Subgroups — C₂×C₃×C₇×D₄ #66736

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₇	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×D₄	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×D₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₇	—	2	4	no	—
6	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₇	C₂²	1	4	yes	complemented
8	C₃×C₄×C₇	C₂²	1	2	yes	complemented
9	C₂³×C₇	C₂×C₃	1	2	yes	complemented
10	C₇×D₄	C₂×C₃	1	4	yes	direct product factor
11	C₂×C₄×C₇	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂×C₃×D₄	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₇	—	2	4	no	—
14	C₂×C₃×C₇	D₄	1	2	yes	direct product factor
15	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
16	C₂²×C₇	—	2	4	no	—
17	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
18	C₂²×C₇	C₂²×C₃	1	4	yes	complemented
19	C₄×C₇	C₂²×C₃	1	2	yes	complemented
20	C₂³×C₃	C₂×C₇	1	2	yes	complemented
21	C₃×D₄	C₂×C₇	1	4	yes	direct product factor
22	C₂×C₃×C₄	C₂×C₇	1	1	yes	characteristic, complemented
23	C₃×C₇	C₂×D₄	1	1	yes	characteristic, direct product factor, Hall
24	C₂×D₄	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
25	C₂×C₇	—	2	4	no	—
26	C₂×C₇	C₃×D₄	1	2	yes	direct product factor
27	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
28	C₂²×C₃	—	2	4	no	—
29	C₂²×C₃	C₂²×C₇	1	1	yes	characteristic
30	C₂²×C₃	C₂²×C₇	1	4	yes	complemented
31	C₃×C₄	C₂²×C₇	1	2	yes	complemented
32	C₂³	C₂×C₃×C₇	1	2	yes	complemented
33	D₄	C₂×C₃×C₇	1	4	yes	direct product factor
34	C₂×C₄	C₂×C₃×C₇	1	1	yes	characteristic, complemented
35	C₇	C₂×C₃×D₄	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₃	—	2	4	no	—
37	C₂×C₃	C₇×D₄	1	2	yes	direct product factor
38	C₂×C₃	C₂³×C₇	1	1	yes	characteristic
39	C₂²	—	2	4	no	—
40	C₂²	C₂²×C₃×C₇	1	1	yes	characteristic
41	C₂²	C₂²×C₃×C₇	1	4	yes	complemented
42	C₄	C₂²×C₃×C₇	1	2	yes	complemented
43	C₃	C₂×C₇×D₄	1	1	yes	characteristic, direct product factor, Sylow
44	C₂	—	2	4	no	—
45	C₂	C₃×C₇×D₄	1	2	yes	direct product factor
46	C₂	C₂³×C₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂²
commutator, Frattini, cosocle	C₂	C₂³×C₃×C₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₇	C₂	semidirect
2. C₃×C₇×D₄	C₂	direct (N × H)
3. C₂×C₃×C₄×C₇	C₂	semidirect
4. C₂×C₇×D₄	C₃	direct (N × H)
7. C₂²×C₃×C₇	C₂²	semidirect
8. C₃×C₄×C₇	C₂²	semidirect
9. C₂³×C₇	C₂×C₃	semidirect
10. C₇×D₄	C₂×C₃	direct (N × H)
11. C₂×C₄×C₇	C₂×C₃	semidirect
12. C₂×C₃×D₄	C₇	direct (N × H)
14. C₂×C₃×C₇	D₄	direct (N × H)
18. C₂²×C₇	C₂²×C₃	semidirect
19. C₄×C₇	C₂²×C₃	semidirect
20. C₂³×C₃	C₂×C₇	semidirect
21. C₃×D₄	C₂×C₇	direct (N × H)
22. C₂×C₃×C₄	C₂×C₇	semidirect
23. C₃×C₇	C₂×D₄	direct (N × H)
24. C₂×D₄	C₃×C₇	direct (N × H)
26. C₂×C₇	C₃×D₄	direct (N × H)
30. C₂²×C₃	C₂²×C₇	semidirect
31. C₃×C₄	C₂²×C₇	semidirect
32. C₂³	C₂×C₃×C₇	semidirect
33. D₄	C₂×C₃×C₇	direct (N × H)
34. C₂×C₄	C₂×C₃×C₇	semidirect
35. C₇	C₂×C₃×D₄	direct (N × H)
37. C₂×C₃	C₇×D₄	direct (N × H)
41. C₂²	C₂²×C₃×C₇	semidirect
42. C₄	C₂²×C₃×C₇	semidirect
43. C₃	C₂×C₇×D₄	direct (N × H)
45. C₂	C₃×C₇×D₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₇	2
C₃×C₇×D₄	2
C₂×C₃×C₄×C₇	2
C₂×C₇×D₄	3
C₂×C₃×D₄	7

Subgroups — C2×C3×C7×D4 #66736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×D₄ #66736