Group #29514019 — subgroups

Subgroups — C₆₁×#10586 #29514019

Order 15616.

350 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2845	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1836	C₂	1	3	yes	maximal
3	C₆₁×#1295	C₂	1	3	yes	maximal
4	C₂×#97×C₆₁	C₂²	1	3	yes	—
5	C₆₁×#398	—	2	6	no	—
6	C₆₁×#384	C₂²	1	1	yes	characteristic
7	C₆₁×#382	C₂²	1	3	yes	—
8	C₆₁×#340	—	2	6	no	—
9	C₂²×C₈×C₆₁	—	2	3	no	—
10	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
11	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
12	C₂×C₄⋊C₄×C₆₁	D₄	1	6	yes	—
13	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
14	#97×C₆₁	—	2	3	no	—
15	#97×C₆₁	Q₈	1	6	yes	—
16	#95×C₆₁	—	2	2	no	—
17	#95×C₆₁	—	2	6	no	—
18	C₂²×C₄×C₆₁	—	2	6	no	—
19	C₂²×C₄×C₆₁	—	4	1	no	—
20	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
21	C₂×C₈×C₆₁	—	2	6	no	—
22	C₂×C₈×C₆₁	—	4	3	no	—
23	C₂×C₈×C₆₁	—	4	3	no	—
24	C₄⋊C₄×C₆₁	—	4	6	no	—
25	C₄⋊C₄×C₆₁	—	4	6	no	—
26	C₄⋊C₄×C₆₁	—	8	6	no	—
27	C₄²×C₆₁	C₂×Q₈	1	3	yes	—
28	C₄²×C₆₁	#41	1	1	yes	characteristic
29	C₂³×C₆₁	#120	1	1	yes	characteristic
30	C₂×C₄×C₆₁	—	2	3	no	—
31	C₂×C₄×C₆₁	—	2	3	no	—
32	C₂×C₄×C₆₁	—	4	6	no	—
33	C₂×C₄×C₆₁	—	4	6	no	—
34	C₂×C₄×C₆₁	—	4	6	no	—
35	C₂×C₄×C₆₁	—	8	3	no	—
36	C₂×C₄×C₆₁	C₂²⋊Q₈	1	3	yes	—
37	C₂×C₄×C₆₁	C₄⋊Q₈	1	3	yes	—
38	C₈×C₆₁	—	4	6	no	—
39	#10586	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₆₁	#395	1	1	yes	characteristic
41	C₂²×C₆₁	#454	1	3	yes	—
42	C₂²×C₆₁	#455	1	3	yes	—
43	C₄×C₆₁	—	2	3	no	—
44	C₄×C₆₁	—	2	3	no	—
45	C₄×C₆₁	—	8	6	no	—
46	C₄×C₆₁	—	16	1	no	—
47	#2845	C₂×C₆₁	1	1	yes	characteristic
48	#1836	C₂×C₆₁	1	3	yes	—
49	#1295	C₂×C₆₁	1	3	yes	—
50	C₂×C₆₁	#1658	1	1	yes	characteristic
51	C₂×C₆₁	#1662	1	3	yes	—
52	C₂×C₆₁	#2061	1	3	yes	—
53	C₂×#97	C₂²×C₆₁	1	3	yes	—
54	#398	—	2	6	no	—
55	#384	C₂²×C₆₁	1	1	yes	characteristic
56	#382	C₂²×C₆₁	1	3	yes	—
57	#340	—	2	6	no	—
58	C₆₁	#10586	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²×C₈	—	2	3	no	—
60	C₂×C₄⋊C₄	—	2	3	no	—
61	C₂×C₄⋊C₄	—	2	6	no	—
62	C₂×C₄⋊C₄	D₄×C₆₁	1	6	yes	—
63	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
64	#97	—	2	3	no	—
65	#97	Q₈×C₆₁	1	6	yes	—
66	#95	—	2	2	no	—
67	#95	—	2	6	no	—
68	C₂²×C₄	—	2	6	no	—
69	C₂²×C₄	—	4	1	no	—
70	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	—	4	3	no	—
73	C₂×C₈	—	4	3	no	—
74	C₄⋊C₄	—	4	6	no	—
75	C₄⋊C₄	—	4	6	no	—
76	C₄⋊C₄	—	8	6	no	—
77	C₄²	C₂×Q₈×C₆₁	1	3	yes	—
78	C₄²	#41×C₆₁	1	1	yes	characteristic
79	C₂³	#120×C₆₁	1	1	yes	characteristic
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	4	6	no	—
83	C₂×C₄	—	4	6	no	—
84	C₂×C₄	—	4	6	no	—
85	C₂×C₄	—	8	3	no	—
86	C₂×C₄	C₂²⋊Q₈×C₆₁	1	3	yes	—
87	C₂×C₄	C₄⋊Q₈×C₆₁	1	3	yes	—
88	C₈	—	4	6	no	—
89	C₂²	C₆₁×#395	1	1	yes	characteristic
90	C₂²	C₆₁×#454	1	3	yes	—
91	C₂²	C₆₁×#455	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	2	3	no	—
94	C₄	—	8	6	no	—
95	C₄	—	16	1	no	—
96	C₂	C₆₁×#1658	1	1	yes	characteristic
97	C₂	C₆₁×#1662	1	3	yes	—
98	C₂	C₆₁×#2061	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#2845

C₂

yes

maximal, characteristic

C₆₁×#1836

C₂

yes

maximal

C₆₁×#1295

C₂

yes

maximal

C₂×#97×C₆₁

C₂²

yes

—

C₆₁×#398

—

C₆₁×#384

C₂²

yes

characteristic

C₆₁×#382

C₂²

yes

—

C₆₁×#340

—

C₂²×C₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#97×C₆₁

—

#97×C₆₁

Q₈

yes

—

#95×C₆₁

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×Q₈

yes

—

C₄²×C₆₁

yes

C₂³×C₆₁

yes

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

C₄⋊Q₈

yes

—

C₈×C₆₁

—

#10586

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

#395

yes

characteristic

C₂²×C₆₁

#454

yes

—

C₂²×C₆₁

#455

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

#2845

C₂×C₆₁

yes

characteristic

#1836

C₂×C₆₁

yes

—

#1295

C₂×C₆₁

yes

—

C₂×C₆₁

#1658

yes

characteristic

C₂×C₆₁

#1662

yes

—

C₂×C₆₁

#2061

yes

—

C₂×#97

C₂²×C₆₁

yes

—

#398

—

#384

C₂²×C₆₁

yes

characteristic

#382

C₂²×C₆₁

yes

—

#340

—

C₆₁

#10586

yes

characteristic, direct product factor, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₆₁

yes

—

C₂×C₄²

C₂³×C₆₁

yes

characteristic

#97

—

#97

Q₈×C₆₁

yes

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₆₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Q₈×C₆₁

yes

—

C₄²

#41×C₆₁

yes

characteristic

C₂³

#120×C₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Q₈×C₆₁

yes

—

C₂×C₄

C₄⋊Q₈×C₆₁

yes

—

C₈

—

C₂²

C₆₁×#395

yes

characteristic

C₂²

C₆₁×#454

yes

—

C₂²

C₆₁×#455

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

C₆₁×#1658

yes

characteristic

C₂

C₆₁×#1662

yes

—

C₂

C₆₁×#2061

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₆₁

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₆₁

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

N	H ≅ G/N	type
39. #10586	C₆₁	direct (N × H)
58. C₆₁	#10586	direct (N × H)

H ≅ G/N

type

39. #10586

C₆₁

direct (N × H)

58. C₆₁

#10586

direct (N × H)

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

subgroup	index
C₆₁×#2845	2
C₆₁×#1836	2
C₆₁×#1295	2
#10586	61

subgroup

index

C₆₁×#2845

C₆₁×#1836

C₆₁×#1295

#10586

Subgroups — C₆₁×#10586 #29514019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#10586 #29514019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#10586 #29514019