Subgroups — C₂×Q₁₂₈×C₁₃₇ #46863126

Order 35072.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄×C₁₃₇	C₂	1	2	yes	maximal
2	Q₁₂₈×C₁₃₇	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₁₃₇	—	2	2	no	—
5	Q₆₄×C₁₃₇	—	2	2	no	—
6	Q₆₄×C₁₃₇	C₂²	1	4	yes	—
7	C₂×C₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₆₄×C₁₃₇	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₃₇	—	4	2	no	—
10	Q₃₂×C₁₃₇	—	2	4	no	—
11	Q₃₂×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×Q₈×C₁₃₇	—	8	2	no	—
16	Q₁₆×C₁₃₇	—	4	4	no	—
17	Q₁₆×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	Q₈×C₁₃₇	—	8	4	no	—
22	Q₈×C₁₃₇	—	16	2	no	—
23	C₂×C₄×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	D₃₂	1	1	yes	characteristic
28	C₄×C₁₃₇	—	16	4	no	—
29	C₄×C₁₃₇	D₃₂	1	1	yes	characteristic
30	C₄×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
31	C₂×C₁₃₇	Q₁₂₈	1	2	yes	direct product factor
32	C₂×C₁₃₇	C₂×D₃₂	1	1	yes	characteristic
33	C₂×Q₁₂₈	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₁₃₇	C₂×Q₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×Q₆₄	C₂×C₁₃₇	1	2	yes	—
36	Q₁₂₈	C₂×C₁₃₇	1	4	yes	direct product factor
37	C₂×C₆₄	C₂×C₁₃₇	1	1	yes	characteristic
38	C₂×Q₃₂	—	2	2	no	—
39	Q₆₄	—	2	2	no	—
40	Q₆₄	C₂²×C₁₃₇	1	4	yes	—
41	C₂×C₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
42	C₆₄	C₂²×C₁₃₇	1	2	yes	—
43	C₂×Q₁₆	—	4	2	no	—
44	Q₃₂	—	2	4	no	—
45	Q₃₂	—	4	2	no	—
46	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
47	C₃₂	D₄×C₁₃₇	1	1	yes	characteristic
48	C₃₂	C₂³×C₁₃₇	1	1	yes	characteristic
49	C₂×Q₈	—	8	2	no	—
50	Q₁₆	—	4	4	no	—
51	Q₁₆	—	8	2	no	—
52	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
53	C₁₆	D₈×C₁₃₇	1	1	yes	characteristic
54	C₁₆	C₂×D₄×C₁₃₇	1	1	yes	characteristic
55	Q₈	—	8	4	no	—
56	Q₈	—	16	2	no	—
57	C₂×C₄	—	16	2	no	—
58	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
59	C₈	D₁₆×C₁₃₇	1	1	yes	characteristic
60	C₈	C₂×D₈×C₁₃₇	1	1	yes	characteristic
61	C₂²	D₃₂×C₁₃₇	1	1	yes	characteristic
62	C₄	—	16	4	no	—
63	C₄	D₃₂×C₁₃₇	1	1	yes	characteristic
64	C₄	C₂×D₁₆×C₁₃₇	1	1	yes	characteristic
65	C₂	Q₁₂₈×C₁₃₇	1	2	yes	direct product factor
66	C₂	C₂×D₃₂×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	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
2. Q₁₂₈×C₁₃₇	C₂	direct (N × H)
31. C₂×C₁₃₇	Q₁₂₈	direct (N × H)
33. C₂×Q₁₂₈	C₁₃₇	direct (N × H)
34. C₁₃₇	C₂×Q₁₂₈	direct (N × H)
36. Q₁₂₈	C₂×C₁₃₇	direct (N × H)
65. C₂	Q₁₂₈×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×Q₆₄×C₁₃₇	2
Q₁₂₈×C₁₃₇	2
C₂×C₆₄×C₁₃₇	2
C₂×Q₁₂₈	137

Subgroups — C2×Q128×C137 #46863126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₁₂₈×C₁₃₇ #46863126