Subgroups — C₂²×C₃₁×Q₃₂ #16843343

Order 3968.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₁₆×C₃₁	C₂	1	2	yes	maximal
2	C₂×C₃₁×Q₃₂	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₃₁	—	2	2	no	—
5	C₂×Q₁₆×C₃₁	—	2	6	no	—
6	C₂×Q₁₆×C₃₁	C₂²	1	12	yes	—
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×Q₃₂	C₂²	1	16	yes	direct product factor
9	C₂×C₁₆×C₃₁	C₂²	1	6	yes	—
10	C₂×Q₈×C₃₁	—	2	12	no	—
11	C₂×Q₈×C₃₁	—	4	6	no	—
12	C₂²×C₄×C₃₁	—	4	2	no	—
13	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
14	Q₁₆×C₃₁	—	2	12	no	—
15	Q₁₆×C₃₁	C₂³	1	8	yes	—
16	C₂×C₈×C₃₁	D₄	1	3	yes	—
17	C₂×C₈×C₃₁	C₂³	1	3	yes	—
18	C₁₆×C₃₁	C₂³	1	4	yes	—
19	C₂³×C₃₁	D₈	1	1	yes	characteristic
20	Q₈×C₃₁	—	2	8	no	—
21	Q₈×C₃₁	—	4	12	no	—
22	C₂×C₄×C₃₁	—	4	12	no	—
23	C₂×C₄×C₃₁	D₈	1	3	yes	—
24	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
25	C₈×C₃₁	C₂×D₄	1	3	yes	—
26	C₈×C₃₁	C₂⁴	1	1	yes	characteristic
27	C₂²×Q₃₂	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₃₁	Q₃₂	1	4	yes	direct product factor
29	C₂²×C₃₁	C₂×D₈	1	3	yes	—
30	C₄×C₃₁	—	4	8	no	—
31	C₄×C₃₁	C₂×D₈	1	3	yes	—
32	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
33	C₂²×Q₁₆	C₂×C₃₁	1	2	yes	—
34	C₂×Q₃₂	C₂×C₃₁	1	12	yes	direct product factor
35	C₂²×C₁₆	C₂×C₃₁	1	1	yes	characteristic
36	C₂×C₃₁	C₂×Q₃₂	1	6	yes	direct product factor
37	C₂×C₃₁	C₂²×D₈	1	1	yes	characteristic
38	C₂²×Q₈	—	2	2	no	—
39	C₂×Q₁₆	—	2	6	no	—
40	C₂×Q₁₆	C₂²×C₃₁	1	12	yes	—
41	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
42	Q₃₂	C₂²×C₃₁	1	16	yes	direct product factor
43	C₂×C₁₆	C₂²×C₃₁	1	6	yes	—
44	C₃₁	C₂²×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×Q₈	—	2	12	no	—
46	C₂×Q₈	—	4	6	no	—
47	C₂²×C₄	—	4	2	no	—
48	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
49	Q₁₆	—	2	12	no	—
50	Q₁₆	C₂³×C₃₁	1	8	yes	—
51	C₂×C₈	D₄×C₃₁	1	3	yes	—
52	C₂×C₈	C₂³×C₃₁	1	3	yes	—
53	C₁₆	C₂³×C₃₁	1	4	yes	—
54	C₂³	D₈×C₃₁	1	1	yes	characteristic
55	Q₈	—	2	8	no	—
56	Q₈	—	4	12	no	—
57	C₂×C₄	—	4	12	no	—
58	C₂×C₄	D₈×C₃₁	1	3	yes	—
59	C₂×C₄	C₂×D₄×C₃₁	1	3	yes	—
60	C₈	C₂×D₄×C₃₁	1	3	yes	—
61	C₈	C₂⁴×C₃₁	1	1	yes	characteristic
62	C₂²	C₃₁×Q₃₂	1	4	yes	direct product factor
63	C₂²	C₂×D₈×C₃₁	1	3	yes	—
64	C₄	—	4	8	no	—
65	C₄	C₂×D₈×C₃₁	1	3	yes	—
66	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
67	C₂	C₂×C₃₁×Q₃₂	1	6	yes	direct product factor
68	C₂	C₂²×D₈×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₃₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×Q₃₂	C₂	direct (N × H)
8. C₃₁×Q₃₂	C₂²	direct (N × H)
27. C₂²×Q₃₂	C₃₁	direct (N × H)
28. C₂²×C₃₁	Q₃₂	direct (N × H)
34. C₂×Q₃₂	C₂×C₃₁	direct (N × H)
36. C₂×C₃₁	C₂×Q₃₂	direct (N × H)
42. Q₃₂	C₂²×C₃₁	direct (N × H)
44. C₃₁	C₂²×Q₃₂	direct (N × H)
62. C₂²	C₃₁×Q₃₂	direct (N × H)
67. C₂	C₂×C₃₁×Q₃₂	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×Q₁₆×C₃₁	2
C₂×C₃₁×Q₃₂	2
C₂²×C₁₆×C₃₁	2
C₂²×Q₃₂	31

Subgroups — C22×C31×Q32 #16843343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×Q₃₂ #16843343