Subgroups — C₁₃₇×#7546 #46856932

Order 35072.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1423×C₁₃₇	C₂	1	2	yes	maximal
3	C₂×C₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
4	#366×C₁₃₇	—	2	2	no	—
5	C₆₄×C₁₃₇	C₄	1	2	yes	complemented
6	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
7	#107×C₁₃₇	—	4	2	no	—
8	C₃₂×C₁₃₇	C₂×C₄	1	1	yes	characteristic
9	C₃₂×C₁₃₇	Q₈	1	1	yes	characteristic
10	C₂×C₈×C₁₃₇	D₈	1	1	yes	characteristic
11	C₄⋊C₄×C₁₃₇	—	8	2	no	—
12	C₁₆×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
13	C₁₆×C₁₃₇	Q₁₆	1	1	yes	characteristic
14	C₂×C₄×C₁₃₇	—	16	2	no	—
15	C₂×C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
16	C₈×C₁₃₇	#107	1	1	yes	characteristic
17	C₈×C₁₃₇	Q₃₂	1	1	yes	characteristic
18	C₂²×C₁₃₇	D₃₂	1	1	yes	characteristic
19	C₄×C₁₃₇	—	32	2	no	—
20	C₄×C₁₃₇	#366	1	1	yes	characteristic
21	C₄×C₁₃₇	Q₆₄	1	1	yes	characteristic
22	C₂×C₁₃₇	#1423	1	1	yes	characteristic
23	C₂×C₁₃₇	SD₁₂₈	1	2	yes	—
24	#7546	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₁₃₇	#7546	1	1	yes	characteristic, direct product factor, Sylow
26	C₂×C₆₄	C₂×C₁₃₇	1	1	yes	characteristic
27	#1423	C₂×C₁₃₇	1	2	yes	—
28	C₂×C₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
29	#366	—	2	2	no	—
30	C₆₄	C₄×C₁₃₇	1	2	yes	complemented
31	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
32	#107	—	4	2	no	—
33	C₃₂	C₂×C₄×C₁₃₇	1	1	yes	characteristic
34	C₃₂	Q₈×C₁₃₇	1	1	yes	characteristic
35	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
36	C₄⋊C₄	—	8	2	no	—
37	C₁₆	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
38	C₁₆	Q₁₆×C₁₃₇	1	1	yes	characteristic
39	C₂×C₄	—	16	2	no	—
40	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
41	C₈	#107×C₁₃₇	1	1	yes	characteristic
42	C₈	Q₃₂×C₁₃₇	1	1	yes	characteristic
43	C₂²	D₃₂×C₁₃₇	1	1	yes	characteristic
44	C₄	—	32	2	no	—
45	C₄	#366×C₁₃₇	1	1	yes	characteristic
46	C₄	Q₆₄×C₁₃₇	1	1	yes	characteristic
47	C₂	#1423×C₁₃₇	1	1	yes	characteristic
48	C₂	SD₁₂₈×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₇	D₃₂
commutator	C₃₂	C₂×C₄×C₁₃₇
Frattini, cosocle	C₂×C₃₂	C₂²×C₁₃₇
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₈×C₁₃₇ ◁ C₂×C₁₆×C₁₃₇ ◁ C₂×C₃₂×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₆₄×C₁₃₇	C₄	semidirect
24. #7546	C₁₃₇	direct (N × H)
25. C₁₃₇	#7546	direct (N × H)
30. C₆₄	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₄×C₁₃₇	2
#1423×C₁₃₇	2
#7546	137

Subgroups — C137×#7546 #46856932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7546 #46856932