Group #29512258 — subgroups

Subgroups — C₆₁×#8825 #29512258

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×#1566	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×#334	—	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	SD₁₆	1	8	yes	—
28	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₆₁	C₂×D₄	1	2	yes	—
30	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
31	C₂³×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
32	C₂×C₄×C₆₁	—	2	4	no	—
33	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
38	C₈×C₆₁	—	2	8	no	—
39	C₈×C₆₁	—	8	4	no	—
40	C₈×C₆₁	#106	1	8	yes	—
41	#8825	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂²×C₆₁	—	2	2	no	—
43	C₂²×C₆₁	#384	1	1	yes	characteristic
44	C₂²×C₆₁	#498	1	2	yes	—
45	C₄×C₆₁	—	2	2	no	—
46	C₄×C₆₁	C₂×#106	1	2	yes	—
47	C₄×C₆₁	#430	1	4	yes	—
48	C₄×C₆₁	#492	1	2	yes	—
49	#1848	C₂×C₆₁	1	2	yes	—
50	#1566	C₂×C₆₁	1	4	yes	—
51	C₂×#322	C₂×C₆₁	1	1	yes	characteristic
52	C₂×C₆₁	—	2	2	no	—
53	C₂×C₆₁	#1844	1	1	yes	characteristic
54	C₂×C₆₁	#1849	1	2	yes	—
55	C₂×#108	—	2	4	no	—
56	#423	C₂²×C₆₁	1	4	yes	—
57	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
58	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
59	#334	—	2	8	no	—
60	#322	C₄×C₆₁	1	4	yes	—
61	C₆₁	#8825	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×M₁₆	—	2	4	no	—
63	C₂²×C₈	D₄×C₆₁	1	2	yes	—
64	C₂²×C₈	Q₈×C₆₁	1	4	yes	—
65	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
66	#108	—	4	8	no	—
67	C₄⋊C₈	—	2	8	no	—
68	C₄×C₈	C₂×C₄×C₆₁	1	4	yes	—
69	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
70	C₄×C₈	D₄×C₆₁	1	4	yes	—
71	C₄×C₈	Q₈×C₆₁	1	2	yes	—
72	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
73	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
74	M₁₆	—	4	8	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	4	4	no	—
77	C₂×C₈	C₄⋊C₄×C₆₁	1	8	yes	—
78	C₂×C₈	C₄⋊C₄×C₆₁	1	4	yes	—
79	C₂×C₈	SD₁₆×C₆₁	1	8	yes	—
80	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
81	C₄²	C₂×D₄×C₆₁	1	2	yes	—
82	C₄²	C₂×Q₈×C₆₁	1	1	yes	characteristic
83	C₂³	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
86	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
87	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
89	C₂×C₄	C₂×SD₁₆×C₆₁	1	4	yes	—
90	C₈	—	2	8	no	—
91	C₈	—	8	4	no	—
92	C₈	#106×C₆₁	1	8	yes	—
93	C₂²	—	2	2	no	—
94	C₂²	C₆₁×#384	1	1	yes	characteristic
95	C₂²	C₆₁×#498	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	C₂×#106×C₆₁	1	2	yes	—
98	C₄	C₆₁×#430	1	4	yes	—
99	C₄	C₆₁×#492	1	2	yes	—
100	C₂	—	2	2	no	—
101	C₂	C₆₁×#1844	1	1	yes	characteristic
102	C₂	C₆₁×#1849	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#1848

C₂

yes

maximal

C₆₁×#1566

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×#334

—

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

SD₁₆

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×SD₁₆

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#106

yes

—

#8825

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#384

yes

characteristic

C₂²×C₆₁

#498

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#106

yes

—

C₄×C₆₁

#430

yes

—

C₄×C₆₁

#492

yes

—

#1848

C₂×C₆₁

yes

—

#1566

C₂×C₆₁

yes

—

C₂×#322

C₂×C₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#1844

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

#334

—

#322

C₄×C₆₁

yes

—

C₆₁

#8825

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

SD₁₆×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×SD₁₆×C₆₁

yes

—

C₈

—

C₈

—

C₈

#106×C₆₁

yes

—

C₂²

—

C₂²

C₆₁×#384

yes

characteristic

C₂²

C₆₁×#498

yes

—

C₄

—

C₄

C₂×#106×C₆₁

yes

—

C₄

C₆₁×#430

yes

—

C₄

C₆₁×#492

yes

—

100

C₂

—

101

C₂

C₆₁×#1844

yes

characteristic

102

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
41. #8825	C₆₁	direct (N × H)
61. C₆₁	#8825	direct (N × H)

H ≅ G/N

type

41. #8825

C₆₁

direct (N × H)

61. C₆₁

#8825

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×#1566	2
C₂×C₆₁×#322	2
#8825	61

subgroup

index

C₆₁×#1848

C₆₁×#1566

C₂×C₆₁×#322

#8825

Subgroups — C₆₁×#8825 #29512258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#8825 #29512258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8825 #29512258