Subgroups — C₂×D₆₄×C₁₃₇ #46863124

Order 35072.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂²×C₁₃₇	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³×C₁₃₇
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
1. C₂×D₃₂×C₁₃₇	C₂	semidirect
2. D₆₄×C₁₃₇	C₂	direct (N × H)
3. C₂×C₆₄×C₁₃₇	C₂	semidirect
6. D₃₂×C₁₃₇	C₂²	semidirect
8. C₆₄×C₁₃₇	C₂²	semidirect
33. C₂×C₁₃₇	D₆₄	direct (N × H)
35. C₂×D₆₄	C₁₃₇	direct (N × H)
36. C₁₃₇	C₂×D₆₄	direct (N × H)
37. C₂×D₃₂	C₂×C₁₃₇	semidirect
38. D₆₄	C₂×C₁₃₇	direct (N × H)
39. C₂×C₆₄	C₂×C₁₃₇	semidirect
42. D₃₂	C₂²×C₁₃₇	semidirect
44. C₆₄	C₂²×C₁₃₇	semidirect
69. C₂	D₆₄×C₁₃₇	direct (N × H)

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
D₆₄×C₁₃₇	2
C₂×C₆₄×C₁₃₇	2
C₂×D₆₄	137

Subgroups — C2×D64×C137 #46863124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₆₄×C₁₃₇ #46863124