Subgroups — #30668236

Order 16768.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553717	C₂	1	2	yes	maximal
2	#22553138	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#16862246	—	2	2	no	—
5	#16862180	—	2	2	no	—
6	#16862180	C₂²	1	4	yes	—
7	C₂×C₁₆×C₁₃₁	C₂²	1	2	yes	—
8	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₆₂	—	4	2	no	—
10	C₄⋊Dic₁₃₁	—	2	4	no	—
11	C₄⋊Dic₁₃₁	—	4	2	no	—
12	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
14	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
15	C₁₆×C₁₃₁	Q₈	1	4	yes	complemented
16	Dic₂₆₂	—	8	4	no	—
17	C₂×Dic₁₃₁	—	4	4	no	—
18	C₂×C₄×C₁₃₁	D₈	1	1	yes	characteristic
19	C₂×C₄×C₁₃₁	D₈	1	1	yes	characteristic
20	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
21	C₈×C₁₃₁	Q₁₆	1	2	yes	—
22	C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
23	C₂²×C₁₃₁	C₂×D₈	1	1	yes	characteristic
24	C₄×C₁₃₁	D₁₆	1	2	yes	—
25	C₄×C₁₃₁	Q₃₂	1	2	yes	—
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	C₂×D₁₆	1	1	yes	characteristic
31	C₂×C₁₃₁	C₂×Q₃₂	1	1	yes	characteristic
32	C₁₃₁	#2252	1	1	yes	characteristic, complemented, Sylow
33	#2252	—	131	1	no	maximal, Sylow
34	#500	—	131	2	no	—
35	#366	—	131	4	no	—
36	C₄×C₁₆	D₁₃₁	1	1	yes	characteristic
37	C₄⋊Q₈	—	262	2	no	—
38	C₂×C₁₆	C₂×D₁₃₁	1	2	yes	—
39	#107	—	262	2	no	—
40	#107	—	131	4	no	—
41	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
42	C₂×Q₈	—	524	2	no	—
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	2	no	—
47	C₄²	D₅₂₄	1	1	yes	characteristic
48	C₁₆	Dic₂₆₂	1	4	yes	complemented
49	Q₈	—	1048	4	no	—
50	C₂×C₄	—	524	4	no	—
51	C₂×C₄	D₁₀₄₈	1	1	yes	characteristic
52	C₂×C₄	D₁₀₄₈	1	1	yes	characteristic
53	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
54	C₈	Dic₅₂₄	1	2	yes	—
55	C₈	C₂×Dic₂₆₂	1	2	yes	—
56	C₂²	C₂×D₁₀₄₈	1	1	yes	characteristic
57	C₄	—	1048	4	no	—
58	C₄	D₂₀₉₆	1	2	yes	—
59	C₄	Dic₁₀₄₈	1	2	yes	—
60	C₄	#16862246	1	1	yes	characteristic
61	C₄	C₂×Dic₅₂₄	1	1	yes	characteristic
62	C₂	#22553717	1	1	yes	characteristic
63	C₂	C₂×D₂₀₉₆	1	1	yes	characteristic
64	C₂	C₂×Dic₁₀₄₈	1	1	yes	characteristic

	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	#2252
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
15. C₁₆×C₁₃₁	Q₈	semidirect
32. C₁₃₁	#2252	semidirect
48. C₁₆	Dic₂₆₂	semidirect

Maximal subgroups

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

subgroup	index
#22553717	2
#22553138	2
C₄×C₁₆×C₁₃₁	2
#2252	131