Subgroups — C₁₃₇×#7540 #46856926

Order 35072.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2262×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	M₁₂₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#1424×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#508×C₁₃₇	—	2	1	no	—
5	Q₆₄×C₁₃₇	C₄	1	1	yes	characteristic, complemented
6	SD₆₄×C₁₃₇	—	2	1	no	—
7	D₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
9	#366×C₁₃₇	—	2	1	no	—
10	C₆₄×C₁₃₇	—	2	1	no	—
11	#135×C₁₃₇	—	4	1	no	—
12	Q₃₂×C₁₃₇	—	2	1	no	—
13	SD₃₂×C₁₃₇	—	4	1	no	—
14	D₁₆×C₁₃₇	—	2	1	no	—
15	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
16	#107×C₁₃₇	—	4	1	no	—
17	C₃₂×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₁₃₇	D₄	1	1	yes	characteristic
19	#41×C₁₃₇	—	8	1	no	—
20	Q₁₆×C₁₃₇	—	4	1	no	—
21	SD₁₆×C₁₃₇	—	8	1	no	—
22	D₈×C₁₃₇	—	4	1	no	—
23	C₂×C₈×C₁₃₇	D₈	1	1	yes	characteristic
24	C₄⋊C₄×C₁₃₇	—	8	1	no	—
25	C₁₆×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₁₃₇	SD₁₆	1	1	yes	characteristic
27	Q₈×C₁₃₇	—	8	1	no	—
28	D₄×C₁₃₇	—	8	1	no	—
29	D₄×C₁₃₇	—	16	1	no	—
30	C₂×C₄×C₁₃₇	—	16	1	no	—
31	C₂×C₄×C₁₃₇	—	16	1	no	—
32	C₂×C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
33	C₈×C₁₃₇	#102	1	1	yes	characteristic
34	C₈×C₁₃₇	SD₃₂	1	1	yes	characteristic
35	C₂²×C₁₃₇	—	16	1	no	—
36	C₂²×C₁₃₇	D₃₂	1	1	yes	characteristic
37	C₄×C₁₃₇	—	16	1	no	—
38	C₄×C₁₃₇	—	32	1	no	—
39	C₄×C₁₃₇	#357	1	1	yes	characteristic
40	C₄×C₁₃₇	SD₆₄	1	1	yes	characteristic
41	C₂×C₁₃₇	—	2	1	no	—
42	C₂×C₁₃₇	—	32	1	no	—
43	C₂×C₁₃₇	#1415	1	1	yes	characteristic
44	#7540	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₁₃₇	#7540	1	1	yes	characteristic, direct product factor, Sylow
46	#2262	C₂×C₁₃₇	1	1	yes	characteristic
47	M₁₂₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
48	#1424	C₂×C₁₃₇	1	1	yes	characteristic, complemented
49	#508	—	2	1	no	—
50	Q₆₄	C₄×C₁₃₇	1	1	yes	characteristic, complemented
51	SD₆₄	—	2	1	no	—
52	D₃₂	C₄×C₁₃₇	1	1	yes	characteristic, complemented
53	C₂×C₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
54	#366	—	2	1	no	—
55	C₆₄	—	2	1	no	—
56	#135	—	4	1	no	—
57	Q₃₂	—	2	1	no	—
58	SD₃₂	—	4	1	no	—
59	D₁₆	—	2	1	no	—
60	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
61	#107	—	4	1	no	—
62	C₃₂	C₂×C₄×C₁₃₇	1	1	yes	characteristic
63	C₃₂	D₄×C₁₃₇	1	1	yes	characteristic
64	#41	—	8	1	no	—
65	Q₁₆	—	4	1	no	—
66	SD₁₆	—	8	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
69	C₄⋊C₄	—	8	1	no	—
70	C₁₆	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
71	C₁₆	SD₁₆×C₁₃₇	1	1	yes	characteristic
72	Q₈	—	8	1	no	—
73	D₄	—	8	1	no	—
74	D₄	—	16	1	no	—
75	C₂×C₄	—	16	1	no	—
76	C₂×C₄	—	16	1	no	—
77	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
78	C₈	#102×C₁₃₇	1	1	yes	characteristic
79	C₈	SD₃₂×C₁₃₇	1	1	yes	characteristic
80	C₂²	—	16	1	no	—
81	C₂²	D₃₂×C₁₃₇	1	1	yes	characteristic
82	C₄	—	16	1	no	—
83	C₄	—	32	1	no	—
84	C₄	#357×C₁₃₇	1	1	yes	characteristic
85	C₄	SD₆₄×C₁₃₇	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	32	1	no	—
88	C₂	#1415×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₇	#1415
commutator	C₃₂	C₂×C₄×C₁₃₇
Frattini, cosocle	C₂×C₃₂	C₂²×C₁₃₇
prime radical	#2262×C₁₃₇	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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
2. M₁₂₈×C₁₃₇	C₂	semidirect
3. #1424×C₁₃₇	C₂	semidirect
5. Q₆₄×C₁₃₇	C₄	semidirect
7. D₃₂×C₁₃₇	C₄	semidirect
44. #7540	C₁₃₇	direct (N × H)
45. C₁₃₇	#7540	direct (N × H)
47. M₁₂₈	C₂×C₁₃₇	semidirect
48. #1424	C₂×C₁₃₇	semidirect
50. Q₆₄	C₄×C₁₃₇	semidirect
52. D₃₂	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2262×C₁₃₇	2
M₁₂₈×C₁₃₇	2
#1424×C₁₃₇	2
#7540	137

Subgroups — C137×#7540 #46856926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7540 #46856926