Subgroups — D₄×D₂₁ #66729

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₁	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#4268	C₂	1	2	yes	maximal, complemented
4	D₈₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×D₇	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₇	C₂²	1	2	yes	complemented
8	C₂×D₂₁	—	2	4	no	—
9	C₂×D₂₁	C₂²	1	1	yes	characteristic
10	C₂×D₂₁	C₂²	1	2	yes	complemented
11	C₃×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
12	Dic₂₁	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₇	—	3	2	no	—
14	C₇×D₄	D₃	1	1	yes	characteristic, complemented
15	C₂²⋊D₇	—	3	2	no	—
16	D₂₈	—	3	1	no	—
17	C₄×D₇	—	3	1	no	—
18	D₃×D₄	—	7	1	no	maximal, Hall
19	C₂×C₃×C₇	—	2	2	no	—
20	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
21	D₂₁	—	2	2	no	—
22	D₂₁	D₄	1	2	yes	direct product factor
23	C₂²×C₇	C₂×D₃	1	2	yes	complemented
24	C₂×D₇	—	3	1	no	—
25	C₂×D₇	—	3	2	no	—
26	C₂×D₇	—	6	4	no	—
27	C₄×C₇	C₂×D₃	1	1	yes	characteristic, complemented
28	Dic₇	—	3	1	no	—
29	C₂²×D₃	—	7	2	no	—
30	C₃×D₄	D₇	1	1	yes	characteristic, complemented
31	#67	—	7	2	no	—
32	D₁₂	—	7	1	no	—
33	C₄×D₃	—	7	1	no	—
34	C₃×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
35	C₂×D₄	—	21	1	no	Sylow
36	C₂×C₇	—	2	2	no	—
37	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
38	D₇	—	3	2	no	—
39	D₇	—	6	2	no	—
40	C₂²×C₃	C₂×D₇	1	2	yes	complemented
41	C₂×D₃	—	7	1	no	—
42	C₂×D₃	—	7	2	no	—
43	C₂×D₃	—	14	4	no	—
44	C₃×C₄	C₂×D₇	1	1	yes	characteristic, complemented
45	Dic₃	—	7	1	no	—
46	C₂³	—	21	2	no	—
47	D₄	—	21	2	no	—
48	D₄	—	21	1	no	—
49	D₄	D₂₁	1	1	yes	characteristic, direct product factor
50	C₂×C₄	—	21	1	no	—
51	C₇	D₃×D₄	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃	—	2	2	no	—
53	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
54	D₃	—	7	2	no	—
55	D₃	—	14	2	no	—
56	C₂²	—	21	1	no	—
57	C₂²	—	21	2	no	—
58	C₂²	—	42	4	no	—
59	C₂²	C₂×D₂₁	1	2	yes	complemented
60	C₄	—	21	1	no	—
61	C₄	C₂×D₂₁	1	1	yes	characteristic, complemented
62	C₃	D₄×D₇	1	1	yes	characteristic, complemented, Sylow
63	C₂	—	2	2	no	—
64	C₂	—	21	2	no	—
65	C₂	—	42	2	no	—
66	C₂	C₂²×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₂₁
commutator, socle, cosocle	C₂×C₃×C₇	C₂³
Fitting	C₃×C₇×D₄	C₂
hypercenter	D₄	D₂₁
hypocenter	C₃×C₇	C₂×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×D₂₁	C₂	semidirect
2. C₃×C₇×D₄	C₂	semidirect
3. #4268	C₂	semidirect
4. D₈₄	C₂	semidirect
5. C₄×D₂₁	C₂	semidirect
7. C₂²×C₃×C₇	C₂²	semidirect
10. C₂×D₂₁	C₂²	semidirect
11. C₃×C₄×C₇	C₂²	semidirect
12. Dic₂₁	C₂²	semidirect
14. C₇×D₄	D₃	semidirect
22. D₂₁	D₄	direct (N × H)
23. C₂²×C₇	C₂×D₃	semidirect
27. C₄×C₇	C₂×D₃	semidirect
30. C₃×D₄	D₇	semidirect
34. C₃×C₇	C₂×D₄	semidirect
40. C₂²×C₃	C₂×D₇	semidirect
44. C₃×C₄	C₂×D₇	semidirect
49. D₄	D₂₁	direct (N × H)
51. C₇	D₃×D₄	semidirect
59. C₂²	C₂×D₂₁	semidirect
61. C₄	C₂×D₂₁	semidirect
62. C₃	D₄×D₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×D₂₁	2
C₃×C₇×D₄	2
#4268	2
D₈₄	2
C₄×D₂₁	2
D₄×D₇	3
D₃×D₄	7

Subgroups — D4×D21 #66729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×D₂₁ #66729