Subgroups — C₁₃₇×#7458 #46856844

Order 35072.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₃₂×C₁₃₇	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₁₃₇	—	2	2	no	—
4	C₄×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₈²×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₃₇	—	2	1	no	—
8	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
9	C₂×C₁₆×C₁₃₇	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₁₃₇	C₈	1	1	yes	characteristic
11	C₄×C₈×C₁₃₇	C₈	1	1	yes	characteristic
12	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₁₃₇	—	4	2	no	—
14	C₂×C₈×C₁₃₇	—	2	1	no	—
15	C₂×C₈×C₁₃₇	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₁₃₇	C₁₆	1	1	yes	characteristic
17	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
19	C₄²×C₁₃₇	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₁₃₇	—	2	2	no	—
21	C₁₆×C₁₃₇	—	2	1	no	—
22	C₁₆×C₁₃₇	D₈	1	1	yes	characteristic
23	C₁₆×C₁₃₇	Q₁₆	1	1	yes	characteristic
24	C₂×C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	M₃₂	1	1	yes	characteristic
27	C₈×C₁₃₇	—	2	2	no	—
28	C₈×C₁₃₇	—	2	1	no	—
29	C₈×C₁₃₇	—	2	1	no	—
30	C₈×C₁₃₇	C₃₂	1	2	yes	complemented
31	C₈×C₁₃₇	#107	1	1	yes	characteristic
32	C₈×C₁₃₇	#108	1	1	yes	characteristic
33	C₂²×C₁₃₇	C₄⋊C₁₆	1	1	yes	characteristic
34	C₄×C₁₃₇	—	2	1	no	—
35	C₄×C₁₃₇	#335	1	1	yes	characteristic
36	C₄×C₁₃₇	#364	1	1	yes	characteristic
37	C₄×C₁₃₇	C₂×C₃₂	1	1	yes	characteristic
38	C₄×C₁₃₇	M₆₄	1	1	yes	characteristic
39	C₂×C₁₃₇	#1370	1	1	yes	characteristic
40	C₂×C₁₃₇	C₄⋊C₃₂	1	1	yes	characteristic
41	C₂×C₁₃₇	#1422	1	1	yes	characteristic
42	#7458	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₁₃₇	#7458	1	1	yes	characteristic, direct product factor, Sylow
44	C₄⋊C₃₂	C₂×C₁₃₇	1	2	yes	—
45	C₈×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
46	C₂×C₃₂	—	2	2	no	—
47	C₄×C₁₆	C₄×C₁₃₇	1	1	yes	characteristic
48	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
49	C₈²	C₄×C₁₃₇	1	1	yes	characteristic
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
52	C₂×C₁₆	Q₈×C₁₃₇	1	1	yes	characteristic
53	C₄×C₈	C₈×C₁₃₇	1	1	yes	characteristic
54	C₄×C₈	C₈×C₁₃₇	1	1	yes	characteristic
55	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
56	C₃₂	—	4	2	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	C₁₆×C₁₃₇	1	1	yes	characteristic
59	C₂×C₈	C₁₆×C₁₃₇	1	1	yes	characteristic
60	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
61	C₂×C₈	M₁₆×C₁₃₇	1	1	yes	characteristic
62	C₄²	C₂×C₈×C₁₃₇	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	D₈×C₁₃₇	1	1	yes	characteristic
66	C₁₆	Q₁₆×C₁₃₇	1	1	yes	characteristic
67	C₂×C₄	C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
68	C₂×C₄	C₂×C₁₆×C₁₃₇	1	1	yes	characteristic
69	C₂×C₄	M₃₂×C₁₃₇	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	C₃₂×C₁₃₇	1	2	yes	complemented
74	C₈	#107×C₁₃₇	1	1	yes	characteristic
75	C₈	#108×C₁₃₇	1	1	yes	characteristic
76	C₂²	C₄⋊C₁₆×C₁₃₇	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	#335×C₁₃₇	1	1	yes	characteristic
79	C₄	#364×C₁₃₇	1	1	yes	characteristic
80	C₄	C₂×C₃₂×C₁₃₇	1	1	yes	characteristic
81	C₄	M₆₄×C₁₃₇	1	1	yes	characteristic
82	C₂	#1370×C₁₃₇	1	1	yes	characteristic
83	C₂	C₄⋊C₃₂×C₁₃₇	1	1	yes	characteristic
84	C₂	#1422×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆×C₁₃₇	D₄
commutator	C₄	C₂×C₃₂×C₁₃₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₃₇
socle, prime radical	C₂²×C₁₃₇	C₄⋊C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₃₇ ◁ C₄×C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
30. C₈×C₁₃₇	C₃₂	semidirect
42. #7458	C₁₃₇	direct (N × H)
43. C₁₃₇	#7458	direct (N × H)
73. C₈	C₃₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₃₂×C₁₃₇	2
C₈×C₁₆×C₁₃₇	2
#7458	137

Subgroups — C137×#7458 #46856844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7458 #46856844