Subgroups — #49089900

Order 35868.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁⋊#3862	C₂	1	1	yes	maximal, characteristic
2	C₇⋊(C₇⋊#6923)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₆₁⋊#3862	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇²×D₆₁	C₂×C₃	1	1	yes	characteristic
5	#19103980	—	7	1	no	maximal
6	#19103979	—	7	1	no	maximal
7	C₇²×C₆₁	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	#14579421	—	7	1	no	—
9	#14579420	—	7	1	no	—
10	C₇⋊#6923	—	7	1	no	—
11	C₇⋊#6923	—	7	1	no	—
12	C₇⋊#6923	—	21	2	no	—
13	#13046197	—	7	1	no	—
14	#13046196	—	7	1	no	—
15	C₇×D₆₁	—	3	2	no	—
16	C₇×D₆₁	AGL(1,𝔽₇)	1	1	yes	characteristic
17	C₇×D₆₁	AGL(1,𝔽₇)	1	1	yes	characteristic
18	#6923⋊C₃	—	49	1	no	Hall
19	#10596875	—	61	1	no	maximal, Hall
20	C₇×C₆₁	—	3	2	no	—
21	C₇×C₆₁	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
22	C₇×C₆₁	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
23	D₆₁⋊C₃	—	49	1	no	—
24	C₂×#3862	—	61	1	no	—
25	#6923	—	49	1	no	Hall
26	C₇⋊Dic₇	—	61	1	no	Hall
27	C₆₁⋊C₃	—	49	1	no	Hall
28	#3862	—	61	1	no	Hall
29	D₆₁	#64425	1	1	yes	characteristic
30	C₂×C₇²	—	61	1	no	—
31	ΓL(1,𝔽₈)⋊C₄	—	427	1	no	—
32	ΓL(1,𝔽₈)⋊C₄	—	427	1	no	—
33	C₆₁	#10596875	1	1	yes	characteristic, complemented, Sylow
34	C₇²	#6923⋊C₃	1	1	yes	characteristic, complemented, Sylow
35	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
36	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
37	Dic₇	—	1281	2	no	—
38	Dic₇	—	427	1	no	—
39	Dic₇	—	427	1	no	—
40	ΓL(1,𝔽₈)	—	427	1	no	—
41	ΓL(1,𝔽₈)	—	427	1	no	—
42	C₂×C₇	—	61	1	no	—
43	C₂×C₇	—	61	1	no	—
44	C₂×C₇	—	183	2	no	—
45	C₃×C₄	—	2989	1	no	Hall
46	C₇	—	3	2	no	—
47	C₇	#19103979	1	1	yes	characteristic, complemented
48	C₇	#19103980	1	1	yes	characteristic, complemented
49	C₂×C₃	—	2989	1	no	—
50	C₄	—	2989	1	no	Sylow
51	C₃	—	2989	1	no	Sylow
52	C₂	—	61	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇²×C₆₁	C₃×C₄
prime radical	D₆₁⋊#3862	C₂
cosocle	C₇²×D₆₁	C₂×C₃
LCS	G ▷ C₇²×C₆₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇⋊(C₇⋊#6923)	C₃	semidirect
3. C₆₁⋊#3862	C₄	semidirect
7. C₇²×C₆₁	C₃×C₄	semidirect
21. C₇×C₆₁	ΓL(1,𝔽₈)⋊C₄	semidirect
22. C₇×C₆₁	ΓL(1,𝔽₈)⋊C₄	semidirect
33. C₆₁	#10596875	semidirect
34. C₇²	#6923⋊C₃	semidirect
47. C₇	#19103979	semidirect
48. C₇	#19103980	semidirect

Maximal subgroups

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

subgroup	index
D₆₁⋊#3862	2
C₇⋊(C₇⋊#6923)	3
#19103980	7
#19103979	7
#10596875	61