Subgroups — C₁₃₇×Q₂₅₆ #46856939

Order 35072.

42 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₂₈×C₁₃₇	C₂	1	2	yes	maximal
2	C₁₂₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	Q₆₄×C₁₃₇	—	2	2	no	—
4	C₆₄×C₁₃₇	C₂²	1	1	yes	characteristic
5	Q₃₂×C₁₃₇	—	4	2	no	—
6	C₃₂×C₁₃₇	D₄	1	1	yes	characteristic
7	Q₁₆×C₁₃₇	—	8	2	no	—
8	C₁₆×C₁₃₇	D₈	1	1	yes	characteristic
9	Q₈×C₁₃₇	—	16	2	no	—
10	C₈×C₁₃₇	D₁₆	1	1	yes	characteristic
11	C₄×C₁₃₇	—	32	2	no	—
12	C₄×C₁₃₇	D₃₂	1	1	yes	characteristic
13	C₂×C₁₃₇	D₆₄	1	1	yes	characteristic
14	Q₂₅₆	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₁₃₇	Q₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
16	Q₁₂₈	C₂×C₁₃₇	1	2	yes	—
17	C₁₂₈	C₂×C₁₃₇	1	1	yes	characteristic
18	Q₆₄	—	2	2	no	—
19	C₆₄	C₂²×C₁₃₇	1	1	yes	characteristic
20	Q₃₂	—	4	2	no	—
21	C₃₂	D₄×C₁₃₇	1	1	yes	characteristic
22	Q₁₆	—	8	2	no	—
23	C₁₆	D₈×C₁₃₇	1	1	yes	characteristic
24	Q₈	—	16	2	no	—
25	C₈	D₁₆×C₁₃₇	1	1	yes	characteristic
26	C₄	—	32	2	no	—
27	C₄	D₃₂×C₁₃₇	1	1	yes	characteristic
28	C₂	D₆₄×C₁₃₇	1	1	yes	characteristic

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

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

N	H ≅ G/N	type
14. Q₂₅₆	C₁₃₇	direct (N × H)
15. C₁₃₇	Q₂₅₆	direct (N × H)

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