Subgroups — Q₈×C₂₇₁⋊C₁₆ #46838268

Order 34688.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₇₁⋊C₁₆)	C₂	1	3	yes	maximal
2	C₄×C₂₇₁⋊C₁₆	C₂	1	3	yes	maximal
3	C₈×Q₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₇₁⋊C₁₆	C₂²	1	3	yes	—
5	C₂×C₂₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
6	C₄×Q₈×C₂₇₁	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₂₇₁	C₄	1	3	yes	—
8	C₄×C₈×C₂₇₁	C₂²	1	3	yes	—
9	C₂₇₁⋊C₁₆	—	2	3	no	—
10	C₂₇₁⋊C₁₆	Q₈	1	2	yes	direct product factor
11	C₂×Q₈×C₂₇₁	C₈	1	1	yes	characteristic
12	C₂×C₈×C₂₇₁	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
14	C₄⋊C₄×C₂₇₁	C₈	1	3	yes	—
15	C₄²×C₂₇₁	C₂×C₄	1	3	yes	—
16	Q₈×C₂₇₁	C₁₆	1	4	yes	complemented
17	C₂×C₄×C₂₇₁	C₂×C₈	1	3	yes	—
18	C₂×C₄×C₂₇₁	C₂×C₈	1	3	yes	—
19	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
20	C₈×C₂₇₁	—	2	3	no	—
21	C₈×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
22	C₈×C₂₇₁	#41	1	1	yes	characteristic
23	C₂²×C₂₇₁	C₂²×C₈	1	1	yes	characteristic
24	C₄×C₂₇₁	—	2	3	no	—
25	C₄×C₂₇₁	C₂×C₁₆	1	6	yes	—
26	C₄×C₂₇₁	C₄×Q₈	1	1	yes	characteristic
27	C₄×C₂₇₁	#131	1	1	yes	characteristic
28	C₂×C₂₇₁	C₈×Q₈	1	1	yes	characteristic
29	C₂×C₂₇₁	C₂²×C₁₆	1	1	yes	characteristic
30	C₂×C₂₇₁	#504	1	1	yes	characteristic
31	C₂₇₁	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
32	Q₈×C₁₆	—	271	1	no	maximal, Sylow
33	C₈×Q₈	D₂₇₁	1	1	yes	characteristic
34	C₄⋊C₁₆	—	271	3	no	—
35	C₄×C₁₆	—	271	3	no	—
36	C₄×Q₈	Dic₂₇₁	1	1	yes	characteristic
37	C₂×C₁₆	—	271	1	no	—
38	C₂×C₁₆	—	271	3	no	—
39	C₄⋊C₈	Dic₂₇₁	1	3	yes	—
40	C₄×C₈	C₂×D₂₇₁	1	3	yes	—
41	C₂×Q₈	C₂₇₁⋊C₈	1	1	yes	characteristic
42	C₂×C₈	C₂×Dic₂₇₁	1	3	yes	—
43	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
44	C₄⋊C₄	C₂₇₁⋊C₈	1	3	yes	—
45	C₄²	C₂×Dic₂₇₁	1	3	yes	—
46	C₁₆	—	271	2	no	—
47	C₁₆	—	542	3	no	—
48	Q₈	C₂₇₁⋊C₁₆	1	4	yes	direct product factor
49	C₂×C₄	C₂×C₂₇₁⋊C₈	1	3	yes	—
50	C₂×C₄	C₂×C₂₇₁⋊C₈	1	3	yes	—
51	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
52	C₈	—	2	3	no	—
53	C₈	Q₈×D₂₇₁	1	1	yes	characteristic
54	C₈	#16864655	1	1	yes	characteristic
55	C₂²	C₂²×C₂₇₁⋊C₈	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×C₂₇₁⋊C₁₆	1	6	yes	—
58	C₄	Q₈×Dic₂₇₁	1	1	yes	characteristic
59	C₄	#22576167	1	1	yes	characteristic
60	C₂	Q₈×C₂₇₁⋊C₈	1	1	yes	characteristic
61	C₂	C₂²×C₂₇₁⋊C₁₆	1	1	yes	characteristic
62	C₂	#31774314	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₇₁
commutator	C₂×C₂₇₁	C₂²×C₁₆
Fitting	C₈×Q₈×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	C₂²×C₈
cosocle	C₂×C₈×C₂₇₁	C₂³
hypercenter	C₈×Q₈	D₂₇₁
hypocenter	C₂₇₁	Q₈×C₁₆
LCS	G ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
10. C₂₇₁⋊C₁₆	Q₈	direct (N × H)
16. Q₈×C₂₇₁	C₁₆	semidirect
31. C₂₇₁	Q₈×C₁₆	semidirect
48. Q₈	C₂₇₁⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₇₁⋊C₁₆)	2
C₄×C₂₇₁⋊C₁₆	2
C₈×Q₈×C₂₇₁	2
Q₈×C₁₆	271

Subgroups — Q8×C271⋊C16 #46838268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂₇₁⋊C₁₆ #46838268