Subgroups — C₃²×C₄×C₇×C₆₇ #30683096

Order 16884.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₇×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇×C₆₇	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₇×C₆₇	C₄	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₃×C₇×C₆₇	C₂×C₃	1	4	yes	—
5	C₃²×C₄×C₆₇	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₄×C₇×C₆₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃×C₇×C₆₇	C₃×C₄	1	4	yes	direct product factor
8	C₂×C₃²×C₆₇	C₂×C₇	1	1	yes	characteristic
9	C₂×C₇×C₆₇	C₂×C₃²	1	1	yes	characteristic
10	C₃×C₄×C₆₇	C₃×C₇	1	4	yes	direct product factor
11	C₃²×C₆₇	C₄×C₇	1	1	yes	characteristic, direct product factor, Hall
12	C₇×C₆₇	C₃²×C₄	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₃×C₆₇	C₂×C₃×C₇	1	4	yes	—
14	C₄×C₆₇	C₃²×C₇	1	1	yes	characteristic, direct product factor, Hall
15	C₃²×C₄×C₇	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₃×C₆₇	C₃×C₄×C₇	1	4	yes	direct product factor
17	C₂×C₆₇	C₂×C₃²×C₇	1	1	yes	characteristic
18	C₂×C₃²×C₇	C₂×C₆₇	1	1	yes	characteristic
19	C₃×C₄×C₇	C₃×C₆₇	1	4	yes	direct product factor
20	C₆₇	C₃²×C₄×C₇	1	1	yes	characteristic, direct product factor, Sylow
21	C₃²×C₇	C₄×C₆₇	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₃×C₇	C₂×C₃×C₆₇	1	4	yes	—
23	C₃²×C₄	C₇×C₆₇	1	1	yes	characteristic, direct product factor, Hall
24	C₄×C₇	C₃²×C₆₇	1	1	yes	characteristic, direct product factor, Hall
25	C₃×C₇	C₃×C₄×C₆₇	1	4	yes	direct product factor
26	C₂×C₃²	C₂×C₇×C₆₇	1	1	yes	characteristic
27	C₂×C₇	C₂×C₃²×C₆₇	1	1	yes	characteristic
28	C₃×C₄	C₃×C₇×C₆₇	1	4	yes	direct product factor
29	C₃²	C₄×C₇×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
30	C₇	C₃²×C₄×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃	C₂×C₃×C₇×C₆₇	1	4	yes	—
32	C₄	C₃²×C₇×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	C₃×C₄×C₇×C₆₇	1	4	yes	direct product factor
34	C₂	C₂×C₃²×C₇×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃²×C₇×C₆₇
socle, prime radical	C₂×C₃²×C₇×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×C₇×C₆₇	C₃	direct (N × H)
3. C₃²×C₇×C₆₇	C₄	direct (N × H)
5. C₃²×C₄×C₆₇	C₇	direct (N × H)
6. C₄×C₇×C₆₇	C₃²	direct (N × H)
7. C₃×C₇×C₆₇	C₃×C₄	direct (N × H)
10. C₃×C₄×C₆₇	C₃×C₇	direct (N × H)
11. C₃²×C₆₇	C₄×C₇	direct (N × H)
12. C₇×C₆₇	C₃²×C₄	direct (N × H)
14. C₄×C₆₇	C₃²×C₇	direct (N × H)
15. C₃²×C₄×C₇	C₆₇	direct (N × H)
16. C₃×C₆₇	C₃×C₄×C₇	direct (N × H)
19. C₃×C₄×C₇	C₃×C₆₇	direct (N × H)
20. C₆₇	C₃²×C₄×C₇	direct (N × H)
21. C₃²×C₇	C₄×C₆₇	direct (N × H)
23. C₃²×C₄	C₇×C₆₇	direct (N × H)
24. C₄×C₇	C₃²×C₆₇	direct (N × H)
25. C₃×C₇	C₃×C₄×C₆₇	direct (N × H)
28. C₃×C₄	C₃×C₇×C₆₇	direct (N × H)
29. C₃²	C₄×C₇×C₆₇	direct (N × H)
30. C₇	C₃²×C₄×C₆₇	direct (N × H)
32. C₄	C₃²×C₇×C₆₇	direct (N × H)
33. C₃	C₃×C₄×C₇×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×C₇×C₆₇	2
C₃×C₄×C₇×C₆₇	3
C₃²×C₄×C₆₇	7
C₃²×C₄×C₇	67

Subgroups — C32×C4×C7×C67 #30683096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₇×C₆₇ #30683096