Subgroups — C₄²×C₇²×C₈ #72185444

Order 6272.

2040 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄³×C₇²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇²×C₈	C₂	1	6	yes	maximal
3	C₂²×C₇²×C₈	C₂²	1	4	yes	—
4	C₂×C₄²×C₇²	C₄	1	4	yes	—
5	C₂×C₄²×C₇²	C₂²	1	3	yes	—
6	C₄×C₇²×C₈	C₄	1	24	yes	direct product factor
7	C₄²×C₇×C₈	C₇	1	8	yes	maximal, direct product factor
8	C₂²×C₄×C₇²	C₂×C₄	1	6	yes	—
9	C₂²×C₄×C₇²	C₂³	1	1	yes	characteristic
10	C₂×C₇²×C₈	C₂×C₄	1	24	yes	—
11	C₄²×C₇²	C₈	1	16	yes	direct product factor
12	C₄²×C₇²	C₂×C₄	1	12	yes	—
13	C₂×C₄×C₇×C₈	C₂×C₇	1	48	yes	—
14	C₄³×C₇	C₂×C₇	1	8	yes	—
15	C₂³×C₇²	C₂²×C₄	1	1	yes	characteristic
16	C₂×C₄×C₇²	C₄²	1	12	yes	—
17	C₂×C₄×C₇²	C₂×C₈	1	24	yes	—
18	C₂×C₄×C₇²	C₂²×C₄	1	6	yes	—
19	C₇²×C₈	C₄²	1	16	yes	direct product factor
20	C₂²×C₇×C₈	C₂²×C₇	1	32	yes	—
21	C₂×C₄²×C₇	C₄×C₇	1	32	yes	—
22	C₂×C₄²×C₇	C₂²×C₇	1	24	yes	—
23	C₄×C₇×C₈	C₄×C₇	1	192	yes	direct product factor
24	C₂²×C₇²	C₂×C₄²	1	3	yes	—
25	C₂²×C₇²	C₂²×C₈	1	4	yes	—
26	C₄×C₇²	C₄×C₈	1	24	yes	direct product factor
27	C₄×C₇²	C₂×C₄²	1	4	yes	—
28	C₄²×C₈	C₇²	1	1	yes	characteristic, direct product factor, Sylow
29	C₂²×C₄×C₇	C₂×C₄×C₇	1	48	yes	—
30	C₂²×C₄×C₇	C₂³×C₇	1	8	yes	—
31	C₂×C₇×C₈	C₂×C₄×C₇	1	192	yes	—
32	C₄²×C₇	C₇×C₈	1	128	yes	direct product factor
33	C₄²×C₇	C₂×C₄×C₇	1	96	yes	—
34	C₂×C₇²	C₄³	1	1	yes	characteristic
35	C₂×C₇²	C₂×C₄×C₈	1	6	yes	—
36	C₂×C₄×C₈	C₂×C₇²	1	6	yes	—
37	C₄³	C₂×C₇²	1	1	yes	characteristic
38	C₂³×C₇	C₂²×C₄×C₇	1	8	yes	—
39	C₂×C₄×C₇	C₄²×C₇	1	96	yes	—
40	C₂×C₄×C₇	C₂×C₇×C₈	1	192	yes	—
41	C₂×C₄×C₇	C₂²×C₄×C₇	1	48	yes	—
42	C₇×C₈	C₄²×C₇	1	128	yes	direct product factor
43	C₇²	C₄²×C₈	1	1	yes	characteristic, direct product factor, Sylow
44	C₂²×C₈	C₂²×C₇²	1	4	yes	—
45	C₂×C₄²	C₄×C₇²	1	4	yes	—
46	C₂×C₄²	C₂²×C₇²	1	3	yes	—
47	C₄×C₈	C₄×C₇²	1	24	yes	direct product factor
48	C₂²×C₇	C₂×C₄²×C₇	1	24	yes	—
49	C₂²×C₇	C₂²×C₇×C₈	1	32	yes	—
50	C₄×C₇	C₄×C₇×C₈	1	192	yes	direct product factor
51	C₄×C₇	C₂×C₄²×C₇	1	32	yes	—
52	C₂²×C₄	C₂×C₄×C₇²	1	6	yes	—
53	C₂²×C₄	C₂³×C₇²	1	1	yes	characteristic
54	C₂×C₈	C₂×C₄×C₇²	1	24	yes	—
55	C₄²	C₇²×C₈	1	16	yes	direct product factor
56	C₄²	C₂×C₄×C₇²	1	12	yes	—
57	C₂×C₇	C₄³×C₇	1	8	yes	—
58	C₂×C₇	C₂×C₄×C₇×C₈	1	48	yes	—
59	C₂³	C₂²×C₄×C₇²	1	1	yes	characteristic
60	C₂×C₄	C₄²×C₇²	1	12	yes	—
61	C₂×C₄	C₂×C₇²×C₈	1	24	yes	—
62	C₂×C₄	C₂²×C₄×C₇²	1	6	yes	—
63	C₈	C₄²×C₇²	1	16	yes	direct product factor
64	C₇	C₄²×C₇×C₈	1	8	yes	direct product factor
65	C₂²	C₂×C₄²×C₇²	1	3	yes	—
66	C₂²	C₂²×C₇²×C₈	1	4	yes	—
67	C₄	C₄×C₇²×C₈	1	24	yes	direct product factor
68	C₄	C₂×C₄²×C₇²	1	4	yes	—
69	C₂	C₂×C₄×C₇²×C₈	1	6	yes	—
70	C₂	C₄³×C₇²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₄	C₂³×C₇²
socle, prime radical	C₂³×C₇²	C₂²×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₄×C₇²×C₈	C₄	direct (N × H)
7. C₄²×C₇×C₈	C₇	direct (N × H)
11. C₄²×C₇²	C₈	direct (N × H)
19. C₇²×C₈	C₄²	direct (N × H)
23. C₄×C₇×C₈	C₄×C₇	direct (N × H)
26. C₄×C₇²	C₄×C₈	direct (N × H)
28. C₄²×C₈	C₇²	direct (N × H)
32. C₄²×C₇	C₇×C₈	direct (N × H)
42. C₇×C₈	C₄²×C₇	direct (N × H)
43. C₇²	C₄²×C₈	direct (N × H)
47. C₄×C₈	C₄×C₇²	direct (N × H)
50. C₄×C₇	C₄×C₇×C₈	direct (N × H)
55. C₄²	C₇²×C₈	direct (N × H)
63. C₈	C₄²×C₇²	direct (N × H)
64. C₇	C₄²×C₇×C₈	direct (N × H)
67. C₄	C₄×C₇²×C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄³×C₇²	2
C₂×C₄×C₇²×C₈	2
C₄²×C₇×C₈	7

Subgroups — C42×C72×C8 #72185444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₇²×C₈ #72185444