Subgroups — Q₈×C₁₆×C₆₇ #22556613

Order 8576.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₁₆×C₆₇	C₂	1	3	yes	maximal
3	C₄×C₁₆×C₆₇	C₂	1	3	yes	maximal
4	C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₆₇	C₂²	1	3	yes	—
6	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
7	C₄⋊C₈×C₆₇	C₄	1	3	yes	—
8	C₄×C₈×C₆₇	C₂²	1	3	yes	—
9	C₂×Q₈×C₆₇	C₈	1	1	yes	characteristic
10	C₂×C₈×C₆₇	C₂×C₄	1	3	yes	—
11	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
12	C₄⋊C₄×C₆₇	C₈	1	3	yes	—
13	C₄²×C₆₇	C₂×C₄	1	3	yes	—
14	C₁₆×C₆₇	—	2	3	no	—
15	C₁₆×C₆₇	Q₈	1	2	yes	direct product factor
16	Q₈×C₆₇	C₁₆	1	4	yes	direct product factor
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	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₆₇	Q₈×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
33	C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
34	C₄⋊C₁₆	C₂×C₆₇	1	3	yes	—
35	C₄×C₁₆	C₂×C₆₇	1	3	yes	—
36	C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
37	C₂×C₁₆	C₂²×C₆₇	1	3	yes	—
38	C₂×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
39	C₄⋊C₈	C₄×C₆₇	1	3	yes	—
40	C₄×C₈	C₂²×C₆₇	1	3	yes	—
41	C₂×Q₈	C₈×C₆₇	1	1	yes	characteristic
42	C₂×C₈	C₂×C₄×C₆₇	1	3	yes	—
43	C₂×C₈	C₂³×C₆₇	1	1	yes	characteristic
44	C₄⋊C₄	C₈×C₆₇	1	3	yes	—
45	C₄²	C₂×C₄×C₆₇	1	3	yes	—
46	C₁₆	—	2	3	no	—
47	C₁₆	Q₈×C₆₇	1	2	yes	direct product factor
48	Q₈	C₁₆×C₆₇	1	4	yes	direct product factor
49	C₂×C₄	C₂×C₈×C₆₇	1	3	yes	—
50	C₂×C₄	C₂×C₈×C₆₇	1	3	yes	—
51	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
52	C₈	—	2	3	no	—
53	C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
54	C₈	#41×C₆₇	1	1	yes	characteristic
55	C₂²	C₂²×C₈×C₆₇	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×C₁₆×C₆₇	1	6	yes	—
58	C₄	C₄×Q₈×C₆₇	1	1	yes	characteristic
59	C₄	#131×C₆₇	1	1	yes	characteristic
60	C₂	C₈×Q₈×C₆₇	1	1	yes	characteristic
61	C₂	C₂²×C₁₆×C₆₇	1	1	yes	characteristic
62	C₂	#504×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆×C₆₇	C₂²
commutator	C₂	C₂²×C₁₆×C₆₇
Frattini, cosocle	C₂×C₈	C₂³×C₆₇
socle, prime radical	C₂²×C₆₇	C₂²×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₆₇	Q₈	direct (N × H)
16. Q₈×C₆₇	C₁₆	direct (N × H)
31. Q₈×C₁₆	C₆₇	direct (N × H)
32. C₆₇	Q₈×C₁₆	direct (N × H)
47. C₁₆	Q₈×C₆₇	direct (N × H)
48. Q₈	C₁₆×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×Q₈×C₆₇	2
C₄⋊C₁₆×C₆₇	2
C₄×C₁₆×C₆₇	2
Q₈×C₁₆	67

Subgroups — Q8×C16×C67 #22556613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₁₆×C₆₇ #22556613