Subgroups — C₂⁴×C₃⋊D₂₁ #72246713

Order 2016.

4488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃⋊D₂₁	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×D₂₁	—	3	4	no	maximal
4	C₂³×C₃²×C₇	C₂²	1	15	yes	complemented
5	C₂²×C₃⋊D₂₁	C₂²	1	140	yes	direct product factor
6	C₂⁴×C₃×C₇	D₃	1	4	yes	complemented
7	C₂³×D₂₁	—	3	120	no	—
8	C₂⁴×C₃⋊D₃	—	7	1	no	maximal, Hall
9	C₂²×C₃²×C₇	C₂³	1	35	yes	complemented
10	C₂×C₃⋊D₂₁	C₂³	1	120	yes	direct product factor
11	C₂⁴×D₇	—	9	1	no	Hall
12	C₂³×C₃×C₇	C₂×D₃	1	60	yes	complemented
13	C₂²×D₂₁	—	3	560	no	—
14	C₂⁴×C₃²	D₇	1	1	yes	characteristic, complemented
15	C₂³×C₃⋊D₃	—	7	30	no	—
16	C₂×C₃²×C₇	C₂⁴	1	15	yes	complemented
17	C₃⋊D₂₁	C₂⁴	1	16	yes	direct product factor
18	C₂⁴×C₇	C₃⋊D₃	1	1	yes	characteristic, complemented
19	C₂³×D₇	—	9	30	no	—
20	C₂⁴×D₃	—	21	4	no	—
21	C₂²×C₃×C₇	C₂²×D₃	1	140	yes	complemented
22	C₂×D₂₁	—	3	480	no	—
23	C₂³×C₃²	C₂×D₇	1	15	yes	complemented
24	C₂²×C₃⋊D₃	—	7	140	no	—
25	C₃²×C₇	C₂⁵	1	1	yes	characteristic, complemented, Hall
26	C₂³×C₇	C₂×C₃⋊D₃	1	15	yes	complemented
27	C₂²×D₇	—	9	140	no	—
28	C₂⁴×C₃	D₂₁	1	4	yes	complemented
29	C₂³×D₃	—	21	120	no	—
30	C₂×C₃×C₇	C₂³×D₃	1	60	yes	complemented
31	D₂₁	—	3	64	no	—
32	C₂²×C₃²	C₂²×D₇	1	35	yes	complemented
33	C₂×C₃⋊D₃	—	7	120	no	—
34	C₂⁵	—	63	1	no	Sylow
35	C₂²×C₇	C₂²×C₃⋊D₃	1	35	yes	complemented
36	C₂×D₇	—	9	120	no	—
37	C₂³×C₃	C₂×D₂₁	1	60	yes	complemented
38	C₂²×D₃	—	21	560	no	—
39	C₃×C₇	C₂⁴×D₃	1	4	yes	complemented
40	C₂×C₃²	C₂³×D₇	1	15	yes	complemented
41	C₃⋊D₃	—	7	16	no	—
42	C₂⁴	—	63	30	no	—
43	C₂⁴	C₃⋊D₂₁	1	1	yes	characteristic, direct product factor
44	C₂×C₇	C₂³×C₃⋊D₃	1	15	yes	complemented
45	D₇	—	9	16	no	—
46	C₂²×C₃	C₂²×D₂₁	1	140	yes	complemented
47	C₂×D₃	—	21	480	no	—
48	C₃²	C₂⁴×D₇	1	1	yes	characteristic, complemented, Sylow
49	C₂³	—	63	140	no	—
50	C₂³	C₂×C₃⋊D₂₁	1	15	yes	direct product factor
51	C₇	C₂⁴×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃	C₂³×D₂₁	1	60	yes	complemented
53	D₃	—	21	64	no	—
54	C₂²	—	63	120	no	—
55	C₂²	C₂²×C₃⋊D₂₁	1	35	yes	direct product factor
56	C₃	C₂⁴×D₂₁	1	4	yes	complemented
57	C₂	—	63	16	no	—
58	C₂	C₂³×C₃⋊D₂₁	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	C₃⋊D₂₁
commutator, cosocle, hypocenter	C₃²×C₇	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃²×C₇	C₂
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃²×C₇	C₂	semidirect
2. C₂³×C₃⋊D₂₁	C₂	direct (N × H)
4. C₂³×C₃²×C₇	C₂²	semidirect
5. C₂²×C₃⋊D₂₁	C₂²	direct (N × H)
6. C₂⁴×C₃×C₇	D₃	semidirect
9. C₂²×C₃²×C₇	C₂³	semidirect
10. C₂×C₃⋊D₂₁	C₂³	direct (N × H)
12. C₂³×C₃×C₇	C₂×D₃	semidirect
14. C₂⁴×C₃²	D₇	semidirect
16. C₂×C₃²×C₇	C₂⁴	semidirect
17. C₃⋊D₂₁	C₂⁴	direct (N × H)
18. C₂⁴×C₇	C₃⋊D₃	semidirect
21. C₂²×C₃×C₇	C₂²×D₃	semidirect
23. C₂³×C₃²	C₂×D₇	semidirect
25. C₃²×C₇	C₂⁵	semidirect
26. C₂³×C₇	C₂×C₃⋊D₃	semidirect
28. C₂⁴×C₃	D₂₁	semidirect
30. C₂×C₃×C₇	C₂³×D₃	semidirect
32. C₂²×C₃²	C₂²×D₇	semidirect
35. C₂²×C₇	C₂²×C₃⋊D₃	semidirect
37. C₂³×C₃	C₂×D₂₁	semidirect
39. C₃×C₇	C₂⁴×D₃	semidirect
40. C₂×C₃²	C₂³×D₇	semidirect
43. C₂⁴	C₃⋊D₂₁	direct (N × H)
44. C₂×C₇	C₂³×C₃⋊D₃	semidirect
46. C₂²×C₃	C₂²×D₂₁	semidirect
48. C₃²	C₂⁴×D₇	semidirect
50. C₂³	C₂×C₃⋊D₂₁	direct (N × H)
51. C₇	C₂⁴×C₃⋊D₃	semidirect
52. C₂×C₃	C₂³×D₂₁	semidirect
55. C₂²	C₂²×C₃⋊D₂₁	direct (N × H)
56. C₃	C₂⁴×D₂₁	semidirect
58. C₂	C₂³×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
C₂⁴×C₃²×C₇	2
C₂³×C₃⋊D₂₁	2
C₂⁴×D₂₁	3
C₂⁴×C₃⋊D₃	7

Subgroups — C24×C3⋊D21 #72246713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃⋊D₂₁ #72246713