Subgroups — C₂×Dic₃×D₇ #66682

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₃×D₇	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×D₇	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×D₇	C₄	1	4	yes	complemented
8	C₂×C₃×D₇	C₂²	1	2	yes	—
9	Dic₂₁	C₂²	1	2	yes	complemented
10	C₇×Dic₃	C₂²	1	2	yes	complemented
11	C₂²×D₇	D₃	1	1	yes	characteristic
12	C₂×C₄×C₇	—	3	1	no	—
13	C₂×Dic₇	—	3	1	no	—
14	C₄×D₇	—	3	4	no	—
15	C₂²×Dic₃	—	7	1	no	maximal, Hall
16	C₂×C₃×C₇	C₂×C₄	1	2	yes	complemented
17	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
18	C₃×D₇	C₂×C₄	1	4	yes	complemented
19	C₂²×C₇	C₂×D₃	1	1	yes	characteristic
20	C₂×D₇	Dic₃	1	4	yes	direct product factor
21	C₂×D₇	C₂×D₃	1	2	yes	—
22	C₄×C₇	—	3	2	no	—
23	Dic₇	—	3	2	no	—
24	C₂³×C₃	—	7	1	no	—
25	C₂×Dic₃	—	7	1	no	—
26	C₂×Dic₃	—	7	4	no	—
27	C₂×Dic₃	D₇	1	1	yes	characteristic, direct product factor
28	C₃×C₇	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×C₄	—	21	1	no	Sylow
30	C₂×C₇	C₂×Dic₃	1	2	yes	complemented
31	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
32	D₇	C₂×Dic₃	1	4	yes	direct product factor
33	C₂²×C₃	—	7	4	no	—
34	C₂²×C₃	—	7	2	no	—
35	C₂²×C₃	C₂×D₇	1	1	yes	characteristic
36	Dic₃	—	7	2	no	—
37	Dic₃	C₂×D₇	1	2	yes	direct product factor
38	C₂³	—	7	1	no	—
39	C₂×C₄	—	3	1	no	—
40	C₂×C₄	—	21	4	no	—
41	C₂×C₄	—	21	1	no	—
42	C₇	C₂²×Dic₃	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	7	4	no	—
44	C₂×C₃	C₄×D₇	1	2	yes	complemented
45	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
46	C₂²	—	7	4	no	—
47	C₂²	—	7	2	no	—
48	C₂²	D₃×D₇	1	1	yes	characteristic
49	C₄	—	3	2	no	—
50	C₄	—	21	2	no	—
51	C₃	C₂×C₄×D₇	1	1	yes	characteristic, complemented, Sylow
52	C₂	—	7	4	no	—
53	C₂	Dic₃×D₇	1	2	yes	direct product factor
54	C₂	C₂×D₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	D₃×D₇
commutator, hypocenter	C₃×C₇	C₂²×C₄
Frattini	C₂	C₂×D₃×D₇
Fitting, socle	C₂²×C₃×C₇	C₂²
prime radical	C₂²×C₃×D₇	C₂
cosocle	C₂×C₃×C₇	C₂³
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₂₁	C₂	semidirect
3. C₂×C₇×Dic₃	C₂	semidirect
4. Dic₃×D₇	C₂	direct (N × H)
7. C₂×C₃×D₇	C₄	semidirect
9. Dic₂₁	C₂²	semidirect
10. C₇×Dic₃	C₂²	semidirect
16. C₂×C₃×C₇	C₂×C₄	semidirect
18. C₃×D₇	C₂×C₄	semidirect
20. C₂×D₇	Dic₃	direct (N × H)
27. C₂×Dic₃	D₇	direct (N × H)
28. C₃×C₇	C₂²×C₄	semidirect
30. C₂×C₇	C₂×Dic₃	semidirect
32. D₇	C₂×Dic₃	direct (N × H)
37. Dic₃	C₂×D₇	direct (N × H)
42. C₇	C₂²×Dic₃	semidirect
44. C₂×C₃	C₄×D₇	semidirect
51. C₃	C₂×C₄×D₇	semidirect
53. C₂	Dic₃×D₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₇	2
C₂×Dic₂₁	2
C₂×C₇×Dic₃	2
Dic₃×D₇	2
C₂×C₄×D₇	3
C₂²×Dic₃	7

Subgroups — C2×Dic3×D7 #66682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₃×D₇ #66682