Subgroups — C₂×C₄×D₂₁ #66726

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₂₁	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×D₇	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₂×D₂₁	C₄	1	4	yes	direct product factor
8	C₂×D₂₁	C₂²	1	2	yes	—
9	C₃×C₄×C₇	C₂²	1	2	yes	complemented
10	Dic₂₁	C₂²	1	2	yes	complemented
11	C₂²×D₇	—	3	1	no	—
12	C₂×C₄×C₇	D₃	1	1	yes	characteristic, complemented
13	C₂×Dic₇	—	3	1	no	—
14	C₄×D₇	—	3	4	no	—
15	C₂×C₄×D₃	—	7	1	no	maximal, Hall
16	C₂×C₃×C₇	C₂×C₄	1	2	yes	complemented
17	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
18	D₂₁	C₂×C₄	1	4	yes	direct product factor
19	C₂²×C₇	C₂×D₃	1	1	yes	characteristic
20	C₂×D₇	—	3	4	no	—
21	C₂×D₇	—	3	2	no	—
22	C₄×C₇	C₂×D₃	1	2	yes	complemented
23	Dic₇	—	3	2	no	—
24	C₂²×D₃	—	7	1	no	—
25	C₂×C₃×C₄	D₇	1	1	yes	characteristic, complemented
26	C₂×Dic₃	—	7	1	no	—
27	C₄×D₃	—	7	4	no	—
28	C₃×C₇	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×C₄	—	21	1	no	Sylow
30	C₂×C₇	C₄×D₃	1	2	yes	complemented
31	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
32	D₇	—	3	4	no	—
33	C₂²×C₃	C₂×D₇	1	1	yes	characteristic
34	C₂×D₃	—	7	4	no	—
35	C₂×D₃	—	7	2	no	—
36	C₃×C₄	C₂×D₇	1	2	yes	complemented
37	Dic₃	—	7	2	no	—
38	C₂³	—	21	1	no	—
39	C₂×C₄	—	21	1	no	—
40	C₂×C₄	—	21	4	no	—
41	C₂×C₄	D₂₁	1	1	yes	characteristic, direct product factor
42	C₇	C₂×C₄×D₃	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	C₄×D₇	1	2	yes	complemented
44	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
45	D₃	—	7	4	no	—
46	C₂²	—	21	4	no	—
47	C₂²	—	21	2	no	—
48	C₂²	C₂×D₂₁	1	1	yes	characteristic
49	C₄	—	21	2	no	—
50	C₄	C₂×D₂₁	1	2	yes	direct product factor
51	C₃	C₂×C₄×D₇	1	1	yes	characteristic, complemented, Sylow
52	C₂	—	21	4	no	—
53	C₂	C₄×D₂₁	1	2	yes	direct product factor
54	C₂	C₂²×D₂₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×C₇	C₂	semidirect
3. C₂×Dic₂₁	C₂	semidirect
4. C₄×D₂₁	C₂	direct (N × H)
7. C₂×D₂₁	C₄	direct (N × H)
9. C₃×C₄×C₇	C₂²	semidirect
10. Dic₂₁	C₂²	semidirect
12. C₂×C₄×C₇	D₃	semidirect
16. C₂×C₃×C₇	C₂×C₄	semidirect
18. D₂₁	C₂×C₄	direct (N × H)
22. C₄×C₇	C₂×D₃	semidirect
25. C₂×C₃×C₄	D₇	semidirect
28. C₃×C₇	C₂²×C₄	semidirect
30. C₂×C₇	C₄×D₃	semidirect
36. C₃×C₄	C₂×D₇	semidirect
41. C₂×C₄	D₂₁	direct (N × H)
42. C₇	C₂×C₄×D₃	semidirect
43. C₂×C₃	C₄×D₇	semidirect
50. C₄	C₂×D₂₁	direct (N × H)
51. C₃	C₂×C₄×D₇	semidirect
53. C₂	C₄×D₂₁	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
C₂×Dic₂₁	2
C₄×D₂₁	2
C₂×C₄×D₇	3
C₂×C₄×D₃	7

Subgroups — C2×C4×D21 #66726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₂₁ #66726