Subgroups — C₃⁴×C₄×C₇₃ #72201002

Order 23652.

1272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⁴×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄×C₇₃	C₃	1	40	yes	maximal, direct product factor
3	C₂×C₃³×C₇₃	C₂×C₃	1	40	yes	—
4	C₃⁴×C₇₃	C₄	1	1	yes	characteristic, direct product factor, Hall
5	C₃²×C₄×C₇₃	C₃²	1	130	yes	direct product factor
6	C₃³×C₇₃	C₃×C₄	1	40	yes	direct product factor
7	C₂×C₃²×C₇₃	C₂×C₃²	1	130	yes	—
8	C₃×C₄×C₇₃	C₃³	1	40	yes	direct product factor
9	C₃²×C₇₃	C₃²×C₄	1	130	yes	direct product factor
10	C₂×C₃×C₇₃	C₂×C₃³	1	40	yes	—
11	C₃⁴×C₄	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₄×C₇₃	C₃⁴	1	1	yes	characteristic, direct product factor, Hall
13	C₃×C₇₃	C₃³×C₄	1	40	yes	direct product factor
14	C₂×C₃⁴	C₂×C₇₃	1	1	yes	characteristic
15	C₂×C₇₃	C₂×C₃⁴	1	1	yes	characteristic
16	C₃³×C₄	C₃×C₇₃	1	40	yes	direct product factor
17	C₃⁴	C₄×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
18	C₇₃	C₃⁴×C₄	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₃³	C₂×C₃×C₇₃	1	40	yes	—
20	C₃²×C₄	C₃²×C₇₃	1	130	yes	direct product factor
21	C₃³	C₃×C₄×C₇₃	1	40	yes	direct product factor
22	C₂×C₃²	C₂×C₃²×C₇₃	1	130	yes	—
23	C₃×C₄	C₃³×C₇₃	1	40	yes	direct product factor
24	C₃²	C₃²×C₄×C₇₃	1	130	yes	direct product factor
25	C₂×C₃	C₂×C₃³×C₇₃	1	40	yes	—
26	C₄	C₃⁴×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
27	C₃	C₃³×C₄×C₇₃	1	40	yes	direct product factor
28	C₂	C₂×C₃⁴×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃⁴×C₇₃
socle, prime radical	C₂×C₃⁴×C₇₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃³×C₄×C₇₃	C₃	direct (N × H)
4. C₃⁴×C₇₃	C₄	direct (N × H)
5. C₃²×C₄×C₇₃	C₃²	direct (N × H)
6. C₃³×C₇₃	C₃×C₄	direct (N × H)
8. C₃×C₄×C₇₃	C₃³	direct (N × H)
9. C₃²×C₇₃	C₃²×C₄	direct (N × H)
11. C₃⁴×C₄	C₇₃	direct (N × H)
12. C₄×C₇₃	C₃⁴	direct (N × H)
13. C₃×C₇₃	C₃³×C₄	direct (N × H)
16. C₃³×C₄	C₃×C₇₃	direct (N × H)
17. C₃⁴	C₄×C₇₃	direct (N × H)
18. C₇₃	C₃⁴×C₄	direct (N × H)
20. C₃²×C₄	C₃²×C₇₃	direct (N × H)
21. C₃³	C₃×C₄×C₇₃	direct (N × H)
23. C₃×C₄	C₃³×C₇₃	direct (N × H)
24. C₃²	C₃²×C₄×C₇₃	direct (N × H)
26. C₄	C₃⁴×C₇₃	direct (N × H)
27. C₃	C₃³×C₄×C₇₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃⁴×C₇₃	2
C₃³×C₄×C₇₃	3
C₃⁴×C₄	73

Subgroups — C34×C4×C73 #72201002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⁴×C₄×C₇₃ #72201002