Subgroups — C₃²×C₇×C₁₆×C₂₃ #72200513

Order 23184.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₇×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₁₆×C₂₃	C₃	1	4	yes	maximal, direct product factor
3	C₃×C₇×C₈×C₂₃	C₂×C₃	1	4	yes	—
4	C₃²×C₁₆×C₂₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₇×C₁₆×C₂₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
6	C₃²×C₄×C₇×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₃²×C₇×C₂₃	C₈	1	1	yes	characteristic
8	C₃×C₄×C₇×C₂₃	C₃×C₄	1	4	yes	—
9	C₃²×C₈×C₂₃	C₂×C₇	1	1	yes	characteristic
10	C₃²×C₇×C₂₃	C₁₆	1	1	yes	characteristic, direct product factor, Hall
11	C₇×C₈×C₂₃	C₂×C₃²	1	1	yes	characteristic
12	C₃×C₁₆×C₂₃	C₃×C₇	1	4	yes	direct product factor
13	C₃²×C₇×C₁₆	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₃×C₇×C₂₃	C₃×C₈	1	4	yes	—
15	C₃²×C₄×C₂₃	C₄×C₇	1	1	yes	characteristic
16	C₄×C₇×C₂₃	C₃²×C₄	1	1	yes	characteristic
17	C₃×C₈×C₂₃	C₂×C₃×C₇	1	4	yes	—
18	C₃²×C₇×C₈	C₂×C₂₃	1	1	yes	characteristic
19	C₃×C₇×C₂₃	C₃×C₁₆	1	4	yes	direct product factor
20	C₂×C₃²×C₂₃	C₇×C₈	1	1	yes	characteristic
21	C₁₆×C₂₃	C₃²×C₇	1	1	yes	characteristic, direct product factor, Hall
22	C₃×C₇×C₁₆	C₃×C₂₃	1	4	yes	direct product factor
23	C₂×C₇×C₂₃	C₃²×C₈	1	1	yes	characteristic
24	C₃×C₄×C₂₃	C₃×C₄×C₇	1	4	yes	—
25	C₃²×C₄×C₇	C₄×C₂₃	1	1	yes	characteristic
26	C₃²×C₂₃	C₇×C₁₆	1	1	yes	characteristic, direct product factor, Hall
27	C₈×C₂₃	C₂×C₃²×C₇	1	1	yes	characteristic
28	C₃×C₇×C₈	C₂×C₃×C₂₃	1	4	yes	—
29	C₇×C₂₃	C₃²×C₁₆	1	1	yes	characteristic, direct product factor, Hall
30	C₃²×C₁₆	C₇×C₂₃	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃×C₂₃	C₃×C₇×C₈	1	4	yes	—
32	C₂×C₃²×C₇	C₈×C₂₃	1	1	yes	characteristic
33	C₇×C₁₆	C₃²×C₂₃	1	1	yes	characteristic, direct product factor, Hall
34	C₄×C₂₃	C₃²×C₄×C₇	1	1	yes	characteristic
35	C₃×C₄×C₇	C₃×C₄×C₂₃	1	4	yes	—
36	C₃²×C₈	C₂×C₇×C₂₃	1	1	yes	characteristic
37	C₃×C₂₃	C₃×C₇×C₁₆	1	4	yes	direct product factor
38	C₃²×C₇	C₁₆×C₂₃	1	1	yes	characteristic, direct product factor, Hall
39	C₇×C₈	C₂×C₃²×C₂₃	1	1	yes	characteristic
40	C₃×C₁₆	C₃×C₇×C₂₃	1	4	yes	direct product factor
41	C₂×C₂₃	C₃²×C₇×C₈	1	1	yes	characteristic
42	C₂×C₃×C₇	C₃×C₈×C₂₃	1	4	yes	—
43	C₃²×C₄	C₄×C₇×C₂₃	1	1	yes	characteristic
44	C₄×C₇	C₃²×C₄×C₂₃	1	1	yes	characteristic
45	C₃×C₈	C₂×C₃×C₇×C₂₃	1	4	yes	—
46	C₂₃	C₃²×C₇×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
47	C₃×C₇	C₃×C₁₆×C₂₃	1	4	yes	direct product factor
48	C₂×C₃²	C₇×C₈×C₂₃	1	1	yes	characteristic
49	C₁₆	C₃²×C₇×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×C₇	C₃²×C₈×C₂₃	1	1	yes	characteristic
51	C₃×C₄	C₃×C₄×C₇×C₂₃	1	4	yes	—
52	C₃²	C₇×C₁₆×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
53	C₈	C₂×C₃²×C₇×C₂₃	1	1	yes	characteristic
54	C₇	C₃²×C₁₆×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×C₃	C₃×C₇×C₈×C₂₃	1	4	yes	—
56	C₄	C₃²×C₄×C₇×C₂₃	1	1	yes	characteristic
57	C₃	C₃×C₇×C₁₆×C₂₃	1	4	yes	direct product factor
58	C₂	C₃²×C₇×C₈×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₃²×C₇×C₂₃
socle, prime radical	C₂×C₃²×C₇×C₂₃	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×C₁₆×C₂₃	C₃	direct (N × H)
4. C₃²×C₁₆×C₂₃	C₇	direct (N × H)
5. C₇×C₁₆×C₂₃	C₃²	direct (N × H)
10. C₃²×C₇×C₂₃	C₁₆	direct (N × H)
12. C₃×C₁₆×C₂₃	C₃×C₇	direct (N × H)
13. C₃²×C₇×C₁₆	C₂₃	direct (N × H)
19. C₃×C₇×C₂₃	C₃×C₁₆	direct (N × H)
21. C₁₆×C₂₃	C₃²×C₇	direct (N × H)
22. C₃×C₇×C₁₆	C₃×C₂₃	direct (N × H)
26. C₃²×C₂₃	C₇×C₁₆	direct (N × H)
29. C₇×C₂₃	C₃²×C₁₆	direct (N × H)
30. C₃²×C₁₆	C₇×C₂₃	direct (N × H)
33. C₇×C₁₆	C₃²×C₂₃	direct (N × H)
37. C₃×C₂₃	C₃×C₇×C₁₆	direct (N × H)
38. C₃²×C₇	C₁₆×C₂₃	direct (N × H)
40. C₃×C₁₆	C₃×C₇×C₂₃	direct (N × H)
46. C₂₃	C₃²×C₇×C₁₆	direct (N × H)
47. C₃×C₇	C₃×C₁₆×C₂₃	direct (N × H)
49. C₁₆	C₃²×C₇×C₂₃	direct (N × H)
52. C₃²	C₇×C₁₆×C₂₃	direct (N × H)
54. C₇	C₃²×C₁₆×C₂₃	direct (N × H)
57. C₃	C₃×C₇×C₁₆×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
C₃²×C₇×C₈×C₂₃	2
C₃×C₇×C₁₆×C₂₃	3
C₃²×C₁₆×C₂₃	7
C₃²×C₇×C₁₆	23

Subgroups — C32×C7×C16×C23 #72200513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₇×C₁₆×C₂₃ #72200513