Subgroups — C₁₃₇×SD₂₅₆ #46856938

Order 35072.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₂₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₆₄×C₁₃₇	—	2	1	no	—
5	D₃₂×C₁₃₇	—	2	1	no	—
6	C₆₄×C₁₃₇	C₂²	1	1	yes	characteristic
7	Q₃₂×C₁₃₇	—	4	1	no	—
8	D₁₆×C₁₃₇	—	4	1	no	—
9	C₃₂×C₁₃₇	D₄	1	1	yes	characteristic
10	Q₁₆×C₁₃₇	—	8	1	no	—
11	D₈×C₁₃₇	—	8	1	no	—
12	C₁₆×C₁₃₇	D₈	1	1	yes	characteristic
13	Q₈×C₁₃₇	—	16	1	no	—
14	D₄×C₁₃₇	—	16	1	no	—
15	C₈×C₁₃₇	D₁₆	1	1	yes	characteristic
16	C₂²×C₁₃₇	—	32	1	no	—
17	C₄×C₁₃₇	—	32	1	no	—
18	C₄×C₁₃₇	D₃₂	1	1	yes	characteristic
19	C₂×C₁₃₇	—	64	1	no	—
20	C₂×C₁₃₇	D₆₄	1	1	yes	characteristic
21	SD₂₅₆	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₁₃₇	SD₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
23	Q₁₂₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
24	D₆₄	C₂×C₁₃₇	1	1	yes	characteristic
25	C₁₂₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
26	Q₆₄	—	2	1	no	—
27	D₃₂	—	2	1	no	—
28	C₆₄	C₂²×C₁₃₇	1	1	yes	characteristic
29	Q₃₂	—	4	1	no	—
30	D₁₆	—	4	1	no	—
31	C₃₂	D₄×C₁₃₇	1	1	yes	characteristic
32	Q₁₆	—	8	1	no	—
33	D₈	—	8	1	no	—
34	C₁₆	D₈×C₁₃₇	1	1	yes	characteristic
35	Q₈	—	16	1	no	—
36	D₄	—	16	1	no	—
37	C₈	D₁₆×C₁₃₇	1	1	yes	characteristic
38	C₂²	—	32	1	no	—
39	C₄	—	32	1	no	—
40	C₄	D₃₂×C₁₃₇	1	1	yes	characteristic
41	C₂	—	64	1	no	—
42	C₂	D₆₄×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₇	D₆₄
commutator, Frattini, cosocle	C₆₄	C₂²×C₁₃₇
prime radical	D₆₄×C₁₃₇	C₂
LCS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₇ ◁ C₄×C₁₃₇ ◁ C₈×C₁₃₇ ◁ C₁₆×C₁₃₇ ◁ C₃₂×C₁₃₇ ◁ C₆₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₁₂₈×C₁₃₇	C₂	semidirect
3. C₁₂₈×C₁₃₇	C₂	semidirect
21. SD₂₅₆	C₁₃₇	direct (N × H)
22. C₁₃₇	SD₂₅₆	direct (N × H)
23. Q₁₂₈	C₂×C₁₃₇	semidirect
25. C₁₂₈	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
Q₁₂₈×C₁₃₇	2
D₆₄×C₁₃₇	2
C₁₂₈×C₁₃₇	2
SD₂₅₆	137

Subgroups — C137×SD256 #46856938

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×SD₂₅₆ #46856938