Subgroups — C₂×C₄²×C₁₇×C₄₁ #72199749

Order 22304.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₇×C₄₁	C₂	1	3	yes	maximal
2	C₄²×C₁₇×C₄₁	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₁₇×C₄₁	C₂²	1	1	yes	characteristic
4	C₂×C₄×C₁₇×C₄₁	C₄	1	12	yes	direct product factor
5	C₂×C₄×C₁₇×C₄₁	C₂²	1	6	yes	—
6	C₂²×C₁₇×C₄₁	C₂×C₄	1	6	yes	—
7	C₂²×C₁₇×C₄₁	C₂³	1	1	yes	characteristic
8	C₄×C₁₇×C₄₁	C₂×C₄	1	12	yes	direct product factor
9	C₂×C₁₇×C₄₁	C₄²	1	4	yes	direct product factor
10	C₂×C₁₇×C₄₁	C₂²×C₄	1	3	yes	—
11	C₂×C₄²×C₄₁	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₁₇×C₄₁	C₂×C₄²	1	1	yes	characteristic, direct product factor, Hall
13	C₂²×C₄×C₄₁	C₂×C₁₇	1	3	yes	—
14	C₄²×C₄₁	C₂×C₁₇	1	4	yes	direct product factor
15	C₂×C₄²×C₁₇	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂³×C₄₁	C₂²×C₁₇	1	1	yes	characteristic
17	C₂×C₄×C₄₁	C₄×C₁₇	1	12	yes	direct product factor
18	C₂×C₄×C₄₁	C₂²×C₁₇	1	6	yes	—
19	C₂²×C₄×C₁₇	C₂×C₄₁	1	3	yes	—
20	C₄²×C₁₇	C₂×C₄₁	1	4	yes	direct product factor
21	C₂²×C₄₁	C₂×C₄×C₁₇	1	6	yes	—
22	C₂²×C₄₁	C₂³×C₁₇	1	1	yes	characteristic
23	C₄×C₄₁	C₂×C₄×C₁₇	1	12	yes	direct product factor
24	C₂³×C₁₇	C₂²×C₄₁	1	1	yes	characteristic
25	C₂×C₄×C₁₇	C₄×C₄₁	1	12	yes	direct product factor
26	C₂×C₄×C₁₇	C₂²×C₄₁	1	6	yes	—
27	C₂×C₄₁	C₄²×C₁₇	1	4	yes	direct product factor
28	C₂×C₄₁	C₂²×C₄×C₁₇	1	3	yes	—
29	C₂²×C₁₇	C₂×C₄×C₄₁	1	6	yes	—
30	C₂²×C₁₇	C₂³×C₄₁	1	1	yes	characteristic
31	C₄×C₁₇	C₂×C₄×C₄₁	1	12	yes	direct product factor
32	C₄₁	C₂×C₄²×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×C₁₇	C₄²×C₄₁	1	4	yes	direct product factor
34	C₂×C₁₇	C₂²×C₄×C₄₁	1	3	yes	—
35	C₂×C₄²	C₁₇×C₄₁	1	1	yes	characteristic, direct product factor, Sylow
36	C₁₇	C₂×C₄²×C₄₁	1	1	yes	characteristic, direct product factor, Sylow
37	C₂²×C₄	C₂×C₁₇×C₄₁	1	3	yes	—
38	C₄²	C₂×C₁₇×C₄₁	1	4	yes	direct product factor
39	C₂³	C₂²×C₁₇×C₄₁	1	1	yes	characteristic
40	C₂×C₄	C₄×C₁₇×C₄₁	1	12	yes	direct product factor
41	C₂×C₄	C₂²×C₁₇×C₄₁	1	6	yes	—
42	C₂²	C₂×C₄×C₁₇×C₄₁	1	6	yes	—
43	C₂²	C₂³×C₁₇×C₄₁	1	1	yes	characteristic
44	C₄	C₂×C₄×C₁₇×C₄₁	1	12	yes	direct product factor
45	C₂	C₄²×C₁₇×C₄₁	1	4	yes	direct product factor
46	C₂	C₂²×C₄×C₁₇×C₄₁	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂³×C₁₇×C₄₁
socle, prime radical	C₂³×C₁₇×C₄₁	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×C₁₇×C₄₁	C₂	direct (N × H)
4. C₂×C₄×C₁₇×C₄₁	C₄	direct (N × H)
8. C₄×C₁₇×C₄₁	C₂×C₄	direct (N × H)
9. C₂×C₁₇×C₄₁	C₄²	direct (N × H)
11. C₂×C₄²×C₄₁	C₁₇	direct (N × H)
12. C₁₇×C₄₁	C₂×C₄²	direct (N × H)
14. C₄²×C₄₁	C₂×C₁₇	direct (N × H)
15. C₂×C₄²×C₁₇	C₄₁	direct (N × H)
17. C₂×C₄×C₄₁	C₄×C₁₇	direct (N × H)
20. C₄²×C₁₇	C₂×C₄₁	direct (N × H)
23. C₄×C₄₁	C₂×C₄×C₁₇	direct (N × H)
25. C₂×C₄×C₁₇	C₄×C₄₁	direct (N × H)
27. C₂×C₄₁	C₄²×C₁₇	direct (N × H)
31. C₄×C₁₇	C₂×C₄×C₄₁	direct (N × H)
32. C₄₁	C₂×C₄²×C₁₇	direct (N × H)
33. C₂×C₁₇	C₄²×C₄₁	direct (N × H)
35. C₂×C₄²	C₁₇×C₄₁	direct (N × H)
36. C₁₇	C₂×C₄²×C₄₁	direct (N × H)
38. C₄²	C₂×C₁₇×C₄₁	direct (N × H)
40. C₂×C₄	C₄×C₁₇×C₄₁	direct (N × H)
44. C₄	C₂×C₄×C₁₇×C₄₁	direct (N × H)
45. C₂	C₄²×C₁₇×C₄₁	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×C₄×C₁₇×C₄₁	2
C₄²×C₁₇×C₄₁	2
C₂×C₄²×C₄₁	17
C₂×C₄²×C₁₇	41

Subgroups — C2×C42×C17×C41 #72199749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×C₁₇×C₄₁ #72199749