Subgroups — C₄²×C₂₇×C₂₉ #72190870

Order 12528.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂₇×C₂₉	C₂	1	3	yes	maximal
2	C₄²×C₉×C₂₉	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₂₇×C₂₉	C₂²	1	1	yes	characteristic
4	C₄×C₂₇×C₂₉	C₄	1	6	yes	direct product factor
5	C₂×C₄×C₉×C₂₉	C₂×C₃	1	3	yes	—
6	C₂×C₂₇×C₂₉	C₂×C₄	1	3	yes	—
7	C₃×C₄²×C₂₉	C₉	1	1	yes	characteristic
8	C₂²×C₉×C₂₉	C₂²×C₃	1	1	yes	characteristic
9	C₄×C₉×C₂₉	C₃×C₄	1	6	yes	—
10	C₂₇×C₂₉	C₄²	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₃×C₄×C₂₉	C₂×C₉	1	3	yes	—
12	C₂×C₉×C₂₉	C₂×C₃×C₄	1	3	yes	—
13	C₄²×C₂₉	C₂₇	1	1	yes	characteristic, direct product factor, Hall
14	C₄²×C₂₇	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₃×C₂₉	C₂²×C₉	1	1	yes	characteristic
16	C₃×C₄×C₂₉	C₄×C₉	1	6	yes	—
17	C₉×C₂₉	C₃×C₄²	1	1	yes	characteristic
18	C₂×C₄×C₂₉	C₂×C₂₇	1	3	yes	—
19	C₂×C₄×C₂₇	C₂×C₂₉	1	3	yes	—
20	C₂×C₃×C₂₉	C₂×C₄×C₉	1	3	yes	—
21	C₄²×C₉	C₃×C₂₉	1	1	yes	characteristic
22	C₂²×C₂₉	C₂²×C₂₇	1	1	yes	characteristic
23	C₄×C₂₉	C₄×C₂₇	1	6	yes	direct product factor
24	C₂²×C₂₇	C₂²×C₂₉	1	1	yes	characteristic
25	C₄×C₂₇	C₄×C₂₉	1	6	yes	direct product factor
26	C₃×C₂₉	C₄²×C₉	1	1	yes	characteristic
27	C₂×C₄×C₉	C₂×C₃×C₂₉	1	3	yes	—
28	C₂×C₂₉	C₂×C₄×C₂₇	1	3	yes	—
29	C₂×C₂₇	C₂×C₄×C₂₉	1	3	yes	—
30	C₃×C₄²	C₉×C₂₉	1	1	yes	characteristic
31	C₂²×C₉	C₂²×C₃×C₂₉	1	1	yes	characteristic
32	C₄×C₉	C₃×C₄×C₂₉	1	6	yes	—
33	C₂₉	C₄²×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
34	C₂₇	C₄²×C₂₉	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₃×C₄	C₂×C₉×C₂₉	1	3	yes	—
36	C₂×C₉	C₂×C₃×C₄×C₂₉	1	3	yes	—
37	C₄²	C₂₇×C₂₉	1	1	yes	characteristic, direct product factor, Sylow
38	C₂²×C₃	C₂²×C₉×C₂₉	1	1	yes	characteristic
39	C₃×C₄	C₄×C₉×C₂₉	1	6	yes	—
40	C₉	C₃×C₄²×C₂₉	1	1	yes	characteristic
41	C₂×C₄	C₂×C₂₇×C₂₉	1	3	yes	—
42	C₂×C₃	C₂×C₄×C₉×C₂₉	1	3	yes	—
43	C₂²	C₂²×C₂₇×C₂₉	1	1	yes	characteristic
44	C₄	C₄×C₂₇×C₂₉	1	6	yes	direct product factor
45	C₃	C₄²×C₉×C₂₉	1	1	yes	characteristic
46	C₂	C₂×C₄×C₂₇×C₂₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₉	C₂²×C₃×C₂₉
socle, prime radical	C₂²×C₃×C₂₉	C₂²×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₂₇×C₂₉	C₄	direct (N × H)
10. C₂₇×C₂₉	C₄²	direct (N × H)
13. C₄²×C₂₉	C₂₇	direct (N × H)
14. C₄²×C₂₇	C₂₉	direct (N × H)
23. C₄×C₂₉	C₄×C₂₇	direct (N × H)
25. C₄×C₂₇	C₄×C₂₉	direct (N × H)
33. C₂₉	C₄²×C₂₇	direct (N × H)
34. C₂₇	C₄²×C₂₉	direct (N × H)
37. C₄²	C₂₇×C₂₉	direct (N × H)
44. C₄	C₄×C₂₇×C₂₉	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×C₄×C₂₇×C₂₉	2
C₄²×C₉×C₂₉	3
C₄²×C₂₇	29

Subgroups — C42×C27×C29 #72190870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₂₇×C₂₉ #72190870