Group #29512259 — subgroups

Subgroups — C₆₁×#8826 #29512259

Order 15616.

338 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 198 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×#1848	C₂	1	2	yes	maximal
2	C₆₁×#1567	C₂	1	4	yes	maximal
3	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×#108×C₆₁	—	2	4	no	—
5	C₆₁×#423	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×#335	—	2	8	no	—
9	C₆₁×#322	C₄	1	4	yes	—
10	C₂×M₁₆×C₆₁	—	2	4	no	—
11	C₂²×C₈×C₆₁	D₄	1	2	yes	—
12	C₂²×C₈×C₆₁	Q₈	1	4	yes	—
13	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
14	#108×C₆₁	—	4	8	no	—
15	C₄⋊C₈×C₆₁	—	2	8	no	—
16	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₆₁	D₄	1	4	yes	—
19	C₄×C₈×C₆₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
22	M₁₆×C₆₁	—	4	8	no	—
23	C₂×C₈×C₆₁	—	4	4	no	—
24	C₂×C₈×C₆₁	—	4	4	no	—
25	C₂×C₈×C₆₁	C₄⋊C₄	1	8	yes	—
26	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
27	C₂×C₈×C₆₁	D₈	1	4	yes	—
28	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
29	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₆₁	C₂×D₄	1	2	yes	—
31	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
32	C₂³×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
33	C₂×C₄×C₆₁	—	2	4	no	—
34	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
37	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
38	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
39	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
40	C₈×C₆₁	—	2	8	no	—
41	C₈×C₆₁	—	8	4	no	—
42	C₈×C₆₁	#107	1	8	yes	—
43	#8826	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₆₁	—	2	2	no	—
45	C₂²×C₆₁	#384	1	1	yes	characteristic
46	C₂²×C₆₁	#500	1	2	yes	—
47	C₄×C₆₁	—	2	2	no	—
48	C₄×C₆₁	C₂×#107	1	2	yes	—
49	C₄×C₆₁	#430	1	4	yes	—
50	C₄×C₆₁	#493	1	1	yes	characteristic
51	C₄×C₆₁	#494	1	1	yes	characteristic
52	#1848	C₂×C₆₁	1	2	yes	—
53	#1567	C₂×C₆₁	1	4	yes	—
54	C₂×#322	C₂×C₆₁	1	1	yes	characteristic
55	C₂×C₆₁	—	2	2	no	—
56	C₂×C₆₁	#1845	1	1	yes	characteristic
57	C₂×C₆₁	#1849	1	2	yes	—
58	C₂×#108	—	2	4	no	—
59	#423	C₂²×C₆₁	1	4	yes	—
60	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
61	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
62	#335	—	2	8	no	—
63	#322	C₄×C₆₁	1	4	yes	—
64	C₆₁	#8826	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×M₁₆	—	2	4	no	—
66	C₂²×C₈	D₄×C₆₁	1	2	yes	—
67	C₂²×C₈	Q₈×C₆₁	1	4	yes	—
68	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
69	#108	—	4	8	no	—
70	C₄⋊C₈	—	2	8	no	—
71	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
72	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
73	C₄×C₈	D₄×C₆₁	1	4	yes	—
74	C₄×C₈	Q₈×C₆₁	1	2	yes	—
75	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
76	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
77	M₁₆	—	4	8	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	—	4	4	no	—
80	C₂×C₈	C₄⋊C₄×C₆₁	1	8	yes	—
81	C₂×C₈	C₄⋊C₄×C₆₁	1	4	yes	—
82	C₂×C₈	D₈×C₆₁	1	4	yes	—
83	C₂×C₈	Q₁₆×C₆₁	1	4	yes	—
84	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
85	C₄²	C₂×D₄×C₆₁	1	2	yes	—
86	C₄²	C₂×Q₈×C₆₁	1	1	yes	characteristic
87	C₂³	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
91	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
92	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
93	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
94	C₂×C₄	C₂×Q₁₆×C₆₁	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	—	8	4	no	—
97	C₈	#107×C₆₁	1	8	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	C₆₁×#384	1	1	yes	characteristic
100	C₂²	C₆₁×#500	1	2	yes	—
101	C₄	—	2	2	no	—
102	C₄	C₂×#107×C₆₁	1	2	yes	—
103	C₄	C₆₁×#430	1	4	yes	—
104	C₄	C₆₁×#493	1	1	yes	characteristic
105	C₄	C₆₁×#494	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	C₆₁×#1845	1	1	yes	characteristic
108	C₂	C₆₁×#1849	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#1848

C₂

yes

maximal

C₆₁×#1567

C₂

yes

maximal

C₂×C₆₁×#322

C₂

yes

maximal, characteristic

C₂×#108×C₆₁

—

C₆₁×#423

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#335

—

C₆₁×#322

C₄

yes

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

Q₈

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#108×C₆₁

—

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×Q₈

yes

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

Q₁₆

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₂³×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×C₄×C₆₁

C₂×Q₁₆

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#107

yes

—

#8826

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#384

yes

characteristic

C₂²×C₆₁

#500

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#107

yes

—

C₄×C₆₁

#430

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₂×C₆₁

yes

—

#1567

C₂×C₆₁

yes

—

C₂×#322

C₂×C₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#1845

yes

characteristic

C₂×C₆₁

#1849

yes

—

C₂×#108

—

#423

C₂²×C₆₁

yes

—

C₂×C₄×C₈

C₂²×C₆₁

yes

—

C₂×C₄×C₈

C₂²×C₆₁

yes

characteristic

#335

—

#322

C₄×C₆₁

yes

—

C₆₁

#8826

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

D₄×C₆₁

yes

—

C₂²×C₈

Q₈×C₆₁

yes

—

C₂×C₄²

C₂³×C₆₁

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₆₁

yes

—

C₄×C₈

C₂×C₄×C₆₁

yes

—

C₄×C₈

D₄×C₆₁

yes

—

C₄×C₈

Q₈×C₆₁

yes

—

C₂²×C₄

C₂×D₄×C₆₁

yes

characteristic

C₂²×C₄

C₂×Q₈×C₆₁

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₆₁

yes

—

C₂×C₈

C₄⋊C₄×C₆₁

yes

—

C₂×C₈

D₈×C₆₁

yes

—

C₂×C₈

Q₁₆×C₆₁

yes

—

C₄²

C₂²×C₄×C₆₁

yes

characteristic

C₄²

C₂×D₄×C₆₁

yes

—

C₄²

C₂×Q₈×C₆₁

yes

characteristic

C₂³

C₄⋊Q₈×C₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₆₁

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄×C₆₁

yes

—

C₂×C₄

#127×C₆₁

yes

characteristic

C₂×C₄

C₄⋊Q₈×C₆₁

yes

—

C₂×C₄

C₂×D₈×C₆₁

yes

—

C₂×C₄

C₂×Q₁₆×C₆₁

yes

—

C₈

—

C₈

—

C₈

#107×C₆₁

yes

—

C₂²

—

C₂²

C₆₁×#384

yes

characteristic

100

C₂²

C₆₁×#500

yes

—

101

C₄

—

102

C₄

C₂×#107×C₆₁

yes

—

103

C₄

C₆₁×#430

yes

—

104

C₄

C₆₁×#493

yes

characteristic

105

C₄

C₆₁×#494

yes

characteristic

106

C₂

—

107

C₂

C₆₁×#1845

yes

characteristic

108

C₂

C₆₁×#1849

yes

—

	subgroup	G/H
center	C₂×C₄×C₆₁	#127
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#384
prime radical	C₂³×C₆₁	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₁

#127

commutator

C₄²

C₂²×C₄×C₆₁

Frattini, cosocle

C₂×C₄²

C₂³×C₆₁

socle

C₂²×C₆₁

#384

prime radical

C₂³×C₆₁

C₄⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

N	H ≅ G/N	type
43. #8826	C₆₁	direct (N × H)
64. C₆₁	#8826	direct (N × H)

H ≅ G/N

type

43. #8826

C₆₁

direct (N × H)

64. C₆₁

#8826

direct (N × H)

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

subgroup	index
C₆₁×#1848	2
C₆₁×#1567	2
C₂×C₆₁×#322	2
#8826	61

subgroup

index

C₆₁×#1848

C₆₁×#1567

C₂×C₆₁×#322

#8826

Subgroups — C₆₁×#8826 #29512259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#8826 #29512259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8826 #29512259