Subgroups — Q₁₆×D₂₁ #10621400

Order 672.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₂₁	C₂	1	2	yes	maximal
2	C₃×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#66635	C₂	1	2	yes	maximal, complemented
4	Dic₈₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₂₁	C₂	1	1	yes	maximal, characteristic
6	D₇×Q₁₆	—	3	1	no	maximal, Hall
7	C₃×C₇×Q₈	C₂²	1	2	yes	—
8	C₄×D₂₁	—	2	2	no	—
9	C₄×D₂₁	C₂²	1	1	yes	characteristic
10	Dic₄₂	—	2	2	no	—
11	Dic₄₂	C₂²	1	2	yes	—
12	C₃×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
14	Q₈×D₇	—	3	2	no	—
15	C₇×Q₁₆	D₃	1	1	yes	characteristic, complemented
16	#1144	—	3	2	no	—
17	Dic₂₈	—	3	1	no	—
18	C₈×D₇	—	3	1	no	—
19	D₃×Q₁₆	—	7	1	no	maximal, Hall
20	C₂×D₂₁	D₄	1	1	yes	characteristic
21	C₃×C₄×C₇	—	2	2	no	—
22	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
23	Dic₂₁	—	2	2	no	—
24	Dic₂₁	D₄	1	1	yes	characteristic
25	C₇×Q₈	C₂×D₃	1	2	yes	—
26	C₄×D₇	—	3	1	no	—
27	C₄×D₇	—	6	2	no	—
28	Dic₁₄	—	3	2	no	—
29	Dic₁₄	—	6	2	no	—
30	C₇×C₈	C₂×D₃	1	1	yes	characteristic
31	C₇⋊C₈	—	3	1	no	—
32	D₃×Q₈	—	7	2	no	—
33	C₃×Q₁₆	D₇	1	1	yes	characteristic, complemented
34	#216	—	7	2	no	—
35	Dic₁₂	—	7	1	no	—
36	D₃×C₈	—	7	1	no	—
37	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
38	D₂₁	Q₁₆	1	2	yes	direct product factor
39	C₂×Q₁₆	—	21	1	no	Sylow
40	C₂×D₇	—	3	1	no	—
41	C₄×C₇	—	2	2	no	—
42	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
43	Dic₇	—	3	1	no	—
44	Dic₇	—	6	2	no	—
45	C₃×Q₈	C₂×D₇	1	2	yes	—
46	C₄×D₃	—	7	1	no	—
47	C₄×D₃	—	14	2	no	—
48	Dic₆	—	7	2	no	—
49	Dic₆	—	14	2	no	—
50	C₃×C₈	C₂×D₇	1	1	yes	characteristic
51	C₃⋊C₈	—	7	1	no	—
52	C₃×C₇	C₂×Q₁₆	1	1	yes	characteristic, complemented, Hall
53	C₂×Q₈	—	21	2	no	—
54	Q₁₆	—	21	2	no	—
55	Q₁₆	—	21	1	no	—
56	Q₁₆	D₂₁	1	1	yes	characteristic, direct product factor
57	C₂×C₈	—	21	1	no	—
58	C₂×C₇	D₃×D₄	1	1	yes	characteristic
59	D₇	—	3	2	no	—
60	C₂×D₃	—	7	1	no	—
61	C₃×C₄	—	2	2	no	—
62	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
63	Dic₃	—	7	1	no	—
64	Dic₃	—	14	2	no	—
65	Q₈	—	21	2	no	—
66	Q₈	—	42	2	no	—
67	Q₈	C₂×D₂₁	1	2	yes	—
68	C₂×C₄	—	21	1	no	—
69	C₂×C₄	—	42	2	no	—
70	C₈	—	21	1	no	—
71	C₈	C₂×D₂₁	1	1	yes	characteristic
72	C₇	D₃×Q₁₆	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	D₄×D₇	1	1	yes	characteristic
74	D₃	—	7	2	no	—
75	C₂²	—	21	1	no	—
76	C₄	—	2	2	no	—
77	C₄	—	21	1	no	—
78	C₄	—	42	2	no	—
79	C₄	C₂²×D₂₁	1	1	yes	characteristic
80	C₃	D₇×Q₁₆	1	1	yes	characteristic, complemented, Sylow
81	C₂	—	21	2	no	—
82	C₂	D₄×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₂₁
commutator, cosocle	C₃×C₄×C₇	C₂³
Frattini	C₄	C₂²×D₂₁
Fitting	C₃×C₇×Q₁₆	C₂
socle	C₂×C₃×C₇	C₂×D₄
prime radical	C₂×D₂₁	D₄
hypercenter	Q₁₆	D₂₁
hypocenter	C₃×C₇	C₂×Q₁₆
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×Q₁₆	C₂	semidirect
3. #66635	C₂	semidirect
4. Dic₈₄	C₂	semidirect
15. C₇×Q₁₆	D₃	semidirect
33. C₃×Q₁₆	D₇	semidirect
38. D₂₁	Q₁₆	direct (N × H)
52. C₃×C₇	C₂×Q₁₆	semidirect
56. Q₁₆	D₂₁	direct (N × H)
72. C₇	D₃×Q₁₆	semidirect
80. C₃	D₇×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₂₁	2
C₃×C₇×Q₁₆	2
#66635	2
Dic₈₄	2
C₈×D₂₁	2
D₇×Q₁₆	3
D₃×Q₁₆	7

Subgroups — Q16×D21 #10621400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×D₂₁ #10621400