Subgroups — C₁₃₇×#7461 #46856847

Order 35072.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1371×C₁₃₇	C₂	1	2	yes	maximal
3	C₂×C₃₂×C₁₃₇	C₄	1	1	yes	characteristic
4	C₂×C₃₂×C₁₃₇	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
6	#335×C₁₃₇	—	2	2	no	—
7	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
8	C₂×C₁₆×C₁₃₇	Q₈	1	1	yes	characteristic
9	C₄⋊C₈×C₁₃₇	—	4	2	no	—
10	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
11	C₃₂×C₁₃₇	—	2	1	no	—
12	C₃₂×C₁₃₇	C₈	1	2	yes	complemented
13	C₂×C₈×C₁₃₇	—	8	2	no	—
14	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₃₇	Q₁₆	1	1	yes	characteristic
16	C₄²×C₁₃₇	D₈	1	1	yes	characteristic
17	C₁₆×C₁₃₇	—	2	1	no	—
18	C₁₆×C₁₃₇	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₁₃₇	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₁₃₇	#107	1	1	yes	characteristic
21	C₂×C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
22	C₂×C₄×C₁₃₇	Q₃₂	1	1	yes	characteristic
23	C₈×C₁₃₇	—	2	1	no	—
24	C₈×C₁₃₇	—	16	2	no	—
25	C₈×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
26	C₈×C₁₃₇	#108	1	1	yes	characteristic
27	C₂²×C₁₃₇	#366	1	1	yes	characteristic
28	C₄×C₁₃₇	—	2	1	no	—
29	C₄×C₁₃₇	#335	1	1	yes	characteristic
30	C₄×C₁₃₇	#368	1	1	yes	characteristic
31	C₄×C₁₃₇	SD₆₄	1	2	yes	—
32	C₂×C₁₃₇	#1371	1	1	yes	characteristic
33	C₂×C₁₃₇	#1424	1	1	yes	characteristic
34	C₂×C₁₃₇	#1425	1	1	yes	characteristic
35	#7461	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₁₃₇	#7461	1	1	yes	characteristic, direct product factor, Sylow
37	C₄×C₃₂	C₂×C₁₃₇	1	1	yes	characteristic
38	#1371	C₂×C₁₃₇	1	2	yes	—
39	C₂×C₃₂	C₄×C₁₃₇	1	1	yes	characteristic
40	C₂×C₃₂	C₄×C₁₃₇	1	1	yes	characteristic
41	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
42	#335	—	2	2	no	—
43	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
44	C₂×C₁₆	Q₈×C₁₃₇	1	1	yes	characteristic
45	C₄⋊C₈	—	4	2	no	—
46	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
47	C₃₂	—	2	1	no	—
48	C₃₂	C₈×C₁₃₇	1	2	yes	complemented
49	C₂×C₈	—	8	2	no	—
50	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
51	C₂×C₈	Q₁₆×C₁₃₇	1	1	yes	characteristic
52	C₄²	D₈×C₁₃₇	1	1	yes	characteristic
53	C₁₆	—	2	1	no	—
54	C₁₆	C₂×C₈×C₁₃₇	1	1	yes	characteristic
55	C₁₆	M₁₆×C₁₃₇	1	1	yes	characteristic
56	C₂×C₄	#107×C₁₃₇	1	1	yes	characteristic
57	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
58	C₂×C₄	Q₃₂×C₁₃₇	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	—	16	2	no	—
61	C₈	C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
62	C₈	#108×C₁₃₇	1	1	yes	characteristic
63	C₂²	#366×C₁₃₇	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	#335×C₁₃₇	1	1	yes	characteristic
66	C₄	#368×C₁₃₇	1	1	yes	characteristic
67	C₄	SD₆₄×C₁₃₇	1	2	yes	—
68	C₂	#1371×C₁₃₇	1	1	yes	characteristic
69	C₂	#1424×C₁₃₇	1	1	yes	characteristic
70	C₂	#1425×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	D₁₆
commutator	C₁₆	C₂×C₈×C₁₃₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₃₇
socle, prime radical	C₂²×C₁₃₇	#366
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ C₄×C₈×C₁₃₇ ◁ C₄×C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₁₃₇	C₈	semidirect
35. #7461	C₁₃₇	direct (N × H)
36. C₁₃₇	#7461	direct (N × H)
48. C₃₂	C₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₂×C₁₃₇	2
#1371×C₁₃₇	2
#7461	137

Subgroups — C137×#7461 #46856847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7461 #46856847