Subgroups — C₅×C₇×Dic₁₂ #13132309

Order 1680.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×Dic₆	C₂	1	2	yes	maximal
2	C₃×C₅×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×C₇×Q₁₆	—	3	1	no	maximal, Hall
4	C₃×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
5	C₅×C₇×Dic₃	—	2	2	no	—
6	C₇×Dic₁₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅×C₇×Q₈	—	3	2	no	—
8	C₅×C₇×C₈	D₃	1	1	yes	characteristic
9	C₅×Dic₁₂	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×C₇	D₄	1	1	yes	characteristic
11	C₇×Dic₆	C₂×C₅	1	2	yes	—
12	C₃×C₇×C₈	C₂×C₅	1	1	yes	characteristic
13	C₄×C₅×C₇	—	6	2	no	—
14	C₄×C₅×C₇	C₂×D₃	1	1	yes	characteristic
15	C₅×Dic₆	C₂×C₇	1	2	yes	—
16	C₃×C₅×C₈	C₂×C₇	1	1	yes	characteristic
17	C₇×Q₁₆	—	3	1	no	Hall
18	C₃×C₅×C₇	Q₁₆	1	1	yes	characteristic, complemented, Hall
19	C₃×C₄×C₇	C₂²×C₅	1	1	yes	characteristic
20	C₇×Dic₃	—	2	2	no	—
21	C₅×Q₁₆	—	3	1	no	Hall
22	C₂×C₅×C₇	D₁₂	1	1	yes	characteristic
23	C₃×C₄×C₅	C₂²×C₇	1	1	yes	characteristic
24	C₅×Dic₃	—	2	2	no	—
25	C₇×Q₈	—	3	2	no	—
26	C₇×C₈	C₅×D₃	1	1	yes	characteristic
27	Dic₁₂	C₅×C₇	1	1	yes	characteristic, direct product factor, Hall
28	C₂×C₃×C₇	C₅×D₄	1	1	yes	characteristic
29	C₅×Q₈	—	3	2	no	—
30	C₅×C₈	D₃×C₇	1	1	yes	characteristic
31	C₅×C₇	Dic₁₂	1	1	yes	characteristic, direct product factor, Hall
32	C₂×C₃×C₅	C₇×D₄	1	1	yes	characteristic
33	C₄×C₇	—	6	2	no	—
34	C₄×C₇	C₂×C₅×D₃	1	1	yes	characteristic
35	Dic₆	C₂×C₅×C₇	1	2	yes	—
36	C₃×C₈	C₂×C₅×C₇	1	1	yes	characteristic
37	C₃×C₇	C₅×Q₁₆	1	1	yes	characteristic, complemented, Hall
38	C₄×C₅	—	6	2	no	—
39	C₄×C₅	C₂×D₃×C₇	1	1	yes	characteristic
40	Q₁₆	—	3	1	no	Sylow
41	C₃×C₅	C₇×Q₁₆	1	1	yes	characteristic, complemented, Hall
42	C₂×C₇	C₅×D₁₂	1	1	yes	characteristic
43	C₃×C₄	C₂²×C₅×C₇	1	1	yes	characteristic
44	Dic₃	—	2	2	no	—
45	C₂×C₅	C₇×D₁₂	1	1	yes	characteristic
46	Q₈	—	3	2	no	—
47	C₈	C₅×D₃×C₇	1	1	yes	characteristic
48	C₇	C₅×Dic₁₂	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×C₃	C₅×C₇×D₄	1	1	yes	characteristic
50	C₅	C₇×Dic₁₂	1	1	yes	characteristic, direct product factor, Sylow
51	C₄	—	6	2	no	—
52	C₄	C₂×C₅×D₃×C₇	1	1	yes	characteristic
53	C₃	C₅×C₇×Q₁₆	1	1	yes	characteristic, complemented, Sylow
54	C₂	C₅×C₇×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₇	D₁₂
commutator, cosocle	C₃×C₄	C₂²×C₅×C₇
Frattini	C₄	C₂×C₅×D₃×C₇
Fitting	C₃×C₅×C₇×C₈	C₂
socle, prime radical	C₂×C₃×C₅×C₇	D₄
hypercenter	C₅×C₇×C₈	D₃
hypocenter	C₃	C₅×C₇×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₇ ◁ C₄×C₅×C₇ ◁ C₅×C₇×C₈

Complements

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

N	H ≅ G/N	type
6. C₇×Dic₁₂	C₅	direct (N × H)
9. C₅×Dic₁₂	C₇	direct (N × H)
18. C₃×C₅×C₇	Q₁₆	semidirect
27. Dic₁₂	C₅×C₇	direct (N × H)
31. C₅×C₇	Dic₁₂	direct (N × H)
37. C₃×C₇	C₅×Q₁₆	semidirect
41. C₃×C₅	C₇×Q₁₆	semidirect
48. C₇	C₅×Dic₁₂	direct (N × H)
50. C₅	C₇×Dic₁₂	direct (N × H)
53. C₃	C₅×C₇×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₇×Dic₆	2
C₃×C₅×C₇×C₈	2
C₅×C₇×Q₁₆	3
C₇×Dic₁₂	5
C₅×Dic₁₂	7

Subgroups — C5×C7×Dic12 #13132309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇×Dic₁₂ #13132309