Subgroups — C₄₁×C₄⋊C₆₄ #23783744

Order 10496.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×C₆₄	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₄₁	C₄	1	1	yes	characteristic
4	C₂×C₃₂×C₄₁	C₂²	1	1	yes	characteristic
5	C₄×C₁₆×C₄₁	C₄	1	1	yes	characteristic
6	C₄₁×C₆₄	—	2	2	no	—
7	C₂×C₁₆×C₄₁	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₄₁	C₈	1	1	yes	characteristic
10	C₃₂×C₄₁	—	2	1	no	—
11	C₃₂×C₄₁	D₄	1	1	yes	characteristic
12	C₃₂×C₄₁	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₄₁	C₁₆	1	1	yes	characteristic
14	C₂×C₈×C₄₁	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₄₁	C₁₆	1	1	yes	characteristic
16	C₁₆×C₄₁	—	2	1	no	—
17	C₁₆×C₄₁	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₄₁	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₄₁	C₃₂	1	1	yes	characteristic
20	C₂×C₄×C₄₁	C₃₂	1	1	yes	characteristic
21	C₂×C₄×C₄₁	C₂×C₁₆	1	1	yes	characteristic
22	C₈×C₄₁	—	2	1	no	—
23	C₈×C₄₁	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₄₁	M₃₂	1	1	yes	characteristic
25	C₄⋊C₆₄	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₄₁	C₂×C₃₂	1	1	yes	characteristic
27	C₄×C₄₁	—	2	1	no	—
28	C₄×C₄₁	C₆₄	1	2	yes	complemented
29	C₄×C₄₁	C₄⋊C₁₆	1	1	yes	characteristic
30	C₄×C₄₁	M₆₄	1	1	yes	characteristic
31	C₂×C₆₄	C₂×C₄₁	1	2	yes	—
32	C₄×C₃₂	C₂×C₄₁	1	1	yes	characteristic
33	C₂×C₄₁	C₄⋊C₃₂	1	1	yes	characteristic
34	C₂×C₄₁	C₂×C₆₄	1	1	yes	characteristic
35	C₂×C₄₁	M₁₂₈	1	1	yes	characteristic
36	C₂×C₃₂	C₄×C₄₁	1	1	yes	characteristic
37	C₂×C₃₂	C₂²×C₄₁	1	1	yes	characteristic
38	C₄×C₁₆	C₄×C₄₁	1	1	yes	characteristic
39	C₆₄	—	2	2	no	—
40	C₄₁	C₄⋊C₆₄	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₁₆	C₈×C₄₁	1	1	yes	characteristic
42	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
43	C₄×C₈	C₈×C₄₁	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	D₄×C₄₁	1	1	yes	characteristic
46	C₃₂	Q₈×C₄₁	1	1	yes	characteristic
47	C₂×C₈	C₁₆×C₄₁	1	1	yes	characteristic
48	C₂×C₈	C₂×C₈×C₄₁	1	1	yes	characteristic
49	C₄²	C₁₆×C₄₁	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₄⋊C₄×C₄₁	1	1	yes	characteristic
52	C₁₆	M₁₆×C₄₁	1	1	yes	characteristic
53	C₂×C₄	C₃₂×C₄₁	1	1	yes	characteristic
54	C₂×C₄	C₃₂×C₄₁	1	1	yes	characteristic
55	C₂×C₄	C₂×C₁₆×C₄₁	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	C₄⋊C₈×C₄₁	1	1	yes	characteristic
58	C₈	M₃₂×C₄₁	1	1	yes	characteristic
59	C₂²	C₂×C₃₂×C₄₁	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	C₄₁×C₆₄	1	2	yes	complemented
62	C₄	C₄₁×C₄⋊C₁₆	1	1	yes	characteristic
63	C₄	C₄₁×M₆₄	1	1	yes	characteristic
64	C₂	C₄₁×C₄⋊C₃₂	1	1	yes	characteristic
65	C₂	C₂×C₄₁×C₆₄	1	1	yes	characteristic
66	C₂	C₄₁×M₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂×C₄₁	C₂²
commutator	C₂	C₂×C₄₁×C₆₄
Frattini, cosocle	C₂×C₃₂	C₂²×C₄₁
socle, prime radical	C₂²×C₄₁	C₂×C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₄⋊C₆₄	C₄₁	direct (N × H)
28. C₄×C₄₁	C₆₄	semidirect
40. C₄₁	C₄⋊C₆₄	direct (N × H)
61. C₄	C₄₁×C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×C₆₄	2
C₄×C₃₂×C₄₁	2
C₄⋊C₆₄	41

Subgroups — C41×C4⋊C64 #23783744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×C₄⋊C₆₄ #23783744