Subgroups — C₂×C₃₁×C₅₁₂ #72208463

Order 31744.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₂₅₆	C₂	1	1	yes	maximal, characteristic
2	C₃₁×C₅₁₂	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅₁₂	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂×C₃₁×C₁₂₈	C₄	1	1	yes	characteristic
5	C₃₁×C₂₅₆	C₄	1	1	yes	characteristic
6	C₃₁×C₂₅₆	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×C₆₄	C₈	1	1	yes	characteristic
8	C₃₁×C₁₂₈	C₈	1	1	yes	characteristic
9	C₃₁×C₁₂₈	C₂×C₄	1	1	yes	characteristic
10	C₂×C₃₁×C₃₂	C₁₆	1	1	yes	characteristic
11	C₃₁×C₆₄	C₁₆	1	1	yes	characteristic
12	C₃₁×C₆₄	C₂×C₈	1	1	yes	characteristic
13	C₂×C₁₆×C₃₁	C₃₂	1	1	yes	characteristic
14	C₃₁×C₃₂	C₃₂	1	1	yes	characteristic
15	C₃₁×C₃₂	C₂×C₁₆	1	1	yes	characteristic
16	C₂×C₂₅₆	C₂×C₃₁	1	1	yes	characteristic
17	C₅₁₂	C₂×C₃₁	1	2	yes	direct product factor
18	C₂×C₈×C₃₁	C₆₄	1	1	yes	characteristic
19	C₁₆×C₃₁	C₆₄	1	1	yes	characteristic
20	C₁₆×C₃₁	C₂×C₃₂	1	1	yes	characteristic
21	C₂×C₁₂₈	C₄×C₃₁	1	1	yes	characteristic
22	C₂₅₆	C₄×C₃₁	1	1	yes	characteristic
23	C₂₅₆	C₂²×C₃₁	1	1	yes	characteristic
24	C₂×C₄×C₃₁	C₁₂₈	1	1	yes	characteristic
25	C₈×C₃₁	C₁₂₈	1	1	yes	characteristic
26	C₈×C₃₁	C₂×C₆₄	1	1	yes	characteristic
27	C₂×C₆₄	C₈×C₃₁	1	1	yes	characteristic
28	C₁₂₈	C₈×C₃₁	1	1	yes	characteristic
29	C₁₂₈	C₂×C₄×C₃₁	1	1	yes	characteristic
30	C₂²×C₃₁	C₂₅₆	1	1	yes	characteristic
31	C₄×C₃₁	C₂₅₆	1	1	yes	characteristic
32	C₄×C₃₁	C₂×C₁₂₈	1	1	yes	characteristic
33	C₂×C₃₂	C₁₆×C₃₁	1	1	yes	characteristic
34	C₆₄	C₁₆×C₃₁	1	1	yes	characteristic
35	C₆₄	C₂×C₈×C₃₁	1	1	yes	characteristic
36	C₂×C₃₁	C₅₁₂	1	2	yes	direct product factor
37	C₂×C₃₁	C₂×C₂₅₆	1	1	yes	characteristic
38	C₂×C₁₆	C₃₁×C₃₂	1	1	yes	characteristic
39	C₃₂	C₃₁×C₃₂	1	1	yes	characteristic
40	C₃₂	C₂×C₁₆×C₃₁	1	1	yes	characteristic
41	C₃₁	C₂×C₅₁₂	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₈	C₃₁×C₆₄	1	1	yes	characteristic
43	C₁₆	C₃₁×C₆₄	1	1	yes	characteristic
44	C₁₆	C₂×C₃₁×C₃₂	1	1	yes	characteristic
45	C₂×C₄	C₃₁×C₁₂₈	1	1	yes	characteristic
46	C₈	C₃₁×C₁₂₈	1	1	yes	characteristic
47	C₈	C₂×C₃₁×C₆₄	1	1	yes	characteristic
48	C₂²	C₃₁×C₂₅₆	1	1	yes	characteristic
49	C₄	C₃₁×C₂₅₆	1	1	yes	characteristic
50	C₄	C₂×C₃₁×C₁₂₈	1	1	yes	characteristic
51	C₂	C₃₁×C₅₁₂	1	2	yes	direct product factor
52	C₂	C₂×C₃₁×C₂₅₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₅₆	C₂²×C₃₁
socle, prime radical	C₂²×C₃₁	C₂₅₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×C₅₁₂	C₂	direct (N × H)
3. C₂×C₅₁₂	C₃₁	direct (N × H)
17. C₅₁₂	C₂×C₃₁	direct (N × H)
36. C₂×C₃₁	C₅₁₂	direct (N × H)
41. C₃₁	C₂×C₅₁₂	direct (N × H)
51. C₂	C₃₁×C₅₁₂	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×C₂₅₆	2
C₃₁×C₅₁₂	2
C₂×C₅₁₂	31

Subgroups — C2×C31×C512 #72208463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×C₅₁₂ #72208463