Subgroups — C₃×C₇×C₄⋊C₄ #66639

Order 336.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₇	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
5	C₃×C₄×C₇	—	2	2	no	—
6	C₃×C₄×C₇	C₄	1	2	yes	complemented
7	C₂×C₄×C₇	C₂×C₃	1	2	yes	—
8	C₂×C₄×C₇	C₂×C₃	1	1	yes	characteristic
9	C₃×C₄⋊C₄	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₃×C₇	D₄	1	1	yes	characteristic
12	C₂×C₃×C₇	Q₈	1	1	yes	characteristic
13	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
14	C₄×C₇	—	2	2	no	—
15	C₄×C₇	C₃×C₄	1	2	yes	complemented
16	C₂×C₃×C₄	C₂×C₇	1	2	yes	—
17	C₂×C₃×C₄	C₂×C₇	1	1	yes	characteristic
18	C₃×C₇	C₄⋊C₄	1	1	yes	characteristic, direct product factor, Hall
19	C₄⋊C₄	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
20	C₂×C₇	C₂×C₃×C₄	1	1	yes	characteristic
21	C₂×C₇	C₃×D₄	1	1	yes	characteristic
22	C₂×C₇	C₃×Q₈	1	1	yes	characteristic
23	C₂²×C₃	C₂²×C₇	1	1	yes	characteristic
24	C₃×C₄	—	2	2	no	—
25	C₃×C₄	C₄×C₇	1	2	yes	complemented
26	C₂×C₄	C₂×C₃×C₇	1	2	yes	—
27	C₂×C₄	C₂×C₃×C₇	1	1	yes	characteristic
28	C₇	C₃×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
29	C₂×C₃	C₂×C₄×C₇	1	1	yes	characteristic
30	C₂×C₃	C₇×D₄	1	1	yes	characteristic
31	C₂×C₃	C₇×Q₈	1	1	yes	characteristic
32	C₂²	C₂²×C₃×C₇	1	1	yes	characteristic
33	C₄	—	2	2	no	—
34	C₄	C₃×C₄×C₇	1	2	yes	complemented
35	C₃	C₇×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
36	C₂	C₂×C₃×C₄×C₇	1	1	yes	characteristic
37	C₂	C₃×C₇×D₄	1	1	yes	characteristic
38	C₂	C₃×C₇×Q₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₇	C₂²
commutator	C₂	C₂×C₃×C₄×C₇
Frattini, cosocle	C₂²	C₂²×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
3. C₇×C₄⋊C₄	C₃	direct (N × H)
6. C₃×C₄×C₇	C₄	semidirect
9. C₃×C₄⋊C₄	C₇	direct (N × H)
15. C₄×C₇	C₃×C₄	semidirect
18. C₃×C₇	C₄⋊C₄	direct (N × H)
19. C₄⋊C₄	C₃×C₇	direct (N × H)
25. C₃×C₄	C₄×C₇	semidirect
28. C₇	C₃×C₄⋊C₄	direct (N × H)
34. C₄	C₃×C₄×C₇	semidirect
35. C₃	C₇×C₄⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₇	2
C₇×C₄⋊C₄	3
C₃×C₄⋊C₄	7

Subgroups — C3×C7×C4⋊C4 #66639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×C₄⋊C₄ #66639