Subgroups — C₄²×C₇×D₅ #11765115

Order 1120.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₅	C₂	1	3	yes	maximal
2	C₄²×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇×Dic₅	C₂	1	3	yes	maximal, complemented
4	C₂²×C₇×D₅	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
6	C₂×C₇×Dic₅	C₂²	1	3	yes	—
7	C₄×C₇×D₅	C₄	1	12	yes	direct product factor
8	C₂×C₄²×C₇	—	5	1	no	maximal, Hall
9	C₄²×D₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
11	C₂×C₇×D₅	C₂×C₄	1	6	yes	—
12	C₄×C₅×C₇	C₂×C₄	1	6	yes	complemented
13	C₇×Dic₅	C₂×C₄	1	6	yes	complemented
14	C₂²×C₄×C₇	—	5	3	no	—
15	C₄²×C₇	—	5	3	no	—
16	C₄²×C₇	D₅	1	1	yes	characteristic, direct product factor
17	C₂×C₄×D₅	C₂×C₇	1	3	yes	—
18	C₄²×C₅	C₂×C₇	1	1	yes	characteristic, complemented
19	C₄×Dic₅	C₂×C₇	1	3	yes	complemented
20	C₂×C₅×C₇	C₂²×C₄	1	3	yes	—
21	C₇×D₅	C₄²	1	4	yes	direct product factor
22	C₂³×C₇	—	5	1	no	—
23	C₂×C₄×C₇	—	5	3	no	—
24	C₂×C₄×C₇	—	5	12	no	—
25	C₂×C₄×C₇	C₂×D₅	1	3	yes	—
26	C₂²×D₅	C₂²×C₇	1	1	yes	characteristic
27	C₂×C₄×C₅	C₂²×C₇	1	3	yes	—
28	C₂×Dic₅	C₂²×C₇	1	3	yes	—
29	C₄×D₅	C₄×C₇	1	12	yes	direct product factor
30	C₅×C₇	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
31	C₂×C₄²	—	5	1	no	Sylow
32	C₂²×C₇	—	5	6	no	—
33	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
34	C₄×C₇	—	5	6	no	—
35	C₄×C₇	C₄×D₅	1	6	yes	direct product factor
36	C₂²×C₅	C₂³×C₇	1	1	yes	characteristic
37	C₂×D₅	C₂×C₄×C₇	1	6	yes	—
38	C₄×C₅	C₂×C₄×C₇	1	6	yes	complemented
39	Dic₅	C₂×C₄×C₇	1	6	yes	complemented
40	C₂²×C₄	—	5	3	no	—
41	C₄²	—	5	3	no	—
42	C₄²	C₇×D₅	1	1	yes	characteristic, direct product factor
43	C₂×C₇	—	5	4	no	—
44	C₂×C₇	C₂×C₄×D₅	1	3	yes	—
45	C₂×C₅	C₂²×C₄×C₇	1	3	yes	—
46	D₅	C₄²×C₇	1	4	yes	direct product factor
47	C₂³	—	5	1	no	—
48	C₂×C₄	—	5	3	no	—
49	C₂×C₄	—	5	12	no	—
50	C₂×C₄	C₂×C₇×D₅	1	3	yes	—
51	C₇	C₄²×D₅	1	1	yes	characteristic, direct product factor, Sylow
52	C₅	C₂×C₄²×C₇	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	5	6	no	—
54	C₂²	C₂²×C₇×D₅	1	1	yes	characteristic
55	C₄	—	5	6	no	—
56	C₄	C₄×C₇×D₅	1	6	yes	direct product factor
57	C₂	—	5	4	no	—
58	C₂	C₂×C₄×C₇×D₅	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²×C₇	D₅
commutator, hypocenter	C₅	C₂×C₄²×C₇
Frattini	C₂²	C₂²×C₇×D₅
Fitting	C₄²×C₅×C₇	C₂
socle	C₂²×C₅×C₇	C₂³
prime radical	C₂²×C₇×D₅	C₂²
cosocle	C₂²×C₅	C₂³×C₇
LCS	G ▷ C₅
UCS	C₁ ◁ C₄²×C₇

Complements

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

N	H ≅ G/N	type
2. C₄²×C₅×C₇	C₂	semidirect
3. C₄×C₇×Dic₅	C₂	semidirect
7. C₄×C₇×D₅	C₄	direct (N × H)
9. C₄²×D₅	C₇	direct (N × H)
12. C₄×C₅×C₇	C₂×C₄	semidirect
13. C₇×Dic₅	C₂×C₄	semidirect
16. C₄²×C₇	D₅	direct (N × H)
18. C₄²×C₅	C₂×C₇	semidirect
19. C₄×Dic₅	C₂×C₇	semidirect
21. C₇×D₅	C₄²	direct (N × H)
29. C₄×D₅	C₄×C₇	direct (N × H)
30. C₅×C₇	C₂×C₄²	semidirect
35. C₄×C₇	C₄×D₅	direct (N × H)
38. C₄×C₅	C₂×C₄×C₇	semidirect
39. Dic₅	C₂×C₄×C₇	semidirect
42. C₄²	C₇×D₅	direct (N × H)
46. D₅	C₄²×C₇	direct (N × H)
51. C₇	C₄²×D₅	direct (N × H)
52. C₅	C₂×C₄²×C₇	semidirect
56. C₄	C₄×C₇×D₅	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×C₇×D₅	2
C₄²×C₅×C₇	2
C₄×C₇×Dic₅	2
C₂×C₄²×C₇	5
C₄²×D₅	7

Subgroups — C42×C7×D5 #11765115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₇×D₅ #11765115