Subgroups — C₁₃₇×#13731 #46863117

Order 35072.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2252×C₁₃₇	C₂	1	2	yes	maximal
2	#1423×C₁₃₇	C₂	1	4	yes	maximal
3	C₄×C₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₃₇	—	2	2	no	—
5	C₂×C₃₂×C₁₃₇	C₂²	1	2	yes	—
6	#366×C₁₃₇	—	2	2	no	—
7	#366×C₁₃₇	C₂²	1	4	yes	—
8	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₁₃₇	—	4	2	no	—
10	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
12	#107×C₁₃₇	—	2	4	no	—
13	#107×C₁₃₇	—	4	2	no	—
14	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
15	C₃₂×C₁₃₇	Q₈	1	4	yes	complemented
16	C₂×Q₈×C₁₃₇	—	8	2	no	—
17	C₂×C₈×C₁₃₇	D₈	1	1	yes	characteristic
18	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
19	C₄⋊C₄×C₁₃₇	—	4	4	no	—
20	C₄⋊C₄×C₁₃₇	—	8	2	no	—
21	C₄²×C₁₃₇	D₈	1	1	yes	characteristic
22	C₁₆×C₁₃₇	Q₁₆	1	2	yes	—
23	C₁₆×C₁₃₇	C₂×Q₈	1	2	yes	—
24	Q₈×C₁₃₇	—	16	4	no	—
25	C₂×C₄×C₁₃₇	—	8	4	no	—
26	C₂×C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
27	C₂×C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
29	C₈×C₁₃₇	Q₃₂	1	2	yes	—
30	C₈×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
31	C₈×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
32	C₂²×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
33	C₄×C₁₃₇	—	16	4	no	—
34	C₄×C₁₃₇	D₃₂	1	2	yes	—
35	C₄×C₁₃₇	Q₆₄	1	2	yes	—
36	C₄×C₁₃₇	#500	1	1	yes	characteristic
37	C₄×C₁₃₇	C₂×Q₃₂	1	1	yes	characteristic
38	C₂×C₁₃₇	#2252	1	1	yes	characteristic
39	C₂×C₁₃₇	C₂×D₃₂	1	1	yes	characteristic
40	C₂×C₁₃₇	C₂×Q₆₄	1	1	yes	characteristic
41	#13731	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₁₃₇	#13731	1	1	yes	characteristic, direct product factor, Sylow
43	#2252	C₂×C₁₃₇	1	2	yes	—
44	#1423	C₂×C₁₃₇	1	4	yes	—
45	C₄×C₃₂	C₂×C₁₃₇	1	1	yes	characteristic
46	#500	—	2	2	no	—
47	C₂×C₃₂	C₂²×C₁₃₇	1	2	yes	—
48	#366	—	2	2	no	—
49	#366	C₂²×C₁₃₇	1	4	yes	—
50	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
51	C₄⋊Q₈	—	4	2	no	—
52	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
53	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
54	#107	—	2	4	no	—
55	#107	—	4	2	no	—
56	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
57	C₃₂	Q₈×C₁₃₇	1	4	yes	complemented
58	C₂×Q₈	—	8	2	no	—
59	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
60	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
61	C₄⋊C₄	—	4	4	no	—
62	C₄⋊C₄	—	8	2	no	—
63	C₄²	D₈×C₁₃₇	1	1	yes	characteristic
64	C₁₆	Q₁₆×C₁₃₇	1	2	yes	—
65	C₁₆	C₂×Q₈×C₁₃₇	1	2	yes	—
66	Q₈	—	16	4	no	—
67	C₂×C₄	—	8	4	no	—
68	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
69	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
70	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
71	C₈	Q₃₂×C₁₃₇	1	2	yes	—
72	C₈	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
73	C₈	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
74	C₂²	C₂×D₁₆×C₁₃₇	1	1	yes	characteristic
75	C₄	—	16	4	no	—
76	C₄	D₃₂×C₁₃₇	1	2	yes	—
77	C₄	Q₆₄×C₁₃₇	1	2	yes	—
78	C₄	#500×C₁₃₇	1	1	yes	characteristic
79	C₄	C₂×Q₃₂×C₁₃₇	1	1	yes	characteristic
80	C₂	#2252×C₁₃₇	1	1	yes	characteristic
81	C₂	C₂×D₃₂×C₁₃₇	1	1	yes	characteristic
82	C₂	C₂×Q₆₄×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃₇
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
15. C₃₂×C₁₃₇	Q₈	semidirect
41. #13731	C₁₃₇	direct (N × H)
42. C₁₃₇	#13731	direct (N × H)
57. C₃₂	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2252×C₁₃₇	2
#1423×C₁₃₇	2
C₄×C₃₂×C₁₃₇	2
#13731	137

Subgroups — C137×#13731 #46863117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13731 #46863117