Subgroups — C₂×C₄×C₅×D₇ #10587965

Order 560.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×D₇	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₅×D₇	C₄	1	4	yes	direct product factor
7	C₂×C₅×D₇	C₂²	1	2	yes	—
8	C₄×C₅×C₇	C₂²	1	2	yes	complemented
9	C₅×Dic₇	C₂²	1	2	yes	complemented
10	C₂×C₄×D₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₄×C₅	—	7	1	no	maximal, Hall
12	C₂×C₅×C₇	C₂×C₄	1	2	yes	complemented
13	C₂×C₅×C₇	C₂³	1	1	yes	characteristic
14	C₅×D₇	C₂×C₄	1	4	yes	direct product factor
15	C₂²×D₇	C₂×C₅	1	1	yes	characteristic
16	C₂×C₄×C₇	C₂×C₅	1	1	yes	characteristic, complemented
17	C₂×Dic₇	C₂×C₅	1	1	yes	characteristic, complemented
18	C₄×D₇	C₂×C₅	1	4	yes	direct product factor
19	C₂³×C₅	—	7	1	no	—
20	C₂×C₄×C₅	—	7	1	no	—
21	C₂×C₄×C₅	—	7	4	no	—
22	C₂×C₄×C₅	D₇	1	1	yes	characteristic, direct product factor
23	C₅×C₇	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
24	C₂²×C₇	C₂²×C₅	1	1	yes	characteristic
25	C₂×D₇	C₄×C₅	1	4	yes	direct product factor
26	C₂×D₇	C₂²×C₅	1	2	yes	—
27	C₄×C₇	C₂²×C₅	1	2	yes	complemented
28	Dic₇	C₂²×C₅	1	2	yes	complemented
29	C₂²×C₅	—	7	4	no	—
30	C₂²×C₅	—	7	2	no	—
31	C₂²×C₅	C₂×D₇	1	1	yes	characteristic
32	C₄×C₅	—	7	2	no	—
33	C₄×C₅	C₂×D₇	1	2	yes	direct product factor
34	C₂²×C₄	—	7	1	no	Sylow
35	C₂×C₇	C₂×C₄×C₅	1	2	yes	complemented
36	C₂×C₇	C₂³×C₅	1	1	yes	characteristic
37	D₇	C₂×C₄×C₅	1	4	yes	direct product factor
38	C₂×C₅	—	7	4	no	—
39	C₂×C₅	C₄×D₇	1	2	yes	direct product factor
40	C₂×C₅	C₂²×D₇	1	1	yes	characteristic
41	C₂³	—	7	1	no	—
42	C₂×C₄	—	7	1	no	—
43	C₂×C₄	—	7	4	no	—
44	C₂×C₄	C₅×D₇	1	1	yes	characteristic, direct product factor
45	C₇	C₂²×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
46	C₅	C₂×C₄×D₇	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²	—	7	4	no	—
48	C₂²	—	7	2	no	—
49	C₂²	C₂×C₅×D₇	1	1	yes	characteristic
50	C₄	—	7	2	no	—
51	C₄	C₂×C₅×D₇	1	2	yes	direct product factor
52	C₂	—	7	4	no	—
53	C₂	C₄×C₅×D₇	1	2	yes	direct product factor
54	C₂	C₂²×C₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₅	D₇
commutator, hypocenter	C₇	C₂²×C₄×C₅
Frattini	C₂	C₂²×C₅×D₇
Fitting	C₂×C₄×C₅×C₇	C₂
socle	C₂²×C₅×C₇	C₂²
prime radical	C₂²×C₅×D₇	C₂
cosocle	C₂×C₇	C₂³×C₅
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×C₇	C₂	semidirect
3. C₂×C₅×Dic₇	C₂	semidirect
4. C₄×C₅×D₇	C₂	direct (N × H)
6. C₂×C₅×D₇	C₄	direct (N × H)
8. C₄×C₅×C₇	C₂²	semidirect
9. C₅×Dic₇	C₂²	semidirect
10. C₂×C₄×D₇	C₅	direct (N × H)
12. C₂×C₅×C₇	C₂×C₄	semidirect
14. C₅×D₇	C₂×C₄	direct (N × H)
16. C₂×C₄×C₇	C₂×C₅	semidirect
17. C₂×Dic₇	C₂×C₅	semidirect
18. C₄×D₇	C₂×C₅	direct (N × H)
22. C₂×C₄×C₅	D₇	direct (N × H)
23. C₅×C₇	C₂²×C₄	semidirect
25. C₂×D₇	C₄×C₅	direct (N × H)
27. C₄×C₇	C₂²×C₅	semidirect
28. Dic₇	C₂²×C₅	semidirect
33. C₄×C₅	C₂×D₇	direct (N × H)
35. C₂×C₇	C₂×C₄×C₅	semidirect
37. D₇	C₂×C₄×C₅	direct (N × H)
39. C₂×C₅	C₄×D₇	direct (N × H)
44. C₂×C₄	C₅×D₇	direct (N × H)
45. C₇	C₂²×C₄×C₅	semidirect
46. C₅	C₂×C₄×D₇	direct (N × H)
51. C₄	C₂×C₅×D₇	direct (N × H)
53. C₂	C₄×C₅×D₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₇	2
C₂×C₄×C₅×C₇	2
C₂×C₅×Dic₇	2
C₄×C₅×D₇	2
C₂×C₄×D₇	5
C₂²×C₄×C₅	7

Subgroups — C2×C4×C5×D7 #10587965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×D₇ #10587965