Subgroups — C₂×C₄×C₉²×C₄₇ #72207239

Order 30456.

368 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉²×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉²×C₄₇	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₄×C₉×C₄₇	C₃	1	4	yes	maximal
4	C₂×C₉²×C₄₇	C₄	1	2	yes	direct product factor
5	C₂×C₉²×C₄₇	C₂²	1	1	yes	characteristic
6	C₂²×C₃×C₉×C₄₇	C₂×C₃	1	4	yes	—
7	C₃×C₄×C₉×C₄₇	C₂×C₃	1	8	yes	—
8	C₂×C₃²×C₄×C₄₇	C₃²	1	1	yes	characteristic
9	C₂×C₄×C₉×C₄₇	C₉	1	12	yes	direct product factor
10	C₂×C₃×C₉×C₄₇	C₃×C₄	1	8	yes	—
11	C₂×C₃×C₉×C₄₇	C₂²×C₃	1	4	yes	—
12	C₉²×C₄₇	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
13	C₂²×C₃²×C₄₇	C₂×C₃²	1	1	yes	characteristic
14	C₃²×C₄×C₄₇	C₂×C₃²	1	2	yes	—
15	C₂²×C₉×C₄₇	C₂×C₉	1	12	yes	—
16	C₄×C₉×C₄₇	C₂×C₉	1	24	yes	direct product factor
17	C₃×C₉×C₄₇	C₂×C₃×C₄	1	4	yes	—
18	C₂×C₃×C₄×C₄₇	C₃×C₉	1	4	yes	—
19	C₂×C₃²×C₄₇	C₃²×C₄	1	2	yes	—
20	C₂×C₃²×C₄₇	C₂²×C₃²	1	1	yes	characteristic
21	C₂×C₉×C₄₇	C₄×C₉	1	24	yes	direct product factor
22	C₂×C₉×C₄₇	C₂²×C₉	1	12	yes	—
23	C₂×C₄×C₉²	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂²×C₃×C₄₇	C₂×C₃×C₉	1	4	yes	—
25	C₃×C₄×C₄₇	C₂×C₃×C₉	1	8	yes	—
26	C₃²×C₄₇	C₂×C₃²×C₄	1	1	yes	characteristic
27	C₉×C₄₇	C₂×C₄×C₉	1	12	yes	direct product factor
28	C₂×C₄×C₄₇	C₉²	1	1	yes	characteristic, direct product factor, Hall
29	C₂²×C₉²	C₂×C₄₇	1	1	yes	characteristic
30	C₄×C₉²	C₂×C₄₇	1	2	yes	direct product factor
31	C₂×C₃×C₄₇	C₃×C₄×C₉	1	8	yes	—
32	C₂×C₃×C₄₇	C₂²×C₃×C₉	1	4	yes	—
33	C₂×C₃×C₄×C₉	C₃×C₄₇	1	4	yes	—
34	C₂²×C₄₇	C₂×C₉²	1	1	yes	characteristic
35	C₄×C₄₇	C₂×C₉²	1	2	yes	direct product factor
36	C₂×C₉²	C₄×C₄₇	1	2	yes	direct product factor
37	C₂×C₉²	C₂²×C₄₇	1	1	yes	characteristic
38	C₃×C₄₇	C₂×C₃×C₄×C₉	1	4	yes	—
39	C₂²×C₃×C₉	C₂×C₃×C₄₇	1	4	yes	—
40	C₃×C₄×C₉	C₂×C₃×C₄₇	1	8	yes	—
41	C₂×C₄₇	C₄×C₉²	1	2	yes	direct product factor
42	C₂×C₄₇	C₂²×C₉²	1	1	yes	characteristic
43	C₉²	C₂×C₄×C₄₇	1	1	yes	characteristic, direct product factor, Sylow
44	C₂×C₃²×C₄	C₃²×C₄₇	1	1	yes	characteristic
45	C₂×C₄×C₉	C₉×C₄₇	1	12	yes	direct product factor
46	C₂×C₃×C₉	C₃×C₄×C₄₇	1	8	yes	—
47	C₂×C₃×C₉	C₂²×C₃×C₄₇	1	4	yes	—
48	C₄₇	C₂×C₄×C₉²	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×C₃²	C₂×C₃²×C₄₇	1	1	yes	characteristic
50	C₃²×C₄	C₂×C₃²×C₄₇	1	2	yes	—
51	C₂²×C₉	C₂×C₉×C₄₇	1	12	yes	—
52	C₄×C₉	C₂×C₉×C₄₇	1	24	yes	direct product factor
53	C₃×C₉	C₂×C₃×C₄×C₄₇	1	4	yes	—
54	C₂×C₃×C₄	C₃×C₉×C₄₇	1	4	yes	—
55	C₂×C₃²	C₃²×C₄×C₄₇	1	2	yes	—
56	C₂×C₃²	C₂²×C₃²×C₄₇	1	1	yes	characteristic
57	C₂×C₉	C₄×C₉×C₄₇	1	24	yes	direct product factor
58	C₂×C₉	C₂²×C₉×C₄₇	1	12	yes	—
59	C₂²×C₃	C₂×C₃×C₉×C₄₇	1	4	yes	—
60	C₃×C₄	C₂×C₃×C₉×C₄₇	1	8	yes	—
61	C₃²	C₂×C₃²×C₄×C₄₇	1	1	yes	characteristic
62	C₉	C₂×C₄×C₉×C₄₇	1	12	yes	direct product factor
63	C₂×C₄	C₉²×C₄₇	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃	C₃×C₄×C₉×C₄₇	1	8	yes	—
65	C₂×C₃	C₂²×C₃×C₉×C₄₇	1	4	yes	—
66	C₂²	C₂×C₉²×C₄₇	1	1	yes	characteristic
67	C₄	C₂×C₉²×C₄₇	1	2	yes	direct product factor
68	C₃	C₂×C₃×C₄×C₉×C₄₇	1	4	yes	—
69	C₂	C₄×C₉²×C₄₇	1	2	yes	direct product factor
70	C₂	C₂²×C₉²×C₄₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃²	C₂²×C₃²×C₄₇
socle, prime radical	C₂²×C₃²×C₄₇	C₂×C₃²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₉²×C₄₇	C₂	direct (N × H)
4. C₂×C₉²×C₄₇	C₄	direct (N × H)
9. C₂×C₄×C₉×C₄₇	C₉	direct (N × H)
12. C₉²×C₄₇	C₂×C₄	direct (N × H)
16. C₄×C₉×C₄₇	C₂×C₉	direct (N × H)
21. C₂×C₉×C₄₇	C₄×C₉	direct (N × H)
23. C₂×C₄×C₉²	C₄₇	direct (N × H)
27. C₉×C₄₇	C₂×C₄×C₉	direct (N × H)
28. C₂×C₄×C₄₇	C₉²	direct (N × H)
30. C₄×C₉²	C₂×C₄₇	direct (N × H)
35. C₄×C₄₇	C₂×C₉²	direct (N × H)
36. C₂×C₉²	C₄×C₄₇	direct (N × H)
41. C₂×C₄₇	C₄×C₉²	direct (N × H)
43. C₉²	C₂×C₄×C₄₇	direct (N × H)
45. C₂×C₄×C₉	C₉×C₄₇	direct (N × H)
48. C₄₇	C₂×C₄×C₉²	direct (N × H)
52. C₄×C₉	C₂×C₉×C₄₇	direct (N × H)
57. C₂×C₉	C₄×C₉×C₄₇	direct (N × H)
62. C₉	C₂×C₄×C₉×C₄₇	direct (N × H)
63. C₂×C₄	C₉²×C₄₇	direct (N × H)
67. C₄	C₂×C₉²×C₄₇	direct (N × H)
69. C₂	C₄×C₉²×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
C₂²×C₉²×C₄₇	2
C₄×C₉²×C₄₇	2
C₂×C₃×C₄×C₉×C₄₇	3
C₂×C₄×C₉²	47

Subgroups — C2×C4×C92×C47 #72207239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₉²×C₄₇ #72207239