Subgroups — C₁₁₃×M₂₅₆ #43275570

Order 28928.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×C₁₂₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₂×C₁₁₃	C₄	1	1	yes	characteristic
4	C₆₄×C₁₁₃	C₄	1	1	yes	characteristic
5	C₆₄×C₁₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₁₁₃	C₈	1	1	yes	characteristic
7	C₃₂×C₁₁₃	C₈	1	1	yes	characteristic
8	C₃₂×C₁₁₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₁₁₃	C₁₆	1	1	yes	characteristic
10	C₁₆×C₁₁₃	C₁₆	1	1	yes	characteristic
11	C₁₆×C₁₁₃	C₂×C₈	1	1	yes	characteristic
12	C₂×C₄×C₁₁₃	C₃₂	1	1	yes	characteristic
13	C₈×C₁₁₃	C₃₂	1	1	yes	characteristic
14	C₈×C₁₁₃	C₂×C₁₆	1	1	yes	characteristic
15	C₂²×C₁₁₃	C₆₄	1	1	yes	characteristic
16	C₄×C₁₁₃	C₆₄	1	1	yes	characteristic
17	C₄×C₁₁₃	C₂×C₃₂	1	1	yes	characteristic
18	M₂₅₆	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₁₁₃	—	2	1	no	—
20	C₂×C₁₁₃	C₂×C₆₄	1	1	yes	characteristic
21	C₂×C₆₄	C₂×C₁₁₃	1	1	yes	characteristic
22	C₁₂₈	C₂×C₁₁₃	1	2	yes	complemented
23	C₁₁₃	M₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₃₂	C₄×C₁₁₃	1	1	yes	characteristic
25	C₆₄	C₄×C₁₁₃	1	1	yes	characteristic
26	C₆₄	C₂²×C₁₁₃	1	1	yes	characteristic
27	C₂×C₁₆	C₈×C₁₁₃	1	1	yes	characteristic
28	C₃₂	C₈×C₁₁₃	1	1	yes	characteristic
29	C₃₂	C₂×C₄×C₁₁₃	1	1	yes	characteristic
30	C₂×C₈	C₁₆×C₁₁₃	1	1	yes	characteristic
31	C₁₆	C₁₆×C₁₁₃	1	1	yes	characteristic
32	C₁₆	C₂×C₈×C₁₁₃	1	1	yes	characteristic
33	C₂×C₄	C₃₂×C₁₁₃	1	1	yes	characteristic
34	C₈	C₃₂×C₁₁₃	1	1	yes	characteristic
35	C₈	C₂×C₁₆×C₁₁₃	1	1	yes	characteristic
36	C₂²	C₆₄×C₁₁₃	1	1	yes	characteristic
37	C₄	C₆₄×C₁₁₃	1	1	yes	characteristic
38	C₄	C₂×C₃₂×C₁₁₃	1	1	yes	characteristic
39	C₂	—	2	1	no	—
40	C₂	C₂×C₆₄×C₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₆₄×C₁₁₃	C₂²
commutator	C₂	C₂×C₆₄×C₁₁₃
Frattini, cosocle	C₆₄	C₂²×C₁₁₃
socle	C₂×C₁₁₃	C₂×C₆₄
prime radical	C₂²×C₁₁₃	C₆₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₆₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×C₁₂₈	C₂	semidirect
18. M₂₅₆	C₁₁₃	direct (N × H)
22. C₁₂₈	C₂×C₁₁₃	semidirect
23. C₁₁₃	M₂₅₆	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×C₆₄×C₁₁₃	2
C₁₁₃×C₁₂₈	2
M₂₅₆	113

Subgroups — C113×M256 #43275570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×M₂₅₆ #43275570