Subgroups — C₇×A₄×C₁₆ #13055057

Order 1344.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₈×A₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₁₆	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄×C₇×A₄	C₄	1	1	yes	characteristic
4	C₃×C₇×C₁₆	—	4	1	no	maximal
5	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
6	C₂×C₇×C₁₆	—	3	1	no	—
7	C₂×C₇×C₁₆	—	3	1	no	—
8	A₄×C₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₇×A₄	C₈	1	1	yes	characteristic
10	C₃×C₇×C₈	—	4	1	no	—
11	C₂²×C₄×C₇	C₃×C₄	1	1	yes	characteristic
12	C₂×C₇×C₈	—	3	1	no	—
13	C₂×C₇×C₈	—	3	1	no	—
14	C₇×C₁₆	—	3	1	no	—
15	C₇×C₁₆	A₄	1	1	yes	characteristic, direct product factor
16	C₈×A₄	C₂×C₇	1	1	yes	characteristic
17	C₇×A₄	C₁₆	1	1	yes	characteristic, direct product factor
18	C₃×C₄×C₇	—	4	1	no	—
19	C₂²×C₁₆	C₃×C₇	1	1	yes	characteristic, complemented, Sylow
20	C₂³×C₇	C₃×C₈	1	1	yes	characteristic
21	C₂×C₄×C₇	—	3	1	no	—
22	C₂×C₄×C₇	—	3	1	no	—
23	C₇×C₈	—	3	1	no	—
24	C₇×C₈	C₂×A₄	1	1	yes	characteristic
25	C₄×A₄	C₄×C₇	1	1	yes	characteristic
26	C₃×C₁₆	—	4	1	no	—
27	C₂×C₃×C₇	—	4	1	no	—
28	C₂²×C₈	C₂×C₃×C₇	1	1	yes	characteristic
29	C₂×C₁₆	—	3	1	no	—
30	C₂×C₁₆	—	3	1	no	—
31	C₂²×C₇	—	3	1	no	—
32	C₂²×C₇	—	3	1	no	—
33	C₂²×C₇	C₃×C₁₆	1	1	yes	characteristic, complemented
34	C₄×C₇	—	3	1	no	—
35	C₄×C₇	C₄×A₄	1	1	yes	characteristic
36	C₂×A₄	C₇×C₈	1	1	yes	characteristic
37	C₃×C₈	—	4	1	no	—
38	C₃×C₇	—	4	1	no	Hall
39	C₂²×C₄	C₃×C₄×C₇	1	1	yes	characteristic
40	C₂×C₈	—	3	1	no	—
41	C₂×C₈	—	3	1	no	—
42	C₁₆	—	3	1	no	—
43	C₁₆	C₇×A₄	1	1	yes	characteristic, direct product factor
44	C₂×C₇	—	3	1	no	—
45	C₂×C₇	—	3	1	no	—
46	C₂×C₇	C₈×A₄	1	1	yes	characteristic
47	A₄	C₇×C₁₆	1	1	yes	characteristic, direct product factor
48	C₃×C₄	—	4	1	no	—
49	C₂³	C₃×C₇×C₈	1	1	yes	characteristic
50	C₂×C₄	—	3	1	no	—
51	C₂×C₄	—	3	1	no	—
52	C₈	—	3	1	no	—
53	C₈	C₂×C₇×A₄	1	1	yes	characteristic
54	C₇	A₄×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×C₃	—	4	1	no	—
56	C₂²	—	3	1	no	—
57	C₂²	—	3	1	no	—
58	C₂²	C₃×C₇×C₁₆	1	1	yes	characteristic, complemented
59	C₄	—	3	1	no	—
60	C₄	C₄×C₇×A₄	1	1	yes	characteristic
61	C₃	—	4	1	no	Sylow
62	C₂	—	3	1	no	—
63	C₂	—	3	1	no	—
64	C₂	C₇×C₈×A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₇×C₁₆	A₄
commutator, hypocenter	C₂²	C₃×C₇×C₁₆
Frattini	C₈	C₂×C₇×A₄
Fitting	C₂²×C₇×C₁₆	C₃
socle	C₂³×C₇	C₃×C₈
prime radical	C₂×C₇×A₄	C₈
cosocle	C₂²×C₈	C₂×C₃×C₇
LCS	G ▷ C₂²
UCS	C₁ ◁ C₇×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×C₁₆	C₃	semidirect
8. A₄×C₁₆	C₇	direct (N × H)
15. C₇×C₁₆	A₄	direct (N × H)
17. C₇×A₄	C₁₆	direct (N × H)
19. C₂²×C₁₆	C₃×C₇	semidirect
33. C₂²×C₇	C₃×C₁₆	semidirect
43. C₁₆	C₇×A₄	direct (N × H)
47. A₄	C₇×C₁₆	direct (N × H)
54. C₇	A₄×C₁₆	direct (N × H)
58. C₂²	C₃×C₇×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₈×A₄	2
C₂²×C₇×C₁₆	3
C₃×C₇×C₁₆	4
A₄×C₁₆	7

Subgroups — C7×A4×C16 #13055057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×A₄×C₁₆ #13055057