Subgroups — C₂³×C₃²×C₃₄₃ #72202004

Order 24696.

384 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₃₄₃	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₃×C₃₄₃	C₃	1	4	yes	maximal, direct product factor
3	C₂×C₃²×C₃₄₃	C₂²	1	7	yes	direct product factor
4	C₂²×C₃×C₃₄₃	C₂×C₃	1	28	yes	direct product factor
5	C₂³×C₃²×C₄₉	C₇	1	1	yes	maximal, characteristic
6	C₃²×C₃₄₃	C₂³	1	1	yes	characteristic, direct product factor, Hall
7	C₂³×C₃₄₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₂×C₃×C₃₄₃	C₂²×C₃	1	28	yes	direct product factor
9	C₂²×C₃²×C₄₉	C₂×C₇	1	7	yes	—
10	C₂²×C₃₄₃	C₂×C₃²	1	7	yes	direct product factor
11	C₂³×C₃×C₄₉	C₃×C₇	1	4	yes	—
12	C₃×C₃₄₃	C₂³×C₃	1	4	yes	direct product factor
13	C₂×C₃²×C₄₉	C₂²×C₇	1	7	yes	—
14	C₂×C₃₄₃	C₂²×C₃²	1	7	yes	direct product factor
15	C₂²×C₃×C₄₉	C₂×C₃×C₇	1	28	yes	—
16	C₂³×C₃²×C₇	C₄₉	1	1	yes	characteristic
17	C₃²×C₄₉	C₂³×C₇	1	1	yes	characteristic
18	C₂³×C₄₉	C₃²×C₇	1	1	yes	characteristic
19	C₃₄₃	C₂³×C₃²	1	1	yes	characteristic, direct product factor, Sylow
20	C₂×C₃×C₄₉	C₂²×C₃×C₇	1	28	yes	—
21	C₂²×C₃²×C₇	C₂×C₄₉	1	7	yes	—
22	C₂²×C₄₉	C₂×C₃²×C₇	1	7	yes	—
23	C₂³×C₃×C₇	C₃×C₄₉	1	4	yes	—
24	C₃×C₄₉	C₂³×C₃×C₇	1	4	yes	—
25	C₂×C₃²×C₇	C₂²×C₄₉	1	7	yes	—
26	C₂×C₄₉	C₂²×C₃²×C₇	1	7	yes	—
27	C₂²×C₃×C₇	C₂×C₃×C₄₉	1	28	yes	—
28	C₂³×C₃²	C₃₄₃	1	1	yes	characteristic, direct product factor, Hall
29	C₃²×C₇	C₂³×C₄₉	1	1	yes	characteristic
30	C₂³×C₇	C₃²×C₄₉	1	1	yes	characteristic
31	C₄₉	C₂³×C₃²×C₇	1	1	yes	characteristic
32	C₂×C₃×C₇	C₂²×C₃×C₄₉	1	28	yes	—
33	C₂²×C₃²	C₂×C₃₄₃	1	7	yes	direct product factor
34	C₂²×C₇	C₂×C₃²×C₄₉	1	7	yes	—
35	C₂³×C₃	C₃×C₃₄₃	1	4	yes	direct product factor
36	C₃×C₇	C₂³×C₃×C₄₉	1	4	yes	—
37	C₂×C₃²	C₂²×C₃₄₃	1	7	yes	direct product factor
38	C₂×C₇	C₂²×C₃²×C₄₉	1	7	yes	—
39	C₂²×C₃	C₂×C₃×C₃₄₃	1	28	yes	direct product factor
40	C₃²	C₂³×C₃₄₃	1	1	yes	characteristic, direct product factor, Sylow
41	C₂³	C₃²×C₃₄₃	1	1	yes	characteristic, direct product factor, Sylow
42	C₇	C₂³×C₃²×C₄₉	1	1	yes	characteristic
43	C₂×C₃	C₂²×C₃×C₃₄₃	1	28	yes	direct product factor
44	C₂²	C₂×C₃²×C₃₄₃	1	7	yes	direct product factor
45	C₃	C₂³×C₃×C₃₄₃	1	4	yes	direct product factor
46	C₂	C₂²×C₃²×C₃₄₃	1	7	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄₉	C₂³×C₃²×C₇
socle, prime radical	C₂³×C₃²×C₇	C₄₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₃₄₃	C₂	direct (N × H)
2. C₂³×C₃×C₃₄₃	C₃	direct (N × H)
3. C₂×C₃²×C₃₄₃	C₂²	direct (N × H)
4. C₂²×C₃×C₃₄₃	C₂×C₃	direct (N × H)
6. C₃²×C₃₄₃	C₂³	direct (N × H)
7. C₂³×C₃₄₃	C₃²	direct (N × H)
8. C₂×C₃×C₃₄₃	C₂²×C₃	direct (N × H)
10. C₂²×C₃₄₃	C₂×C₃²	direct (N × H)
12. C₃×C₃₄₃	C₂³×C₃	direct (N × H)
14. C₂×C₃₄₃	C₂²×C₃²	direct (N × H)
19. C₃₄₃	C₂³×C₃²	direct (N × H)
28. C₂³×C₃²	C₃₄₃	direct (N × H)
33. C₂²×C₃²	C₂×C₃₄₃	direct (N × H)
35. C₂³×C₃	C₃×C₃₄₃	direct (N × H)
37. C₂×C₃²	C₂²×C₃₄₃	direct (N × H)
39. C₂²×C₃	C₂×C₃×C₃₄₃	direct (N × H)
40. C₃²	C₂³×C₃₄₃	direct (N × H)
41. C₂³	C₃²×C₃₄₃	direct (N × H)
43. C₂×C₃	C₂²×C₃×C₃₄₃	direct (N × H)
44. C₂²	C₂×C₃²×C₃₄₃	direct (N × H)
45. C₃	C₂³×C₃×C₃₄₃	direct (N × H)
46. C₂	C₂²×C₃²×C₃₄₃	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×C₃²×C₃₄₃	2
C₂³×C₃×C₃₄₃	3
C₂³×C₃²×C₄₉	7

Subgroups — C23×C32×C343 #72202004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃²×C₃₄₃ #72202004