Subgroups — C₃×D₄×D₇ #66709

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	2	yes	maximal, complemented
2	C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
4	C₃×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×D₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₇	C₂²	1	2	yes	complemented
8	C₂×C₃×D₇	—	2	4	no	—
9	C₂×C₃×D₇	C₂²	1	1	yes	characteristic
10	C₂×C₃×D₇	C₂²	1	2	yes	complemented
11	C₃×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
12	C₃×Dic₇	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₇	C₂×C₃	1	2	yes	complemented
14	C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
15	C₂²⋊D₇	C₂×C₃	1	2	yes	complemented
16	D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
17	C₄×D₇	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂×C₃×D₄	—	7	1	no	maximal, Hall
19	C₂×C₃×C₇	—	2	2	no	—
20	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
21	C₃×D₇	—	2	2	no	—
22	C₃×D₇	D₄	1	2	yes	direct product factor
23	C₂²×C₇	C₂²×C₃	1	2	yes	complemented
24	C₂×D₇	—	2	4	no	—
25	C₂×D₇	C₂²×C₃	1	1	yes	characteristic
26	C₂×D₇	C₂²×C₃	1	2	yes	complemented
27	C₄×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
28	Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
29	C₂³×C₃	—	7	2	no	—
30	C₃×D₄	—	7	2	no	—
31	C₃×D₄	—	7	1	no	—
32	C₃×D₄	D₇	1	1	yes	characteristic, direct product factor
33	C₂×C₃×C₄	—	7	1	no	—
34	C₃×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
35	C₂×D₄	—	7	1	no	Sylow
36	C₂×C₇	—	2	2	no	—
37	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
38	D₇	—	2	2	no	—
39	D₇	C₃×D₄	1	2	yes	direct product factor
40	C₂²×C₃	—	7	1	no	—
41	C₂²×C₃	—	7	2	no	—
42	C₂²×C₃	—	14	4	no	—
43	C₂²×C₃	C₂×D₇	1	2	yes	complemented
44	C₃×C₄	—	7	1	no	—
45	C₃×C₄	C₂×D₇	1	1	yes	characteristic, complemented
46	C₂³	—	7	2	no	—
47	D₄	—	7	2	no	—
48	D₄	—	7	1	no	—
49	D₄	C₃×D₇	1	1	yes	characteristic, direct product factor
50	C₂×C₄	—	7	1	no	—
51	C₇	C₂×C₃×D₄	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃	—	2	2	no	—
53	C₂×C₃	—	7	2	no	—
54	C₂×C₃	—	14	2	no	—
55	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
56	C₂²	—	7	1	no	—
57	C₂²	—	7	2	no	—
58	C₂²	—	14	4	no	—
59	C₂²	C₂×C₃×D₇	1	2	yes	complemented
60	C₄	—	7	1	no	—
61	C₄	C₂×C₃×D₇	1	1	yes	characteristic, complemented
62	C₃	D₄×D₇	1	1	yes	characteristic, direct product factor, Sylow
63	C₂	—	2	2	no	—
64	C₂	—	7	2	no	—
65	C₂	—	14	2	no	—
66	C₂	C₂²×C₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×D₇
commutator, cosocle	C₂×C₇	C₂³×C₃
Frattini	C₂	C₂²×C₃×D₇
Fitting	C₃×C₇×D₄	C₂
socle	C₂×C₃×C₇	C₂³
hypercenter	C₃×D₄	D₇
hypocenter	C₇	C₂×C₃×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₇	C₂	semidirect
2. C₃×C₇×D₄	C₂	semidirect
3. C₃×C₂²⋊D₇	C₂	semidirect
4. C₃×D₂₈	C₂	semidirect
5. C₃×C₄×D₇	C₂	semidirect
6. D₄×D₇	C₃	direct (N × H)
7. C₂²×C₃×C₇	C₂²	semidirect
10. C₂×C₃×D₇	C₂²	semidirect
11. C₃×C₄×C₇	C₂²	semidirect
12. C₃×Dic₇	C₂²	semidirect
13. C₂²×D₇	C₂×C₃	semidirect
14. C₇×D₄	C₂×C₃	semidirect
15. C₂²⋊D₇	C₂×C₃	semidirect
16. D₂₈	C₂×C₃	semidirect
17. C₄×D₇	C₂×C₃	semidirect
22. C₃×D₇	D₄	direct (N × H)
23. C₂²×C₇	C₂²×C₃	semidirect
26. C₂×D₇	C₂²×C₃	semidirect
27. C₄×C₇	C₂²×C₃	semidirect
28. Dic₇	C₂²×C₃	semidirect
32. C₃×D₄	D₇	direct (N × H)
34. C₃×C₇	C₂×D₄	semidirect
39. D₇	C₃×D₄	direct (N × H)
43. C₂²×C₃	C₂×D₇	semidirect
45. C₃×C₄	C₂×D₇	semidirect
49. D₄	C₃×D₇	direct (N × H)
51. C₇	C₂×C₃×D₄	semidirect
59. C₂²	C₂×C₃×D₇	semidirect
61. C₄	C₂×C₃×D₇	semidirect
62. C₃	D₄×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
C₃×C₇×D₄	2
C₃×C₂²⋊D₇	2
C₃×D₂₈	2
C₃×C₄×D₇	2
D₄×D₇	3
C₂×C₃×D₄	7

Subgroups — C3×D4×D7 #66709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×D₇ #66709