Subgroups — C₈²×C₉×C₄₇ #72204136

Order 27072.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×C₉×C₄₇	C₂	1	3	yes	maximal
2	C₃×C₈²×C₄₇	C₃	1	1	yes	maximal, characteristic
3	C₄²×C₉×C₄₇	C₂²	1	1	yes	characteristic
4	C₂×C₈×C₉×C₄₇	C₄	1	6	yes	—
5	C₃×C₄×C₈×C₄₇	C₂×C₃	1	3	yes	—
6	C₂×C₄×C₉×C₄₇	C₂×C₄	1	3	yes	—
7	C₈×C₉×C₄₇	C₈	1	12	yes	direct product factor
8	C₃×C₄²×C₄₇	C₂²×C₃	1	1	yes	characteristic
9	C₂×C₃×C₈×C₄₇	C₃×C₄	1	6	yes	—
10	C₈²×C₄₇	C₉	1	1	yes	characteristic, direct product factor, Hall
11	C₂²×C₉×C₄₇	C₄²	1	1	yes	characteristic
12	C₄×C₉×C₄₇	C₂×C₈	1	6	yes	—
13	C₄×C₈×C₄₇	C₂×C₉	1	3	yes	—
14	C₂×C₃×C₄×C₄₇	C₂×C₃×C₄	1	3	yes	—
15	C₃×C₈×C₄₇	C₃×C₈	1	12	yes	—
16	C₂×C₉×C₄₇	C₄×C₈	1	3	yes	—
17	C₂×C₈×C₄₇	C₄×C₉	1	6	yes	—
18	C₄²×C₄₇	C₂²×C₉	1	1	yes	characteristic
19	C₈²×C₉	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂²×C₃×C₄₇	C₃×C₄²	1	1	yes	characteristic
21	C₃×C₄×C₄₇	C₂×C₃×C₈	1	6	yes	—
22	C₉×C₄₇	C₈²	1	1	yes	characteristic, direct product factor, Hall
23	C₂×C₄×C₄₇	C₂×C₄×C₉	1	3	yes	—
24	C₈×C₄₇	C₈×C₉	1	12	yes	direct product factor
25	C₄×C₈×C₉	C₂×C₄₇	1	3	yes	—
26	C₂×C₃×C₄₇	C₃×C₄×C₈	1	3	yes	—
27	C₃×C₈²	C₃×C₄₇	1	1	yes	characteristic
28	C₂²×C₄₇	C₄²×C₉	1	1	yes	characteristic
29	C₄×C₄₇	C₂×C₈×C₉	1	6	yes	—
30	C₂×C₈×C₉	C₄×C₄₇	1	6	yes	—
31	C₄²×C₉	C₂²×C₄₇	1	1	yes	characteristic
32	C₃×C₄₇	C₃×C₈²	1	1	yes	characteristic
33	C₃×C₄×C₈	C₂×C₃×C₄₇	1	3	yes	—
34	C₂×C₄₇	C₄×C₈×C₉	1	3	yes	—
35	C₂×C₄×C₉	C₂×C₄×C₄₇	1	3	yes	—
36	C₈×C₉	C₈×C₄₇	1	12	yes	direct product factor
37	C₈²	C₉×C₄₇	1	1	yes	characteristic, direct product factor, Sylow
38	C₂×C₃×C₈	C₃×C₄×C₄₇	1	6	yes	—
39	C₃×C₄²	C₂²×C₃×C₄₇	1	1	yes	characteristic
40	C₄₇	C₈²×C₉	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²×C₉	C₄²×C₄₇	1	1	yes	characteristic
42	C₄×C₉	C₂×C₈×C₄₇	1	6	yes	—
43	C₄×C₈	C₂×C₉×C₄₇	1	3	yes	—
44	C₂×C₃×C₄	C₂×C₃×C₄×C₄₇	1	3	yes	—
45	C₃×C₈	C₃×C₈×C₄₇	1	12	yes	—
46	C₂×C₉	C₄×C₈×C₄₇	1	3	yes	—
47	C₂×C₈	C₄×C₉×C₄₇	1	6	yes	—
48	C₄²	C₂²×C₉×C₄₇	1	1	yes	characteristic
49	C₂²×C₃	C₃×C₄²×C₄₇	1	1	yes	characteristic
50	C₃×C₄	C₂×C₃×C₈×C₄₇	1	6	yes	—
51	C₉	C₈²×C₄₇	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₄	C₂×C₄×C₉×C₄₇	1	3	yes	—
53	C₈	C₈×C₉×C₄₇	1	12	yes	direct product factor
54	C₂×C₃	C₃×C₄×C₈×C₄₇	1	3	yes	—
55	C₂²	C₄²×C₉×C₄₇	1	1	yes	characteristic
56	C₄	C₂×C₈×C₉×C₄₇	1	6	yes	—
57	C₃	C₃×C₈²×C₄₇	1	1	yes	characteristic
58	C₂	C₄×C₈×C₉×C₄₇	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄²	C₂²×C₃×C₄₇
socle, prime radical	C₂²×C₃×C₄₇	C₃×C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₈×C₉×C₄₇	C₈	direct (N × H)
10. C₈²×C₄₇	C₉	direct (N × H)
19. C₈²×C₉	C₄₇	direct (N × H)
22. C₉×C₄₇	C₈²	direct (N × H)
24. C₈×C₄₇	C₈×C₉	direct (N × H)
36. C₈×C₉	C₈×C₄₇	direct (N × H)
37. C₈²	C₉×C₄₇	direct (N × H)
40. C₄₇	C₈²×C₉	direct (N × H)
51. C₉	C₈²×C₄₇	direct (N × H)
53. C₈	C₈×C₉×C₄₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₈×C₉×C₄₇	2
C₃×C₈²×C₄₇	3
C₈²×C₉	47

Subgroups — C82×C9×C47 #72204136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈²×C₉×C₄₇ #72204136