Subgroups — #30668240

Order 16768.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553714	C₂	1	2	yes	maximal
2	#22553140	C₂	1	2	yes	maximal
3	#22553138	C₂	1	2	yes	maximal
4	C₄×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#16862238	—	2	2	no	—
6	#16862180	C₂²	1	4	yes	—
7	#16862178	—	2	2	no	—
8	C₂×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₃₁	—	2	4	no	—
12	Dic₁₃₁⋊C₄	—	4	4	no	—
13	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
15	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
16	C₁₆×C₁₃₁	Q₈	1	2	yes	—
17	C₁₆×C₁₃₁	Q₈	1	2	yes	—
18	C₂×Dic₁₃₁	—	4	4	no	—
19	C₂×C₄×C₁₃₁	D₈	1	2	yes	—
20	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
21	C₈×C₁₃₁	Q₁₆	1	2	yes	—
22	C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
23	C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₁₃₁	C₂×D₈	1	1	yes	characteristic
25	C₄×C₁₃₁	—	2	2	no	—
26	C₄×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₁₃₁	—	8	4	no	—
29	C₂×C₁₃₁	#500	1	1	yes	characteristic
30	C₂×C₁₃₁	#508	1	2	yes	—
31	C₁₃₁	#2253	1	1	yes	characteristic, complemented, Sylow
32	#2253	—	131	1	no	maximal, Sylow
33	#499	—	131	2	no	—
34	#367	—	131	2	no	—
35	#366	—	131	2	no	—
36	C₄×C₁₆	D₁₃₁	1	1	yes	characteristic
37	#125	—	262	2	no	—
38	C₂×C₁₆	C₂×D₁₃₁	1	1	yes	characteristic
39	C₂×C₁₆	C₂×D₁₃₁	1	1	yes	characteristic
40	#107	—	131	4	no	—
41	#106	—	262	2	no	—
42	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
43	C₂×C₈	D₅₂₄	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
45	C₄⋊C₄	—	262	4	no	—
46	C₄⋊C₄	—	524	4	no	—
47	C₄²	D₅₂₄	1	1	yes	characteristic
48	C₁₆	Dic₂₆₂	1	2	yes	—
49	C₁₆	Dic₂₆₂	1	2	yes	—
50	C₂×C₄	—	524	4	no	—
51	C₂×C₄	D₁₀₄₈	1	2	yes	—
52	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
53	C₈	Dic₅₂₄	1	2	yes	—
54	C₈	C₂×Dic₂₆₂	1	1	yes	characteristic
55	C₈	C₂×Dic₂₆₂	1	1	yes	characteristic
56	C₂²	C₂×D₁₀₄₈	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	1048	4	no	—
59	C₄	#16862246	1	1	yes	characteristic
60	C₄	C₂×Dic₅₂₄	1	1	yes	characteristic
61	C₂	#22553717	1	1	yes	characteristic
62	C₂	#22553744	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₄₈
commutator, cosocle	C₂×C₈×C₁₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₁
Fitting	C₄×C₁₆×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	C₂×D₈
hypercenter	C₄×C₁₆	D₁₃₁
hypocenter	C₁₃₁	#2253
LCS	G ▷ C₂×C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
31. C₁₃₁	#2253	semidirect

Maximal subgroups

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

subgroup	index
#22553714	2
#22553140	2
#22553138	2
C₄×C₁₆×C₁₃₁	2
#2253	131