Subgroups — C₇³×C₈×C₉ #72201993

Order 24696.

1392 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇³×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇³×C₈	C₃	1	1	yes	maximal, characteristic
3	C₂×C₇³×C₉	C₄	1	1	yes	characteristic
4	C₃×C₄×C₇³	C₂×C₃	1	1	yes	characteristic
5	C₇²×C₈×C₉	C₇	1	57	yes	maximal, direct product factor
6	C₇³×C₉	C₈	1	1	yes	characteristic, direct product factor, Hall
7	C₇³×C₈	C₉	1	1	yes	characteristic, direct product factor, Hall
8	C₂×C₃×C₇³	C₃×C₄	1	1	yes	characteristic
9	C₄×C₇²×C₉	C₂×C₇	1	57	yes	—
10	C₄×C₇³	C₂×C₉	1	1	yes	characteristic
11	C₃×C₇²×C₈	C₃×C₇	1	57	yes	—
12	C₃×C₇³	C₃×C₈	1	1	yes	characteristic
13	C₂×C₇²×C₉	C₄×C₇	1	57	yes	—
14	C₂×C₇³	C₄×C₉	1	1	yes	characteristic
15	C₃×C₄×C₇²	C₂×C₃×C₇	1	57	yes	—
16	C₇×C₈×C₉	C₇²	1	57	yes	direct product factor
17	C₇²×C₉	C₇×C₈	1	57	yes	direct product factor
18	C₇²×C₈	C₇×C₉	1	57	yes	direct product factor
19	C₇³	C₈×C₉	1	1	yes	characteristic, direct product factor, Sylow
20	C₂×C₃×C₇²	C₃×C₄×C₇	1	57	yes	—
21	C₄×C₇×C₉	C₂×C₇²	1	57	yes	—
22	C₄×C₇²	C₂×C₇×C₉	1	57	yes	—
23	C₃×C₇×C₈	C₃×C₇²	1	57	yes	—
24	C₃×C₇²	C₃×C₇×C₈	1	57	yes	—
25	C₂×C₇×C₉	C₄×C₇²	1	57	yes	—
26	C₂×C₇²	C₄×C₇×C₉	1	57	yes	—
27	C₃×C₄×C₇	C₂×C₃×C₇²	1	57	yes	—
28	C₈×C₉	C₇³	1	1	yes	characteristic, direct product factor, Hall
29	C₇×C₉	C₇²×C₈	1	57	yes	direct product factor
30	C₇×C₈	C₇²×C₉	1	57	yes	direct product factor
31	C₇²	C₇×C₈×C₉	1	57	yes	direct product factor
32	C₂×C₃×C₇	C₃×C₄×C₇²	1	57	yes	—
33	C₄×C₉	C₂×C₇³	1	1	yes	characteristic
34	C₄×C₇	C₂×C₇²×C₉	1	57	yes	—
35	C₃×C₈	C₃×C₇³	1	1	yes	characteristic
36	C₃×C₇	C₃×C₇²×C₈	1	57	yes	—
37	C₂×C₉	C₄×C₇³	1	1	yes	characteristic
38	C₂×C₇	C₄×C₇²×C₉	1	57	yes	—
39	C₃×C₄	C₂×C₃×C₇³	1	1	yes	characteristic
40	C₉	C₇³×C₈	1	1	yes	characteristic, direct product factor, Sylow
41	C₈	C₇³×C₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₇	C₇²×C₈×C₉	1	57	yes	direct product factor
43	C₂×C₃	C₃×C₄×C₇³	1	1	yes	characteristic
44	C₄	C₂×C₇³×C₉	1	1	yes	characteristic
45	C₃	C₃×C₇³×C₈	1	1	yes	characteristic
46	C₂	C₄×C₇³×C₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄	C₂×C₃×C₇³
socle, prime radical	C₂×C₃×C₇³	C₃×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₇²×C₈×C₉	C₇	direct (N × H)
6. C₇³×C₉	C₈	direct (N × H)
7. C₇³×C₈	C₉	direct (N × H)
16. C₇×C₈×C₉	C₇²	direct (N × H)
17. C₇²×C₉	C₇×C₈	direct (N × H)
18. C₇²×C₈	C₇×C₉	direct (N × H)
19. C₇³	C₈×C₉	direct (N × H)
28. C₈×C₉	C₇³	direct (N × H)
29. C₇×C₉	C₇²×C₈	direct (N × H)
30. C₇×C₈	C₇²×C₉	direct (N × H)
31. C₇²	C₇×C₈×C₉	direct (N × H)
40. C₉	C₇³×C₈	direct (N × H)
41. C₈	C₇³×C₉	direct (N × H)
42. C₇	C₇²×C₈×C₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇³×C₉	2
C₃×C₇³×C₈	3
C₇²×C₈×C₉	7

Subgroups — C73×C8×C9 #72201993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇³×C₈×C₉ #72201993