Subgroups — C₂×C₃×C₂²⋊D₇ #66714

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

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

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₇	C₂	semidirect
2. C₂²×C₃×D₇	C₂	semidirect
3. C₃×C₂²⋊D₇	C₂	direct (N × H)
4. C₂×C₃×Dic₇	C₂	semidirect
5. C₂×C₂²⋊D₇	C₃	direct (N × H)
8. C₂²×C₃×C₇	C₂²	semidirect
10. C₂×C₃×D₇	C₂²	semidirect
11. C₃×Dic₇	C₂²	semidirect
12. C₂³×C₇	C₂×C₃	semidirect
13. C₂²×D₇	C₂×C₃	semidirect
14. C₂²⋊D₇	C₂×C₃	direct (N × H)
15. C₂×Dic₇	C₂×C₃	semidirect
18. C₂×C₃×C₇	D₄	semidirect
23. C₂²×C₇	C₂²×C₃	semidirect
25. C₂×D₇	C₂²×C₃	semidirect
26. Dic₇	C₂²×C₃	semidirect
28. C₂³×C₃	D₇	semidirect
31. C₃×C₇	C₂×D₄	semidirect
34. C₂×C₇	C₃×D₄	semidirect
41. C₂²×C₃	C₂×D₇	semidirect
44. C₂³	C₃×D₇	semidirect
47. C₇	C₂×C₃×D₄	semidirect
50. C₂×C₃	C₂²⋊D₇	direct (N × H)
56. C₂²	C₂×C₃×D₇	semidirect
58. C₃	C₂×C₂²⋊D₇	direct (N × H)
61. C₂	C₃×C₂²⋊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
C₂³×C₃×C₇	2
C₂²×C₃×D₇	2
C₃×C₂²⋊D₇	2
C₂×C₃×Dic₇	2
C₂×C₂²⋊D₇	3
C₂×C₃×D₄	7

Subgroups — C2×C3×C22⋊D7 #66714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₂²⋊D₇ #66714