Subgroups — C₁₃₇×#7537 #46856923

Order 35072.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#1423×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆×C₁₃₇	—	2	1	no	—
5	D₃₂×C₁₃₇	—	2	1	no	—
6	D₃₂×C₁₃₇	C₄	1	2	yes	complemented
7	C₂×C₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
8	#366×C₁₃₇	—	2	1	no	—
9	C₆₄×C₁₃₇	—	2	1	no	—
10	C₂×D₈×C₁₃₇	—	4	1	no	—
11	D₁₆×C₁₃₇	—	2	2	no	—
12	D₁₆×C₁₃₇	—	4	1	no	—
13	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
14	#107×C₁₃₇	—	4	1	no	—
15	C₃₂×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₁₃₇	D₄	1	1	yes	characteristic
17	C₂×D₄×C₁₃₇	—	8	1	no	—
18	D₈×C₁₃₇	—	4	2	no	—
19	D₈×C₁₃₇	—	8	1	no	—
20	C₂×C₈×C₁₃₇	D₈	1	1	yes	characteristic
21	C₄⋊C₄×C₁₃₇	—	8	1	no	—
22	C₁₆×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₁₃₇	SD₁₆	1	1	yes	characteristic
24	C₂³×C₁₃₇	—	16	1	no	—
25	D₄×C₁₃₇	—	8	2	no	—
26	D₄×C₁₃₇	—	16	1	no	—
27	C₂×C₄×C₁₃₇	—	16	1	no	—
28	C₂×C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
29	C₈×C₁₃₇	#102	1	1	yes	characteristic
30	C₈×C₁₃₇	SD₃₂	1	1	yes	characteristic
31	C₂²×C₁₃₇	—	16	2	no	—
32	C₂²×C₁₃₇	—	32	1	no	—
33	C₂²×C₁₃₇	—	32	1	no	—
34	C₂²×C₁₃₇	D₃₂	1	1	yes	characteristic
35	C₄×C₁₃₇	—	32	1	no	—
36	C₄×C₁₃₇	#357	1	1	yes	characteristic
37	C₄×C₁₃₇	SD₆₄	1	1	yes	characteristic
38	C₂×C₁₃₇	—	32	2	no	—
39	C₂×C₁₃₇	#1415	1	1	yes	characteristic
40	C₂×C₁₃₇	D₆₄	1	1	yes	characteristic
41	C₂×C₁₃₇	SD₁₂₈	1	1	yes	characteristic
42	#7537	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₁₃₇	#7537	1	1	yes	characteristic, direct product factor, Sylow
44	C₂×D₃₂	C₂×C₁₃₇	1	1	yes	characteristic
45	C₂×C₆₄	C₂×C₁₃₇	1	1	yes	characteristic, complemented
46	#1423	C₂×C₁₃₇	1	1	yes	characteristic, complemented
47	C₂×D₁₆	—	2	1	no	—
48	D₃₂	—	2	1	no	—
49	D₃₂	C₄×C₁₃₇	1	2	yes	complemented
50	C₂×C₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
51	#366	—	2	1	no	—
52	C₆₄	—	2	1	no	—
53	C₂×D₈	—	4	1	no	—
54	D₁₆	—	2	2	no	—
55	D₁₆	—	4	1	no	—
56	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
57	#107	—	4	1	no	—
58	C₃₂	C₂×C₄×C₁₃₇	1	1	yes	characteristic
59	C₃₂	D₄×C₁₃₇	1	1	yes	characteristic
60	C₂×D₄	—	8	1	no	—
61	D₈	—	4	2	no	—
62	D₈	—	8	1	no	—
63	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
64	C₄⋊C₄	—	8	1	no	—
65	C₁₆	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
66	C₁₆	SD₁₆×C₁₃₇	1	1	yes	characteristic
67	C₂³	—	16	1	no	—
68	D₄	—	8	2	no	—
69	D₄	—	16	1	no	—
70	C₂×C₄	—	16	1	no	—
71	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
72	C₈	#102×C₁₃₇	1	1	yes	characteristic
73	C₈	SD₃₂×C₁₃₇	1	1	yes	characteristic
74	C₂²	—	16	2	no	—
75	C₂²	—	32	1	no	—
76	C₂²	—	32	1	no	—
77	C₂²	D₃₂×C₁₃₇	1	1	yes	characteristic
78	C₄	—	32	1	no	—
79	C₄	#357×C₁₃₇	1	1	yes	characteristic
80	C₄	SD₆₄×C₁₃₇	1	1	yes	characteristic
81	C₂	—	32	2	no	—
82	C₂	#1415×C₁₃₇	1	1	yes	characteristic
83	C₂	D₆₄×C₁₃₇	1	1	yes	characteristic
84	C₂	SD₁₂₈×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	D₃₂
commutator	C₃₂	C₂×C₄×C₁₃₇
Frattini, cosocle	C₂×C₃₂	C₂²×C₁₃₇
prime radical	C₂×D₃₂×C₁₃₇	C₂
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
2. C₂×C₆₄×C₁₃₇	C₂	semidirect
3. #1423×C₁₃₇	C₂	semidirect
6. D₃₂×C₁₃₇	C₄	semidirect
42. #7537	C₁₃₇	direct (N × H)
43. C₁₃₇	#7537	direct (N × H)
45. C₂×C₆₄	C₂×C₁₃₇	semidirect
46. #1423	C₂×C₁₃₇	semidirect
49. 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
C₂×D₃₂×C₁₃₇	2
C₂×C₆₄×C₁₃₇	2
#1423×C₁₃₇	2
#7537	137

Subgroups — C137×#7537 #46856923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7537 #46856923