Subgroups — C₂²×C₃×Dic₇ #66713

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
2	C₂×C₃×Dic₇	C₂	1	6	yes	maximal, direct product factor
3	C₂²×Dic₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₇	C₄	1	4	yes	complemented
5	C₂²×C₃×C₇	C₂²	1	3	yes	—
6	C₃×Dic₇	C₂²	1	4	yes	direct product factor
7	C₂³×C₇	C₂×C₃	1	1	yes	characteristic
8	C₂×Dic₇	C₂×C₃	1	6	yes	direct product factor
9	C₂²×C₃×C₄	—	7	1	no	maximal, Hall
10	C₂×C₃×C₇	C₂×C₄	1	6	yes	complemented
11	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
12	C₂²×C₇	C₃×C₄	1	4	yes	complemented
13	C₂²×C₇	C₂²×C₃	1	3	yes	—
14	Dic₇	C₂²×C₃	1	4	yes	direct product factor
15	C₂³×C₃	D₇	1	1	yes	characteristic
16	C₂×C₃×C₄	—	7	6	no	—
17	C₃×C₇	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
18	C₂²×C₄	—	7	1	no	Sylow
19	C₂×C₇	C₂×C₃×C₄	1	6	yes	complemented
20	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
21	C₂²×C₃	Dic₇	1	4	yes	direct product factor
22	C₂²×C₃	C₂×D₇	1	3	yes	—
23	C₃×C₄	—	7	4	no	—
24	C₂³	C₃×D₇	1	1	yes	characteristic
25	C₂×C₄	—	7	6	no	—
26	C₇	C₂²×C₃×C₄	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₃	C₂×Dic₇	1	6	yes	direct product factor
28	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
29	C₂²	C₃×Dic₇	1	4	yes	direct product factor
30	C₂²	C₂×C₃×D₇	1	3	yes	—
31	C₄	—	7	4	no	—
32	C₃	C₂²×Dic₇	1	1	yes	characteristic, direct product factor, Sylow
33	C₂	C₂×C₃×Dic₇	1	6	yes	direct product factor
34	C₂	C₂²×C₃×D₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×Dic₇	C₂	direct (N × H)
3. C₂²×Dic₇	C₃	direct (N × H)
4. C₂²×C₃×C₇	C₄	semidirect
6. C₃×Dic₇	C₂²	direct (N × H)
8. C₂×Dic₇	C₂×C₃	direct (N × H)
10. C₂×C₃×C₇	C₂×C₄	semidirect
12. C₂²×C₇	C₃×C₄	semidirect
14. Dic₇	C₂²×C₃	direct (N × H)
17. C₃×C₇	C₂²×C₄	semidirect
19. C₂×C₇	C₂×C₃×C₄	semidirect
21. C₂²×C₃	Dic₇	direct (N × H)
26. C₇	C₂²×C₃×C₄	semidirect
27. C₂×C₃	C₂×Dic₇	direct (N × H)
29. C₂²	C₃×Dic₇	direct (N × H)
32. C₃	C₂²×Dic₇	direct (N × H)
33. C₂	C₂×C₃×Dic₇	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
C₂×C₃×Dic₇	2
C₂²×Dic₇	3
C₂²×C₃×C₄	7

Subgroups — C22×C3×Dic7 #66713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×Dic₇ #66713