Subgroups — C₃²×C₇×C₈×C₃₁ #72193643

Order 15624.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₈×C₃₁	C₃	1	4	yes	maximal, direct product factor
2	C₃²×C₈×C₃₁	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃²×C₄×C₇×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃²×C₇×C₃₁	C₄	1	1	yes	characteristic
5	C₃×C₄×C₇×C₃₁	C₂×C₃	1	4	yes	—
6	C₃²×C₇×C₃₁	C₈	1	1	yes	characteristic, direct product factor, Hall
7	C₇×C₈×C₃₁	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₂×C₃×C₇×C₃₁	C₃×C₄	1	4	yes	—
9	C₃²×C₄×C₃₁	C₂×C₇	1	1	yes	characteristic
10	C₄×C₇×C₃₁	C₂×C₃²	1	1	yes	characteristic
11	C₃×C₈×C₃₁	C₃×C₇	1	4	yes	direct product factor
12	C₃×C₇×C₃₁	C₃×C₈	1	4	yes	direct product factor
13	C₂×C₃²×C₃₁	C₄×C₇	1	1	yes	characteristic
14	C₃²×C₇×C₈	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₇×C₃₁	C₃²×C₄	1	1	yes	characteristic
16	C₃×C₄×C₃₁	C₂×C₃×C₇	1	4	yes	—
17	C₃²×C₃₁	C₇×C₈	1	1	yes	characteristic, direct product factor, Hall
18	C₃²×C₄×C₇	C₂×C₃₁	1	1	yes	characteristic
19	C₈×C₃₁	C₃²×C₇	1	1	yes	characteristic, direct product factor, Hall
20	C₇×C₃₁	C₃²×C₈	1	1	yes	characteristic, direct product factor, Hall
21	C₂×C₃×C₃₁	C₃×C₄×C₇	1	4	yes	—
22	C₃×C₇×C₈	C₃×C₃₁	1	4	yes	direct product factor
23	C₂×C₃²×C₇	C₄×C₃₁	1	1	yes	characteristic
24	C₄×C₃₁	C₂×C₃²×C₇	1	1	yes	characteristic
25	C₃×C₃₁	C₃×C₇×C₈	1	4	yes	direct product factor
26	C₃×C₄×C₇	C₂×C₃×C₃₁	1	4	yes	—
27	C₃²×C₈	C₇×C₃₁	1	1	yes	characteristic, direct product factor, Hall
28	C₃²×C₇	C₈×C₃₁	1	1	yes	characteristic, direct product factor, Hall
29	C₂×C₃₁	C₃²×C₄×C₇	1	1	yes	characteristic
30	C₇×C₈	C₃²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃×C₇	C₃×C₄×C₃₁	1	4	yes	—
32	C₃²×C₄	C₂×C₇×C₃₁	1	1	yes	characteristic
33	C₃₁	C₃²×C₇×C₈	1	1	yes	characteristic, direct product factor, Sylow
34	C₄×C₇	C₂×C₃²×C₃₁	1	1	yes	characteristic
35	C₃×C₈	C₃×C₇×C₃₁	1	4	yes	direct product factor
36	C₃×C₇	C₃×C₈×C₃₁	1	4	yes	direct product factor
37	C₂×C₃²	C₄×C₇×C₃₁	1	1	yes	characteristic
38	C₂×C₇	C₃²×C₄×C₃₁	1	1	yes	characteristic
39	C₃×C₄	C₂×C₃×C₇×C₃₁	1	4	yes	—
40	C₃²	C₇×C₈×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
41	C₈	C₃²×C₇×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
42	C₇	C₃²×C₈×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃	C₃×C₄×C₇×C₃₁	1	4	yes	—
44	C₄	C₂×C₃²×C₇×C₃₁	1	1	yes	characteristic
45	C₃	C₃×C₇×C₈×C₃₁	1	4	yes	direct product factor
46	C₂	C₃²×C₄×C₇×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₃²×C₇×C₃₁
socle, prime radical	C₂×C₃²×C₇×C₃₁	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₈×C₃₁	C₃	direct (N × H)
2. C₃²×C₈×C₃₁	C₇	direct (N × H)
6. C₃²×C₇×C₃₁	C₈	direct (N × H)
7. C₇×C₈×C₃₁	C₃²	direct (N × H)
11. C₃×C₈×C₃₁	C₃×C₇	direct (N × H)
12. C₃×C₇×C₃₁	C₃×C₈	direct (N × H)
14. C₃²×C₇×C₈	C₃₁	direct (N × H)
17. C₃²×C₃₁	C₇×C₈	direct (N × H)
19. C₈×C₃₁	C₃²×C₇	direct (N × H)
20. C₇×C₃₁	C₃²×C₈	direct (N × H)
22. C₃×C₇×C₈	C₃×C₃₁	direct (N × H)
25. C₃×C₃₁	C₃×C₇×C₈	direct (N × H)
27. C₃²×C₈	C₇×C₃₁	direct (N × H)
28. C₃²×C₇	C₈×C₃₁	direct (N × H)
30. C₇×C₈	C₃²×C₃₁	direct (N × H)
33. C₃₁	C₃²×C₇×C₈	direct (N × H)
35. C₃×C₈	C₃×C₇×C₃₁	direct (N × H)
36. C₃×C₇	C₃×C₈×C₃₁	direct (N × H)
40. C₃²	C₇×C₈×C₃₁	direct (N × H)
41. C₈	C₃²×C₇×C₃₁	direct (N × H)
42. C₇	C₃²×C₈×C₃₁	direct (N × H)
45. C₃	C₃×C₇×C₈×C₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇×C₈×C₃₁	3
C₃²×C₈×C₃₁	7
C₃²×C₄×C₇×C₃₁	2
C₃²×C₇×C₈	31

Subgroups — C32×C7×C8×C31 #72193643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₇×C₈×C₃₁ #72193643