Subgroups — C₁₃₇×#7532 #46856918

Order 35072.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1713×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1411×C₁₃₇	C₂	1	2	yes	maximal
3	#500×C₁₃₇	—	2	2	no	—
4	#494×C₁₃₇	C₄	1	1	yes	characteristic
5	#494×C₁₃₇	C₂²	1	1	yes	characteristic
6	#356×C₁₃₇	—	2	2	no	—
7	C₈²×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂×Q₁₆×C₁₃₇	—	4	1	no	—
9	C₂×Q₁₆×C₁₃₇	—	4	1	no	—
10	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
11	C₄⋊Q₈×C₁₃₇	D₄	1	2	yes	—
12	#107×C₁₃₇	—	4	4	no	—
13	#101×C₁₃₇	—	4	2	no	—
14	C₄×C₈×C₁₃₇	—	2	1	no	—
15	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×Q₈×C₁₃₇	—	2	2	no	—
17	C₂×Q₈×C₁₃₇	—	4	1	no	—
18	Q₁₆×C₁₃₇	—	8	2	no	—
19	Q₁₆×C₁₃₇	—	8	2	no	—
20	M₁₆×C₁₃₇	—	8	2	no	—
21	C₂×C₈×C₁₃₇	—	2	2	no	—
22	C₂×C₈×C₁₃₇	—	2	1	no	—
23	C₄⋊C₄×C₁₃₇	—	4	2	no	—
24	C₄⋊C₄×C₁₃₇	—	4	2	no	—
25	C₄²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	Q₈×C₁₃₇	—	8	2	no	—
27	Q₈×C₁₃₇	—	8	2	no	—
28	C₂×C₄×C₁₃₇	—	2	1	no	—
29	C₂×C₄×C₁₃₇	—	4	2	no	—
30	C₂×C₄×C₁₃₇	—	8	1	no	—
31	C₂×C₄×C₁₃₇	#99	1	1	yes	characteristic
32	C₈×C₁₃₇	—	2	4	no	—
33	C₈×C₁₃₇	—	2	2	no	—
34	C₈×C₁₃₇	—	16	2	no	—
35	C₂²×C₁₃₇	#353	1	1	yes	characteristic
36	C₄×C₁₃₇	—	2	2	no	—
37	C₄×C₁₃₇	—	2	1	no	—
38	C₄×C₁₃₇	—	8	2	no	—
39	C₄×C₁₃₇	—	16	1	no	—
40	C₂×C₁₃₇	—	2	1	no	—
41	C₂×C₁₃₇	#1408	1	1	yes	characteristic
42	#7532	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₁₃₇	#7532	1	1	yes	characteristic, direct product factor, Sylow
44	#1713	C₂×C₁₃₇	1	1	yes	characteristic
45	#1411	C₂×C₁₃₇	1	2	yes	—
46	#500	—	2	2	no	—
47	#494	C₄×C₁₃₇	1	1	yes	characteristic
48	#494	C₂²×C₁₃₇	1	1	yes	characteristic
49	#356	—	2	2	no	—
50	C₈²	C₄×C₁₃₇	1	1	yes	characteristic
51	C₂×Q₁₆	—	4	1	no	—
52	C₂×Q₁₆	—	4	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	D₄×C₁₃₇	1	2	yes	—
55	#107	—	4	4	no	—
56	#101	—	4	2	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	4	1	no	—
61	Q₁₆	—	8	2	no	—
62	Q₁₆	—	8	2	no	—
63	M₁₆	—	8	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄²	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
69	Q₈	—	8	2	no	—
70	Q₈	—	8	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	8	1	no	—
74	C₂×C₄	#99×C₁₃₇	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	2	4	no	—
77	C₈	—	16	2	no	—
78	C₂²	#353×C₁₃₇	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	8	2	no	—
82	C₄	—	16	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#1408×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₇	#1408
commutator	C₄×C₈	C₂×C₄×C₁₃₇
Frattini, cosocle	#494	C₂²×C₁₃₇
prime radical	C₂²×C₁₃₇	#353
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₇ ◁ C₂²×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ #494×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
42. #7532	C₁₃₇	direct (N × H)
43. C₁₃₇	#7532	direct (N × H)

Maximal subgroups

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

subgroup	index
#1713×C₁₃₇	2
#1411×C₁₃₇	2
#7532	137

Subgroups — C137×#7532 #46856918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7532 #46856918