Subgroups — C₂×C₃₇×Q₁₂₈ #22634993

Order 9472.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×Q₆₄	C₂	1	2	yes	maximal
2	C₃₇×Q₁₂₈	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃₇×C₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₃₇	—	2	2	no	—
5	C₃₇×Q₆₄	—	2	2	no	—
6	C₃₇×Q₆₄	C₂²	1	4	yes	—
7	C₂×C₃₂×C₃₇	C₂²	1	1	yes	characteristic
8	C₃₇×C₆₄	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₃₇	—	4	2	no	—
10	Q₃₂×C₃₇	—	2	4	no	—
11	Q₃₂×C₃₇	—	4	2	no	—
12	C₂×C₁₆×C₃₇	D₄	1	1	yes	characteristic
13	C₃₂×C₃₇	D₄	1	1	yes	characteristic
14	C₃₂×C₃₇	C₂³	1	1	yes	characteristic
15	C₂×Q₈×C₃₇	—	8	2	no	—
16	Q₁₆×C₃₇	—	4	4	no	—
17	Q₁₆×C₃₇	—	8	2	no	—
18	C₂×C₈×C₃₇	D₈	1	1	yes	characteristic
19	C₁₆×C₃₇	D₈	1	1	yes	characteristic
20	C₁₆×C₃₇	C₂×D₄	1	1	yes	characteristic
21	Q₈×C₃₇	—	8	4	no	—
22	Q₈×C₃₇	—	16	2	no	—
23	C₂×C₄×C₃₇	—	16	2	no	—
24	C₂×C₄×C₃₇	D₁₆	1	1	yes	characteristic
25	C₈×C₃₇	D₁₆	1	1	yes	characteristic
26	C₈×C₃₇	C₂×D₈	1	1	yes	characteristic
27	C₂×Q₁₂₈	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₃₇	D₃₂	1	1	yes	characteristic
29	C₄×C₃₇	—	16	4	no	—
30	C₄×C₃₇	D₃₂	1	1	yes	characteristic
31	C₄×C₃₇	C₂×D₁₆	1	1	yes	characteristic
32	C₂×Q₆₄	C₂×C₃₇	1	2	yes	—
33	Q₁₂₈	C₂×C₃₇	1	4	yes	direct product factor
34	C₂×C₆₄	C₂×C₃₇	1	1	yes	characteristic
35	C₂×C₃₇	Q₁₂₈	1	2	yes	direct product factor
36	C₂×C₃₇	C₂×D₃₂	1	1	yes	characteristic
37	C₂×Q₃₂	—	2	2	no	—
38	Q₆₄	—	2	2	no	—
39	Q₆₄	C₂²×C₃₇	1	4	yes	—
40	C₂×C₃₂	C₂²×C₃₇	1	1	yes	characteristic
41	C₆₄	C₂²×C₃₇	1	2	yes	—
42	C₃₇	C₂×Q₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×Q₁₆	—	4	2	no	—
44	Q₃₂	—	2	4	no	—
45	Q₃₂	—	4	2	no	—
46	C₂×C₁₆	D₄×C₃₇	1	1	yes	characteristic
47	C₃₂	D₄×C₃₇	1	1	yes	characteristic
48	C₃₂	C₂³×C₃₇	1	1	yes	characteristic
49	C₂×Q₈	—	8	2	no	—
50	Q₁₆	—	4	4	no	—
51	Q₁₆	—	8	2	no	—
52	C₂×C₈	D₈×C₃₇	1	1	yes	characteristic
53	C₁₆	D₈×C₃₇	1	1	yes	characteristic
54	C₁₆	C₂×D₄×C₃₇	1	1	yes	characteristic
55	Q₈	—	8	4	no	—
56	Q₈	—	16	2	no	—
57	C₂×C₄	—	16	2	no	—
58	C₂×C₄	D₁₆×C₃₇	1	1	yes	characteristic
59	C₈	D₁₆×C₃₇	1	1	yes	characteristic
60	C₈	C₂×D₈×C₃₇	1	1	yes	characteristic
61	C₂²	C₃₇×D₃₂	1	1	yes	characteristic
62	C₄	—	16	4	no	—
63	C₄	C₃₇×D₃₂	1	1	yes	characteristic
64	C₄	C₂×D₁₆×C₃₇	1	1	yes	characteristic
65	C₂	C₃₇×Q₁₂₈	1	2	yes	direct product factor
66	C₂	C₂×C₃₇×D₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₇	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³×C₃₇
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄×C₃₇ ◁ C₂×C₈×C₃₇ ◁ C₂×C₁₆×C₃₇ ◁ C₂×C₃₂×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×Q₁₂₈	C₂	direct (N × H)
27. C₂×Q₁₂₈	C₃₇	direct (N × H)
33. Q₁₂₈	C₂×C₃₇	direct (N × H)
35. C₂×C₃₇	Q₁₂₈	direct (N × H)
42. C₃₇	C₂×Q₁₂₈	direct (N × H)
65. C₂	C₃₇×Q₁₂₈	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×C₃₇×Q₆₄	2
C₃₇×Q₁₂₈	2
C₂×C₃₇×C₆₄	2
C₂×Q₁₂₈	37

Subgroups — C2×C37×Q128 #22634993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×Q₁₂₈ #22634993