Subgroups — C₂×C₅×C₇×Q₁₆ #11765304

Order 1120.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇×Q₈	C₂	1	2	yes	maximal
2	C₅×C₇×Q₁₆	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×C₇×Q₈	—	2	2	no	—
5	C₅×C₇×Q₈	C₂²	1	4	yes	—
6	C₂×C₄×C₅×C₇	—	2	2	no	—
7	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
8	C₅×C₇×C₈	C₂²	1	2	yes	—
9	C₂×C₇×Q₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₅×Q₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₅×C₇	D₄	1	1	yes	characteristic
12	C₄×C₅×C₇	—	2	4	no	—
13	C₄×C₅×C₇	D₄	1	1	yes	characteristic
14	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₂×C₇×Q₈	C₂×C₅	1	2	yes	—
16	C₇×Q₁₆	C₂×C₅	1	4	yes	direct product factor
17	C₂×C₇×C₈	C₂×C₅	1	1	yes	characteristic
18	C₂×C₅×Q₈	C₂×C₇	1	2	yes	—
19	C₅×Q₁₆	C₂×C₇	1	4	yes	direct product factor
20	C₂×C₅×C₈	C₂×C₇	1	1	yes	characteristic
21	C₂×C₅×C₇	Q₁₆	1	2	yes	direct product factor
22	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
23	C₇×Q₈	—	2	2	no	—
24	C₇×Q₈	C₂²×C₅	1	4	yes	—
25	C₂×C₄×C₇	—	2	2	no	—
26	C₂×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
27	C₇×C₈	C₂²×C₅	1	2	yes	—
28	C₅×Q₈	—	2	2	no	—
29	C₅×Q₈	C₂²×C₇	1	4	yes	—
30	C₂×C₄×C₅	—	2	2	no	—
31	C₂×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
32	C₅×C₈	C₂²×C₇	1	2	yes	—
33	C₅×C₇	C₂×Q₁₆	1	1	yes	characteristic, direct product factor, Hall
34	C₂×Q₁₆	C₅×C₇	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²×C₇	C₅×D₄	1	1	yes	characteristic
36	C₄×C₇	—	2	4	no	—
37	C₄×C₇	C₅×D₄	1	1	yes	characteristic
38	C₄×C₇	C₂³×C₅	1	1	yes	characteristic
39	C₂²×C₅	C₇×D₄	1	1	yes	characteristic
40	C₄×C₅	—	2	4	no	—
41	C₄×C₅	C₇×D₄	1	1	yes	characteristic
42	C₄×C₅	C₂³×C₇	1	1	yes	characteristic
43	C₂×Q₈	C₂×C₅×C₇	1	2	yes	—
44	Q₁₆	C₂×C₅×C₇	1	4	yes	direct product factor
45	C₂×C₈	C₂×C₅×C₇	1	1	yes	characteristic
46	C₂×C₇	C₅×Q₁₆	1	2	yes	direct product factor
47	C₂×C₇	C₂×C₅×D₄	1	1	yes	characteristic
48	C₂×C₅	C₇×Q₁₆	1	2	yes	direct product factor
49	C₂×C₅	C₂×C₇×D₄	1	1	yes	characteristic
50	Q₈	—	2	2	no	—
51	Q₈	C₂²×C₅×C₇	1	4	yes	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	C₂²×C₅×C₇	1	1	yes	characteristic
54	C₈	C₂²×C₅×C₇	1	2	yes	—
55	C₇	C₂×C₅×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
56	C₅	C₂×C₇×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²	C₅×C₇×D₄	1	1	yes	characteristic
58	C₄	—	2	4	no	—
59	C₄	C₅×C₇×D₄	1	1	yes	characteristic
60	C₄	C₂³×C₅×C₇	1	1	yes	characteristic
61	C₂	C₅×C₇×Q₁₆	1	2	yes	direct product factor
62	C₂	C₂×C₅×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₅×C₇	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₅×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₇ ◁ C₂×C₄×C₅×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×C₇×Q₁₆	C₂	direct (N × H)
9. C₂×C₇×Q₁₆	C₅	direct (N × H)
10. C₂×C₅×Q₁₆	C₇	direct (N × H)
16. C₇×Q₁₆	C₂×C₅	direct (N × H)
19. C₅×Q₁₆	C₂×C₇	direct (N × H)
21. C₂×C₅×C₇	Q₁₆	direct (N × H)
33. C₅×C₇	C₂×Q₁₆	direct (N × H)
34. C₂×Q₁₆	C₅×C₇	direct (N × H)
44. Q₁₆	C₂×C₅×C₇	direct (N × H)
46. C₂×C₇	C₅×Q₁₆	direct (N × H)
48. C₂×C₅	C₇×Q₁₆	direct (N × H)
55. C₇	C₂×C₅×Q₁₆	direct (N × H)
56. C₅	C₂×C₇×Q₁₆	direct (N × H)
61. C₂	C₅×C₇×Q₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₇×Q₈	2
C₅×C₇×Q₁₆	2
C₂×C₅×C₇×C₈	2
C₂×C₇×Q₁₆	5
C₂×C₅×Q₁₆	7

Subgroups — C2×C5×C7×Q16 #11765304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₇×Q₁₆ #11765304