Subgroups — C₇×#17987954 #45657396

Order 32634.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#14556422	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₇×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₇²×C₃₇⋊C₃	C₂×C₃	1	1	yes	characteristic
4	C₂×C₇×#66526	—	7	1	no	maximal
5	#17987954	C₇	1	1	yes	maximal, characteristic, direct product factor
6	C₇×D₇×C₃₇	C₉	1	1	yes	characteristic, complemented, Hall
7	#14556422	C₂×C₇	1	1	yes	characteristic, complemented
8	C₇×#66526	—	7	1	no	—
9	C₇²×C₃₇	C₂×C₉	1	1	yes	characteristic, complemented, Hall
10	C₂×C₇×C₃₇⋊C₃	—	7	1	no	—
11	D₇×C₃₇⋊C₃	C₃×C₇	1	1	yes	characteristic
12	C₇×D₇⋊C₉	—	37	1	no	maximal, Hall
13	C₇×C₃₇⋊C₃	—	6	1	no	—
14	C₇×C₃₇⋊C₃	AGL(1,𝔽₇)	1	1	yes	characteristic
15	C₇×C₃₇⋊C₃	C₂×C₃×C₇	1	1	yes	characteristic
16	C₂×#66526	—	7	1	no	Hall
17	C₂×C₇×C₃₇	—	7	1	no	—
18	D₇×C₃₇	C₇×C₉	1	1	yes	characteristic, complemented
19	C₇×C₇⋊C₉	—	37	1	no	Hall
20	#66526	—	7	1	no	Hall
21	C₃×C₇×D₇	—	37	1	no	—
22	C₇×C₃₇	—	6	1	no	—
23	C₇×C₃₇	D₇⋊C₉	1	1	yes	characteristic, complemented
24	C₇×C₃₇	C₂×C₇×C₉	1	1	yes	characteristic, complemented
25	C₂×C₃₇⋊C₃	—	7	1	no	—
26	C₃×C₇²	—	37	1	no	—
27	C₂×C₇×C₉	—	259	1	no	—
28	D₇⋊C₉	—	37	1	no	—
29	C₃₇⋊C₃	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic
30	C₇×D₇	#66526	1	1	yes	characteristic, complemented, Hall
31	C₂×C₃₇	—	7	1	no	Hall
32	C₇×C₉	—	259	1	no	—
33	C₇⋊C₉	—	37	1	no	—
34	C₇²	C₂×#66526	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₃×C₇	—	259	1	no	—
36	C₃×D₇	—	37	1	no	—
37	C₃₇	C₇×D₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
38	C₃×C₇	—	37	1	no	—
39	C₃×C₇	—	37	1	no	—
40	C₃×C₇	—	222	1	no	—
41	C₂×C₉	—	259	1	no	Hall
42	C₂×C₇	—	7	1	no	—
43	D₇	C₇×#66526	1	1	yes	characteristic, complemented
44	C₉	—	259	1	no	Sylow
45	C₇	—	6	1	no	—
46	C₇	#17987954	1	1	yes	characteristic, direct product factor
47	C₇	C₂×C₇×#66526	1	1	yes	characteristic, complemented
48	C₂×C₃	—	259	1	no	—
49	C₃	—	37	1	no	—
50	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	#17987954
commutator, hypocenter	C₇×C₃₇	C₂×C₇×C₉
Frattini	C₁	G
Fitting, socle	C₇²×C₃₇	C₂×C₉
prime radical	C₇×D₇×C₃₇⋊C₃	C₃
cosocle	C₇×C₃₇⋊C₃	C₂×C₃×C₇
LCS	G ▷ C₇×C₃₇
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₇×#14556422	C₂	semidirect
5. #17987954	C₇	direct (N × H)
6. C₇×D₇×C₃₇	C₉	semidirect
7. #14556422	C₂×C₇	semidirect
9. C₇²×C₃₇	C₂×C₉	semidirect
18. D₇×C₃₇	C₇×C₉	semidirect
23. C₇×C₃₇	D₇⋊C₉	semidirect
24. C₇×C₃₇	C₂×C₇×C₉	semidirect
30. C₇×D₇	#66526	semidirect
34. C₇²	C₂×#66526	semidirect
37. C₃₇	C₇×D₇⋊C₉	semidirect
43. D₇	C₇×#66526	semidirect
46. C₇	#17987954	direct (N × H)
47. C₇	C₂×C₇×#66526	semidirect

Maximal subgroups

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

subgroup	index
C₇×#14556422	2
C₇×D₇×C₃₇⋊C₃	3
C₂×C₇×#66526	7
#17987954	7
C₇×D₇⋊C₉	37

Subgroups — C7×#17987954 #45657396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17987954 #45657396