Subgroups — C₂²×C₅×D₂₈ #11765423

Order 1120.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×D₇	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×D₂₈	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₅×C₇	C₂²	1	1	yes	characteristic
5	C₂²×C₅×D₇	—	2	8	no	—
6	C₂²×C₅×D₇	C₂²	1	12	yes	complemented
7	C₂×C₄×C₅×C₇	C₂²	1	6	yes	complemented
8	C₅×D₂₈	C₂²	1	16	yes	direct product factor
9	C₂²×D₂₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₅×D₄	—	7	1	no	maximal, Hall
11	C₂²×C₅×C₇	D₄	1	4	yes	complemented
12	C₂²×C₅×C₇	C₂³	1	3	yes	—
13	C₂×C₅×D₇	—	2	24	no	—
14	C₂×C₅×D₇	C₂³	1	8	yes	complemented
15	C₄×C₅×C₇	C₂³	1	4	yes	complemented
16	C₂³×D₇	C₂×C₅	1	2	yes	complemented
17	C₂²×C₄×C₇	C₂×C₅	1	1	yes	characteristic, complemented
18	C₂×D₂₈	C₂×C₅	1	12	yes	direct product factor
19	C₂⁴×C₅	—	7	2	no	—
20	C₂×C₅×D₄	—	7	12	no	—
21	C₂²×C₄×C₅	D₇	1	1	yes	characteristic, complemented
22	C₂×C₅×C₇	C₂×D₄	1	6	yes	complemented
23	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
24	C₅×D₇	—	2	8	no	—
25	C₂³×C₇	C₂²×C₅	1	1	yes	characteristic
26	C₂²×D₇	—	2	8	no	—
27	C₂²×D₇	C₂²×C₅	1	12	yes	complemented
28	C₂×C₄×C₇	C₂²×C₅	1	6	yes	complemented
29	D₂₈	C₂²×C₅	1	16	yes	direct product factor
30	C₂³×C₅	—	7	12	no	—
31	C₂³×C₅	—	14	8	no	—
32	C₂³×C₅	C₂×D₇	1	1	yes	characteristic
33	C₅×D₄	—	7	16	no	—
34	C₂×C₄×C₅	C₂×D₇	1	6	yes	complemented
35	C₅×C₇	C₂²×D₄	1	1	yes	characteristic, complemented, Hall
36	C₂²×D₄	—	7	1	no	Sylow
37	C₂²×C₇	C₅×D₄	1	4	yes	complemented
38	C₂²×C₇	C₂³×C₅	1	3	yes	—
39	C₂×D₇	—	2	24	no	—
40	C₂×D₇	C₂³×C₅	1	8	yes	complemented
41	C₄×C₇	C₂³×C₅	1	4	yes	complemented
42	C₂²×C₅	—	7	8	no	—
43	C₂²×C₅	—	14	24	no	—
44	C₂²×C₅	D₂₈	1	4	yes	direct product factor
45	C₂²×C₅	C₂²×D₇	1	3	yes	—
46	C₄×C₅	C₂²×D₇	1	4	yes	complemented
47	C₂⁴	—	7	2	no	—
48	C₂×D₄	—	7	12	no	—
49	C₂²×C₄	C₅×D₇	1	1	yes	characteristic, complemented
50	C₂×C₇	C₂×C₅×D₄	1	6	yes	complemented
51	C₂×C₇	C₂⁴×C₅	1	1	yes	characteristic
52	D₇	—	2	8	no	—
53	C₂×C₅	—	14	8	no	—
54	C₂×C₅	C₂×D₂₈	1	6	yes	direct product factor
55	C₂×C₅	C₂³×D₇	1	1	yes	characteristic
56	C₂³	—	7	12	no	—
57	C₂³	—	14	8	no	—
58	C₂³	C₂×C₅×D₇	1	1	yes	characteristic
59	D₄	—	7	16	no	—
60	C₂×C₄	C₂×C₅×D₇	1	6	yes	complemented
61	C₇	C₂²×C₅×D₄	1	1	yes	characteristic, complemented, Sylow
62	C₅	C₂²×D₂₈	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²	—	7	8	no	—
64	C₂²	—	14	24	no	—
65	C₂²	C₅×D₂₈	1	4	yes	direct product factor
66	C₂²	C₂²×C₅×D₇	1	3	yes	—
67	C₄	C₂²×C₅×D₇	1	4	yes	complemented
68	C₂	—	14	8	no	—
69	C₂	C₂×C₅×D₂₈	1	6	yes	direct product factor
70	C₂	C₂³×C₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×D₇
commutator, cosocle	C₂×C₇	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×D₇
Fitting	C₂²×C₄×C₅×C₇	C₂
socle	C₂³×C₅×C₇	C₂²
hypercenter	C₂²×C₄×C₅	D₇
hypocenter	C₇	C₂²×C₅×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅

Complements

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

N	H ≅ G/N	type
1. C₂³×C₅×D₇	C₂	semidirect
2. C₂²×C₄×C₅×C₇	C₂	semidirect
3. C₂×C₅×D₂₈	C₂	direct (N × H)
6. C₂²×C₅×D₇	C₂²	semidirect
7. C₂×C₄×C₅×C₇	C₂²	semidirect
8. C₅×D₂₈	C₂²	direct (N × H)
9. C₂²×D₂₈	C₅	direct (N × H)
11. C₂²×C₅×C₇	D₄	semidirect
14. C₂×C₅×D₇	C₂³	semidirect
15. C₄×C₅×C₇	C₂³	semidirect
16. C₂³×D₇	C₂×C₅	semidirect
17. C₂²×C₄×C₇	C₂×C₅	semidirect
18. C₂×D₂₈	C₂×C₅	direct (N × H)
21. C₂²×C₄×C₅	D₇	semidirect
22. C₂×C₅×C₇	C₂×D₄	semidirect
27. C₂²×D₇	C₂²×C₅	semidirect
28. C₂×C₄×C₇	C₂²×C₅	semidirect
29. D₂₈	C₂²×C₅	direct (N × H)
34. C₂×C₄×C₅	C₂×D₇	semidirect
35. C₅×C₇	C₂²×D₄	semidirect
37. C₂²×C₇	C₅×D₄	semidirect
40. C₂×D₇	C₂³×C₅	semidirect
41. C₄×C₇	C₂³×C₅	semidirect
44. C₂²×C₅	D₂₈	direct (N × H)
46. C₄×C₅	C₂²×D₇	semidirect
49. C₂²×C₄	C₅×D₇	semidirect
50. C₂×C₇	C₂×C₅×D₄	semidirect
54. C₂×C₅	C₂×D₂₈	direct (N × H)
60. C₂×C₄	C₂×C₅×D₇	semidirect
61. C₇	C₂²×C₅×D₄	semidirect
62. C₅	C₂²×D₂₈	direct (N × H)
65. C₂²	C₅×D₂₈	direct (N × H)
67. C₄	C₂²×C₅×D₇	semidirect
69. C₂	C₂×C₅×D₂₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×D₇	2
C₂²×C₄×C₅×C₇	2
C₂×C₅×D₂₈	2
C₂²×D₂₈	5
C₂²×C₅×D₄	7

Subgroups — C22×C5×D28 #11765423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×D₂₈ #11765423