Subgroups — #44548425

Order 32128.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632089	C₂	1	1	yes	maximal, characteristic
2	#30631508	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#22548707	C₄	1	2	yes	—
5	#22548655	—	2	1	no	—
6	#22548649	—	2	1	no	—
7	#22548647	—	2	1	no	—
8	M₃₂×C₂₅₁	—	2	1	no	—
9	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₅₁	—	2	2	no	—
11	Dic₂₅₁⋊C₄	—	4	2	no	—
12	C₂₅₁⋊C₁₆	—	4	1	no	—
13	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
14	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
15	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₅₁	—	4	1	no	—
17	C₂×Dic₂₅₁	—	4	2	no	—
18	C₂×C₄×C₂₅₁	—	2	1	no	—
19	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₂₅₁	SD₁₆	1	1	yes	characteristic
21	C₈×C₂₅₁	SD₁₆	1	1	yes	characteristic
22	C₈×C₂₅₁	Q₁₆	1	1	yes	characteristic
23	C₈×C₂₅₁	Q₁₆	1	1	yes	characteristic
24	C₂²×C₂₅₁	#100	1	1	yes	characteristic
25	C₄×C₂₅₁	—	4	1	no	—
26	C₄×C₂₅₁	#103	1	1	yes	characteristic
27	C₄×C₂₅₁	#103	1	1	yes	characteristic
28	Dic₂₅₁	—	8	2	no	—
29	C₂×C₂₅₁	—	2	1	no	—
30	C₂×C₂₅₁	#333	1	1	yes	characteristic
31	C₂₅₁	#1366	1	1	yes	characteristic, complemented, Sylow
32	#1366	—	251	1	no	maximal, Sylow
33	#499	—	251	1	no	—
34	#364	—	251	1	no	—
35	#364	D₂₅₁	1	1	yes	characteristic
36	#125	—	251	2	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	—	502	1	no	—
39	#107	—	502	1	no	—
40	#106	—	502	1	no	—
41	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
42	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
43	C₂×C₈	D₁₀₀₄	1	1	yes	characteristic
44	C₄⋊C₄	—	1004	2	no	—
45	C₄⋊C₄	—	502	2	no	—
46	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
47	C₁₆	—	4	1	no	—
48	C₁₆	—	1004	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	1004	2	no	—
51	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
52	C₈	#16860798	1	1	yes	characteristic
53	C₈	#16860799	1	1	yes	characteristic
54	C₈	#16860800	1	1	yes	characteristic
55	C₈	Dic₁₀₀₄	1	1	yes	characteristic
56	C₂²	#22548631	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	2008	2	no	—
59	C₄	#22548638	1	1	yes	characteristic
60	C₄	#22548640	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#30631435	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30631435
commutator	C₄²×C₂₅₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₅₁
Fitting	#364×C₂₅₁	C₂
socle	C₂×C₂₅₁	#333
prime radical	C₂²×C₂₅₁	#100
cosocle	C₄×C₈×C₂₅₁	C₂²
hypercenter	#364	D₂₅₁
hypocenter	C₂₅₁	#1366
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
31. C₂₅₁	#1366	semidirect

Maximal subgroups

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

subgroup	index
#30632089	2
#30631508	2
#364×C₂₅₁	2
#1366	251