Subgroups — C₂×C₃×D₂₈ #66707

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₂₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×D₂₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₃×D₇	—	2	4	no	—
7	C₂×C₃×D₇	C₂²	1	4	yes	complemented
8	C₃×C₄×C₇	C₂²	1	2	yes	complemented
9	C₂²×D₇	C₂×C₃	1	2	yes	complemented
10	C₂×C₄×C₇	C₂×C₃	1	1	yes	characteristic, complemented
11	D₂₈	C₂×C₃	1	4	yes	direct product factor
12	C₂×C₃×D₄	—	7	1	no	maximal, Hall
13	C₂×C₃×C₇	D₄	1	2	yes	complemented
14	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
15	C₃×D₇	—	2	4	no	—
16	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
17	C₂×D₇	—	2	4	no	—
18	C₂×D₇	C₂²×C₃	1	4	yes	complemented
19	C₄×C₇	C₂²×C₃	1	2	yes	complemented
20	C₂³×C₃	—	7	2	no	—
21	C₃×D₄	—	7	4	no	—
22	C₂×C₃×C₄	D₇	1	1	yes	characteristic, complemented
23	C₃×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
24	C₂×D₄	—	7	1	no	Sylow
25	C₂×C₇	C₃×D₄	1	2	yes	complemented
26	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
27	D₇	—	2	4	no	—
28	C₂²×C₃	—	7	4	no	—
29	C₂²×C₃	—	14	4	no	—
30	C₂²×C₃	C₂×D₇	1	1	yes	characteristic
31	C₃×C₄	C₂×D₇	1	2	yes	complemented
32	C₂³	—	7	2	no	—
33	D₄	—	7	4	no	—
34	C₂×C₄	C₃×D₇	1	1	yes	characteristic, complemented
35	C₇	C₂×C₃×D₄	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃	—	14	4	no	—
37	C₂×C₃	D₂₈	1	2	yes	direct product factor
38	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
39	C₂²	—	7	4	no	—
40	C₂²	—	14	4	no	—
41	C₂²	C₂×C₃×D₇	1	1	yes	characteristic
42	C₄	C₂×C₃×D₇	1	2	yes	complemented
43	C₃	C₂×D₂₈	1	1	yes	characteristic, direct product factor, Sylow
44	C₂	—	14	4	no	—
45	C₂	C₃×D₂₈	1	2	yes	direct product factor
46	C₂	C₂²×C₃×D₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₇	C₂	semidirect
2. C₂×C₃×C₄×C₇	C₂	semidirect
3. C₃×D₂₈	C₂	direct (N × H)
4. C₂×D₂₈	C₃	direct (N × H)
7. C₂×C₃×D₇	C₂²	semidirect
8. C₃×C₄×C₇	C₂²	semidirect
9. C₂²×D₇	C₂×C₃	semidirect
10. C₂×C₄×C₇	C₂×C₃	semidirect
11. D₂₈	C₂×C₃	direct (N × H)
13. C₂×C₃×C₇	D₄	semidirect
18. C₂×D₇	C₂²×C₃	semidirect
19. C₄×C₇	C₂²×C₃	semidirect
22. C₂×C₃×C₄	D₇	semidirect
23. C₃×C₇	C₂×D₄	semidirect
25. C₂×C₇	C₃×D₄	semidirect
31. C₃×C₄	C₂×D₇	semidirect
34. C₂×C₄	C₃×D₇	semidirect
35. C₇	C₂×C₃×D₄	semidirect
37. C₂×C₃	D₂₈	direct (N × H)
42. C₄	C₂×C₃×D₇	semidirect
43. C₃	C₂×D₂₈	direct (N × H)
45. C₂	C₃×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×D₇	2
C₂×C₃×C₄×C₇	2
C₃×D₂₈	2
C₂×D₂₈	3
C₂×C₃×D₄	7

Subgroups — C2×C3×D28 #66707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₂₈ #66707