Subgroups — #30664149

Order 16768.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553744	C₂	1	1	yes	maximal, characteristic
2	#22553140	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862268	—	2	1	no	—
5	Dic₁₀₄₈	C₄	1	1	yes	characteristic, complemented
6	#16862191	—	2	1	no	—
7	D₂₀₉₆	C₄	1	1	yes	characteristic, complemented
8	#16862180	—	2	1	no	—
9	C₂×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₃₁	—	2	1	no	—
11	#14523513	—	4	1	no	—
12	Dic₅₂₄	—	2	1	no	—
13	#14523502	—	4	1	no	—
14	D₁₀₄₈	—	2	1	no	—
15	C₄⋊Dic₁₃₁	—	4	1	no	—
16	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
17	C₁₆×C₁₃₁	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₃₁	D₄	1	1	yes	characteristic
19	Dic₂₆₂	—	4	1	no	—
20	D₅₂₄	—	4	1	no	—
21	C₂²⋊D₁₃₁	—	8	1	no	—
22	C₂×Dic₁₃₁	—	8	1	no	—
23	C₄×D₁₃₁	—	8	1	no	—
24	C₂×C₄×C₁₃₁	D₈	1	1	yes	characteristic
25	C₈×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₃₁	D₁₆	1	1	yes	characteristic
28	C₂×D₁₃₁	—	8	1	no	—
29	C₄×C₁₃₁	#102	1	1	yes	characteristic
30	C₄×C₁₃₁	SD₃₂	1	1	yes	characteristic
31	Dic₁₃₁	—	8	1	no	—
32	Dic₁₃₁	—	16	1	no	—
33	C₂×C₁₃₁	—	2	1	no	—
34	C₂×C₁₃₁	#357	1	1	yes	characteristic
35	D₁₃₁	—	16	1	no	—
36	C₁₃₁	#1418	1	1	yes	characteristic, complemented, Sylow
37	#1418	—	131	1	no	maximal, Sylow
38	#508	—	131	1	no	—
39	M₆₄	D₁₃₁	1	1	yes	characteristic, complemented
40	#367	—	131	1	no	—
41	#135	—	262	1	no	—
42	Q₃₂	—	131	1	no	—
43	SD₃₂	—	262	1	no	—
44	D₁₆	—	131	1	no	—
45	C₂×C₁₆	C₂×D₁₃₁	1	1	yes	characteristic
46	#107	—	262	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	524	1	no	—
49	Q₁₆	—	262	1	no	—
50	SD₁₆	—	524	1	no	—
51	D₈	—	262	1	no	—
52	C₂×C₈	D₅₂₄	1	1	yes	characteristic
53	C₄⋊C₄	—	524	1	no	—
54	C₁₆	C₄×D₁₃₁	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₃₁	1	1	yes	characteristic
56	Q₈	—	524	1	no	—
57	D₄	—	524	1	no	—
58	D₄	—	1048	1	no	—
59	C₂×C₄	—	1048	1	no	—
60	C₂×C₄	—	1048	1	no	—
61	C₂×C₄	D₁₀₄₈	1	1	yes	characteristic
62	C₈	(C₂×D₁₃₁)⋊C₄	1	1	yes	characteristic
63	C₈	#14523502	1	1	yes	characteristic
64	C₂²	—	1048	1	no	—
65	C₂²	D₂₀₉₆	1	1	yes	characteristic
66	C₄	—	1048	1	no	—
67	C₄	—	2096	1	no	—
68	C₄	#16862168	1	1	yes	characteristic
69	C₄	#16862191	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	2096	1	no	—
72	C₂	#22553114	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22553114
commutator	C₁₆×C₁₃₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₃₁
Fitting	M₆₄×C₁₃₁	C₂
socle	C₂×C₁₃₁	#357
prime radical	#22553744	C₂
cosocle	C₂×C₁₆×C₁₃₁	C₂²
hypercenter	M₆₄	D₁₃₁
hypocenter	C₁₃₁	#1418
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
2. #22553140	C₂	semidirect
3. M₆₄×C₁₃₁	C₂	semidirect
5. Dic₁₀₄₈	C₄	semidirect
7. D₂₀₉₆	C₄	semidirect
36. C₁₃₁	#1418	semidirect
39. M₆₄	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553744	2
#22553140	2
M₆₄×C₁₃₁	2
#1418	131