Subgroups — C₃⁴×C₈×C₁₉ #72190708

Order 12312.

1696 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₈×C₁₉	C₃	1	40	yes	maximal, direct product factor
3	C₂×C₃⁴×C₁₉	C₄	1	1	yes	characteristic
4	C₃³×C₄×C₁₉	C₂×C₃	1	40	yes	—
5	C₃⁴×C₁₉	C₈	1	1	yes	characteristic, direct product factor, Hall
6	C₃²×C₈×C₁₉	C₃²	1	130	yes	direct product factor
7	C₂×C₃³×C₁₉	C₃×C₄	1	40	yes	—
8	C₃²×C₄×C₁₉	C₂×C₃²	1	130	yes	—
9	C₃⁴×C₈	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃³×C₁₉	C₃×C₈	1	40	yes	direct product factor
11	C₃×C₈×C₁₉	C₃³	1	40	yes	direct product factor
12	C₂×C₃²×C₁₉	C₃²×C₄	1	130	yes	—
13	C₃⁴×C₄	C₂×C₁₉	1	1	yes	characteristic
14	C₃×C₄×C₁₉	C₂×C₃³	1	40	yes	—
15	C₃³×C₈	C₃×C₁₉	1	40	yes	direct product factor
16	C₃²×C₁₉	C₃²×C₈	1	130	yes	direct product factor
17	C₂×C₃⁴	C₄×C₁₉	1	1	yes	characteristic
18	C₈×C₁₉	C₃⁴	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₃×C₁₉	C₃³×C₄	1	40	yes	—
20	C₃³×C₄	C₂×C₃×C₁₉	1	40	yes	—
21	C₃⁴	C₈×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
22	C₄×C₁₉	C₂×C₃⁴	1	1	yes	characteristic
23	C₃²×C₈	C₃²×C₁₉	1	130	yes	direct product factor
24	C₃×C₁₉	C₃³×C₈	1	40	yes	direct product factor
25	C₂×C₃³	C₃×C₄×C₁₉	1	40	yes	—
26	C₂×C₁₉	C₃⁴×C₄	1	1	yes	characteristic
27	C₃²×C₄	C₂×C₃²×C₁₉	1	130	yes	—
28	C₃³	C₃×C₈×C₁₉	1	40	yes	direct product factor
29	C₃×C₈	C₃³×C₁₉	1	40	yes	direct product factor
30	C₁₉	C₃⁴×C₈	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃²	C₃²×C₄×C₁₉	1	130	yes	—
32	C₃×C₄	C₂×C₃³×C₁₉	1	40	yes	—
33	C₃²	C₃²×C₈×C₁₉	1	130	yes	direct product factor
34	C₈	C₃⁴×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₃	C₃³×C₄×C₁₉	1	40	yes	—
36	C₄	C₂×C₃⁴×C₁₉	1	1	yes	characteristic
37	C₃	C₃³×C₈×C₁₉	1	40	yes	direct product factor
38	C₂	C₃⁴×C₄×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₃⁴×C₁₉
socle, prime radical	C₂×C₃⁴×C₁₉	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃³×C₈×C₁₉	C₃	direct (N × H)
5. C₃⁴×C₁₉	C₈	direct (N × H)
6. C₃²×C₈×C₁₉	C₃²	direct (N × H)
9. C₃⁴×C₈	C₁₉	direct (N × H)
10. C₃³×C₁₉	C₃×C₈	direct (N × H)
11. C₃×C₈×C₁₉	C₃³	direct (N × H)
15. C₃³×C₈	C₃×C₁₉	direct (N × H)
16. C₃²×C₁₉	C₃²×C₈	direct (N × H)
18. C₈×C₁₉	C₃⁴	direct (N × H)
21. C₃⁴	C₈×C₁₉	direct (N × H)
23. C₃²×C₈	C₃²×C₁₉	direct (N × H)
24. C₃×C₁₉	C₃³×C₈	direct (N × H)
28. C₃³	C₃×C₈×C₁₉	direct (N × H)
29. C₃×C₈	C₃³×C₁₉	direct (N × H)
30. C₁₉	C₃⁴×C₈	direct (N × H)
33. C₃²	C₃²×C₈×C₁₉	direct (N × H)
34. C₈	C₃⁴×C₁₉	direct (N × H)
37. C₃	C₃³×C₈×C₁₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴×C₄×C₁₉	2
C₃³×C₈×C₁₉	3
C₃⁴×C₈	19

Subgroups — C34×C8×C19 #72190708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⁴×C₈×C₁₉ #72190708