Subgroups — C₂×C₆₁×C₂₅₆ #72208017

Order 31232.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×C₁₂₈	C₂	1	1	yes	maximal, characteristic
2	C₆₁×C₂₅₆	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₆₁×C₆₄	C₄	1	1	yes	characteristic
4	C₆₁×C₁₂₈	C₄	1	1	yes	characteristic
5	C₆₁×C₁₂₈	C₂²	1	1	yes	characteristic
6	C₂×C₃₂×C₆₁	C₈	1	1	yes	characteristic
7	C₆₁×C₆₄	C₈	1	1	yes	characteristic
8	C₆₁×C₆₄	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₆₁	C₁₆	1	1	yes	characteristic
10	C₃₂×C₆₁	C₁₆	1	1	yes	characteristic
11	C₃₂×C₆₁	C₂×C₈	1	1	yes	characteristic
12	C₂×C₈×C₆₁	C₃₂	1	1	yes	characteristic
13	C₁₆×C₆₁	C₃₂	1	1	yes	characteristic
14	C₁₆×C₆₁	C₂×C₁₆	1	1	yes	characteristic
15	C₂×C₂₅₆	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₄×C₆₁	C₆₄	1	1	yes	characteristic
17	C₈×C₆₁	C₆₄	1	1	yes	characteristic
18	C₈×C₆₁	C₂×C₃₂	1	1	yes	characteristic
19	C₂×C₁₂₈	C₂×C₆₁	1	1	yes	characteristic
20	C₂₅₆	C₂×C₆₁	1	2	yes	direct product factor
21	C₂²×C₆₁	C₁₂₈	1	1	yes	characteristic
22	C₄×C₆₁	C₁₂₈	1	1	yes	characteristic
23	C₄×C₆₁	C₂×C₆₄	1	1	yes	characteristic
24	C₂×C₆₄	C₄×C₆₁	1	1	yes	characteristic
25	C₁₂₈	C₄×C₆₁	1	1	yes	characteristic
26	C₁₂₈	C₂²×C₆₁	1	1	yes	characteristic
27	C₂×C₆₁	C₂₅₆	1	2	yes	direct product factor
28	C₂×C₆₁	C₂×C₁₂₈	1	1	yes	characteristic
29	C₂×C₃₂	C₈×C₆₁	1	1	yes	characteristic
30	C₆₄	C₈×C₆₁	1	1	yes	characteristic
31	C₆₄	C₂×C₄×C₆₁	1	1	yes	characteristic
32	C₆₁	C₂×C₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×C₁₆	C₁₆×C₆₁	1	1	yes	characteristic
34	C₃₂	C₁₆×C₆₁	1	1	yes	characteristic
35	C₃₂	C₂×C₈×C₆₁	1	1	yes	characteristic
36	C₂×C₈	C₃₂×C₆₁	1	1	yes	characteristic
37	C₁₆	C₃₂×C₆₁	1	1	yes	characteristic
38	C₁₆	C₂×C₁₆×C₆₁	1	1	yes	characteristic
39	C₂×C₄	C₆₁×C₆₄	1	1	yes	characteristic
40	C₈	C₆₁×C₆₄	1	1	yes	characteristic
41	C₈	C₂×C₃₂×C₆₁	1	1	yes	characteristic
42	C₂²	C₆₁×C₁₂₈	1	1	yes	characteristic
43	C₄	C₆₁×C₁₂₈	1	1	yes	characteristic
44	C₄	C₂×C₆₁×C₆₄	1	1	yes	characteristic
45	C₂	C₆₁×C₂₅₆	1	2	yes	direct product factor
46	C₂	C₂×C₆₁×C₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₂₈	C₂²×C₆₁
socle, prime radical	C₂²×C₆₁	C₁₂₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×C₂₅₆	C₂	direct (N × H)
15. C₂×C₂₅₆	C₆₁	direct (N × H)
20. C₂₅₆	C₂×C₆₁	direct (N × H)
27. C₂×C₆₁	C₂₅₆	direct (N × H)
32. C₆₁	C₂×C₂₅₆	direct (N × H)
45. C₂	C₆₁×C₂₅₆	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×C₆₁×C₁₂₈	2
C₆₁×C₂₅₆	2
C₂×C₂₅₆	61

Subgroups — C2×C61×C256 #72208017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×C₂₅₆ #72208017