Subgroups — C₃×C₄×C₂₇×C₆₇ #72199209

Order 21708.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂₇×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₉×C₆₇	C₃	1	1	yes	maximal, characteristic
3	C₄×C₂₇×C₆₇	C₃	1	3	yes	maximal, direct product factor
4	C₂×C₃×C₉×C₆₇	C₂×C₃	1	1	yes	characteristic
5	C₂×C₂₇×C₆₇	C₂×C₃	1	3	yes	—
6	C₃×C₂₇×C₆₇	C₄	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×C₄×C₆₇	C₉	1	1	yes	characteristic
8	C₄×C₉×C₆₇	C₉	1	2	yes	—
9	C₄×C₉×C₆₇	C₃²	1	1	yes	characteristic
10	C₃×C₉×C₆₇	C₃×C₄	1	1	yes	characteristic
11	C₂₇×C₆₇	C₃×C₄	1	3	yes	direct product factor
12	C₂×C₃²×C₆₇	C₂×C₉	1	1	yes	characteristic
13	C₂×C₉×C₆₇	C₂×C₉	1	2	yes	—
14	C₂×C₉×C₆₇	C₂×C₃²	1	1	yes	characteristic
15	C₃×C₄×C₆₇	C₂₇	1	3	yes	direct product factor
16	C₃×C₄×C₆₇	C₃×C₉	1	1	yes	characteristic
17	C₃²×C₆₇	C₄×C₉	1	1	yes	characteristic
18	C₉×C₆₇	C₄×C₉	1	2	yes	—
19	C₉×C₆₇	C₃²×C₄	1	1	yes	characteristic
20	C₂×C₃×C₆₇	C₂×C₂₇	1	3	yes	—
21	C₂×C₃×C₆₇	C₂×C₃×C₉	1	1	yes	characteristic
22	C₃×C₄×C₂₇	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₄×C₆₇	C₃×C₂₇	1	1	yes	characteristic, direct product factor, Hall
24	C₃×C₆₇	C₄×C₂₇	1	3	yes	direct product factor
25	C₃×C₆₇	C₃×C₄×C₉	1	1	yes	characteristic
26	C₂×C₃×C₂₇	C₂×C₆₇	1	1	yes	characteristic
27	C₂×C₆₇	C₂×C₃×C₂₇	1	1	yes	characteristic
28	C₃×C₄×C₉	C₃×C₆₇	1	1	yes	characteristic
29	C₄×C₂₇	C₃×C₆₇	1	3	yes	direct product factor
30	C₃×C₂₇	C₄×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
31	C₆₇	C₃×C₄×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×C₃×C₉	C₂×C₃×C₆₇	1	1	yes	characteristic
33	C₂×C₂₇	C₂×C₃×C₆₇	1	3	yes	—
34	C₃²×C₄	C₉×C₆₇	1	1	yes	characteristic
35	C₄×C₉	C₉×C₆₇	1	2	yes	—
36	C₄×C₉	C₃²×C₆₇	1	1	yes	characteristic
37	C₃×C₉	C₃×C₄×C₆₇	1	1	yes	characteristic
38	C₂₇	C₃×C₄×C₆₇	1	3	yes	direct product factor
39	C₂×C₃²	C₂×C₉×C₆₇	1	1	yes	characteristic
40	C₂×C₉	C₂×C₉×C₆₇	1	2	yes	—
41	C₂×C₉	C₂×C₃²×C₆₇	1	1	yes	characteristic
42	C₃×C₄	C₂₇×C₆₇	1	3	yes	direct product factor
43	C₃×C₄	C₃×C₉×C₆₇	1	1	yes	characteristic
44	C₃²	C₄×C₉×C₆₇	1	1	yes	characteristic
45	C₉	C₄×C₉×C₆₇	1	2	yes	—
46	C₉	C₃²×C₄×C₆₇	1	1	yes	characteristic
47	C₂×C₃	C₂×C₂₇×C₆₇	1	3	yes	—
48	C₂×C₃	C₂×C₃×C₉×C₆₇	1	1	yes	characteristic
49	C₄	C₃×C₂₇×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
50	C₃	C₄×C₂₇×C₆₇	1	3	yes	direct product factor
51	C₃	C₃×C₄×C₉×C₆₇	1	1	yes	characteristic
52	C₂	C₂×C₃×C₂₇×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₉	C₂×C₃²×C₆₇
socle, prime radical	C₂×C₃²×C₆₇	C₂×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₂₇×C₆₇	C₃	direct (N × H)
6. C₃×C₂₇×C₆₇	C₄	direct (N × H)
11. C₂₇×C₆₇	C₃×C₄	direct (N × H)
15. C₃×C₄×C₆₇	C₂₇	direct (N × H)
22. C₃×C₄×C₂₇	C₆₇	direct (N × H)
23. C₄×C₆₇	C₃×C₂₇	direct (N × H)
24. C₃×C₆₇	C₄×C₂₇	direct (N × H)
29. C₄×C₂₇	C₃×C₆₇	direct (N × H)
30. C₃×C₂₇	C₄×C₆₇	direct (N × H)
31. C₆₇	C₃×C₄×C₂₇	direct (N × H)
38. C₂₇	C₃×C₄×C₆₇	direct (N × H)
42. C₃×C₄	C₂₇×C₆₇	direct (N × H)
49. C₄	C₃×C₂₇×C₆₇	direct (N × H)
50. C₃	C₄×C₂₇×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×C₃×C₂₇×C₆₇	2
C₃×C₄×C₉×C₆₇	3
C₄×C₂₇×C₆₇	3
C₃×C₄×C₂₇	67

Subgroups — C3×C4×C27×C67 #72199209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₂₇×C₆₇ #72199209