Subgroups — C₂×C₄₁×M₆₄ #19106037

Order 5248.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×M₆₄	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃₂×C₄₁	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×C₄₁	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₄₁	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₄₁	C₄	1	2	yes	—
7	C₂×C₁₆×C₄₁	C₂²	1	2	yes	—
8	C₂×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
9	C₃₂×C₄₁	C₂²	1	4	yes	complemented
10	C₂²×C₄×C₄₁	C₈	1	1	yes	characteristic
11	C₂×C₈×C₄₁	C₈	1	1	yes	characteristic
12	C₂×C₈×C₄₁	C₈	1	2	yes	—
13	C₂×C₈×C₄₁	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₄₁	C₂×C₄	1	2	yes	—
16	C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₄₁	C₂³	1	1	yes	characteristic
18	C₂³×C₄₁	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₄₁	C₁₆	1	2	yes	—
20	C₂×C₄×C₄₁	C₁₆	1	1	yes	characteristic
21	C₂×C₄×C₄₁	C₂×C₈	1	2	yes	—
22	C₂×C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
23	C₈×C₄₁	C₂×C₈	1	2	yes	—
24	C₈×C₄₁	C₂×C₈	1	1	yes	characteristic
25	C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄₁	—	2	2	no	—
27	C₂²×C₄₁	C₂×C₁₆	1	2	yes	—
28	C₂²×C₄₁	C₂×C₁₆	1	1	yes	characteristic
29	C₄×C₄₁	C₂×C₁₆	1	1	yes	characteristic
30	C₄×C₄₁	C₂×C₁₆	1	2	yes	—
31	C₄×C₄₁	C₂²×C₈	1	1	yes	characteristic
32	C₂×M₆₄	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₄₁	—	2	2	no	—
34	C₂×C₄₁	M₆₄	1	2	yes	direct product factor
35	C₂×C₄₁	C₂²×C₁₆	1	1	yes	characteristic
36	C₂²×C₁₆	C₂×C₄₁	1	1	yes	characteristic
37	M₆₄	C₂×C₄₁	1	4	yes	direct product factor
38	C₂×C₃₂	C₂×C₄₁	1	2	yes	complemented
39	C₄₁	C₂×M₆₄	1	1	yes	characteristic, direct product factor, Sylow
40	C₂²×C₈	C₄×C₄₁	1	1	yes	characteristic
41	C₂×C₁₆	C₄×C₄₁	1	2	yes	—
42	C₂×C₁₆	C₄×C₄₁	1	1	yes	characteristic
43	C₂×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
44	C₂×C₁₆	C₂²×C₄₁	1	2	yes	—
45	C₃₂	C₂²×C₄₁	1	4	yes	complemented
46	C₂²×C₄	C₈×C₄₁	1	1	yes	characteristic
47	C₂×C₈	C₈×C₄₁	1	2	yes	—
48	C₂×C₈	C₈×C₄₁	1	1	yes	characteristic
49	C₂×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
50	C₂×C₈	C₂×C₄×C₄₁	1	2	yes	—
51	C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
52	C₁₆	C₂×C₄×C₄₁	1	2	yes	—
53	C₁₆	C₂³×C₄₁	1	1	yes	characteristic
54	C₂³	C₁₆×C₄₁	1	1	yes	characteristic
55	C₂×C₄	C₁₆×C₄₁	1	2	yes	—
56	C₂×C₄	C₁₆×C₄₁	1	1	yes	characteristic
57	C₂×C₄	C₂×C₈×C₄₁	1	2	yes	—
58	C₂×C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
59	C₈	C₂×C₈×C₄₁	1	1	yes	characteristic
60	C₈	C₂×C₈×C₄₁	1	2	yes	—
61	C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	C₂×C₁₆×C₄₁	1	2	yes	—
64	C₂²	C₂×C₁₆×C₄₁	1	1	yes	characteristic
65	C₄	C₂×C₁₆×C₄₁	1	1	yes	characteristic
66	C₄	C₂×C₁₆×C₄₁	1	2	yes	—
67	C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	C₄₁×M₆₄	1	2	yes	direct product factor
70	C₂	C₂²×C₁₆×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆×C₄₁	C₂²
commutator	C₂	C₂²×C₁₆×C₄₁
Frattini, cosocle	C₁₆	C₂³×C₄₁
socle	C₂²×C₄₁	C₂×C₁₆
prime radical	C₂³×C₄₁	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×M₆₄	C₂	direct (N × H)
3. C₂×C₃₂×C₄₁	C₂	semidirect
9. C₃₂×C₄₁	C₂²	semidirect
32. C₂×M₆₄	C₄₁	direct (N × H)
34. C₂×C₄₁	M₆₄	direct (N × H)
37. M₆₄	C₂×C₄₁	direct (N × H)
38. C₂×C₃₂	C₂×C₄₁	semidirect
39. C₄₁	C₂×M₆₄	direct (N × H)
45. C₃₂	C₂²×C₄₁	semidirect
69. C₂	C₄₁×M₆₄	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×C₁₆×C₄₁	2
C₄₁×M₆₄	2
C₂×C₃₂×C₄₁	2
C₂×M₆₄	41

Subgroups — C2×C41×M64 #19106037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×M₆₄ #19106037