Subgroups — C₄⋊(C₄₇⋊C₆₄) #26104174

Order 12032.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇⋊C₆₄	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇⋊C₆₄	—	2	2	no	—
4	C₂×C₃₂×C₄₇	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₄₇	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₄₇	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₄₇	C₈	1	1	yes	characteristic
10	C₃₂×C₄₇	—	2	1	no	—
11	C₃₂×C₄₇	D₄	1	1	yes	characteristic
12	C₃₂×C₄₇	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₄₇	C₁₆	1	1	yes	characteristic
14	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₄₇	C₁₆	1	1	yes	characteristic
16	C₁₆×C₄₇	—	2	1	no	—
17	C₁₆×C₄₇	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₄₇	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₄₇	C₃₂	1	1	yes	characteristic
20	C₂×C₄×C₄₇	C₃₂	1	1	yes	characteristic
21	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
22	C₈×C₄₇	—	2	1	no	—
23	C₈×C₄₇	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₄₇	M₃₂	1	1	yes	characteristic
25	C₄⋊C₆₄	—	47	1	no	maximal, Sylow
26	C₂²×C₄₇	C₂×C₃₂	1	1	yes	characteristic
27	C₄×C₄₇	—	2	1	no	—
28	C₄×C₄₇	C₆₄	1	2	yes	complemented
29	C₄×C₄₇	C₄⋊C₁₆	1	1	yes	characteristic
30	C₄×C₄₇	M₆₄	1	1	yes	characteristic
31	C₂×C₆₄	—	47	2	no	—
32	C₄×C₃₂	D₄₇	1	1	yes	characteristic
33	C₂×C₄₇	C₄⋊C₃₂	1	1	yes	characteristic
34	C₂×C₄₇	C₂×C₆₄	1	1	yes	characteristic
35	C₂×C₄₇	M₁₂₈	1	1	yes	characteristic
36	C₂×C₃₂	Dic₄₇	1	1	yes	characteristic
37	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
38	C₄×C₁₆	Dic₄₇	1	1	yes	characteristic
39	C₆₄	—	94	2	no	—
40	C₄₇	C₄⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₁₆	C₄₇⋊C₈	1	1	yes	characteristic
42	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
43	C₄×C₈	C₄₇⋊C₈	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	Dic₉₄	1	1	yes	characteristic
46	C₃₂	D₁₈₈	1	1	yes	characteristic
47	C₂×C₈	C₄₇⋊C₁₆	1	1	yes	characteristic
48	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
49	C₄²	C₄₇⋊C₁₆	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	#10625329	1	1	yes	characteristic
52	C₁₆	C₄⋊Dic₄₇	1	1	yes	characteristic
53	C₂×C₄	C₄₇⋊C₃₂	1	1	yes	characteristic
54	C₂×C₄	C₄₇⋊C₃₂	1	1	yes	characteristic
55	C₂×C₄	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
58	C₈	#13093964	1	1	yes	characteristic
59	C₂²	C₂×C₄₇⋊C₃₂	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	C₄₇⋊C₆₄	1	2	yes	complemented
62	C₄	C₄⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
63	C₄	#15688957	1	1	yes	characteristic
64	C₂	C₄⋊(C₄₇⋊C₃₂)	1	1	yes	characteristic
65	C₂	C₂×C₄₇⋊C₆₄	1	1	yes	characteristic
66	C₂	#20241599	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₄₇
commutator	C₂×C₄₇	C₂×C₆₄
Fitting	C₄×C₃₂×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₂×C₃₂
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	C₄×C₃₂	D₄₇
hypocenter	C₄₇	C₄⋊C₆₄
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₃₂ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
28. C₄×C₄₇	C₆₄	semidirect
40. C₄₇	C₄⋊C₆₄	semidirect
61. C₄	C₄₇⋊C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₇⋊C₆₄	2
C₄×C₃₂×C₄₇	2
C₄⋊C₆₄	47

Subgroups — C4⋊(C47⋊C64) #26104174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊(C₄₇⋊C₆₄) #26104174