Subgroups — C₄×C₈×C₁₇×C₂₃ #72190857

Order 12512.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₁₇×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₁₇×C₂₃	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₇×C₂₃	C₄	1	2	yes	—
4	C₂×C₄×C₁₇×C₂₃	C₂²	1	1	yes	characteristic
5	C₈×C₁₇×C₂₃	C₄	1	4	yes	direct product factor
6	C₂²×C₁₇×C₂₃	C₂×C₄	1	1	yes	characteristic
7	C₄×C₁₇×C₂₃	C₈	1	4	yes	direct product factor
8	C₄×C₁₇×C₂₃	C₂×C₄	1	2	yes	—
9	C₂×C₁₇×C₂₃	C₄²	1	1	yes	characteristic
10	C₂×C₁₇×C₂₃	C₂×C₈	1	2	yes	—
11	C₄×C₈×C₂₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₄×C₈×C₁₇	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₁₇×C₂₃	C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₈×C₂₃	C₂×C₁₇	1	2	yes	—
15	C₄²×C₂₃	C₂×C₁₇	1	1	yes	characteristic
16	C₂×C₈×C₁₇	C₂×C₂₃	1	2	yes	—
17	C₄²×C₁₇	C₂×C₂₃	1	1	yes	characteristic
18	C₂×C₄×C₂₃	C₄×C₁₇	1	2	yes	—
19	C₂×C₄×C₂₃	C₂²×C₁₇	1	1	yes	characteristic
20	C₈×C₂₃	C₄×C₁₇	1	4	yes	direct product factor
21	C₂×C₄×C₁₇	C₄×C₂₃	1	2	yes	—
22	C₂×C₄×C₁₇	C₂²×C₂₃	1	1	yes	characteristic
23	C₈×C₁₇	C₄×C₂₃	1	4	yes	direct product factor
24	C₂²×C₂₃	C₂×C₄×C₁₇	1	1	yes	characteristic
25	C₄×C₂₃	C₈×C₁₇	1	4	yes	direct product factor
26	C₄×C₂₃	C₂×C₄×C₁₇	1	2	yes	—
27	C₂²×C₁₇	C₂×C₄×C₂₃	1	1	yes	characteristic
28	C₄×C₁₇	C₈×C₂₃	1	4	yes	direct product factor
29	C₄×C₁₇	C₂×C₄×C₂₃	1	2	yes	—
30	C₂×C₂₃	C₄²×C₁₇	1	1	yes	characteristic
31	C₂×C₂₃	C₂×C₈×C₁₇	1	2	yes	—
32	C₂×C₁₇	C₄²×C₂₃	1	1	yes	characteristic
33	C₂×C₁₇	C₂×C₈×C₂₃	1	2	yes	—
34	C₄×C₈	C₁₇×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
35	C₂₃	C₄×C₈×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
36	C₁₇	C₄×C₈×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×C₈	C₂×C₁₇×C₂₃	1	2	yes	—
38	C₄²	C₂×C₁₇×C₂₃	1	1	yes	characteristic
39	C₂×C₄	C₄×C₁₇×C₂₃	1	2	yes	—
40	C₂×C₄	C₂²×C₁₇×C₂₃	1	1	yes	characteristic
41	C₈	C₄×C₁₇×C₂₃	1	4	yes	direct product factor
42	C₂²	C₂×C₄×C₁₇×C₂₃	1	1	yes	characteristic
43	C₄	C₈×C₁₇×C₂₃	1	4	yes	direct product factor
44	C₄	C₂×C₄×C₁₇×C₂₃	1	2	yes	—
45	C₂	C₂×C₈×C₁₇×C₂₃	1	2	yes	—
46	C₂	C₄²×C₁₇×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂²×C₁₇×C₂₃
socle, prime radical	C₂²×C₁₇×C₂₃	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₈×C₁₇×C₂₃	C₄	direct (N × H)
7. C₄×C₁₇×C₂₃	C₈	direct (N × H)
11. C₄×C₈×C₂₃	C₁₇	direct (N × H)
12. C₄×C₈×C₁₇	C₂₃	direct (N × H)
13. C₁₇×C₂₃	C₄×C₈	direct (N × H)
20. C₈×C₂₃	C₄×C₁₇	direct (N × H)
23. C₈×C₁₇	C₄×C₂₃	direct (N × H)
25. C₄×C₂₃	C₈×C₁₇	direct (N × H)
28. C₄×C₁₇	C₈×C₂₃	direct (N × H)
34. C₄×C₈	C₁₇×C₂₃	direct (N × H)
35. C₂₃	C₄×C₈×C₁₇	direct (N × H)
36. C₁₇	C₄×C₈×C₂₃	direct (N × H)
41. C₈	C₄×C₁₇×C₂₃	direct (N × H)
43. C₄	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
C₄²×C₁₇×C₂₃	2
C₂×C₈×C₁₇×C₂₃	2
C₄×C₈×C₂₃	17
C₄×C₈×C₁₇	23

Subgroups — C4×C8×C17×C23 #72190857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×C₁₇×C₂₃ #72190857