Subgroups — #26104176

Order 12032.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241599	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	—	2	1	no	—
20	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
21	C₈×C₄₇	C₃₂	1	2	yes	—
22	C₈×C₄₇	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₄₇	M₃₂	1	1	yes	characteristic
24	#7544	—	47	1	no	maximal, Sylow
25	C₂²×C₄₇	M₆₄	1	1	yes	characteristic
26	C₄×C₄₇	—	2	2	no	—
27	C₄×C₄₇	C₄⋊C₁₆	1	1	yes	characteristic
28	C₄×C₄₇	C₂×C₃₂	1	1	yes	characteristic
29	M₁₂₈	—	47	2	no	—
30	C₄×C₃₂	D₄₇	1	1	yes	characteristic
31	C₂×C₄₇	—	2	1	no	—
32	C₂×C₄₇	C₄⋊C₃₂	1	1	yes	characteristic
33	C₂×C₃₂	Dic₄₇	1	1	yes	characteristic
34	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
35	C₄×C₁₆	Dic₄₇	1	1	yes	characteristic
36	C₆₄	—	94	2	no	—
37	C₄₇	#7544	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₁₆	C₄₇⋊C₈	1	1	yes	characteristic
39	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
40	C₄×C₈	C₄₇⋊C₈	1	1	yes	characteristic
41	C₃₂	—	2	1	no	—
42	C₃₂	Dic₉₄	1	1	yes	characteristic
43	C₃₂	D₁₈₈	1	1	yes	characteristic
44	C₂×C₈	C₄₇⋊C₁₆	1	1	yes	characteristic
45	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
46	C₄²	C₄₇⋊C₁₆	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	#10625329	1	1	yes	characteristic
49	C₁₆	C₄⋊Dic₄₇	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
52	C₈	C₄₇⋊C₃₂	1	2	yes	—
53	C₈	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
54	C₈	#13093964	1	1	yes	characteristic
55	C₂²	#15688957	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	C₄⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
58	C₄	C₂×C₄₇⋊C₃₂	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	C₄⋊(C₄₇⋊C₃₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	D₁₈₈
commutator	C₄×C₄₇	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₄₇
Fitting	C₄×C₃₂×C₄₇	C₂
socle	C₂×C₄₇	C₄⋊C₃₂
prime radical	C₂²×C₄₇	M₆₄
cosocle	C₂×C₃₂×C₄₇	C₂²
hypercenter	C₄×C₃₂	D₄₇
hypocenter	C₄₇	#7544
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
37. C₄₇	#7544	semidirect

Maximal subgroups

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

subgroup	index
#20241599	2
C₄×C₃₂×C₄₇	2
#7544	47