Subgroups — C₂×D₈₄ #66727

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₁	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×D₂₈	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
6	C₂×D₂₁	—	2	4	no	—
7	C₂×D₂₁	C₂²	1	4	yes	complemented
8	C₃×C₄×C₇	C₂²	1	2	yes	complemented
9	C₂²×D₇	—	3	2	no	—
10	C₂×C₄×C₇	D₃	1	1	yes	characteristic, complemented
11	D₂₈	—	3	4	no	—
12	C₂×D₁₂	—	7	1	no	maximal, Hall
13	C₂×C₃×C₇	D₄	1	2	yes	complemented
14	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
15	D₂₁	—	2	4	no	—
16	C₂²×C₇	C₂×D₃	1	1	yes	characteristic
17	C₂×D₇	—	3	4	no	—
18	C₂×D₇	—	6	4	no	—
19	C₄×C₇	C₂×D₃	1	2	yes	complemented
20	C₂²×D₃	—	7	2	no	—
21	C₂×C₃×C₄	D₇	1	1	yes	characteristic, complemented
22	D₁₂	—	7	4	no	—
23	C₃×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
24	C₂×D₄	—	21	1	no	Sylow
25	C₂×C₇	D₁₂	1	2	yes	complemented
26	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
27	D₇	—	6	4	no	—
28	C₂²×C₃	C₂×D₇	1	1	yes	characteristic
29	C₂×D₃	—	7	4	no	—
30	C₂×D₃	—	14	4	no	—
31	C₃×C₄	C₂×D₇	1	2	yes	complemented
32	C₂³	—	21	2	no	—
33	D₄	—	21	4	no	—
34	C₂×C₄	D₂₁	1	1	yes	characteristic, complemented
35	C₇	C₂×D₁₂	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃	D₂₈	1	2	yes	complemented
37	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
38	D₃	—	14	4	no	—
39	C₂²	—	21	4	no	—
40	C₂²	—	42	4	no	—
41	C₂²	C₂×D₂₁	1	1	yes	characteristic
42	C₄	C₂×D₂₁	1	2	yes	complemented
43	C₃	C₂×D₂₈	1	1	yes	characteristic, complemented, Sylow
44	C₂	—	42	4	no	—
45	C₂	D₈₄	1	2	yes	direct product factor
46	C₂	C₂²×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₁
commutator, cosocle	C₂×C₃×C₇	C₂³
Frattini	C₂	C₂²×D₂₁
Fitting	C₂×C₃×C₄×C₇	C₂
socle	C₂²×C₃×C₇	C₂²
hypercenter	C₂×C₄	D₂₁
hypocenter	C₃×C₇	C₂×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×D₂₁	C₂	semidirect
2. C₂×C₃×C₄×C₇	C₂	semidirect
3. D₈₄	C₂	direct (N × H)
7. C₂×D₂₁	C₂²	semidirect
8. C₃×C₄×C₇	C₂²	semidirect
10. C₂×C₄×C₇	D₃	semidirect
13. C₂×C₃×C₇	D₄	semidirect
19. C₄×C₇	C₂×D₃	semidirect
21. C₂×C₃×C₄	D₇	semidirect
23. C₃×C₇	C₂×D₄	semidirect
25. C₂×C₇	D₁₂	semidirect
31. C₃×C₄	C₂×D₇	semidirect
34. C₂×C₄	D₂₁	semidirect
35. C₇	C₂×D₁₂	semidirect
36. C₂×C₃	D₂₈	semidirect
42. C₄	C₂×D₂₁	semidirect
43. C₃	C₂×D₂₈	semidirect
45. C₂	D₈₄	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×D₂₁	2
C₂×C₃×C₄×C₇	2
D₈₄	2
C₂×D₂₈	3
C₂×D₁₂	7

Subgroups — C2×D84 #66727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₈₄ #66727