Subgroups — C₃²×Dic₂₁ #10625513

Order 756.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₇	C₂	1	1	yes	maximal, characteristic
2	C₃×Dic₂₁	C₃	1	4	yes	maximal, direct product factor
3	C₃²×Dic₇	—	3	1	no	maximal
4	C₃³×C₇	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃²×C₇	—	2	4	no	—
6	C₂×C₃²×C₇	D₃	1	1	yes	characteristic
7	C₂×C₃²×C₇	C₂×C₃	1	4	yes	—
8	C₃²×Dic₃	—	7	1	no	maximal, Hall
9	Dic₂₁	C₃²	1	1	yes	characteristic, direct product factor
10	C₃×Dic₇	—	3	4	no	—
11	C₃²×C₇	—	2	4	no	—
12	C₃²×C₇	Dic₃	1	1	yes	characteristic, complemented
13	C₃²×C₇	C₃×C₄	1	4	yes	complemented
14	C₂×C₃³	D₇	1	1	yes	characteristic
15	C₂×C₃×C₇	—	2	4	no	—
16	C₂×C₃×C₇	C₃×D₃	1	4	yes	—
17	C₂×C₃×C₇	C₂×C₃²	1	1	yes	characteristic
18	C₃²×C₄	—	21	1	no	—
19	C₃×Dic₃	—	7	4	no	—
20	Dic₇	—	3	1	no	Hall
21	C₃³	Dic₇	1	1	yes	characteristic, complemented, Sylow
22	C₃×C₇	—	2	4	no	—
23	C₃×C₇	C₃×Dic₃	1	4	yes	complemented
24	C₃×C₇	C₃²×C₄	1	1	yes	characteristic, complemented
25	C₂×C₃²	—	2	4	no	—
26	C₂×C₃²	C₃×D₇	1	4	yes	—
27	C₂×C₃²	D₂₁	1	1	yes	characteristic
28	C₂×C₇	C₃²×D₃	1	1	yes	characteristic
29	C₃×C₄	—	21	4	no	—
30	Dic₃	—	7	1	no	—
31	C₃²	—	2	4	no	—
32	C₃²	C₃×Dic₇	1	4	yes	complemented
33	C₃²	Dic₂₁	1	1	yes	characteristic, direct product factor
34	C₇	C₃²×Dic₃	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₃	—	2	4	no	—
36	C₂×C₃	C₃²×D₇	1	1	yes	characteristic
37	C₂×C₃	C₃×D₂₁	1	4	yes	—
38	C₄	—	21	1	no	Sylow
39	C₃	—	2	4	no	—
40	C₃	C₃²×Dic₇	1	1	yes	characteristic, complemented
41	C₃	C₃×Dic₂₁	1	4	yes	direct product factor
42	C₂	C₃²×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃²	D₂₁
commutator, hypocenter	C₃×C₇	C₃²×C₄
Frattini	C₂	C₃²×D₂₁
Fitting, socle, prime radical	C₂×C₃³×C₇	C₂
cosocle	C₂×C₃×C₇	C₂×C₃²
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₃²

Complements

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

N	H ≅ G/N	type
2. C₃×Dic₂₁	C₃	direct (N × H)
4. C₃³×C₇	C₄	semidirect
9. Dic₂₁	C₃²	direct (N × H)
12. C₃²×C₇	Dic₃	semidirect
13. C₃²×C₇	C₃×C₄	semidirect
21. C₃³	Dic₇	semidirect
23. C₃×C₇	C₃×Dic₃	semidirect
24. C₃×C₇	C₃²×C₄	semidirect
32. C₃²	C₃×Dic₇	semidirect
33. C₃²	Dic₂₁	direct (N × H)
34. C₇	C₃²×Dic₃	semidirect
40. C₃	C₃²×Dic₇	semidirect
41. C₃	C₃×Dic₂₁	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×C₃³×C₇	2
C₃×Dic₂₁	3
C₃²×Dic₇	3
C₃²×Dic₃	7

Subgroups — C32×Dic21 #10625513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×Dic₂₁ #10625513