Subgroups — D₆₁×#64424 #49089866

Order 35868.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁×#3861	C₂	1	1	yes	maximal, characteristic, complemented
2	#31822443	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#64424	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇⋊D₇×D₆₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₆₁×#3861	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇⋊D₄₂₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₆₁×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇²×D₆₁	C₂×C₃	1	1	yes	characteristic, complemented
9	AGL(1,𝔽₇)×D₆₁	—	7	8	no	maximal
10	C₇²×C₆₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
11	ΓL(1,𝔽₈)⋊D₆₁	—	7	8	no	—
12	AGL(1,𝔽₇)×C₆₁	—	7	8	no	—
13	ΓL(1,𝔽₈)×D₆₁	—	7	8	no	—
14	D₇×D₆₁	—	7	8	no	—
15	ΓL(1,𝔽₈)×C₆₁	—	7	8	no	—
16	D₄₂₇	—	7	8	no	—
17	C₇×D₆₁	AGL(1,𝔽₇)	1	8	yes	complemented
18	D₇×C₆₁	—	7	8	no	—
19	C₂×C₃×D₆₁	—	49	1	no	Hall
20	C₂×#64424	—	61	1	no	maximal, Hall
21	C₇×C₆₁	C₂×AGL(1,𝔽₇)	1	8	yes	complemented
22	C₂×C₃×C₆₁	—	49	1	no	—
23	C₃×D₆₁	—	49	1	no	—
24	C₃×D₆₁	—	49	1	no	—
25	C₂×#3861	—	61	1	no	—
26	#64424	—	61	1	no	—
27	#64424	D₆₁	1	1	yes	characteristic, direct product factor
28	C₂×D₆₁	—	49	1	no	Hall
29	C₂×C₇⋊D₇	—	61	1	no	Hall
30	C₃×C₆₁	—	49	1	no	Hall
31	#3861	C₂×D₆₁	1	1	yes	characteristic, complemented, Hall
32	C₂×C₆₁	—	49	1	no	—
33	D₆₁	—	49	1	no	—
34	D₆₁	#64424	1	1	yes	characteristic, direct product factor
35	C₂×C₇²	—	61	1	no	—
36	C₇⋊D₇	—	61	1	no	—
37	C₇⋊D₇	C₃×D₆₁	1	1	yes	characteristic, complemented
38	C₂×AGL(1,𝔽₇)	—	427	8	no	—
39	C₆₁	C₂×#64424	1	1	yes	characteristic, complemented, Sylow
40	C₇²	C₂×C₃×D₆₁	1	1	yes	characteristic, complemented, Sylow
41	C₂×ΓL(1,𝔽₈)	—	427	8	no	—
42	AGL(1,𝔽₇)	—	7	8	no	—
43	AGL(1,𝔽₇)	—	427	8	no	—
44	C₂×D₇	—	427	8	no	—
45	ΓL(1,𝔽₈)	—	7	8	no	—
46	C₂×C₇	—	61	8	no	—
47	D₇	—	7	8	no	—
48	D₇	—	427	8	no	—
49	C₂²×C₃	—	2989	1	no	Hall
50	C₇	AGL(1,𝔽₇)×D₆₁	1	8	yes	complemented
51	C₂×C₃	—	49	1	no	—
52	C₂×C₃	—	2989	1	no	—
53	C₂×C₃	—	2989	1	no	—
54	C₂²	—	2989	1	no	Sylow
55	C₃	—	49	1	no	Sylow
56	C₂	—	49	1	no	—
57	C₂	—	61	1	no	—
58	C₂	—	2989	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₆₁	C₂²×C₃
LCS	G ▷ C₇²×C₆₁
UCS	C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. D₆₁×#3861	C₂	semidirect
2. #31822443	C₂	semidirect
3. C₆₁×#64424	C₂	semidirect
4. C₇⋊D₇×D₆₁	C₃	semidirect
5. C₆₁×#3861	C₂²	semidirect
6. C₇⋊D₄₂₇	C₂×C₃	semidirect
7. C₆₁×C₇⋊D₇	C₂×C₃	semidirect
8. C₇²×D₆₁	C₂×C₃	semidirect
10. C₇²×C₆₁	C₂²×C₃	semidirect
17. C₇×D₆₁	AGL(1,𝔽₇)	semidirect
21. C₇×C₆₁	C₂×AGL(1,𝔽₇)	semidirect
27. #64424	D₆₁	direct (N × H)
31. #3861	C₂×D₆₁	semidirect
34. D₆₁	#64424	direct (N × H)
37. C₇⋊D₇	C₃×D₆₁	semidirect
39. C₆₁	C₂×#64424	semidirect
40. C₇²	C₂×C₃×D₆₁	semidirect
50. C₇	AGL(1,𝔽₇)×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
D₆₁×#3861	2
#31822443	2
C₆₁×#64424	2
C₇⋊D₇×D₆₁	3
AGL(1,𝔽₇)×D₆₁	7
C₂×#64424	61

Subgroups — D61×#64424 #49089866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁×#64424 #49089866