Subgroups — C₂×C₄×C₇×C₈ #90048

Order 448.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	2	yes	maximal
2	C₂×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇×C₈	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄×C₇	C₄	1	2	yes	—
5	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₇×C₈	C₄	1	8	yes	direct product factor
7	C₂×C₇×C₈	C₂²	1	4	yes	—
8	C₄²×C₇	C₄	1	2	yes	—
9	C₄²×C₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₇	C₈	1	8	yes	direct product factor
13	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₇	C₂×C₄	1	4	yes	—
15	C₂×C₄×C₇	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₇×C₈	C₂×C₄	1	8	yes	direct product factor
18	C₂²×C₈	C₂×C₇	1	2	yes	—
19	C₂×C₄²	C₂×C₇	1	1	yes	characteristic
20	C₄×C₈	C₂×C₇	1	4	yes	direct product factor
21	C₂²×C₇	C₄²	1	2	yes	—
22	C₂²×C₇	C₂×C₈	1	4	yes	—
23	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₇	C₄²	1	2	yes	—
25	C₄×C₇	C₂×C₈	1	8	yes	direct product factor
26	C₄×C₇	C₂²×C₄	1	2	yes	—
27	C₂²×C₄	C₄×C₇	1	2	yes	—
28	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
29	C₂×C₈	C₄×C₇	1	8	yes	direct product factor
30	C₂×C₈	C₂²×C₇	1	4	yes	—
31	C₄²	C₄×C₇	1	2	yes	—
32	C₄²	C₂²×C₇	1	2	yes	—
33	C₂×C₇	C₄×C₈	1	4	yes	direct product factor
34	C₂×C₇	C₂×C₄²	1	1	yes	characteristic
35	C₂×C₇	C₂²×C₈	1	2	yes	—
36	C₂³	C₂×C₄×C₇	1	1	yes	characteristic
37	C₂×C₄	C₇×C₈	1	8	yes	direct product factor
38	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
39	C₂×C₄	C₂×C₄×C₇	1	4	yes	—
40	C₂×C₄	C₂×C₄×C₇	1	4	yes	—
41	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
42	C₈	C₂×C₄×C₇	1	8	yes	direct product factor
43	C₇	C₂×C₄×C₈	1	1	yes	characteristic, direct product factor, Sylow
44	C₂²	C₄²×C₇	1	2	yes	—
45	C₂²	C₂×C₇×C₈	1	4	yes	—
46	C₂²	C₂²×C₄×C₇	1	1	yes	characteristic
47	C₄	C₄²×C₇	1	2	yes	—
48	C₄	C₂×C₇×C₈	1	8	yes	direct product factor
49	C₄	C₂²×C₄×C₇	1	2	yes	—
50	C₂	C₄×C₇×C₈	1	4	yes	direct product factor
51	C₂	C₂×C₄²×C₇	1	1	yes	characteristic
52	C₂	C₂²×C₇×C₈	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₄×C₇×C₈	C₂	direct (N × H)
6. C₂×C₇×C₈	C₄	direct (N × H)
10. C₂×C₄×C₈	C₇	direct (N × H)
12. C₂×C₄×C₇	C₈	direct (N × H)
17. C₇×C₈	C₂×C₄	direct (N × H)
20. C₄×C₈	C₂×C₇	direct (N × H)
25. C₄×C₇	C₂×C₈	direct (N × H)
29. C₂×C₈	C₄×C₇	direct (N × H)
33. C₂×C₇	C₄×C₈	direct (N × H)
37. C₂×C₄	C₇×C₈	direct (N × H)
42. C₈	C₂×C₄×C₇	direct (N × H)
43. C₇	C₂×C₄×C₈	direct (N × H)
48. C₄	C₂×C₇×C₈	direct (N × H)
50. C₂	C₄×C₇×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
C₂²×C₇×C₈	2
C₂×C₄²×C₇	2
C₄×C₇×C₈	2
C₂×C₄×C₈	7

Subgroups — C2×C4×C7×C8 #90048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₇×C₈ #90048