Subgroups — C₃²×C₈×C₁₇×C₂₃ #72205129

Order 28152.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×C₁₇×C₂₃	C₃	1	4	yes	maximal, direct product factor
2	C₈×C₁₇×C₂₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
3	C₃²×C₄×C₁₇×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃²×C₁₇×C₂₃	C₄	1	1	yes	characteristic
5	C₃×C₄×C₁₇×C₂₃	C₂×C₃	1	4	yes	—
6	C₃²×C₁₇×C₂₃	C₈	1	1	yes	characteristic, direct product factor, Hall
7	C₂×C₃×C₁₇×C₂₃	C₃×C₄	1	4	yes	—
8	C₃²×C₈×C₂₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₄×C₁₇×C₂₃	C₂×C₃²	1	1	yes	characteristic
10	C₃²×C₈×C₁₇	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₁₇×C₂₃	C₃×C₈	1	4	yes	direct product factor
12	C₃²×C₄×C₂₃	C₂×C₁₇	1	1	yes	characteristic
13	C₂×C₁₇×C₂₃	C₃²×C₄	1	1	yes	characteristic
14	C₃²×C₄×C₁₇	C₂×C₂₃	1	1	yes	characteristic
15	C₃×C₈×C₂₃	C₃×C₁₇	1	4	yes	direct product factor
16	C₂×C₃²×C₂₃	C₄×C₁₇	1	1	yes	characteristic
17	C₃×C₈×C₁₇	C₃×C₂₃	1	4	yes	direct product factor
18	C₁₇×C₂₃	C₃²×C₈	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₃²×C₁₇	C₄×C₂₃	1	1	yes	characteristic
20	C₃×C₄×C₂₃	C₂×C₃×C₁₇	1	4	yes	—
21	C₃²×C₂₃	C₈×C₁₇	1	1	yes	characteristic, direct product factor, Hall
22	C₃×C₄×C₁₇	C₂×C₃×C₂₃	1	4	yes	—
23	C₈×C₂₃	C₃²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
24	C₃²×C₁₇	C₈×C₂₃	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₃×C₂₃	C₃×C₄×C₁₇	1	4	yes	—
26	C₈×C₁₇	C₃²×C₂₃	1	1	yes	characteristic, direct product factor, Hall
27	C₂×C₃×C₁₇	C₃×C₄×C₂₃	1	4	yes	—
28	C₄×C₂₃	C₂×C₃²×C₁₇	1	1	yes	characteristic
29	C₃²×C₈	C₁₇×C₂₃	1	1	yes	characteristic, direct product factor, Hall
30	C₃×C₂₃	C₃×C₈×C₁₇	1	4	yes	direct product factor
31	C₄×C₁₇	C₂×C₃²×C₂₃	1	1	yes	characteristic
32	C₃×C₁₇	C₃×C₈×C₂₃	1	4	yes	direct product factor
33	C₂×C₂₃	C₃²×C₄×C₁₇	1	1	yes	characteristic
34	C₃²×C₄	C₂×C₁₇×C₂₃	1	1	yes	characteristic
35	C₂×C₁₇	C₃²×C₄×C₂₃	1	1	yes	characteristic
36	C₃×C₈	C₃×C₁₇×C₂₃	1	4	yes	direct product factor
37	C₂₃	C₃²×C₈×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
38	C₂×C₃²	C₄×C₁₇×C₂₃	1	1	yes	characteristic
39	C₁₇	C₃²×C₈×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
40	C₃×C₄	C₂×C₃×C₁₇×C₂₃	1	4	yes	—
41	C₃²	C₈×C₁₇×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
42	C₈	C₃²×C₁₇×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃	C₃×C₄×C₁₇×C₂₃	1	4	yes	—
44	C₄	C₂×C₃²×C₁₇×C₂₃	1	1	yes	characteristic
45	C₃	C₃×C₈×C₁₇×C₂₃	1	4	yes	direct product factor
46	C₂	C₃²×C₄×C₁₇×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₃²×C₁₇×C₂₃
socle, prime radical	C₂×C₃²×C₁₇×C₂₃	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₈×C₁₇×C₂₃	C₃	direct (N × H)
2. C₈×C₁₇×C₂₃	C₃²	direct (N × H)
6. C₃²×C₁₇×C₂₃	C₈	direct (N × H)
8. C₃²×C₈×C₂₃	C₁₇	direct (N × H)
10. C₃²×C₈×C₁₇	C₂₃	direct (N × H)
11. C₃×C₁₇×C₂₃	C₃×C₈	direct (N × H)
15. C₃×C₈×C₂₃	C₃×C₁₇	direct (N × H)
17. C₃×C₈×C₁₇	C₃×C₂₃	direct (N × H)
18. C₁₇×C₂₃	C₃²×C₈	direct (N × H)
21. C₃²×C₂₃	C₈×C₁₇	direct (N × H)
23. C₈×C₂₃	C₃²×C₁₇	direct (N × H)
24. C₃²×C₁₇	C₈×C₂₃	direct (N × H)
26. C₈×C₁₇	C₃²×C₂₃	direct (N × H)
29. C₃²×C₈	C₁₇×C₂₃	direct (N × H)
30. C₃×C₂₃	C₃×C₈×C₁₇	direct (N × H)
32. C₃×C₁₇	C₃×C₈×C₂₃	direct (N × H)
36. C₃×C₈	C₃×C₁₇×C₂₃	direct (N × H)
37. C₂₃	C₃²×C₈×C₁₇	direct (N × H)
39. C₁₇	C₃²×C₈×C₂₃	direct (N × H)
41. C₃²	C₈×C₁₇×C₂₃	direct (N × H)
42. C₈	C₃²×C₁₇×C₂₃	direct (N × H)
45. C₃	C₃×C₈×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
C₃×C₈×C₁₇×C₂₃	3
C₃²×C₄×C₁₇×C₂₃	2
C₃²×C₈×C₂₃	17
C₃²×C₈×C₁₇	23

Subgroups — C32×C8×C17×C23 #72205129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₈×C₁₇×C₂₃ #72205129