Subgroups — C₂²×C₈×C₁₇×C₂₃ #72190859

Order 12512.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₇×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₁₇×C₂₃	C₂	1	6	yes	maximal, direct product factor
3	C₂³×C₁₇×C₂₃	C₄	1	1	yes	characteristic
4	C₂×C₄×C₁₇×C₂₃	C₄	1	3	yes	—
5	C₂×C₄×C₁₇×C₂₃	C₂²	1	3	yes	—
6	C₈×C₁₇×C₂₃	C₂²	1	4	yes	direct product factor
7	C₂²×C₁₇×C₂₃	C₈	1	4	yes	direct product factor
8	C₂²×C₁₇×C₂₃	C₂×C₄	1	3	yes	—
9	C₄×C₁₇×C₂₃	C₂×C₄	1	3	yes	—
10	C₄×C₁₇×C₂₃	C₂³	1	1	yes	characteristic
11	C₂×C₁₇×C₂₃	C₂×C₈	1	6	yes	direct product factor
12	C₂×C₁₇×C₂₃	C₂²×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₈×C₁₇	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₁₇×C₂₃	C₂²×C₈	1	1	yes	characteristic, direct product factor, Hall
16	C₂²×C₄×C₂₃	C₂×C₁₇	1	1	yes	characteristic
17	C₂×C₈×C₂₃	C₂×C₁₇	1	6	yes	direct product factor
18	C₂²×C₄×C₁₇	C₂×C₂₃	1	1	yes	characteristic
19	C₂×C₈×C₁₇	C₂×C₂₃	1	6	yes	direct product factor
20	C₂³×C₂₃	C₄×C₁₇	1	1	yes	characteristic
21	C₂×C₄×C₂₃	C₄×C₁₇	1	3	yes	—
22	C₂×C₄×C₂₃	C₂²×C₁₇	1	3	yes	—
23	C₈×C₂₃	C₂²×C₁₇	1	4	yes	direct product factor
24	C₂³×C₁₇	C₄×C₂₃	1	1	yes	characteristic
25	C₂×C₄×C₁₇	C₄×C₂₃	1	3	yes	—
26	C₂×C₄×C₁₇	C₂²×C₂₃	1	3	yes	—
27	C₈×C₁₇	C₂²×C₂₃	1	4	yes	direct product factor
28	C₂²×C₂₃	C₈×C₁₇	1	4	yes	direct product factor
29	C₂²×C₂₃	C₂×C₄×C₁₇	1	3	yes	—
30	C₄×C₂₃	C₂×C₄×C₁₇	1	3	yes	—
31	C₄×C₂₃	C₂³×C₁₇	1	1	yes	characteristic
32	C₂²×C₁₇	C₈×C₂₃	1	4	yes	direct product factor
33	C₂²×C₁₇	C₂×C₄×C₂₃	1	3	yes	—
34	C₄×C₁₇	C₂×C₄×C₂₃	1	3	yes	—
35	C₄×C₁₇	C₂³×C₂₃	1	1	yes	characteristic
36	C₂×C₂₃	C₂×C₈×C₁₇	1	6	yes	direct product factor
37	C₂×C₂₃	C₂²×C₄×C₁₇	1	1	yes	characteristic
38	C₂×C₁₇	C₂×C₈×C₂₃	1	6	yes	direct product factor
39	C₂×C₁₇	C₂²×C₄×C₂₃	1	1	yes	characteristic
40	C₂²×C₈	C₁₇×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
41	C₂₃	C₂²×C₈×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
42	C₁₇	C₂²×C₈×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²×C₄	C₂×C₁₇×C₂₃	1	1	yes	characteristic
44	C₂×C₈	C₂×C₁₇×C₂₃	1	6	yes	direct product factor
45	C₂³	C₄×C₁₇×C₂₃	1	1	yes	characteristic
46	C₂×C₄	C₄×C₁₇×C₂₃	1	3	yes	—
47	C₂×C₄	C₂²×C₁₇×C₂₃	1	3	yes	—
48	C₈	C₂²×C₁₇×C₂₃	1	4	yes	direct product factor
49	C₂²	C₈×C₁₇×C₂₃	1	4	yes	direct product factor
50	C₂²	C₂×C₄×C₁₇×C₂₃	1	3	yes	—
51	C₄	C₂×C₄×C₁₇×C₂₃	1	3	yes	—
52	C₄	C₂³×C₁₇×C₂₃	1	1	yes	characteristic
53	C₂	C₂×C₈×C₁₇×C₂₃	1	6	yes	direct product factor
54	C₂	C₂²×C₄×C₁₇×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂³×C₁₇×C₂₃
socle, prime radical	C₂³×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
2. C₂×C₈×C₁₇×C₂₃	C₂	direct (N × H)
6. C₈×C₁₇×C₂₃	C₂²	direct (N × H)
7. C₂²×C₁₇×C₂₃	C₈	direct (N × H)
11. C₂×C₁₇×C₂₃	C₂×C₈	direct (N × H)
13. C₂²×C₈×C₂₃	C₁₇	direct (N × H)
14. C₂²×C₈×C₁₇	C₂₃	direct (N × H)
15. C₁₇×C₂₃	C₂²×C₈	direct (N × H)
17. C₂×C₈×C₂₃	C₂×C₁₇	direct (N × H)
19. C₂×C₈×C₁₇	C₂×C₂₃	direct (N × H)
23. C₈×C₂₃	C₂²×C₁₇	direct (N × H)
27. C₈×C₁₇	C₂²×C₂₃	direct (N × H)
28. C₂²×C₂₃	C₈×C₁₇	direct (N × H)
32. C₂²×C₁₇	C₈×C₂₃	direct (N × H)
36. C₂×C₂₃	C₂×C₈×C₁₇	direct (N × H)
38. C₂×C₁₇	C₂×C₈×C₂₃	direct (N × H)
40. C₂²×C₈	C₁₇×C₂₃	direct (N × H)
41. C₂₃	C₂²×C₈×C₁₇	direct (N × H)
42. C₁₇	C₂²×C₈×C₂₃	direct (N × H)
44. C₂×C₈	C₂×C₁₇×C₂₃	direct (N × H)
48. C₈	C₂²×C₁₇×C₂₃	direct (N × H)
49. C₂²	C₈×C₁₇×C₂₃	direct (N × H)
53. C₂	C₂×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
C₂²×C₄×C₁₇×C₂₃	2
C₂×C₈×C₁₇×C₂₃	2
C₂²×C₈×C₂₃	17
C₂²×C₈×C₁₇	23

Subgroups — C22×C8×C17×C23 #72190859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×C₁₇×C₂₃ #72190859