Subgroups — C₃₇×M₂₅₆ #22628803

Order 9472.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×C₆₄	C₂	1	1	yes	maximal, characteristic
2	C₃₇×C₁₂₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₂×C₃₇	C₄	1	1	yes	characteristic
4	C₃₇×C₆₄	C₄	1	1	yes	characteristic
5	C₃₇×C₆₄	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₃₇	C₈	1	1	yes	characteristic
7	C₃₂×C₃₇	C₈	1	1	yes	characteristic
8	C₃₂×C₃₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₃₇	C₁₆	1	1	yes	characteristic
10	C₁₆×C₃₇	C₁₆	1	1	yes	characteristic
11	C₁₆×C₃₇	C₂×C₈	1	1	yes	characteristic
12	C₂×C₄×C₃₇	C₃₂	1	1	yes	characteristic
13	C₈×C₃₇	C₃₂	1	1	yes	characteristic
14	C₈×C₃₇	C₂×C₁₆	1	1	yes	characteristic
15	M₂₅₆	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₃₇	C₆₄	1	1	yes	characteristic
17	C₄×C₃₇	C₆₄	1	1	yes	characteristic
18	C₄×C₃₇	C₂×C₃₂	1	1	yes	characteristic
19	C₂×C₆₄	C₂×C₃₇	1	1	yes	characteristic
20	C₁₂₈	C₂×C₃₇	1	2	yes	complemented
21	C₂×C₃₇	—	2	1	no	—
22	C₂×C₃₇	C₂×C₆₄	1	1	yes	characteristic
23	C₂×C₃₂	C₄×C₃₇	1	1	yes	characteristic
24	C₆₄	C₄×C₃₇	1	1	yes	characteristic
25	C₆₄	C₂²×C₃₇	1	1	yes	characteristic
26	C₃₇	M₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₁₆	C₈×C₃₇	1	1	yes	characteristic
28	C₃₂	C₈×C₃₇	1	1	yes	characteristic
29	C₃₂	C₂×C₄×C₃₇	1	1	yes	characteristic
30	C₂×C₈	C₁₆×C₃₇	1	1	yes	characteristic
31	C₁₆	C₁₆×C₃₇	1	1	yes	characteristic
32	C₁₆	C₂×C₈×C₃₇	1	1	yes	characteristic
33	C₂×C₄	C₃₂×C₃₇	1	1	yes	characteristic
34	C₈	C₃₂×C₃₇	1	1	yes	characteristic
35	C₈	C₂×C₁₆×C₃₇	1	1	yes	characteristic
36	C₂²	C₃₇×C₆₄	1	1	yes	characteristic
37	C₄	C₃₇×C₆₄	1	1	yes	characteristic
38	C₄	C₂×C₃₂×C₃₇	1	1	yes	characteristic
39	C₂	—	2	1	no	—
40	C₂	C₂×C₃₇×C₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃₇×C₆₄	C₂²
commutator	C₂	C₂×C₃₇×C₆₄
Frattini, cosocle	C₆₄	C₂²×C₃₇
socle	C₂×C₃₇	C₂×C₆₄
prime radical	C₂²×C₃₇	C₆₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₇×C₆₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×C₁₂₈	C₂	semidirect
15. M₂₅₆	C₃₇	direct (N × H)
20. C₁₂₈	C₂×C₃₇	semidirect
26. C₃₇	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×C₃₇×C₆₄	2
C₃₇×C₁₂₈	2
M₂₅₆	37

Subgroups — C37×M256 #22628803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×M₂₅₆ #22628803