Subgroups — C₃×C₄×D₂₁ #92713

Order 504.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₂₁	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₂₁	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×C₄×D₇	—	3	1	no	maximal
6	C₂×C₃²×C₇	C₂²	1	1	yes	characteristic
7	C₃×D₂₁	C₄	1	2	yes	direct product factor
8	C₂×D₂₁	C₂×C₃	1	1	yes	characteristic
9	C₂×C₃×D₇	—	3	1	no	—
10	C₃×C₄×C₇	—	2	1	no	—
11	C₃×C₄×C₇	D₃	1	1	yes	characteristic, complemented
12	C₃×C₄×C₇	C₂×C₃	1	1	yes	characteristic, complemented
13	Dic₂₁	C₂×C₃	1	1	yes	characteristic, complemented
14	C₃×Dic₇	—	3	1	no	—
15	C₃×C₄×D₃	—	7	1	no	maximal, Hall
16	C₃²×C₇	C₂×C₄	1	1	yes	characteristic, complemented, Hall
17	C₄×D₇	—	3	1	no	Hall
18	C₂×C₃×C₇	—	2	1	no	—
19	C₂×C₃×C₇	C₂×D₃	1	1	yes	characteristic
20	C₂×C₃×C₇	C₂²×C₃	1	1	yes	characteristic
21	D₂₁	C₃×C₄	1	2	yes	direct product factor
22	C₃×D₇	—	3	2	no	—
23	C₂×C₃×D₃	—	7	1	no	—
24	C₃²×C₄	D₇	1	1	yes	characteristic, complemented
25	C₃×Dic₃	—	7	1	no	—
26	C₂×D₇	—	3	1	no	—
27	C₄×C₇	C₃×D₃	1	1	yes	characteristic, complemented
28	Dic₇	—	3	1	no	—
29	C₂×C₃×C₄	—	21	1	no	—
30	C₄×D₃	—	7	1	no	—
31	C₃×C₇	—	2	1	no	—
32	C₃×C₇	C₄×D₃	1	1	yes	characteristic, complemented
33	C₃×C₇	C₂×C₃×C₄	1	1	yes	characteristic, complemented
34	C₂×C₃²	C₂×D₇	1	1	yes	characteristic
35	C₃×D₃	—	7	2	no	—
36	C₂×C₇	C₂×C₃×D₃	1	1	yes	characteristic
37	D₇	—	3	2	no	—
38	C₂²×C₃	—	21	1	no	—
39	C₂×D₃	—	7	1	no	—
40	C₃×C₄	—	2	1	no	—
41	C₃×C₄	—	21	1	no	—
42	C₃×C₄	C₃×D₇	1	1	yes	characteristic, complemented
43	C₃×C₄	D₂₁	1	1	yes	characteristic, direct product factor
44	Dic₃	—	7	1	no	—
45	C₃²	C₄×D₇	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₄	—	21	1	no	Sylow
47	C₇	C₃×C₄×D₃	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₃	—	2	1	no	—
49	C₂×C₃	—	21	2	no	—
50	C₂×C₃	C₂×C₃×D₇	1	1	yes	characteristic
51	C₂×C₃	C₂×D₂₁	1	1	yes	characteristic
52	D₃	—	7	2	no	—
53	C₂²	—	21	1	no	—
54	C₄	—	21	1	no	—
55	C₄	C₃×D₂₁	1	1	yes	characteristic, direct product factor
56	C₃	—	2	1	no	—
57	C₃	C₃×C₄×D₇	1	1	yes	characteristic, complemented
58	C₃	C₄×D₂₁	1	1	yes	characteristic, direct product factor
59	C₂	—	21	2	no	—
60	C₂	C₂×C₃×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃×C₄	D₂₁
commutator, hypocenter	C₃×C₇	C₂×C₃×C₄
Frattini	C₂	C₂×C₃×D₂₁
Fitting	C₃²×C₄×C₇	C₂
socle	C₂×C₃²×C₇	C₂²
prime radical	C₂×C₃×D₂₁	C₂
cosocle	C₂×C₃×C₇	C₂²×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄×C₇	C₂	semidirect
3. C₃×Dic₂₁	C₂	semidirect
4. C₄×D₂₁	C₃	direct (N × H)
7. C₃×D₂₁	C₄	direct (N × H)
11. C₃×C₄×C₇	D₃	semidirect
12. C₃×C₄×C₇	C₂×C₃	semidirect
13. Dic₂₁	C₂×C₃	semidirect
16. C₃²×C₇	C₂×C₄	semidirect
21. D₂₁	C₃×C₄	direct (N × H)
24. C₃²×C₄	D₇	semidirect
27. C₄×C₇	C₃×D₃	semidirect
32. C₃×C₇	C₄×D₃	semidirect
33. C₃×C₇	C₂×C₃×C₄	semidirect
42. C₃×C₄	C₃×D₇	semidirect
43. C₃×C₄	D₂₁	direct (N × H)
45. C₃²	C₄×D₇	semidirect
47. C₇	C₃×C₄×D₃	semidirect
55. C₄	C₃×D₂₁	direct (N × H)
57. C₃	C₃×C₄×D₇	semidirect
58. C₃	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
C₂×C₃×D₂₁	2
C₃²×C₄×C₇	2
C₃×Dic₂₁	2
C₄×D₂₁	3
C₃×C₄×D₇	3
C₃×C₄×D₃	7

Subgroups — C3×C4×D21 #92713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×D₂₁ #92713