Subgroups — Q₈×C₆₇⋊C₁₆ #22562113

Order 8576.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₆₇⋊C₁₆)	C₂	1	3	yes	maximal
2	C₄×C₆₇⋊C₁₆	C₂	1	3	yes	maximal
3	C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₇⋊C₁₆	C₂²	1	3	yes	—
5	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
6	C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₆₇	C₄	1	3	yes	—
8	C₄×C₈×C₆₇	C₂²	1	3	yes	—
9	C₆₇⋊C₁₆	—	2	3	no	—
10	C₆₇⋊C₁₆	Q₈	1	2	yes	direct product factor
11	C₂×Q₈×C₆₇	C₈	1	1	yes	characteristic
12	C₂×C₈×C₆₇	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
14	C₄⋊C₄×C₆₇	C₈	1	3	yes	—
15	C₄²×C₆₇	C₂×C₄	1	3	yes	—
16	Q₈×C₆₇	C₁₆	1	4	yes	complemented
17	C₂×C₄×C₆₇	C₂×C₈	1	3	yes	—
18	C₂×C₄×C₆₇	C₂×C₈	1	3	yes	—
19	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
20	C₈×C₆₇	—	2	3	no	—
21	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
22	C₈×C₆₇	#41	1	1	yes	characteristic
23	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
24	C₄×C₆₇	—	2	3	no	—
25	C₄×C₆₇	C₂×C₁₆	1	6	yes	—
26	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
27	C₄×C₆₇	#131	1	1	yes	characteristic
28	C₂×C₆₇	C₈×Q₈	1	1	yes	characteristic
29	C₂×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
30	C₂×C₆₇	#504	1	1	yes	characteristic
31	Q₈×C₁₆	—	67	1	no	maximal, Sylow
32	C₆₇	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
33	C₈×Q₈	D₆₇	1	1	yes	characteristic
34	C₄⋊C₁₆	—	67	3	no	—
35	C₄×C₁₆	—	67	3	no	—
36	C₄×Q₈	Dic₆₇	1	1	yes	characteristic
37	C₂×C₁₆	—	67	1	no	—
38	C₂×C₁₆	—	67	3	no	—
39	C₄⋊C₈	Dic₆₇	1	3	yes	—
40	C₄×C₈	C₂×D₆₇	1	3	yes	—
41	C₂×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
42	C₂×C₈	C₂×Dic₆₇	1	3	yes	—
43	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
44	C₄⋊C₄	C₆₇⋊C₈	1	3	yes	—
45	C₄²	C₂×Dic₆₇	1	3	yes	—
46	C₁₆	—	67	2	no	—
47	C₁₆	—	134	3	no	—
48	Q₈	C₆₇⋊C₁₆	1	4	yes	direct product factor
49	C₂×C₄	C₂×C₆₇⋊C₈	1	3	yes	—
50	C₂×C₄	C₂×C₆₇⋊C₈	1	3	yes	—
51	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
52	C₈	—	2	3	no	—
53	C₈	Q₈×D₆₇	1	1	yes	characteristic
54	C₈	#11761629	1	1	yes	characteristic
55	C₂²	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×C₆₇⋊C₁₆	1	6	yes	—
58	C₄	Q₈×Dic₆₇	1	1	yes	characteristic
59	C₄	#14524050	1	1	yes	characteristic
60	C₂	Q₈×C₆₇⋊C₈	1	1	yes	characteristic
61	C₂	C₂²×C₆₇⋊C₁₆	1	1	yes	characteristic
62	C₂	#16863813	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂²×C₁₆
Fitting	C₈×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₈
hypercenter	C₈×Q₈	D₆₇
hypocenter	C₆₇	Q₈×C₁₆
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
10. C₆₇⋊C₁₆	Q₈	direct (N × H)
16. Q₈×C₆₇	C₁₆	semidirect
32. C₆₇	Q₈×C₁₆	semidirect
48. Q₈	C₆₇⋊C₁₆	direct (N × H)

Subgroups — Q8×C67⋊C16 #22562113

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×C₆₇⋊C₁₆ #22562113