Subgroups — C₃×C₇×M₁₆ #66641

Order 336.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₈	C₂	1	2	yes	maximal, complemented
3	C₇×M₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₇	C₄	1	1	yes	characteristic
5	C₃×C₄×C₇	C₄	1	1	yes	characteristic
6	C₃×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇	C₂×C₃	1	1	yes	characteristic
8	C₇×C₈	C₂×C₃	1	2	yes	complemented
9	C₃×M₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₇	—	2	1	no	—
11	C₂×C₃×C₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₇	C₃×C₄	1	1	yes	characteristic
13	C₄×C₇	C₃×C₄	1	1	yes	characteristic
14	C₄×C₇	C₂²×C₃	1	1	yes	characteristic
15	C₂×C₃×C₄	C₂×C₇	1	1	yes	characteristic
16	C₃×C₈	C₂×C₇	1	2	yes	complemented
17	C₃×C₇	M₁₆	1	1	yes	characteristic, direct product factor, Hall
18	M₁₆	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₇	—	2	1	no	—
20	C₂×C₇	C₂×C₃×C₄	1	1	yes	characteristic
21	C₂²×C₃	C₄×C₇	1	1	yes	characteristic
22	C₃×C₄	C₄×C₇	1	1	yes	characteristic
23	C₃×C₄	C₂²×C₇	1	1	yes	characteristic
24	C₂×C₄	C₂×C₃×C₇	1	1	yes	characteristic
25	C₈	C₂×C₃×C₇	1	2	yes	complemented
26	C₇	C₃×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₃	—	2	1	no	—
28	C₂×C₃	C₂×C₄×C₇	1	1	yes	characteristic
29	C₂²	C₃×C₄×C₇	1	1	yes	characteristic
30	C₄	C₃×C₄×C₇	1	1	yes	characteristic
31	C₄	C₂²×C₃×C₇	1	1	yes	characteristic
32	C₃	C₇×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
33	C₂	—	2	1	no	—
34	C₂	C₂×C₃×C₄×C₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×C₈	C₂	semidirect
3. C₇×M₁₆	C₃	direct (N × H)
8. C₇×C₈	C₂×C₃	semidirect
9. C₃×M₁₆	C₇	direct (N × H)
16. C₃×C₈	C₂×C₇	semidirect
17. C₃×C₇	M₁₆	direct (N × H)
18. M₁₆	C₃×C₇	direct (N × H)
25. C₈	C₂×C₃×C₇	semidirect
26. C₇	C₃×M₁₆	direct (N × H)
32. C₃	C₇×M₁₆	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
C₃×C₇×C₈	2
C₇×M₁₆	3
C₃×M₁₆	7

Subgroups — C3×C7×M16 #66641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×M₁₆ #66641