Subgroups — C₂²×C₃×D₂₁ #92773

Order 504.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₂₁	C₂	1	6	yes	maximal, direct product factor
3	C₂²×D₂₁	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂²×C₃×D₇	—	3	1	no	maximal
5	C₂×C₃²×C₇	C₂²	1	3	yes	complemented
6	C₃×D₂₁	C₂²	1	4	yes	direct product factor
7	C₂²×C₃×C₇	—	2	1	no	—
8	C₂²×C₃×C₇	D₃	1	1	yes	characteristic, complemented
9	C₂²×C₃×C₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×D₂₁	C₂×C₃	1	6	yes	direct product factor
11	C₂×C₃×D₇	—	3	6	no	—
12	C₂²×C₃×D₃	—	7	1	no	maximal, Hall
13	C₃²×C₇	C₂³	1	1	yes	characteristic, complemented, Hall
14	C₂²×D₇	—	3	1	no	Hall
15	C₂×C₃×C₇	—	2	3	no	—
16	C₂×C₃×C₇	C₂×D₃	1	3	yes	complemented
17	C₂×C₃×C₇	C₂²×C₃	1	3	yes	complemented
18	D₂₁	C₂²×C₃	1	4	yes	direct product factor
19	C₃×D₇	—	3	4	no	—
20	C₂²×C₃²	D₇	1	1	yes	characteristic, complemented
21	C₂×C₃×D₃	—	7	6	no	—
22	C₂²×C₇	C₃×D₃	1	1	yes	characteristic, complemented
23	C₂×D₇	—	3	6	no	—
24	C₂³×C₃	—	21	1	no	—
25	C₂²×D₃	—	7	1	no	—
26	C₃×C₇	—	2	1	no	—
27	C₃×C₇	C₂²×D₃	1	1	yes	characteristic, complemented
28	C₃×C₇	C₂³×C₃	1	1	yes	characteristic, complemented
29	C₂×C₃²	C₂×D₇	1	3	yes	complemented
30	C₃×D₃	—	7	4	no	—
31	C₂×C₇	C₂×C₃×D₃	1	3	yes	complemented
32	D₇	—	3	4	no	—
33	C₂²×C₃	—	2	1	no	—
34	C₂²×C₃	—	21	6	no	—
35	C₂²×C₃	C₃×D₇	1	1	yes	characteristic, complemented
36	C₂²×C₃	D₂₁	1	1	yes	characteristic, direct product factor
37	C₂×D₃	—	7	6	no	—
38	C₃²	C₂²×D₇	1	1	yes	characteristic, complemented, Sylow
39	C₂³	—	21	1	no	Sylow
40	C₇	C₂²×C₃×D₃	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	—	2	3	no	—
42	C₂×C₃	—	21	4	no	—
43	C₂×C₃	C₂×C₃×D₇	1	3	yes	complemented
44	C₂×C₃	C₂×D₂₁	1	3	yes	direct product factor
45	D₃	—	7	4	no	—
46	C₂²	—	21	6	no	—
47	C₂²	C₃×D₂₁	1	1	yes	characteristic, direct product factor
48	C₃	—	2	1	no	—
49	C₃	C₂²×C₃×D₇	1	1	yes	characteristic, complemented
50	C₃	C₂²×D₂₁	1	1	yes	characteristic, direct product factor
51	C₂	—	21	4	no	—
52	C₂	C₂×C₃×D₂₁	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃	D₂₁
commutator, cosocle, hypocenter	C₃×C₇	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₇	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₇	C₂	semidirect
2. C₂×C₃×D₂₁	C₂	direct (N × H)
3. C₂²×D₂₁	C₃	direct (N × H)
5. C₂×C₃²×C₇	C₂²	semidirect
6. C₃×D₂₁	C₂²	direct (N × H)
8. C₂²×C₃×C₇	D₃	semidirect
9. C₂²×C₃×C₇	C₂×C₃	semidirect
10. C₂×D₂₁	C₂×C₃	direct (N × H)
13. C₃²×C₇	C₂³	semidirect
16. C₂×C₃×C₇	C₂×D₃	semidirect
17. C₂×C₃×C₇	C₂²×C₃	semidirect
18. D₂₁	C₂²×C₃	direct (N × H)
20. C₂²×C₃²	D₇	semidirect
22. C₂²×C₇	C₃×D₃	semidirect
27. C₃×C₇	C₂²×D₃	semidirect
28. C₃×C₇	C₂³×C₃	semidirect
29. C₂×C₃²	C₂×D₇	semidirect
31. C₂×C₇	C₂×C₃×D₃	semidirect
35. C₂²×C₃	C₃×D₇	semidirect
36. C₂²×C₃	D₂₁	direct (N × H)
38. C₃²	C₂²×D₇	semidirect
40. C₇	C₂²×C₃×D₃	semidirect
43. C₂×C₃	C₂×C₃×D₇	semidirect
44. C₂×C₃	C₂×D₂₁	direct (N × H)
47. C₂²	C₃×D₂₁	direct (N × H)
49. C₃	C₂²×C₃×D₇	semidirect
50. C₃	C₂²×D₂₁	direct (N × H)
52. C₂	C₂×C₃×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×C₃²×C₇	2
C₂×C₃×D₂₁	2
C₂²×D₂₁	3
C₂²×C₃×D₇	3
C₂²×C₃×D₃	7

Subgroups — C22×C3×D21 #92773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×D₂₁ #92773