Subgroups — C₂×C₁₆²×C₃₇ #72196680

Order 18944.

644 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×C₁₆×C₃₇	C₂	1	3	yes	maximal
2	C₁₆²×C₃₇	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₃₇	C₄	1	6	yes	—
4	C₂×C₈²×C₃₇	C₂²	1	1	yes	characteristic
5	C₈×C₁₆×C₃₇	C₄	1	6	yes	—
6	C₈×C₁₆×C₃₇	C₂²	1	6	yes	—
7	C₂²×C₁₆×C₃₇	C₈	1	12	yes	—
8	C₂×C₄×C₈×C₃₇	C₂×C₄	1	3	yes	—
9	C₄×C₁₆×C₃₇	C₈	1	12	yes	—
10	C₄×C₁₆×C₃₇	C₂×C₄	1	12	yes	—
11	C₈²×C₃₇	C₂×C₄	1	3	yes	—
12	C₈²×C₃₇	C₂³	1	1	yes	characteristic
13	C₂²×C₈×C₃₇	C₂×C₈	1	6	yes	—
14	C₂×C₄²×C₃₇	C₄²	1	1	yes	characteristic
15	C₂×C₁₆×C₃₇	C₁₆	1	48	yes	direct product factor
16	C₂×C₁₆×C₃₇	C₂×C₈	1	24	yes	—
17	C₄×C₈×C₃₇	C₄²	1	3	yes	—
18	C₄×C₈×C₃₇	C₂×C₈	1	6	yes	—
19	C₄×C₈×C₃₇	C₂²×C₄	1	3	yes	—
20	C₂²×C₄×C₃₇	C₄×C₈	1	3	yes	—
21	C₂×C₈×C₃₇	C₄×C₈	1	6	yes	—
22	C₂×C₈×C₃₇	C₂×C₁₆	1	24	yes	—
23	C₂×C₈×C₃₇	C₂²×C₈	1	6	yes	—
24	C₄²×C₃₇	C₄×C₈	1	3	yes	—
25	C₄²×C₃₇	C₂×C₄²	1	1	yes	characteristic
26	C₁₆×C₃₇	C₂×C₁₆	1	48	yes	direct product factor
27	C₂×C₁₆²	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₃₇	C₈²	1	1	yes	characteristic
29	C₂×C₄×C₃₇	C₈²	1	3	yes	—
30	C₂×C₄×C₃₇	C₄×C₁₆	1	12	yes	—
31	C₂×C₄×C₃₇	C₂×C₄×C₈	1	3	yes	—
32	C₈×C₃₇	C₄×C₁₆	1	12	yes	—
33	C₈×C₃₇	C₂²×C₁₆	1	12	yes	—
34	C₂×C₈×C₁₆	C₂×C₃₇	1	3	yes	—
35	C₁₆²	C₂×C₃₇	1	4	yes	direct product factor
36	C₂²×C₃₇	C₈×C₁₆	1	6	yes	—
37	C₂²×C₃₇	C₂×C₈²	1	1	yes	characteristic
38	C₄×C₃₇	C₈×C₁₆	1	6	yes	—
39	C₄×C₃₇	C₂×C₄×C₁₆	1	6	yes	—
40	C₂×C₄×C₁₆	C₄×C₃₇	1	6	yes	—
41	C₂×C₈²	C₂²×C₃₇	1	1	yes	characteristic
42	C₈×C₁₆	C₄×C₃₇	1	6	yes	—
43	C₈×C₁₆	C₂²×C₃₇	1	6	yes	—
44	C₂×C₃₇	C₁₆²	1	4	yes	direct product factor
45	C₂×C₃₇	C₂×C₈×C₁₆	1	3	yes	—
46	C₂²×C₁₆	C₈×C₃₇	1	12	yes	—
47	C₂×C₄×C₈	C₂×C₄×C₃₇	1	3	yes	—
48	C₄×C₁₆	C₈×C₃₇	1	12	yes	—
49	C₄×C₁₆	C₂×C₄×C₃₇	1	12	yes	—
50	C₈²	C₂×C₄×C₃₇	1	3	yes	—
51	C₈²	C₂³×C₃₇	1	1	yes	characteristic
52	C₃₇	C₂×C₁₆²	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²×C₈	C₂×C₈×C₃₇	1	6	yes	—
54	C₂×C₄²	C₄²×C₃₇	1	1	yes	characteristic
55	C₂×C₁₆	C₁₆×C₃₇	1	48	yes	direct product factor
56	C₂×C₁₆	C₂×C₈×C₃₇	1	24	yes	—
57	C₄×C₈	C₄²×C₃₇	1	3	yes	—
58	C₄×C₈	C₂×C₈×C₃₇	1	6	yes	—
59	C₄×C₈	C₂²×C₄×C₃₇	1	3	yes	—
60	C₂²×C₄	C₄×C₈×C₃₇	1	3	yes	—
61	C₂×C₈	C₄×C₈×C₃₇	1	6	yes	—
62	C₂×C₈	C₂×C₁₆×C₃₇	1	24	yes	—
63	C₂×C₈	C₂²×C₈×C₃₇	1	6	yes	—
64	C₄²	C₄×C₈×C₃₇	1	3	yes	—
65	C₄²	C₂×C₄²×C₃₇	1	1	yes	characteristic
66	C₁₆	C₂×C₁₆×C₃₇	1	48	yes	direct product factor
67	C₂³	C₈²×C₃₇	1	1	yes	characteristic
68	C₂×C₄	C₈²×C₃₇	1	3	yes	—
69	C₂×C₄	C₄×C₁₆×C₃₇	1	12	yes	—
70	C₂×C₄	C₂×C₄×C₈×C₃₇	1	3	yes	—
71	C₈	C₄×C₁₆×C₃₇	1	12	yes	—
72	C₈	C₂²×C₁₆×C₃₇	1	12	yes	—
73	C₂²	C₈×C₁₆×C₃₇	1	6	yes	—
74	C₂²	C₂×C₈²×C₃₇	1	1	yes	characteristic
75	C₄	C₈×C₁₆×C₃₇	1	6	yes	—
76	C₄	C₂×C₄×C₁₆×C₃₇	1	6	yes	—
77	C₂	C₁₆²×C₃₇	1	4	yes	direct product factor
78	C₂	C₂×C₈×C₁₆×C₃₇	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈²	C₂³×C₃₇
socle, prime radical	C₂³×C₃₇	C₈²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₆²×C₃₇	C₂	direct (N × H)
15. C₂×C₁₆×C₃₇	C₁₆	direct (N × H)
26. C₁₆×C₃₇	C₂×C₁₆	direct (N × H)
27. C₂×C₁₆²	C₃₇	direct (N × H)
35. C₁₆²	C₂×C₃₇	direct (N × H)
44. C₂×C₃₇	C₁₆²	direct (N × H)
52. C₃₇	C₂×C₁₆²	direct (N × H)
55. C₂×C₁₆	C₁₆×C₃₇	direct (N × H)
66. C₁₆	C₂×C₁₆×C₃₇	direct (N × H)
77. C₂	C₁₆²×C₃₇	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×C₈×C₁₆×C₃₇	2
C₁₆²×C₃₇	2
C₂×C₁₆²	37

Subgroups — C2×C162×C37 #72196680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆²×C₃₇ #72196680