Group #29515946 — subgroups

Subgroups — C₆₁×#12513 #29515946

Order 15616.

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

normal subgroups only

This group has 290 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×#2240	C₂	1	4	yes	maximal, complemented
2	C₆₁×#1712	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#500	C₂²	1	2	yes	—
6	C₆₁×#493	—	2	2	no	—
7	C₆₁×#493	C₂²	1	2	yes	—
8	C₆₁×#487	—	2	2	no	—
9	C₆₁×#357	—	2	8	no	—
10	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
11	C₈²×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×D₈×C₆₁	—	2	4	no	—
13	C₂×D₈×C₆₁	—	4	4	no	—
14	C₄⋊Q₈×C₆₁	—	2	2	no	—
15	#127×C₆₁	—	2	2	no	—
16	C₂×C₁₆×C₆₁	D₄	1	4	yes	—
17	#107×C₆₁	—	2	4	no	—
18	#103×C₆₁	—	4	4	no	—
19	C₄×C₈×C₆₁	D₄	1	2	yes	—
20	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₆₁	—	4	2	no	—
22	C₂×D₄×C₆₁	—	4	2	no	—
23	C₂×D₄×C₆₁	—	4	4	no	—
24	D₈×C₆₁	—	4	8	no	—
25	D₈×C₆₁	—	8	8	no	—
26	C₂×C₈×C₆₁	D₈	1	4	yes	—
27	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
28	C₄⋊C₄×C₆₁	—	4	2	no	—
29	C₄⋊C₄×C₆₁	—	4	2	no	—
30	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
31	C₁₆×C₆₁	—	2	4	no	—
32	C₂³×C₆₁	—	8	2	no	—
33	Q₈×C₆₁	—	8	4	no	—
34	D₄×C₆₁	—	8	4	no	—
35	D₄×C₆₁	—	8	8	no	—
36	C₂×C₄×C₆₁	—	8	2	no	—
37	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
39	C₈×C₆₁	—	2	4	no	—
40	C₈×C₆₁	#135	1	4	yes	—
41	#12513	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂²×C₆₁	—	16	2	no	—
43	C₂²×C₆₁	—	16	4	no	—
44	C₂²×C₆₁	#493	1	1	yes	characteristic
45	C₄×C₆₁	—	16	2	no	—
46	C₄×C₆₁	#495	1	2	yes	—
47	C₄×C₆₁	#509	1	4	yes	—
48	#2240	C₂×C₆₁	1	4	yes	complemented
49	#1712	C₂×C₆₁	1	1	yes	characteristic
50	#1710	C₂×C₆₁	1	1	yes	characteristic, complemented
51	#1311	C₂×C₆₁	1	1	yes	characteristic, complemented
52	C₂×C₆₁	—	32	2	no	—
53	C₂×C₆₁	#1709	1	1	yes	characteristic
54	C₂×C₆₁	#2250	1	2	yes	—
55	#500	C₂²×C₆₁	1	2	yes	—
56	#493	—	2	2	no	—
57	#493	C₂²×C₆₁	1	2	yes	—
58	#487	—	2	2	no	—
59	#357	—	2	8	no	—
60	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
61	C₈²	C₂²×C₆₁	1	1	yes	characteristic
62	C₆₁	#12513	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×D₈	—	2	4	no	—
64	C₂×D₈	—	4	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	#127	—	2	2	no	—
67	C₂×C₁₆	D₄×C₆₁	1	4	yes	—
68	#107	—	2	4	no	—
69	#103	—	4	4	no	—
70	C₄×C₈	D₄×C₆₁	1	2	yes	—
71	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
72	C₂×Q₈	—	4	2	no	—
73	C₂×D₄	—	4	2	no	—
74	C₂×D₄	—	4	4	no	—
75	D₈	—	4	8	no	—
76	D₈	—	8	8	no	—
77	C₂×C₈	D₈×C₆₁	1	4	yes	—
78	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
82	C₁₆	—	2	4	no	—
83	C₂³	—	8	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	8	4	no	—
86	D₄	—	8	8	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
89	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#135×C₆₁	1	4	yes	—
92	C₂²	—	16	2	no	—
93	C₂²	—	16	4	no	—
94	C₂²	C₆₁×#493	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	C₆₁×#495	1	2	yes	—
97	C₄	C₆₁×#509	1	4	yes	—
98	C₂	—	32	2	no	—
99	C₂	C₆₁×#1709	1	1	yes	characteristic
100	C₂	C₆₁×#2250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#2240

C₂

yes

maximal, complemented

C₆₁×#1712

C₂

yes

maximal, characteristic

C₆₁×#1710

C₂

yes

maximal, characteristic, complemented

C₆₁×#1311

C₂

yes

maximal, characteristic, complemented

C₆₁×#500

C₂²

yes

—

C₆₁×#493

—

C₆₁×#493

C₂²

yes

—

C₆₁×#487

—

C₆₁×#357

—

C₄×C₁₆×C₆₁

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

#127×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

#107×C₆₁

—

#103×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

D₈×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₁₆×C₆₁

—

C₂³×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₈×C₆₁

—

C₈×C₆₁

#135

yes

—

#12513

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#493

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#495

yes

—

C₄×C₆₁

#509

yes

—

#2240

C₂×C₆₁

yes

complemented

#1712

C₂×C₆₁

yes

characteristic

#1710

C₂×C₆₁

yes

characteristic, complemented

#1311

C₂×C₆₁

yes

characteristic, complemented

C₂×C₆₁

—

C₂×C₆₁

#1709

yes

characteristic

C₂×C₆₁

#2250

yes

—

#500

C₂²×C₆₁

yes

—

#493

—

#493

C₂²×C₆₁

yes

—

#487

—

#357

—

C₄×C₁₆

C₂²×C₆₁

yes

—

C₈²

C₂²×C₆₁

yes

characteristic

C₆₁

#12513

yes

characteristic, direct product factor, Sylow

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₄×C₆₁

yes

—

#107

—

#103

—

C₄×C₈

D₄×C₆₁

yes

—

C₄×C₈

C₂³×C₆₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

D₈×C₆₁

yes

—

C₂×C₈

C₂×D₄×C₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄×C₆₁

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#127×C₆₁

yes

characteristic

C₂×C₄

C₂×D₈×C₆₁

yes

—

C₈

—

C₈

#135×C₆₁

yes

—

C₂²

—

C₂²

—

C₂²

C₆₁×#493

yes

characteristic

C₄

—

C₄

C₆₁×#495

yes

—

C₄

C₆₁×#509

yes

—

C₂

—

C₂

C₆₁×#1709

yes

characteristic

100

C₂

C₆₁×#2250

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₁	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₆₁
prime radical	C₆₁×#1712	C₂
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×#1712

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ C₈²×C₆₁ ◁ G

N	H ≅ G/N	type
1. C₆₁×#2240	C₂	semidirect
3. C₆₁×#1710	C₂	semidirect
4. C₆₁×#1311	C₂	semidirect
41. #12513	C₆₁	direct (N × H)
48. #2240	C₂×C₆₁	semidirect
50. #1710	C₂×C₆₁	semidirect
51. #1311	C₂×C₆₁	semidirect
62. C₆₁	#12513	direct (N × H)

H ≅ G/N

type

1. C₆₁×#2240

C₂

semidirect

3. C₆₁×#1710

C₂

semidirect

4. C₆₁×#1311

C₂

semidirect

41. #12513

C₆₁

direct (N × H)

48. #2240

C₂×C₆₁

semidirect

50. #1710

C₂×C₆₁

semidirect

51. #1311

C₂×C₆₁

semidirect

62. C₆₁

#12513

direct (N × H)

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

subgroup	index
C₆₁×#2240	2
C₆₁×#1712	2
C₆₁×#1710	2
C₆₁×#1311	2
#12513	61

subgroup

index

C₆₁×#2240

C₆₁×#1712

C₆₁×#1710

C₆₁×#1311

#12513

Subgroups — C₆₁×#12513 #29515946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#12513 #29515946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12513 #29515946