Subgroups — C₃²×C₄×C₁₃×C₆₇ #44535641

Order 31356.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₁₃×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₁₃×C₆₇	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₁₃×C₆₇	C₄	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₃×C₁₃×C₆₇	C₂×C₃	1	4	yes	—
5	C₄×C₁₃×C₆₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
6	C₃×C₁₃×C₆₇	C₃×C₄	1	4	yes	direct product factor
7	C₃²×C₄×C₆₇	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₁₃×C₆₇	C₂×C₃²	1	1	yes	characteristic
9	C₂×C₃²×C₆₇	C₂×C₁₃	1	1	yes	characteristic
10	C₁₃×C₆₇	C₃²×C₄	1	1	yes	characteristic, direct product factor, Hall
11	C₃×C₄×C₆₇	C₃×C₁₃	1	4	yes	direct product factor
12	C₃²×C₆₇	C₄×C₁₃	1	1	yes	characteristic, direct product factor, Hall
13	C₃²×C₄×C₁₃	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₃×C₆₇	C₂×C₃×C₁₃	1	4	yes	—
15	C₄×C₆₇	C₃²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₃²×C₁₃	C₂×C₆₇	1	1	yes	characteristic
17	C₃×C₆₇	C₃×C₄×C₁₃	1	4	yes	direct product factor
18	C₃×C₄×C₁₃	C₃×C₆₇	1	4	yes	direct product factor
19	C₂×C₆₇	C₂×C₃²×C₁₃	1	1	yes	characteristic
20	C₃²×C₁₃	C₄×C₆₇	1	1	yes	characteristic, direct product factor, Hall
21	C₂×C₃×C₁₃	C₂×C₃×C₆₇	1	4	yes	—
22	C₆₇	C₃²×C₄×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
23	C₄×C₁₃	C₃²×C₆₇	1	1	yes	characteristic, direct product factor, Hall
24	C₃×C₁₃	C₃×C₄×C₆₇	1	4	yes	direct product factor
25	C₃²×C₄	C₁₃×C₆₇	1	1	yes	characteristic, direct product factor, Hall
26	C₂×C₁₃	C₂×C₃²×C₆₇	1	1	yes	characteristic
27	C₂×C₃²	C₂×C₁₃×C₆₇	1	1	yes	characteristic
28	C₁₃	C₃²×C₄×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
29	C₃×C₄	C₃×C₁₃×C₆₇	1	4	yes	direct product factor
30	C₃²	C₄×C₁₃×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃	C₂×C₃×C₁₃×C₆₇	1	4	yes	—
32	C₄	C₃²×C₁₃×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	C₃×C₄×C₁₃×C₆₇	1	4	yes	direct product factor
34	C₂	C₂×C₃²×C₁₃×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃²×C₁₃×C₆₇
socle, prime radical	C₂×C₃²×C₁₃×C₆₇	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₁₃×C₆₇	C₃	direct (N × H)
3. C₃²×C₁₃×C₆₇	C₄	direct (N × H)
5. C₄×C₁₃×C₆₇	C₃²	direct (N × H)
6. C₃×C₁₃×C₆₇	C₃×C₄	direct (N × H)
7. C₃²×C₄×C₆₇	C₁₃	direct (N × H)
10. C₁₃×C₆₇	C₃²×C₄	direct (N × H)
11. C₃×C₄×C₆₇	C₃×C₁₃	direct (N × H)
12. C₃²×C₆₇	C₄×C₁₃	direct (N × H)
13. C₃²×C₄×C₁₃	C₆₇	direct (N × H)
15. C₄×C₆₇	C₃²×C₁₃	direct (N × H)
17. C₃×C₆₇	C₃×C₄×C₁₃	direct (N × H)
18. C₃×C₄×C₁₃	C₃×C₆₇	direct (N × H)
20. C₃²×C₁₃	C₄×C₆₇	direct (N × H)
22. C₆₇	C₃²×C₄×C₁₃	direct (N × H)
23. C₄×C₁₃	C₃²×C₆₇	direct (N × H)
24. C₃×C₁₃	C₃×C₄×C₆₇	direct (N × H)
25. C₃²×C₄	C₁₃×C₆₇	direct (N × H)
28. C₁₃	C₃²×C₄×C₆₇	direct (N × H)
29. C₃×C₄	C₃×C₁₃×C₆₇	direct (N × H)
30. C₃²	C₄×C₁₃×C₆₇	direct (N × H)
32. C₄	C₃²×C₁₃×C₆₇	direct (N × H)
33. C₃	C₃×C₄×C₁₃×C₆₇	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×C₃²×C₁₃×C₆₇	2
C₃×C₄×C₁₃×C₆₇	3
C₃²×C₄×C₆₇	13
C₃²×C₄×C₁₃	67

Subgroups — C32×C4×C13×C67 #44535641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₁₃×C₆₇ #44535641