Subgroups — C₂×C₃×C₄×D₇ #66706

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×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×D₇	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×D₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×D₇	C₄	1	4	yes	direct product factor
8	C₂×C₃×D₇	C₂²	1	2	yes	—
9	C₃×C₄×C₇	C₂²	1	2	yes	complemented
10	C₃×Dic₇	C₂²	1	2	yes	complemented
11	C₂²×D₇	C₂×C₃	1	1	yes	characteristic
12	C₂×C₄×C₇	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
14	C₄×D₇	C₂×C₃	1	4	yes	direct product factor
15	C₂²×C₃×C₄	—	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	direct product factor
19	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
20	C₂×D₇	C₃×C₄	1	4	yes	direct product factor
21	C₂×D₇	C₂²×C₃	1	2	yes	—
22	C₄×C₇	C₂²×C₃	1	2	yes	complemented
23	Dic₇	C₂²×C₃	1	2	yes	complemented
24	C₂³×C₃	—	7	1	no	—
25	C₂×C₃×C₄	—	7	1	no	—
26	C₂×C₃×C₄	—	7	4	no	—
27	C₂×C₃×C₄	D₇	1	1	yes	characteristic, direct product factor
28	C₃×C₇	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×C₄	—	7	1	no	Sylow
30	C₂×C₇	C₂×C₃×C₄	1	2	yes	complemented
31	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
32	D₇	C₂×C₃×C₄	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	C₃×C₄	—	7	2	no	—
37	C₃×C₄	C₂×D₇	1	2	yes	direct product factor
38	C₂³	—	7	1	no	—
39	C₂×C₄	—	7	1	no	—
40	C₂×C₄	—	7	4	no	—
41	C₂×C₄	C₃×D₇	1	1	yes	characteristic, direct product factor
42	C₇	C₂²×C₃×C₄	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	7	4	no	—
44	C₂×C₃	C₄×D₇	1	2	yes	direct product factor
45	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
46	C₂²	—	7	4	no	—
47	C₂²	—	7	2	no	—
48	C₂²	C₂×C₃×D₇	1	1	yes	characteristic
49	C₄	—	7	2	no	—
50	C₄	C₂×C₃×D₇	1	2	yes	direct product factor
51	C₃	C₂×C₄×D₇	1	1	yes	characteristic, direct product factor, Sylow
52	C₂	—	7	4	no	—
53	C₂	C₃×C₄×D₇	1	2	yes	direct product factor
54	C₂	C₂²×C₃×D₇	1	1	yes	characteristic

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

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×C₃×Dic₇	C₂	semidirect
4. C₃×C₄×D₇	C₂	direct (N × H)
5. C₂×C₄×D₇	C₃	direct (N × H)
7. C₂×C₃×D₇	C₄	direct (N × H)
9. C₃×C₄×C₇	C₂²	semidirect
10. C₃×Dic₇	C₂²	semidirect
12. C₂×C₄×C₇	C₂×C₃	semidirect
13. C₂×Dic₇	C₂×C₃	semidirect
14. C₄×D₇	C₂×C₃	direct (N × H)
16. C₂×C₃×C₇	C₂×C₄	semidirect
18. C₃×D₇	C₂×C₄	direct (N × H)
20. C₂×D₇	C₃×C₄	direct (N × H)
22. C₄×C₇	C₂²×C₃	semidirect
23. Dic₇	C₂²×C₃	semidirect
27. C₂×C₃×C₄	D₇	direct (N × H)
28. C₃×C₇	C₂²×C₄	semidirect
30. C₂×C₇	C₂×C₃×C₄	semidirect
32. D₇	C₂×C₃×C₄	direct (N × H)
37. C₃×C₄	C₂×D₇	direct (N × H)
41. C₂×C₄	C₃×D₇	direct (N × H)
42. C₇	C₂²×C₃×C₄	semidirect
44. C₂×C₃	C₄×D₇	direct (N × H)
50. C₄	C₂×C₃×D₇	direct (N × H)
51. C₃	C₂×C₄×D₇	direct (N × H)
53. C₂	C₃×C₄×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×C₃×C₄×C₇	2
C₂×C₃×Dic₇	2
C₃×C₄×D₇	2
C₂×C₄×D₇	3
C₂²×C₃×C₄	7

Subgroups — C2×C3×C4×D7 #66706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×D₇ #66706