Subgroups — C₂×C₃×C₄²×C₃₃₁ #72208552

Order 31776.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₃₃₁	C₂	1	3	yes	maximal
2	C₃×C₄²×C₃₃₁	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₃×C₃₃₁	C₂²	1	1	yes	characteristic
4	C₂×C₃×C₄×C₃₃₁	C₄	1	12	yes	direct product factor
5	C₂×C₃×C₄×C₃₃₁	C₂²	1	6	yes	—
6	C₂×C₄²×C₃₃₁	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₄×C₃₃₁	C₂×C₃	1	3	yes	—
8	C₄²×C₃₃₁	C₂×C₃	1	4	yes	direct product factor
9	C₂²×C₃×C₃₃₁	C₂×C₄	1	6	yes	—
10	C₂²×C₃×C₃₃₁	C₂³	1	1	yes	characteristic
11	C₃×C₄×C₃₃₁	C₂×C₄	1	12	yes	direct product factor
12	C₂³×C₃₃₁	C₂²×C₃	1	1	yes	characteristic
13	C₂×C₄×C₃₃₁	C₃×C₄	1	12	yes	direct product factor
14	C₂×C₄×C₃₃₁	C₂²×C₃	1	6	yes	—
15	C₂×C₃×C₃₃₁	C₄²	1	4	yes	direct product factor
16	C₂×C₃×C₃₃₁	C₂²×C₄	1	3	yes	—
17	C₂²×C₃₃₁	C₂×C₃×C₄	1	6	yes	—
18	C₂²×C₃₃₁	C₂³×C₃	1	1	yes	characteristic
19	C₄×C₃₃₁	C₂×C₃×C₄	1	12	yes	direct product factor
20	C₃×C₃₃₁	C₂×C₄²	1	1	yes	characteristic, direct product factor, Hall
21	C₂×C₃₃₁	C₃×C₄²	1	4	yes	direct product factor
22	C₂×C₃₃₁	C₂²×C₃×C₄	1	3	yes	—
23	C₃₃₁	C₂×C₃×C₄²	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₃×C₄²	C₃₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂²×C₃×C₄	C₂×C₃₃₁	1	3	yes	—
26	C₃×C₄²	C₂×C₃₃₁	1	4	yes	direct product factor
27	C₂×C₄²	C₃×C₃₃₁	1	1	yes	characteristic, direct product factor, Sylow
28	C₂³×C₃	C₂²×C₃₃₁	1	1	yes	characteristic
29	C₂×C₃×C₄	C₄×C₃₃₁	1	12	yes	direct product factor
30	C₂×C₃×C₄	C₂²×C₃₃₁	1	6	yes	—
31	C₂²×C₄	C₂×C₃×C₃₃₁	1	3	yes	—
32	C₄²	C₂×C₃×C₃₃₁	1	4	yes	direct product factor
33	C₂²×C₃	C₂×C₄×C₃₃₁	1	6	yes	—
34	C₂²×C₃	C₂³×C₃₃₁	1	1	yes	characteristic
35	C₃×C₄	C₂×C₄×C₃₃₁	1	12	yes	direct product factor
36	C₂³	C₂²×C₃×C₃₃₁	1	1	yes	characteristic
37	C₂×C₄	C₃×C₄×C₃₃₁	1	12	yes	direct product factor
38	C₂×C₄	C₂²×C₃×C₃₃₁	1	6	yes	—
39	C₂×C₃	C₄²×C₃₃₁	1	4	yes	direct product factor
40	C₂×C₃	C₂²×C₄×C₃₃₁	1	3	yes	—
41	C₂²	C₂×C₃×C₄×C₃₃₁	1	6	yes	—
42	C₂²	C₂³×C₃×C₃₃₁	1	1	yes	characteristic
43	C₄	C₂×C₃×C₄×C₃₃₁	1	12	yes	direct product factor
44	C₃	C₂×C₄²×C₃₃₁	1	1	yes	characteristic, direct product factor, Sylow
45	C₂	C₃×C₄²×C₃₃₁	1	4	yes	direct product factor
46	C₂	C₂²×C₃×C₄×C₃₃₁	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂³×C₃×C₃₃₁
socle, prime radical	C₂³×C₃×C₃₃₁	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄²×C₃₃₁	C₂	direct (N × H)
4. C₂×C₃×C₄×C₃₃₁	C₄	direct (N × H)
6. C₂×C₄²×C₃₃₁	C₃	direct (N × H)
8. C₄²×C₃₃₁	C₂×C₃	direct (N × H)
11. C₃×C₄×C₃₃₁	C₂×C₄	direct (N × H)
13. C₂×C₄×C₃₃₁	C₃×C₄	direct (N × H)
15. C₂×C₃×C₃₃₁	C₄²	direct (N × H)
19. C₄×C₃₃₁	C₂×C₃×C₄	direct (N × H)
20. C₃×C₃₃₁	C₂×C₄²	direct (N × H)
21. C₂×C₃₃₁	C₃×C₄²	direct (N × H)
23. C₃₃₁	C₂×C₃×C₄²	direct (N × H)
24. C₂×C₃×C₄²	C₃₃₁	direct (N × H)
26. C₃×C₄²	C₂×C₃₃₁	direct (N × H)
27. C₂×C₄²	C₃×C₃₃₁	direct (N × H)
29. C₂×C₃×C₄	C₄×C₃₃₁	direct (N × H)
32. C₄²	C₂×C₃×C₃₃₁	direct (N × H)
35. C₃×C₄	C₂×C₄×C₃₃₁	direct (N × H)
37. C₂×C₄	C₃×C₄×C₃₃₁	direct (N × H)
39. C₂×C₃	C₄²×C₃₃₁	direct (N × H)
43. C₄	C₂×C₃×C₄×C₃₃₁	direct (N × H)
44. C₃	C₂×C₄²×C₃₃₁	direct (N × H)
45. C₂	C₃×C₄²×C₃₃₁	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×C₃×C₄×C₃₃₁	2
C₃×C₄²×C₃₃₁	2
C₂×C₄²×C₃₃₁	3
C₂×C₃×C₄²	331

Subgroups — C2×C3×C42×C331 #72208552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄²×C₃₃₁ #72208552