Subgroups — C₂×Q₈²×C₃₇ #17992295

Order 4736.

1112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₃₇	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈×C₃₇	C₂	1	9	yes	maximal
3	C₂×C₄×Q₈×C₃₇	C₂	1	6	yes	maximal
4	C₂²×Q₈×C₃₇	C₂²	1	6	yes	—
5	C₂²×Q₈×C₃₇	C₂²	1	2	yes	—
6	C₄⋊Q₈×C₃₇	C₂²	1	72	yes	—
7	C₄×Q₈×C₃₇	C₂²	1	48	yes	—
8	C₂×C₄⋊C₄×C₃₇	C₂²	1	18	yes	—
9	C₂×C₄²×C₃₇	C₂²	1	9	yes	—
10	C₂×Q₈×C₃₇	—	4	12	no	—
11	C₂×Q₈×C₃₇	Q₈	1	16	yes	direct product factor
12	C₂×Q₈×C₃₇	C₂³	1	24	yes	—
13	C₂×Q₈×C₃₇	C₂³	1	8	yes	—
14	C₂²×C₄×C₃₇	C₂³	1	9	yes	—
15	C₂²×C₄×C₃₇	C₂³	1	6	yes	—
16	C₄⋊C₄×C₃₇	C₂³	1	72	yes	—
17	C₄²×C₃₇	C₂³	1	36	yes	—
18	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
19	Q₈×C₃₇	—	4	24	no	—
20	Q₈×C₃₇	C₂×Q₈	1	32	yes	direct product factor
21	C₂×C₄×C₃₇	—	2	18	no	—
22	C₂×C₄×C₃₇	C₂×Q₈	1	24	yes	—
23	C₂×C₄×C₃₇	C₂⁴	1	18	yes	—
24	C₂×C₄×C₃₇	C₂⁴	1	12	yes	—
25	C₂²×C₃₇	C₂²×Q₈	1	4	yes	—
26	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
27	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
28	C₄×C₃₇	—	2	18	no	—
29	C₄×C₃₇	C₂²×Q₈	1	24	yes	—
30	C₂×Q₈²	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₃₇	Q₈²	1	4	yes	direct product factor
32	C₂×C₃₇	C₂³×Q₈	1	2	yes	—
33	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
34	Q₈²	C₂×C₃₇	1	16	yes	direct product factor
35	C₂×C₄⋊Q₈	C₂×C₃₇	1	9	yes	—
36	C₂×C₄×Q₈	C₂×C₃₇	1	6	yes	—
37	C₃₇	C₂×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
38	C₂²×Q₈	C₂²×C₃₇	1	6	yes	—
39	C₂²×Q₈	C₂²×C₃₇	1	2	yes	—
40	C₄⋊Q₈	C₂²×C₃₇	1	72	yes	—
41	C₄×Q₈	C₂²×C₃₇	1	48	yes	—
42	C₂×C₄⋊C₄	C₂²×C₃₇	1	18	yes	—
43	C₂×C₄²	C₂²×C₃₇	1	9	yes	—
44	C₂×Q₈	—	4	12	no	—
45	C₂×Q₈	Q₈×C₃₇	1	16	yes	direct product factor
46	C₂×Q₈	C₂³×C₃₇	1	24	yes	—
47	C₂×Q₈	C₂³×C₃₇	1	8	yes	—
48	C₂²×C₄	C₂³×C₃₇	1	9	yes	—
49	C₂²×C₄	C₂³×C₃₇	1	6	yes	—
50	C₄⋊C₄	C₂³×C₃₇	1	72	yes	—
51	C₄²	C₂³×C₃₇	1	36	yes	—
52	C₂³	C₂⁴×C₃₇	1	1	yes	characteristic
53	Q₈	—	4	24	no	—
54	Q₈	C₂×Q₈×C₃₇	1	32	yes	direct product factor
55	C₂×C₄	—	2	18	no	—
56	C₂×C₄	C₂×Q₈×C₃₇	1	24	yes	—
57	C₂×C₄	C₂⁴×C₃₇	1	18	yes	—
58	C₂×C₄	C₂⁴×C₃₇	1	12	yes	—
59	C₂²	C₂²×Q₈×C₃₇	1	4	yes	—
60	C₂²	H(5,𝔽₂)×C₃₇	1	2	yes	—
61	C₂²	C₂⁵×C₃₇	1	1	yes	characteristic
62	C₄	—	2	18	no	—
63	C₄	C₂²×Q₈×C₃₇	1	24	yes	—
64	C₂	Q₈²×C₃₇	1	4	yes	direct product factor
65	C₂	C₂³×Q₈×C₃₇	1	2	yes	—
66	C₂	C₂×H(5,𝔽₂)×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈²×C₃₇	C₂	direct (N × H)
11. C₂×Q₈×C₃₇	Q₈	direct (N × H)
20. Q₈×C₃₇	C₂×Q₈	direct (N × H)
30. C₂×Q₈²	C₃₇	direct (N × H)
31. C₂×C₃₇	Q₈²	direct (N × H)
34. Q₈²	C₂×C₃₇	direct (N × H)
37. C₃₇	C₂×Q₈²	direct (N × H)
45. C₂×Q₈	Q₈×C₃₇	direct (N × H)
54. Q₈	C₂×Q₈×C₃₇	direct (N × H)
64. C₂	Q₈²×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
Q₈²×C₃₇	2
C₂×C₄⋊Q₈×C₃₇	2
C₂×C₄×Q₈×C₃₇	2
C₂×Q₈²	37

Subgroups — C2×Q82×C37 #17992295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈²×C₃₇ #17992295