Subgroups — C₁₃₇×C₄⋊C₆₄ #46856929

Order 35072.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₄×C₁₃₇	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₁₃₇	C₄	1	1	yes	characteristic
4	C₂×C₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
5	C₄×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
6	C₆₄×C₁₃₇	—	2	2	no	—
7	C₂×C₁₆×C₁₃₇	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₁₃₇	C₈	1	1	yes	characteristic
10	C₃₂×C₁₃₇	—	2	1	no	—
11	C₃₂×C₁₃₇	D₄	1	1	yes	characteristic
12	C₃₂×C₁₃₇	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₁₃₇	C₁₆	1	1	yes	characteristic
14	C₂×C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₁₃₇	C₁₆	1	1	yes	characteristic
16	C₁₆×C₁₃₇	—	2	1	no	—
17	C₁₆×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₁₃₇	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₃₇	C₃₂	1	1	yes	characteristic
20	C₂×C₄×C₁₃₇	C₃₂	1	1	yes	characteristic
21	C₂×C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
22	C₈×C₁₃₇	—	2	1	no	—
23	C₈×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₁₃₇	M₃₂	1	1	yes	characteristic
25	C₂²×C₁₃₇	C₂×C₃₂	1	1	yes	characteristic
26	C₄×C₁₃₇	—	2	1	no	—
27	C₄×C₁₃₇	C₆₄	1	2	yes	complemented
28	C₄×C₁₃₇	C₄⋊C₁₆	1	1	yes	characteristic
29	C₄×C₁₃₇	M₆₄	1	1	yes	characteristic
30	C₂×C₁₃₇	C₄⋊C₃₂	1	1	yes	characteristic
31	C₂×C₁₃₇	C₂×C₆₄	1	1	yes	characteristic
32	C₂×C₁₃₇	M₁₂₈	1	1	yes	characteristic
33	C₄⋊C₆₄	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₁₃₇	C₄⋊C₆₄	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₆₄	C₂×C₁₃₇	1	2	yes	—
36	C₄×C₃₂	C₂×C₁₃₇	1	1	yes	characteristic
37	C₂×C₃₂	C₄×C₁₃₇	1	1	yes	characteristic
38	C₂×C₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
39	C₄×C₁₆	C₄×C₁₃₇	1	1	yes	characteristic
40	C₆₄	—	2	2	no	—
41	C₂×C₁₆	C₈×C₁₃₇	1	1	yes	characteristic
42	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
43	C₄×C₈	C₈×C₁₃₇	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	D₄×C₁₃₇	1	1	yes	characteristic
46	C₃₂	Q₈×C₁₃₇	1	1	yes	characteristic
47	C₂×C₈	C₁₆×C₁₃₇	1	1	yes	characteristic
48	C₂×C₈	C₂×C₈×C₁₃₇	1	1	yes	characteristic
49	C₄²	C₁₆×C₁₃₇	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
52	C₁₆	M₁₆×C₁₃₇	1	1	yes	characteristic
53	C₂×C₄	C₃₂×C₁₃₇	1	1	yes	characteristic
54	C₂×C₄	C₃₂×C₁₃₇	1	1	yes	characteristic
55	C₂×C₄	C₂×C₁₆×C₁₃₇	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
58	C₈	M₃₂×C₁₃₇	1	1	yes	characteristic
59	C₂²	C₂×C₃₂×C₁₃₇	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	C₆₄×C₁₃₇	1	2	yes	complemented
62	C₄	C₄⋊C₁₆×C₁₃₇	1	1	yes	characteristic
63	C₄	M₆₄×C₁₃₇	1	1	yes	characteristic
64	C₂	C₄⋊C₃₂×C₁₃₇	1	1	yes	characteristic
65	C₂	C₂×C₆₄×C₁₃₇	1	1	yes	characteristic
66	C₂	M₁₂₈×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂×C₁₃₇	C₂²
commutator	C₂	C₂×C₆₄×C₁₃₇
Frattini, cosocle	C₂×C₃₂	C₂²×C₁₃₇
socle, prime radical	C₂²×C₁₃₇	C₂×C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
27. C₄×C₁₃₇	C₆₄	semidirect
33. C₄⋊C₆₄	C₁₃₇	direct (N × H)
34. C₁₃₇	C₄⋊C₆₄	direct (N × H)
61. C₄	C₆₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₄×C₁₃₇	2
C₄×C₃₂×C₁₃₇	2
C₄⋊C₆₄	137

Subgroups — C137×C4⋊C64 #46856929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×C₄⋊C₆₄ #46856929