Group #21441607 — subgroups

Subgroups — C₆₁×#1566 #21441607

Order 7808.

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

normal subgroups only

This group has 184 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×#423	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#334	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₆₁	—	2	2	no	—
5	C₂²×C₈×C₆₁	C₄	1	2	yes	—
6	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
7	C₄⋊C₈×C₆₁	—	2	2	no	—
8	C₄⋊C₈×C₆₁	C₂²	1	4	yes	—
9	C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
10	C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
11	C₄×C₈×C₆₁	C₂²	1	2	yes	—
12	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	Q₈	1	1	yes	characteristic
15	M₁₆×C₆₁	—	4	4	no	—
16	C₂×C₈×C₆₁	—	2	4	no	—
17	C₂×C₈×C₆₁	—	2	4	no	—
18	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
20	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₆₁	D₄	1	1	yes	characteristic
22	C₄²×C₆₁	Q₈	1	1	yes	characteristic
23	C₄²×C₆₁	C₂³	1	1	yes	characteristic
24	C₂³×C₆₁	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₆₁	—	2	2	no	—
26	C₂×C₄×C₆₁	—	2	2	no	—
27	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₁	SD₁₆	1	4	yes	—
31	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₆₁	—	2	2	no	—
35	C₈×C₆₁	—	4	4	no	—
36	C₈×C₆₁	M₁₆	1	4	yes	complemented
37	C₂²×C₆₁	—	2	1	no	—
38	C₂²×C₆₁	—	2	1	no	—
39	C₂²×C₆₁	#106	1	2	yes	—
40	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₆₁	—	2	1	no	—
42	C₄×C₆₁	—	2	1	no	—
43	C₄×C₆₁	#106	1	2	yes	—
44	C₄×C₆₁	#108	1	2	yes	—
45	C₄×C₆₁	C₂×M₁₆	1	2	yes	—
46	C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
47	#1566	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₆₁	—	2	2	no	—
49	C₂×C₆₁	#423	1	1	yes	characteristic
50	C₂×C₆₁	C₂×#106	1	1	yes	characteristic
51	C₂×C₆₁	C₂×#108	1	1	yes	characteristic
52	#423	C₂×C₆₁	1	2	yes	—
53	C₂×C₄×C₈	C₂×C₆₁	1	1	yes	characteristic
54	#334	C₂×C₆₁	1	4	yes	complemented
55	C₆₁	#1566	1	1	yes	characteristic, direct product factor, Sylow
56	C₂×M₁₆	—	2	2	no	—
57	C₂²×C₈	C₄×C₆₁	1	2	yes	—
58	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	C₂²×C₆₁	1	4	yes	—
61	C₄×C₈	C₄×C₆₁	1	1	yes	characteristic
62	C₄×C₈	C₄×C₆₁	1	1	yes	characteristic
63	C₄×C₈	C₂²×C₆₁	1	2	yes	—
64	C₂²×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
65	C₂²×C₄	D₄×C₆₁	1	1	yes	characteristic
66	C₂²×C₄	Q₈×C₆₁	1	1	yes	characteristic
67	M₁₆	—	4	4	no	—
68	C₂×C₈	—	2	4	no	—
69	C₂×C₈	—	2	4	no	—
70	C₂×C₈	C₂×C₄×C₆₁	1	2	yes	—
71	C₂×C₈	C₂×C₄×C₆₁	1	2	yes	—
72	C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
73	C₄²	D₄×C₆₁	1	1	yes	characteristic
74	C₄²	Q₈×C₆₁	1	1	yes	characteristic
75	C₄²	C₂³×C₆₁	1	1	yes	characteristic
76	C₂³	C₄⋊C₄×C₆₁	1	1	yes	characteristic
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
80	C₂×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
81	C₂×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
82	C₂×C₄	SD₁₆×C₆₁	1	4	yes	—
83	C₂×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
84	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
85	C₂×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	4	4	no	—
88	C₈	M₁₆×C₆₁	1	4	yes	complemented
89	C₂²	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#106×C₆₁	1	2	yes	—
92	C₂²	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	#106×C₆₁	1	2	yes	—
96	C₄	#108×C₆₁	1	2	yes	—
97	C₄	C₂×M₁₆×C₆₁	1	2	yes	—
98	C₄	C₂×SD₁₆×C₆₁	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	C₆₁×#423	1	1	yes	characteristic
101	C₂	C₂×#106×C₆₁	1	1	yes	characteristic
102	C₂	C₂×#108×C₆₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#423

C₂

yes

maximal

C₂×C₄×C₈×C₆₁

C₂

yes

maximal, characteristic

C₆₁×#334

C₂

yes

maximal, complemented

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₄

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₄

yes

characteristic

C₄×C₈×C₆₁

C₄

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

Q₈

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

Q₈

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₂³×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

M₁₆

yes

complemented

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#106

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#106

yes

—

C₄×C₆₁

#108

yes

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

C₂×SD₁₆

yes

—

#1566

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₄×C₈

C₂×C₆₁

yes

characteristic

#334

C₂×C₆₁

yes

complemented

C₆₁

#1566

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

C₄×C₆₁

yes

—

C₂×C₄²

C₂²×C₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₆₁

yes

—

C₄×C₈

C₄×C₆₁

yes

characteristic

C₄×C₈

C₄×C₆₁

yes

characteristic

C₄×C₈

C₂²×C₆₁

yes

—

C₂²×C₄

C₂×C₄×C₆₁

yes

characteristic

C₂²×C₄

D₄×C₆₁

yes

characteristic

C₂²×C₄

Q₈×C₆₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₆₁

yes

—

C₂×C₈

C₂×C₄×C₆₁

yes

—

C₄²

C₂×C₄×C₆₁

yes

characteristic

C₄²

D₄×C₆₁

yes

characteristic

C₄²

Q₈×C₆₁

yes

characteristic

C₄²

C₂³×C₆₁

yes

characteristic

C₂³

C₄⋊C₄×C₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₄×C₆₁

yes

characteristic

C₂×C₄

C₄⋊C₄×C₆₁

yes

characteristic

C₂×C₄

C₄⋊C₄×C₆₁

yes

characteristic

C₂×C₄

SD₁₆×C₆₁

yes

—

C₂×C₄

C₂²×C₄×C₆₁

yes

characteristic

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

C₂×C₄

C₂×Q₈×C₆₁

yes

characteristic

C₈

—

C₈

—

C₈

M₁₆×C₆₁

yes

complemented

C₂²

—

C₂²

—

C₂²

#106×C₆₁

yes

—

C₂²

C₂×C₄⋊C₄×C₆₁

yes

characteristic

C₄

—

C₄

—

C₄

#106×C₆₁

yes

—

C₄

#108×C₆₁

yes

—

C₄

C₂×M₁₆×C₆₁

yes

—

C₄

C₂×SD₁₆×C₆₁

yes

—

C₂

—

100

C₂

C₆₁×#423

yes

characteristic

101

C₂

C₂×#106×C₆₁

yes

characteristic

102

C₂

C₂×#108×C₆₁

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×C₄⋊C₄
prime radical	C₂³×C₆₁	C₄⋊C₄
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

C₄²

C₂³×C₆₁

socle

C₂²×C₆₁

C₂×C₄⋊C₄

prime radical

C₂³×C₆₁

C₄⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

N	H ≅ G/N	type
3. C₆₁×#334	C₂	semidirect
36. C₈×C₆₁	M₁₆	semidirect
47. #1566	C₆₁	direct (N × H)
54. #334	C₂×C₆₁	semidirect
55. C₆₁	#1566	direct (N × H)
88. C₈	M₁₆×C₆₁	semidirect

H ≅ G/N

type

3. C₆₁×#334

C₂

semidirect

36. C₈×C₆₁

M₁₆

semidirect

47. #1566

C₆₁

direct (N × H)

54. #334

C₂×C₆₁

semidirect

55. C₆₁

#1566

direct (N × H)

88. C₈

M₁₆×C₆₁

semidirect

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

subgroup	index
C₆₁×#423	2
C₂×C₄×C₈×C₆₁	2
C₆₁×#334	2
#1566	61

subgroup

index

C₆₁×#423

C₂×C₄×C₈×C₆₁

C₆₁×#334

#1566

Subgroups — C₆₁×#1566 #21441607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#1566 #21441607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#1566 #21441607