Subgroups — C₁₃₇×#7457 #46856843

Order 35072.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1398×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1369×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1368×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	M₆₄×C₁₃₇	C₄	1	1	yes	characteristic
5	M₆₄×C₁₃₇	C₄	1	1	yes	characteristic
6	#364×C₁₃₇	—	2	1	no	—
7	#346×C₁₃₇	C₂²	1	1	yes	characteristic
8	#334×C₁₃₇	—	2	1	no	—
9	M₃₂×C₁₃₇	—	4	1	no	—
10	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₁₆×C₁₃₇	Q₈	1	1	yes	characteristic
12	C₄⋊C₈×C₁₃₇	—	4	1	no	—
13	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
14	C₃₂×C₁₃₇	—	2	1	no	—
15	C₃₂×C₁₃₇	C₈	1	2	yes	complemented
16	C₂×C₈×C₁₃₇	—	8	1	no	—
17	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
19	C₄²×C₁₃₇	SD₁₆	1	1	yes	characteristic
20	C₁₆×C₁₃₇	—	2	1	no	—
21	C₁₆×C₁₃₇	—	8	1	no	—
22	C₁₆×C₁₃₇	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₁₃₇	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₁₃₇	—	2	1	no	—
25	C₂×C₄×C₁₃₇	#106	1	1	yes	characteristic
26	C₈×C₁₃₇	—	2	1	no	—
27	C₈×C₁₃₇	—	16	1	no	—
28	C₈×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
29	C₈×C₁₃₇	#108	1	1	yes	characteristic
30	C₂²×C₁₃₇	#365	1	1	yes	characteristic
31	C₄×C₁₃₇	—	4	1	no	—
32	C₄×C₁₃₇	#334	1	1	yes	characteristic
33	C₄×C₁₃₇	#365	1	1	yes	characteristic
34	C₂×C₁₃₇	—	2	1	no	—
35	C₂×C₁₃₇	#1368	1	1	yes	characteristic
36	#7457	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₁₃₇	#7457	1	1	yes	characteristic, direct product factor, Sylow
38	#1398	C₂×C₁₃₇	1	1	yes	characteristic
39	#1369	C₂×C₁₃₇	1	1	yes	characteristic
40	#1368	C₂×C₁₃₇	1	1	yes	characteristic
41	M₆₄	C₄×C₁₃₇	1	1	yes	characteristic
42	M₆₄	C₄×C₁₃₇	1	1	yes	characteristic
43	#364	—	2	1	no	—
44	#346	C₂²×C₁₃₇	1	1	yes	characteristic
45	#334	—	2	1	no	—
46	M₃₂	—	4	1	no	—
47	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
48	C₂×C₁₆	Q₈×C₁₃₇	1	1	yes	characteristic
49	C₄⋊C₈	—	4	1	no	—
50	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
51	C₃₂	—	2	1	no	—
52	C₃₂	C₈×C₁₃₇	1	2	yes	complemented
53	C₂×C₈	—	8	1	no	—
54	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
55	C₂×C₈	SD₁₆×C₁₃₇	1	1	yes	characteristic
56	C₄²	SD₁₆×C₁₃₇	1	1	yes	characteristic
57	C₁₆	—	2	1	no	—
58	C₁₆	—	8	1	no	—
59	C₁₆	C₂×C₈×C₁₃₇	1	1	yes	characteristic
60	C₁₆	M₁₆×C₁₃₇	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#106×C₁₃₇	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	16	1	no	—
65	C₈	C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
66	C₈	#108×C₁₃₇	1	1	yes	characteristic
67	C₂²	#365×C₁₃₇	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	#334×C₁₃₇	1	1	yes	characteristic
70	C₄	#365×C₁₃₇	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#1368×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₇	#1368
commutator	C₁₆	C₂×C₈×C₁₃₇
Frattini, cosocle	#346	C₂²×C₁₃₇
prime radical	C₂²×C₁₃₇	#365
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ C₄×C₈×C₁₃₇ ◁ #346×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₁₃₇	C₈	semidirect
36. #7457	C₁₃₇	direct (N × H)
37. C₁₃₇	#7457	direct (N × H)
52. 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
#1398×C₁₃₇	2
#1369×C₁₃₇	2
#1368×C₁₃₇	2
#7457	137

Subgroups — C137×#7457 #46856843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7457 #46856843