Subgroups — C₂×C₄×C₃₇×C₈₁ #72201251

Order 23976.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₇×C₈₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₇×C₈₁	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₄×C₂₇×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₂×C₃₇×C₈₁	C₄	1	2	yes	direct product factor
5	C₂×C₃₇×C₈₁	C₂²	1	1	yes	characteristic
6	C₂²×C₂₇×C₃₇	C₂×C₃	1	1	yes	characteristic
7	C₄×C₂₇×C₃₇	C₂×C₃	1	2	yes	—
8	C₂×C₄×C₉×C₃₇	C₉	1	1	yes	characteristic
9	C₃₇×C₈₁	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₂₇×C₃₇	C₃×C₄	1	2	yes	—
11	C₂×C₂₇×C₃₇	C₂²×C₃	1	1	yes	characteristic
12	C₂²×C₉×C₃₇	C₂×C₉	1	1	yes	characteristic
13	C₄×C₉×C₃₇	C₂×C₉	1	2	yes	—
14	C₂₇×C₃₇	C₂×C₃×C₄	1	1	yes	characteristic
15	C₂×C₃×C₄×C₃₇	C₂₇	1	1	yes	characteristic
16	C₂×C₉×C₃₇	C₄×C₉	1	2	yes	—
17	C₂×C₉×C₃₇	C₂²×C₉	1	1	yes	characteristic
18	C₂×C₄×C₈₁	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₃×C₃₇	C₂×C₂₇	1	1	yes	characteristic
20	C₃×C₄×C₃₇	C₂×C₂₇	1	2	yes	—
21	C₉×C₃₇	C₂×C₄×C₉	1	1	yes	characteristic
22	C₂²×C₈₁	C₂×C₃₇	1	1	yes	characteristic
23	C₄×C₈₁	C₂×C₃₇	1	2	yes	direct product factor
24	C₂×C₄×C₃₇	C₈₁	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₃×C₃₇	C₄×C₂₇	1	2	yes	—
26	C₂×C₃×C₃₇	C₂²×C₂₇	1	1	yes	characteristic
27	C₂×C₄×C₂₇	C₃×C₃₇	1	1	yes	characteristic
28	C₂×C₈₁	C₄×C₃₇	1	2	yes	direct product factor
29	C₂×C₈₁	C₂²×C₃₇	1	1	yes	characteristic
30	C₂²×C₃₇	C₂×C₈₁	1	1	yes	characteristic
31	C₄×C₃₇	C₂×C₈₁	1	2	yes	direct product factor
32	C₃×C₃₇	C₂×C₄×C₂₇	1	1	yes	characteristic
33	C₂²×C₂₇	C₂×C₃×C₃₇	1	1	yes	characteristic
34	C₄×C₂₇	C₂×C₃×C₃₇	1	2	yes	—
35	C₈₁	C₂×C₄×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₃₇	C₄×C₈₁	1	2	yes	direct product factor
37	C₂×C₃₇	C₂²×C₈₁	1	1	yes	characteristic
38	C₂×C₄×C₉	C₉×C₃₇	1	1	yes	characteristic
39	C₂×C₂₇	C₃×C₄×C₃₇	1	2	yes	—
40	C₂×C₂₇	C₂²×C₃×C₃₇	1	1	yes	characteristic
41	C₃₇	C₂×C₄×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×C₉	C₂×C₉×C₃₇	1	1	yes	characteristic
43	C₄×C₉	C₂×C₉×C₃₇	1	2	yes	—
44	C₂₇	C₂×C₃×C₄×C₃₇	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂₇×C₃₇	1	1	yes	characteristic
46	C₂×C₉	C₄×C₉×C₃₇	1	2	yes	—
47	C₂×C₉	C₂²×C₉×C₃₇	1	1	yes	characteristic
48	C₂²×C₃	C₂×C₂₇×C₃₇	1	1	yes	characteristic
49	C₃×C₄	C₂×C₂₇×C₃₇	1	2	yes	—
50	C₉	C₂×C₄×C₉×C₃₇	1	1	yes	characteristic
51	C₂×C₄	C₃₇×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₃	C₄×C₂₇×C₃₇	1	2	yes	—
53	C₂×C₃	C₂²×C₂₇×C₃₇	1	1	yes	characteristic
54	C₂²	C₂×C₃₇×C₈₁	1	1	yes	characteristic
55	C₄	C₂×C₃₇×C₈₁	1	2	yes	direct product factor
56	C₃	C₂×C₄×C₂₇×C₃₇	1	1	yes	characteristic
57	C₂	C₄×C₃₇×C₈₁	1	2	yes	direct product factor
58	C₂	C₂²×C₃₇×C₈₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₂₇	C₂²×C₃×C₃₇
socle, prime radical	C₂²×C₃×C₃₇	C₂×C₂₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₃₇×C₈₁	C₂	direct (N × H)
4. C₂×C₃₇×C₈₁	C₄	direct (N × H)
9. C₃₇×C₈₁	C₂×C₄	direct (N × H)
18. C₂×C₄×C₈₁	C₃₇	direct (N × H)
23. C₄×C₈₁	C₂×C₃₇	direct (N × H)
24. C₂×C₄×C₃₇	C₈₁	direct (N × H)
28. C₂×C₈₁	C₄×C₃₇	direct (N × H)
31. C₄×C₃₇	C₂×C₈₁	direct (N × H)
35. C₈₁	C₂×C₄×C₃₇	direct (N × H)
36. C₂×C₃₇	C₄×C₈₁	direct (N × H)
41. C₃₇	C₂×C₄×C₈₁	direct (N × H)
51. C₂×C₄	C₃₇×C₈₁	direct (N × H)
55. C₄	C₂×C₃₇×C₈₁	direct (N × H)
57. C₂	C₄×C₃₇×C₈₁	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃₇×C₈₁	2
C₄×C₃₇×C₈₁	2
C₂×C₄×C₂₇×C₃₇	3
C₂×C₄×C₈₁	37

Subgroups — C2×C4×C37×C81 #72201251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₃₇×C₈₁ #72201251