Subgroups — C₂³×Q₃₂×C₃₇ #22683880

Order 9472.

2078 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₃₇	C₂	1	2	yes	maximal
2	C₂²×Q₃₂×C₃₇	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Q₈×C₃₇	—	2	2	no	—
5	C₂²×Q₁₆×C₃₇	—	2	14	no	—
6	C₂²×Q₁₆×C₃₇	C₂²	1	28	yes	—
7	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
8	C₂×Q₃₂×C₃₇	C₂²	1	112	yes	direct product factor
9	C₂²×C₁₆×C₃₇	C₂²	1	14	yes	—
10	C₂²×Q₈×C₃₇	—	2	28	no	—
11	C₂²×Q₈×C₃₇	—	4	14	no	—
12	C₂³×C₄×C₃₇	—	4	2	no	—
13	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
14	C₂×Q₁₆×C₃₇	—	2	84	no	—
15	C₂×Q₁₆×C₃₇	C₂³	1	56	yes	—
16	C₂²×C₈×C₃₇	D₄	1	7	yes	—
17	C₂²×C₈×C₃₇	C₂³	1	7	yes	—
18	Q₃₂×C₃₇	C₂³	1	64	yes	direct product factor
19	C₂×C₁₆×C₃₇	C₂³	1	28	yes	—
20	C₂⁴×C₃₇	D₈	1	1	yes	characteristic
21	C₂×Q₈×C₃₇	—	2	56	no	—
22	C₂×Q₈×C₃₇	—	4	84	no	—
23	C₂²×C₄×C₃₇	—	4	28	no	—
24	C₂²×C₄×C₃₇	D₈	1	7	yes	—
25	C₂²×C₄×C₃₇	C₂×D₄	1	7	yes	—
26	Q₁₆×C₃₇	—	2	56	no	—
27	Q₁₆×C₃₇	C₂⁴	1	16	yes	—
28	C₂×C₈×C₃₇	C₂×D₄	1	21	yes	—
29	C₂×C₈×C₃₇	C₂⁴	1	7	yes	—
30	C₁₆×C₃₇	C₂⁴	1	8	yes	—
31	C₂³×C₃₇	Q₃₂	1	8	yes	direct product factor
32	C₂³×C₃₇	C₂×D₈	1	7	yes	—
33	Q₈×C₃₇	—	2	16	no	—
34	Q₈×C₃₇	—	4	56	no	—
35	C₂×C₄×C₃₇	—	4	56	no	—
36	C₂×C₄×C₃₇	C₂×D₈	1	21	yes	—
37	C₂×C₄×C₃₇	C₂²×D₄	1	7	yes	—
38	C₈×C₃₇	C₂²×D₄	1	7	yes	—
39	C₈×C₃₇	C₂⁵	1	1	yes	characteristic
40	C₂³×Q₃₂	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂²×C₃₇	C₂×Q₃₂	1	28	yes	direct product factor
42	C₂²×C₃₇	C₂²×D₈	1	7	yes	—
43	C₄×C₃₇	—	4	16	no	—
44	C₄×C₃₇	C₂²×D₈	1	7	yes	—
45	C₄×C₃₇	C₂³×D₄	1	1	yes	characteristic
46	C₂³×Q₁₆	C₂×C₃₇	1	2	yes	—
47	C₂²×Q₃₂	C₂×C₃₇	1	28	yes	direct product factor
48	C₂³×C₁₆	C₂×C₃₇	1	1	yes	characteristic
49	C₂×C₃₇	C₂²×Q₃₂	1	14	yes	direct product factor
50	C₂×C₃₇	C₂³×D₈	1	1	yes	characteristic
51	C₂³×Q₈	—	2	2	no	—
52	C₂²×Q₁₆	—	2	14	no	—
53	C₂²×Q₁₆	C₂²×C₃₇	1	28	yes	—
54	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
55	C₂×Q₃₂	C₂²×C₃₇	1	112	yes	direct product factor
56	C₂²×C₁₆	C₂²×C₃₇	1	14	yes	—
57	C₃₇	C₂³×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×Q₈	—	2	28	no	—
59	C₂²×Q₈	—	4	14	no	—
60	C₂³×C₄	—	4	2	no	—
61	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
62	C₂×Q₁₆	—	2	84	no	—
63	C₂×Q₁₆	C₂³×C₃₇	1	56	yes	—
64	C₂²×C₈	D₄×C₃₇	1	7	yes	—
65	C₂²×C₈	C₂³×C₃₇	1	7	yes	—
66	Q₃₂	C₂³×C₃₇	1	64	yes	direct product factor
67	C₂×C₁₆	C₂³×C₃₇	1	28	yes	—
68	C₂⁴	D₈×C₃₇	1	1	yes	characteristic
69	C₂×Q₈	—	2	56	no	—
70	C₂×Q₈	—	4	84	no	—
71	C₂²×C₄	—	4	28	no	—
72	C₂²×C₄	D₈×C₃₇	1	7	yes	—
73	C₂²×C₄	C₂×D₄×C₃₇	1	7	yes	—
74	Q₁₆	—	2	56	no	—
75	Q₁₆	C₂⁴×C₃₇	1	16	yes	—
76	C₂×C₈	C₂×D₄×C₃₇	1	21	yes	—
77	C₂×C₈	C₂⁴×C₃₇	1	7	yes	—
78	C₁₆	C₂⁴×C₃₇	1	8	yes	—
79	C₂³	Q₃₂×C₃₇	1	8	yes	direct product factor
80	C₂³	C₂×D₈×C₃₇	1	7	yes	—
81	Q₈	—	2	16	no	—
82	Q₈	—	4	56	no	—
83	C₂×C₄	—	4	56	no	—
84	C₂×C₄	C₂×D₈×C₃₇	1	21	yes	—
85	C₂×C₄	C₂²×D₄×C₃₇	1	7	yes	—
86	C₈	C₂²×D₄×C₃₇	1	7	yes	—
87	C₈	C₂⁵×C₃₇	1	1	yes	characteristic
88	C₂²	C₂×Q₃₂×C₃₇	1	28	yes	direct product factor
89	C₂²	C₂²×D₈×C₃₇	1	7	yes	—
90	C₄	—	4	16	no	—
91	C₄	C₂²×D₈×C₃₇	1	7	yes	—
92	C₄	C₂³×D₄×C₃₇	1	1	yes	characteristic
93	C₂	C₂²×Q₃₂×C₃₇	1	14	yes	direct product factor
94	C₂	C₂³×D₈×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃₇	D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₃₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ C₂³×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₃₂×C₃₇	C₂	direct (N × H)
8. C₂×Q₃₂×C₃₇	C₂²	direct (N × H)
18. Q₃₂×C₃₇	C₂³	direct (N × H)
31. C₂³×C₃₇	Q₃₂	direct (N × H)
40. C₂³×Q₃₂	C₃₇	direct (N × H)
41. C₂²×C₃₇	C₂×Q₃₂	direct (N × H)
47. C₂²×Q₃₂	C₂×C₃₇	direct (N × H)
49. C₂×C₃₇	C₂²×Q₃₂	direct (N × H)
55. C₂×Q₃₂	C₂²×C₃₇	direct (N × H)
57. C₃₇	C₂³×Q₃₂	direct (N × H)
66. Q₃₂	C₂³×C₃₇	direct (N × H)
79. C₂³	Q₃₂×C₃₇	direct (N × H)
88. C₂²	C₂×Q₃₂×C₃₇	direct (N × H)
93. C₂	C₂²×Q₃₂×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
C₂³×Q₁₆×C₃₇	2
C₂²×Q₃₂×C₃₇	2
C₂³×C₁₆×C₃₇	2
C₂³×Q₃₂	37

Subgroups — C23×Q32×C37 #22683880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×Q₃₂×C₃₇ #22683880