Q₈⋊C₈

C₂×C₆₁

semidirect

65. C₆₁

#1630

direct (N × H)

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

subgroup	index
C₆₁×#559	2
C₆₁×#492	2
C₆₁×#477	2
C₆₁×#331	2
C₆₁×Q₈⋊C₈	2
#1630	61

subgroup

index

C₆₁×#559

C₆₁×#492

C₆₁×#477

C₆₁×#331

C₆₁×Q₈⋊C₈

#1630

Subgroups — C₆₁×#1630 #21441671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#1630 #21441671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#1630 #21441671