Subgroups — C₂×C₄²×C₁₇×C₂₃ #72190858

Order 12512.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂³×C₁₇×C₂₃
socle, prime radical	C₂³×C₁₇×C₂₃	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×C₂₃	C₂	direct (N × H)
4. C₂×C₄×C₁₇×C₂₃	C₄	direct (N × H)
8. C₄×C₁₇×C₂₃	C₂×C₄	direct (N × H)
9. C₂×C₁₇×C₂₃	C₄²	direct (N × H)
11. C₂×C₄²×C₂₃	C₁₇	direct (N × H)
12. C₂×C₄²×C₁₇	C₂₃	direct (N × H)
13. C₁₇×C₂₃	C₂×C₄²	direct (N × H)
15. C₄²×C₂₃	C₂×C₁₇	direct (N × H)
17. C₄²×C₁₇	C₂×C₂₃	direct (N × H)
19. C₂×C₄×C₂₃	C₄×C₁₇	direct (N × H)
22. C₂×C₄×C₁₇	C₄×C₂₃	direct (N × H)
26. C₄×C₂₃	C₂×C₄×C₁₇	direct (N × H)
29. C₄×C₁₇	C₂×C₄×C₂₃	direct (N × H)
30. C₂×C₂₃	C₄²×C₁₇	direct (N × H)
32. C₂×C₁₇	C₄²×C₂₃	direct (N × H)
34. C₂×C₄²	C₁₇×C₂₃	direct (N × H)
35. C₂₃	C₂×C₄²×C₁₇	direct (N × H)
36. C₁₇	C₂×C₄²×C₂₃	direct (N × H)
38. C₄²	C₂×C₁₇×C₂₃	direct (N × H)
40. C₂×C₄	C₄×C₁₇×C₂₃	direct (N × H)
44. C₄	C₂×C₄×C₁₇×C₂₃	direct (N × H)
45. C₂	C₄²×C₁₇×C₂₃	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×C₂₃	2
C₄²×C₁₇×C₂₃	2
C₂×C₄²×C₂₃	17
C₂×C₄²×C₁₇	23

Subgroups — C2×C42×C17×C23 #72190858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×C₁₇×C₂₃ #72190858