Subgroups — C₂×C₇×M₁₆ #6533

Order 224.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₇×M₁₆	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₇×C₈	C₂	1	2	yes	maximal, complemented
4	C₂³×C₇	C₄	1	1	yes	characteristic
5	C₂×C₄×C₇	C₄	1	1	yes	characteristic
6	C₂×C₄×C₇	C₄	1	2	yes	—
7	C₂×C₄×C₇	C₂²	1	2	yes	—
8	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₇×C₈	C₂²	1	4	yes	complemented
10	C₂×M₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₇	—	2	2	no	—
12	C₂²×C₇	C₂×C₄	1	2	yes	—
13	C₂²×C₇	C₂×C₄	1	1	yes	characteristic
14	C₄×C₇	C₂×C₄	1	2	yes	—
15	C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₄×C₇	C₂³	1	1	yes	characteristic
17	C₂²×C₄	C₂×C₇	1	1	yes	characteristic
18	M₁₆	C₂×C₇	1	4	yes	direct product factor
19	C₂×C₈	C₂×C₇	1	2	yes	complemented
20	C₂×C₇	—	2	2	no	—
21	C₂×C₇	M₁₆	1	2	yes	direct product factor
22	C₂×C₇	C₂²×C₄	1	1	yes	characteristic
23	C₂³	C₄×C₇	1	1	yes	characteristic
24	C₂×C₄	C₄×C₇	1	2	yes	—
25	C₂×C₄	C₄×C₇	1	1	yes	characteristic
26	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
27	C₂×C₄	C₂²×C₇	1	2	yes	—
28	C₈	C₂²×C₇	1	4	yes	complemented
29	C₇	C₂×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
30	C₂²	—	2	2	no	—
31	C₂²	C₂×C₄×C₇	1	2	yes	—
32	C₂²	C₂×C₄×C₇	1	1	yes	characteristic
33	C₄	C₂×C₄×C₇	1	1	yes	characteristic
34	C₄	C₂×C₄×C₇	1	2	yes	—
35	C₄	C₂³×C₇	1	1	yes	characteristic
36	C₂	—	2	2	no	—
37	C₂	C₇×M₁₆	1	2	yes	direct product factor
38	C₂	C₂²×C₄×C₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₇×M₁₆	C₂	direct (N × H)
3. C₂×C₇×C₈	C₂	semidirect
9. C₇×C₈	C₂²	semidirect
10. C₂×M₁₆	C₇	direct (N × H)
18. M₁₆	C₂×C₇	direct (N × H)
19. C₂×C₈	C₂×C₇	semidirect
21. C₂×C₇	M₁₆	direct (N × H)
28. C₈	C₂²×C₇	semidirect
29. C₇	C₂×M₁₆	direct (N × H)
37. C₂	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
C₂²×C₄×C₇	2
C₇×M₁₆	2
C₂×C₇×C₈	2
C₂×M₁₆	7

Subgroups — C2×C7×M16 #6533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×M₁₆ #6533