Subgroups — C₄²×C₅×D₇ #11765029

Order 1120.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₇	C₂	1	3	yes	maximal
2	C₄²×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×Dic₇	C₂	1	3	yes	maximal, complemented
4	C₂²×C₅×D₇	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
6	C₂×C₅×Dic₇	C₂²	1	3	yes	—
7	C₄×C₅×D₇	C₄	1	12	yes	direct product factor
8	C₄²×D₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₄²×C₅	—	7	1	no	maximal, Hall
10	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
11	C₂×C₅×D₇	C₂×C₄	1	6	yes	—
12	C₄×C₅×C₇	C₂×C₄	1	6	yes	complemented
13	C₅×Dic₇	C₂×C₄	1	6	yes	complemented
14	C₂×C₄×D₇	C₂×C₅	1	3	yes	—
15	C₄²×C₇	C₂×C₅	1	1	yes	characteristic, complemented
16	C₄×Dic₇	C₂×C₅	1	3	yes	complemented
17	C₂²×C₄×C₅	—	7	3	no	—
18	C₄²×C₅	—	7	3	no	—
19	C₄²×C₅	D₇	1	1	yes	characteristic, direct product factor
20	C₂×C₅×C₇	C₂²×C₄	1	3	yes	—
21	C₅×D₇	C₄²	1	4	yes	direct product factor
22	C₂²×D₇	C₂²×C₅	1	1	yes	characteristic
23	C₂×C₄×C₇	C₂²×C₅	1	3	yes	—
24	C₂×Dic₇	C₂²×C₅	1	3	yes	—
25	C₄×D₇	C₄×C₅	1	12	yes	direct product factor
26	C₂³×C₅	—	7	1	no	—
27	C₂×C₄×C₅	—	7	3	no	—
28	C₂×C₄×C₅	—	7	12	no	—
29	C₂×C₄×C₅	C₂×D₇	1	3	yes	—
30	C₅×C₇	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
31	C₂×C₄²	—	7	1	no	Sylow
32	C₂²×C₇	C₂³×C₅	1	1	yes	characteristic
33	C₂×D₇	C₂×C₄×C₅	1	6	yes	—
34	C₄×C₇	C₂×C₄×C₅	1	6	yes	complemented
35	Dic₇	C₂×C₄×C₅	1	6	yes	complemented
36	C₂²×C₅	—	7	6	no	—
37	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
38	C₄×C₅	—	7	6	no	—
39	C₄×C₅	C₄×D₇	1	6	yes	direct product factor
40	C₂²×C₄	—	7	3	no	—
41	C₄²	—	7	3	no	—
42	C₄²	C₅×D₇	1	1	yes	characteristic, direct product factor
43	C₂×C₇	C₂²×C₄×C₅	1	3	yes	—
44	D₇	C₄²×C₅	1	4	yes	direct product factor
45	C₂×C₅	—	7	4	no	—
46	C₂×C₅	C₂×C₄×D₇	1	3	yes	—
47	C₂³	—	7	1	no	—
48	C₂×C₄	—	7	3	no	—
49	C₂×C₄	—	7	12	no	—
50	C₂×C₄	C₂×C₅×D₇	1	3	yes	—
51	C₇	C₂×C₄²×C₅	1	1	yes	characteristic, complemented, Sylow
52	C₅	C₄²×D₇	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	—	7	6	no	—
54	C₂²	C₂²×C₅×D₇	1	1	yes	characteristic
55	C₄	—	7	6	no	—
56	C₄	C₄×C₅×D₇	1	6	yes	direct product factor
57	C₂	—	7	4	no	—
58	C₂	C₂×C₄×C₅×D₇	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²×C₅	D₇
commutator, hypocenter	C₇	C₂×C₄²×C₅
Frattini	C₂²	C₂²×C₅×D₇
Fitting	C₄²×C₅×C₇	C₂
socle	C₂²×C₅×C₇	C₂³
prime radical	C₂²×C₅×D₇	C₂²
cosocle	C₂²×C₇	C₂³×C₅
LCS	G ▷ C₇
UCS	C₁ ◁ C₄²×C₅

Complements

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

N	H ≅ G/N	type
2. C₄²×C₅×C₇	C₂	semidirect
3. C₄×C₅×Dic₇	C₂	semidirect
7. C₄×C₅×D₇	C₄	direct (N × H)
8. C₄²×D₇	C₅	direct (N × H)
12. C₄×C₅×C₇	C₂×C₄	semidirect
13. C₅×Dic₇	C₂×C₄	semidirect
15. C₄²×C₇	C₂×C₅	semidirect
16. C₄×Dic₇	C₂×C₅	semidirect
19. C₄²×C₅	D₇	direct (N × H)
21. C₅×D₇	C₄²	direct (N × H)
25. C₄×D₇	C₄×C₅	direct (N × H)
30. C₅×C₇	C₂×C₄²	semidirect
34. C₄×C₇	C₂×C₄×C₅	semidirect
35. Dic₇	C₂×C₄×C₅	semidirect
39. C₄×C₅	C₄×D₇	direct (N × H)
42. C₄²	C₅×D₇	direct (N × H)
44. D₇	C₄²×C₅	direct (N × H)
51. C₇	C₂×C₄²×C₅	semidirect
52. C₅	C₄²×D₇	direct (N × H)
56. C₄	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
C₂×C₄×C₅×D₇	2
C₄²×C₅×C₇	2
C₄×C₅×Dic₇	2
C₄²×D₇	5
C₂×C₄²×C₅	7

Subgroups — C42×C5×D7 #11765029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₅×D₇ #11765029