Subgroups — #30664167

Order 16768.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553142	C₂	1	1	yes	maximal, characteristic
2	#22553140	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#16862182	—	2	1	no	—
5	#16862180	—	2	1	no	—
6	C₂×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₃₂×C₁₃₁	C₄	1	2	yes	complemented
8	#14523497	—	4	1	no	—
9	C₄⋊Dic₁₃₁	—	4	1	no	—
10	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
11	C₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₃₁	Q₈	1	1	yes	characteristic
13	C₂×Dic₁₃₁	—	8	1	no	—
14	C₁₃₁⋊C₈	—	8	1	no	—
15	C₂×C₄×C₁₃₁	D₈	1	1	yes	characteristic
16	C₈×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₃₁	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₃₁	SD₃₂	1	1	yes	characteristic
19	C₄×C₁₃₁	#107	1	1	yes	characteristic
20	C₄×C₁₃₁	SD₃₂	1	1	yes	characteristic
21	Dic₁₃₁	—	16	1	no	—
22	C₂×C₁₃₁	—	2	1	no	—
23	C₂×C₁₃₁	#367	1	1	yes	characteristic
24	C₁₃₁	#1426	1	1	yes	characteristic, complemented, Sylow
25	#1426	—	131	1	no	maximal, Sylow
26	M₆₄	D₁₃₁	1	1	yes	characteristic
27	#368	—	131	1	no	—
28	#367	—	131	1	no	—
29	C₂×C₁₆	C₂×D₁₃₁	1	1	yes	characteristic
30	#108	—	262	1	no	—
31	#107	—	262	1	no	—
32	C₃₂	Dic₁₃₁	1	2	yes	complemented
33	M₁₆	—	524	1	no	—
34	C₂×C₈	D₅₂₄	1	1	yes	characteristic
35	C₄⋊C₄	—	524	1	no	—
36	C₁₆	C₂×Dic₁₃₁	1	1	yes	characteristic
37	C₁₆	Dic₂₆₂	1	1	yes	characteristic
38	C₂×C₄	—	1048	1	no	—
39	C₂×C₄	D₁₀₄₈	1	1	yes	characteristic
40	C₈	—	1048	1	no	—
41	C₈	C₄⋊Dic₁₃₁	1	1	yes	characteristic
42	C₈	Dic₅₂₄	1	1	yes	characteristic
43	C₂²	#16862191	1	1	yes	characteristic
44	C₄	—	2096	1	no	—
45	C₄	#16862180	1	1	yes	characteristic
46	C₄	#16862191	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#22553140	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22553140
commutator	C₁₆×C₁₃₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₃₁
Fitting	M₆₄×C₁₃₁	C₂
socle	C₂×C₁₃₁	#367
prime radical	C₂²×C₁₃₁	SD₃₂
cosocle	C₂×C₁₆×C₁₃₁	C₂²
hypercenter	M₆₄	D₁₃₁
hypocenter	C₁₃₁	#1426
LCS	G ▷ C₁₆×C₁₃₁ ▷ C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₁₃₁	C₄	semidirect
24. C₁₃₁	#1426	semidirect
32. C₃₂	Dic₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553142	2
#22553140	2
M₆₄×C₁₃₁	2
#1426	131