Subgroups — C₂×C₄×D₃×C₇ #66716

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×C₇	C₂	semidirect
3. C₂×C₇×Dic₃	C₂	semidirect
4. C₄×D₃×C₇	C₂	direct (N × H)
7. C₂×D₃×C₇	C₄	direct (N × H)
9. C₃×C₄×C₇	C₂²	semidirect
10. C₇×Dic₃	C₂²	semidirect
14. C₂×C₄×C₇	D₃	direct (N × H)
15. C₂×C₄×D₃	C₇	direct (N × H)
16. C₂×C₃×C₇	C₂×C₄	semidirect
18. D₃×C₇	C₂×C₄	direct (N × H)
23. C₄×C₇	C₂×D₃	direct (N × H)
25. C₂×C₃×C₄	C₂×C₇	semidirect
26. C₂×Dic₃	C₂×C₇	semidirect
27. C₄×D₃	C₂×C₇	direct (N × H)
28. C₃×C₇	C₂²×C₄	semidirect
31. C₂×C₇	C₄×D₃	direct (N × H)
34. C₂×D₃	C₄×C₇	direct (N × H)
36. C₃×C₄	C₂²×C₇	semidirect
37. Dic₃	C₂²×C₇	semidirect
41. C₂×C₄	D₃×C₇	direct (N × H)
42. C₇	C₂×C₄×D₃	direct (N × H)
43. C₂×C₃	C₂×C₄×C₇	semidirect
45. D₃	C₂×C₄×C₇	direct (N × H)
50. C₄	C₂×D₃×C₇	direct (N × H)
51. C₃	C₂²×C₄×C₇	semidirect
53. C₂	C₄×D₃×C₇	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×D₃×C₇	2
C₂×C₃×C₄×C₇	2
C₂×C₇×Dic₃	2
C₄×D₃×C₇	2
C₂²×C₄×C₇	3
C₂×C₄×D₃	7

Subgroups — C2×C4×D3×C7 #66716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₃×C₇ #66716