Subgroups — C₂×C₇×Dic₆ #66715

Order 336.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Dic₃	C₂	1	2	yes	maximal
3	C₇×Dic₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×Q₈	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₇	C₂²	1	2	yes	—
7	C₇×Dic₃	C₂²	1	4	yes	—
8	C₇×Q₈	—	3	4	no	—
9	C₂×C₄×C₇	—	3	2	no	—
10	C₂×C₄×C₇	D₃	1	1	yes	characteristic
11	C₂×Dic₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₃×C₇	Q₈	1	2	yes	complemented
13	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
14	C₂²×C₇	C₂×D₃	1	1	yes	characteristic
15	C₄×C₇	—	3	4	no	—
16	C₄×C₇	C₂×D₃	1	2	yes	—
17	C₂×C₃×C₄	C₂×C₇	1	1	yes	characteristic
18	C₂×Dic₃	C₂×C₇	1	2	yes	—
19	Dic₆	C₂×C₇	1	4	yes	direct product factor
20	C₃×C₇	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
21	C₂×Q₈	—	3	1	no	Sylow
22	C₂×C₇	Dic₆	1	2	yes	direct product factor
23	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
24	C₂²×C₃	C₂²×C₇	1	1	yes	characteristic
25	C₃×C₄	C₂²×C₇	1	2	yes	—
26	Dic₃	C₂²×C₇	1	4	yes	—
27	Q₈	—	3	4	no	—
28	C₂×C₄	—	3	2	no	—
29	C₂×C₄	D₃×C₇	1	1	yes	characteristic
30	C₇	C₂×Dic₆	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃	C₇×Q₈	1	2	yes	complemented
32	C₂×C₃	C₂³×C₇	1	1	yes	characteristic
33	C₂²	C₂×D₃×C₇	1	1	yes	characteristic
34	C₄	—	3	4	no	—
35	C₄	C₂×D₃×C₇	1	2	yes	—
36	C₃	C₂×C₇×Q₈	1	1	yes	characteristic, complemented, Sylow
37	C₂	C₇×Dic₆	1	2	yes	direct product factor
38	C₂	C₂²×D₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂×D₃
commutator, cosocle	C₂×C₃	C₂³×C₇
Frattini	C₂	C₂²×D₃×C₇
Fitting	C₂×C₃×C₄×C₇	C₂
socle, prime radical	C₂²×C₃×C₇	C₂²
hypercenter	C₂×C₄×C₇	D₃
hypocenter	C₃	C₂×C₇×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇

Complements

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

N	H ≅ G/N	type
3. C₇×Dic₆	C₂	direct (N × H)
11. C₂×Dic₆	C₇	direct (N × H)
12. C₂×C₃×C₇	Q₈	semidirect
19. Dic₆	C₂×C₇	direct (N × H)
20. C₃×C₇	C₂×Q₈	semidirect
22. C₂×C₇	Dic₆	direct (N × H)
30. C₇	C₂×Dic₆	direct (N × H)
31. C₂×C₃	C₇×Q₈	semidirect
36. C₃	C₂×C₇×Q₈	semidirect
37. C₂	C₇×Dic₆	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×C₃×C₄×C₇	2
C₂×C₇×Dic₃	2
C₇×Dic₆	2
C₂×C₇×Q₈	3
C₂×Dic₆	7

Subgroups — C2×C7×Dic6 #66715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×Dic₆ #66715