Subgroups — D₇×SD₁₆ #6476

Order 224.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₇	C₂	1	1	yes	maximal, characteristic
3	C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#1143	C₂	1	1	yes	maximal, characteristic, complemented
5	#1142	C₂	1	1	yes	maximal, characteristic, complemented
6	#1132	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇	—	2	1	no	—
9	C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
10	C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²⋊D₇	—	2	1	no	—
12	D₂₈	C₂²	1	1	yes	characteristic
13	C₄×D₇	—	2	1	no	—
14	C₄×D₇	C₂²	1	1	yes	characteristic
15	Dic₁₄	—	2	1	no	—
16	Dic₁₄	C₂²	1	1	yes	characteristic, complemented
17	C₇×C₈	C₂²	1	1	yes	characteristic, complemented
18	C₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
19	C₂×SD₁₆	—	7	1	no	maximal, Sylow
20	C₂²×C₇	—	2	1	no	—
21	C₂×D₇	—	2	1	no	—
22	C₂×D₇	—	4	2	no	—
23	C₂×D₇	D₄	1	1	yes	characteristic
24	C₄×C₇	—	2	1	no	—
25	C₄×C₇	C₂³	1	1	yes	characteristic
26	Dic₇	—	2	1	no	—
27	Dic₇	D₄	1	1	yes	characteristic
28	C₂×Q₈	—	7	1	no	—
29	C₂×D₄	—	7	1	no	—
30	SD₁₆	—	7	1	no	—
31	SD₁₆	—	7	1	no	—
32	SD₁₆	—	7	1	no	—
33	SD₁₆	D₇	1	1	yes	characteristic, direct product factor
34	C₂×C₈	—	7	1	no	—
35	C₂×C₇	—	4	1	no	—
36	C₂×C₇	C₂×D₄	1	1	yes	characteristic
37	D₇	—	4	1	no	—
38	D₇	SD₁₆	1	2	yes	direct product factor
39	C₂³	—	14	1	no	—
40	Q₈	—	7	1	no	—
41	Q₈	—	14	1	no	—
42	Q₈	C₂×D₇	1	1	yes	characteristic, complemented
43	D₄	—	7	1	no	—
44	D₄	—	14	1	no	—
45	D₄	C₂×D₇	1	1	yes	characteristic
46	C₂×C₄	—	7	1	no	—
47	C₂×C₄	—	14	1	no	—
48	C₈	—	7	1	no	—
49	C₈	C₂×D₇	1	1	yes	characteristic, complemented
50	C₇	C₂×SD₁₆	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	2	1	no	—
52	C₂²	—	7	1	no	—
53	C₂²	—	14	1	no	—
54	C₂²	—	28	2	no	—
55	C₄	—	2	1	no	—
56	C₄	—	7	1	no	—
57	C₄	—	14	1	no	—
58	C₄	C₂²×D₇	1	1	yes	characteristic
59	C₂	—	4	1	no	—
60	C₂	—	7	2	no	—
61	C₂	—	28	1	no	—
62	C₂	D₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₇
commutator, cosocle	C₄×C₇	C₂³
Frattini	C₄	C₂²×D₇
Fitting	C₇×SD₁₆	C₂
socle	C₂×C₇	C₂×D₄
prime radical	D₄×D₇	C₂
hypercenter	SD₁₆	D₇
hypocenter	C₇	C₂×SD₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
1. Q₈×D₇	C₂	semidirect
3. C₇×SD₁₆	C₂	semidirect
4. #1143	C₂	semidirect
5. #1142	C₂	semidirect
6. #1132	C₂	semidirect
7. C₈×D₇	C₂	semidirect
9. C₇×Q₈	C₂²	semidirect
16. Dic₁₄	C₂²	semidirect
17. C₇×C₈	C₂²	semidirect
18. C₇⋊C₈	C₂²	semidirect
33. SD₁₆	D₇	direct (N × H)
38. D₇	SD₁₆	direct (N × H)
42. Q₈	C₂×D₇	semidirect
49. C₈	C₂×D₇	semidirect
50. C₇	C₂×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₇	2
D₄×D₇	2
C₇×SD₁₆	2
#1143	2
#1142	2
#1132	2
C₈×D₇	2
C₂×SD₁₆	7

Subgroups — D7×SD16 #6476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×SD₁₆ #6476