Subgroups — C₂³×C₇×C₈ #90586

Order 448.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₈	C₂	1	14	yes	maximal, direct product factor
3	C₂⁴×C₇	C₄	1	1	yes	characteristic
4	C₂²×C₄×C₇	C₄	1	7	yes	—
5	C₂²×C₄×C₇	C₂²	1	7	yes	—
6	C₂×C₇×C₈	C₂²	1	28	yes	direct product factor
7	C₂³×C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₇	C₈	1	8	yes	direct product factor
9	C₂³×C₇	C₂×C₄	1	7	yes	—
10	C₂×C₄×C₇	C₂×C₄	1	21	yes	—
11	C₂×C₄×C₇	C₂³	1	7	yes	—
12	C₇×C₈	C₂³	1	8	yes	direct product factor
13	C₂³×C₄	C₂×C₇	1	1	yes	characteristic
14	C₂²×C₈	C₂×C₇	1	14	yes	direct product factor
15	C₂²×C₇	C₂×C₈	1	28	yes	direct product factor
16	C₂²×C₇	C₂²×C₄	1	7	yes	—
17	C₄×C₇	C₂²×C₄	1	7	yes	—
18	C₄×C₇	C₂⁴	1	1	yes	characteristic
19	C₂⁴	C₄×C₇	1	1	yes	characteristic
20	C₂²×C₄	C₄×C₇	1	7	yes	—
21	C₂²×C₄	C₂²×C₇	1	7	yes	—
22	C₂×C₈	C₂²×C₇	1	28	yes	direct product factor
23	C₂×C₇	C₂²×C₈	1	14	yes	direct product factor
24	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
25	C₂³	C₇×C₈	1	8	yes	direct product factor
26	C₂³	C₂×C₄×C₇	1	7	yes	—
27	C₂×C₄	C₂×C₄×C₇	1	21	yes	—
28	C₂×C₄	C₂³×C₇	1	7	yes	—
29	C₈	C₂³×C₇	1	8	yes	direct product factor
30	C₇	C₂³×C₈	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²	C₂×C₇×C₈	1	28	yes	direct product factor
32	C₂²	C₂²×C₄×C₇	1	7	yes	—
33	C₄	C₂²×C₄×C₇	1	7	yes	—
34	C₄	C₂⁴×C₇	1	1	yes	characteristic
35	C₂	C₂²×C₇×C₈	1	14	yes	direct product factor
36	C₂	C₂³×C₄×C₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂⁴×C₇
socle, prime radical	C₂⁴×C₇	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×C₈	C₂	direct (N × H)
6. C₂×C₇×C₈	C₂²	direct (N × H)
7. C₂³×C₈	C₇	direct (N × H)
8. C₂³×C₇	C₈	direct (N × H)
12. C₇×C₈	C₂³	direct (N × H)
14. C₂²×C₈	C₂×C₇	direct (N × H)
15. C₂²×C₇	C₂×C₈	direct (N × H)
22. C₂×C₈	C₂²×C₇	direct (N × H)
23. C₂×C₇	C₂²×C₈	direct (N × H)
25. C₂³	C₇×C₈	direct (N × H)
29. C₈	C₂³×C₇	direct (N × H)
30. C₇	C₂³×C₈	direct (N × H)
31. C₂²	C₂×C₇×C₈	direct (N × H)
35. C₂	C₂²×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
C₂³×C₄×C₇	2
C₂²×C₇×C₈	2
C₂³×C₈	7

Subgroups — C23×C7×C8 #90586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×C₈ #90586