Subgroups — C₂×C₃×C₅×D₂₈ #13132832

Order 1680.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×D₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×D₂₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×D₂₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₅×D₇	—	2	4	no	—
7	C₂×C₃×C₅×D₇	C₂²	1	4	yes	complemented
8	C₃×C₄×C₅×C₇	C₂²	1	2	yes	complemented
9	C₂×C₃×D₂₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₅×D₇	C₂×C₃	1	2	yes	complemented
11	C₂×C₄×C₅×C₇	C₂×C₃	1	1	yes	characteristic, complemented
12	C₅×D₂₈	C₂×C₃	1	4	yes	direct product factor
13	C₂×C₃×C₅×D₄	—	7	1	no	maximal, Hall
14	C₂×C₃×C₅×C₇	D₄	1	2	yes	complemented
15	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
16	C₃×C₅×D₇	—	2	4	no	—
17	C₂²×C₃×D₇	C₂×C₅	1	2	yes	complemented
18	C₂×C₃×C₄×C₇	C₂×C₅	1	1	yes	characteristic, complemented
19	C₃×D₂₈	C₂×C₅	1	4	yes	direct product factor
20	C₂²×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
21	C₂×C₅×D₇	—	2	4	no	—
22	C₂×C₅×D₇	C₂²×C₃	1	4	yes	complemented
23	C₄×C₅×C₇	C₂²×C₃	1	2	yes	complemented
24	C₂³×C₃×C₅	—	7	2	no	—
25	C₃×C₅×D₄	—	7	4	no	—
26	C₂×C₃×C₄×C₅	D₇	1	1	yes	characteristic, complemented
27	C₂×D₂₈	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
28	C₃×C₅×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×C₃×C₇	C₂²×C₅	1	1	yes	characteristic
30	C₂×C₃×D₇	—	2	4	no	—
31	C₂×C₃×D₇	C₂²×C₅	1	4	yes	complemented
32	C₃×C₄×C₇	C₂²×C₅	1	2	yes	complemented
33	C₂×C₅×D₄	—	7	1	no	Hall
34	C₂×C₅×C₇	C₃×D₄	1	2	yes	complemented
35	C₂×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
36	C₅×D₇	—	2	4	no	—
37	C₂²×C₃×C₅	—	7	4	no	—
38	C₂²×C₃×C₅	—	14	4	no	—
39	C₂²×C₃×C₅	C₂×D₇	1	1	yes	characteristic
40	C₃×C₄×C₅	C₂×D₇	1	2	yes	complemented
41	C₂²×D₇	C₂×C₃×C₅	1	2	yes	complemented
42	C₂×C₄×C₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
43	D₂₈	C₂×C₃×C₅	1	4	yes	direct product factor
44	C₂×C₃×D₄	—	7	1	no	Hall
45	C₂×C₃×C₇	C₅×D₄	1	2	yes	complemented
46	C₂×C₃×C₇	C₂³×C₅	1	1	yes	characteristic
47	C₃×D₇	—	2	4	no	—
48	C₂³×C₅	—	7	2	no	—
49	C₅×D₄	—	7	4	no	—
50	C₂×C₄×C₅	C₃×D₇	1	1	yes	characteristic, complemented
51	C₅×C₇	C₂×C₃×D₄	1	1	yes	characteristic, complemented, Hall
52	C₂×C₃×C₅	—	14	4	no	—
53	C₂×C₃×C₅	D₂₈	1	2	yes	direct product factor
54	C₂×C₃×C₅	C₂²×D₇	1	1	yes	characteristic
55	C₂²×C₇	C₂²×C₃×C₅	1	1	yes	characteristic
56	C₂×D₇	—	2	4	no	—
57	C₂×D₇	C₂²×C₃×C₅	1	4	yes	complemented
58	C₄×C₇	C₂²×C₃×C₅	1	2	yes	complemented
59	C₂³×C₃	—	7	2	no	—
60	C₃×D₄	—	7	4	no	—
61	C₂×C₃×C₄	C₅×D₇	1	1	yes	characteristic, complemented
62	C₃×C₇	C₂×C₅×D₄	1	1	yes	characteristic, complemented, Hall
63	C₂²×C₅	—	7	4	no	—
64	C₂²×C₅	—	14	4	no	—
65	C₂²×C₅	C₂×C₃×D₇	1	1	yes	characteristic
66	C₄×C₅	C₂×C₃×D₇	1	2	yes	complemented
67	C₂×D₄	—	7	1	no	Sylow
68	C₃×C₅	C₂×D₂₈	1	1	yes	characteristic, direct product factor, Hall
69	C₂×C₇	C₃×C₅×D₄	1	2	yes	complemented
70	C₂×C₇	C₂³×C₃×C₅	1	1	yes	characteristic
71	D₇	—	2	4	no	—
72	C₂²×C₃	—	7	4	no	—
73	C₂²×C₃	—	14	4	no	—
74	C₂²×C₃	C₂×C₅×D₇	1	1	yes	characteristic
75	C₃×C₄	C₂×C₅×D₇	1	2	yes	complemented
76	C₂×C₅	—	14	4	no	—
77	C₂×C₅	C₃×D₂₈	1	2	yes	direct product factor
78	C₂×C₅	C₂²×C₃×D₇	1	1	yes	characteristic
79	C₂³	—	7	2	no	—
80	D₄	—	7	4	no	—
81	C₂×C₄	C₃×C₅×D₇	1	1	yes	characteristic, complemented
82	C₇	C₂×C₃×C₅×D₄	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	—	14	4	no	—
84	C₂×C₃	C₅×D₂₈	1	2	yes	direct product factor
85	C₂×C₃	C₂²×C₅×D₇	1	1	yes	characteristic
86	C₅	C₂×C₃×D₂₈	1	1	yes	characteristic, direct product factor, Sylow
87	C₂²	—	7	4	no	—
88	C₂²	—	14	4	no	—
89	C₂²	C₂×C₃×C₅×D₇	1	1	yes	characteristic
90	C₄	C₂×C₃×C₅×D₇	1	2	yes	complemented
91	C₃	C₂×C₅×D₂₈	1	1	yes	characteristic, direct product factor, Sylow
92	C₂	—	14	4	no	—
93	C₂	C₃×C₅×D₂₈	1	2	yes	direct product factor
94	C₂	C₂²×C₃×C₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₅	C₂×D₇
commutator, cosocle	C₂×C₇	C₂³×C₃×C₅
Frattini	C₂	C₂²×C₃×C₅×D₇
Fitting	C₂×C₃×C₄×C₅×C₇	C₂
socle	C₂²×C₃×C₅×C₇	C₂²
hypercenter	C₂×C₃×C₄×C₅	D₇
hypocenter	C₇	C₂×C₃×C₅×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄×C₅

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×D₇	C₂	semidirect
2. C₂×C₃×C₄×C₅×C₇	C₂	semidirect
3. C₃×C₅×D₂₈	C₂	direct (N × H)
4. C₂×C₅×D₂₈	C₃	direct (N × H)
7. C₂×C₃×C₅×D₇	C₂²	semidirect
8. C₃×C₄×C₅×C₇	C₂²	semidirect
9. C₂×C₃×D₂₈	C₅	direct (N × H)
10. C₂²×C₅×D₇	C₂×C₃	semidirect
11. C₂×C₄×C₅×C₇	C₂×C₃	semidirect
12. C₅×D₂₈	C₂×C₃	direct (N × H)
14. C₂×C₃×C₅×C₇	D₄	semidirect
17. C₂²×C₃×D₇	C₂×C₅	semidirect
18. C₂×C₃×C₄×C₇	C₂×C₅	semidirect
19. C₃×D₂₈	C₂×C₅	direct (N × H)
22. C₂×C₅×D₇	C₂²×C₃	semidirect
23. C₄×C₅×C₇	C₂²×C₃	semidirect
26. C₂×C₃×C₄×C₅	D₇	semidirect
27. C₂×D₂₈	C₃×C₅	direct (N × H)
28. C₃×C₅×C₇	C₂×D₄	semidirect
31. C₂×C₃×D₇	C₂²×C₅	semidirect
32. C₃×C₄×C₇	C₂²×C₅	semidirect
34. C₂×C₅×C₇	C₃×D₄	semidirect
40. C₃×C₄×C₅	C₂×D₇	semidirect
41. C₂²×D₇	C₂×C₃×C₅	semidirect
42. C₂×C₄×C₇	C₂×C₃×C₅	semidirect
43. D₂₈	C₂×C₃×C₅	direct (N × H)
45. C₂×C₃×C₇	C₅×D₄	semidirect
50. C₂×C₄×C₅	C₃×D₇	semidirect
51. C₅×C₇	C₂×C₃×D₄	semidirect
53. C₂×C₃×C₅	D₂₈	direct (N × H)
57. C₂×D₇	C₂²×C₃×C₅	semidirect
58. C₄×C₇	C₂²×C₃×C₅	semidirect
61. C₂×C₃×C₄	C₅×D₇	semidirect
62. C₃×C₇	C₂×C₅×D₄	semidirect
66. C₄×C₅	C₂×C₃×D₇	semidirect
68. C₃×C₅	C₂×D₂₈	direct (N × H)
69. C₂×C₇	C₃×C₅×D₄	semidirect
75. C₃×C₄	C₂×C₅×D₇	semidirect
77. C₂×C₅	C₃×D₂₈	direct (N × H)
81. C₂×C₄	C₃×C₅×D₇	semidirect
82. C₇	C₂×C₃×C₅×D₄	semidirect
84. C₂×C₃	C₅×D₂₈	direct (N × H)
86. C₅	C₂×C₃×D₂₈	direct (N × H)
90. C₄	C₂×C₃×C₅×D₇	semidirect
91. C₃	C₂×C₅×D₂₈	direct (N × H)
93. C₂	C₃×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
C₂²×C₃×C₅×D₇	2
C₂×C₃×C₄×C₅×C₇	2
C₃×C₅×D₂₈	2
C₂×C₅×D₂₈	3
C₂×C₃×D₂₈	5
C₂×C₃×C₅×D₄	7

Subgroups — C2×C3×C5×D28 #13132832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×D₂₈ #13132832