Subgroups — #44548548

Order 32128.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632119	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₅₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#368×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#22548737	—	2	1	no	—
5	Dic₂₀₀₈	C₄	1	1	yes	characteristic
6	#22548660	—	2	1	no	—
7	D₄₀₁₆	C₄	1	1	yes	characteristic
8	C₂₅₁⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
10	#108×C₂₅₁	—	2	1	no	—
11	#16860810	—	4	1	no	—
12	Dic₁₀₀₄	—	2	1	no	—
13	#16860799	—	4	1	no	—
14	D₂₀₀₈	—	2	1	no	—
15	M₁₆×C₂₅₁	—	4	1	no	—
16	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
17	C₁₆×C₂₅₁	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₂₅₁	D₄	1	1	yes	characteristic
19	Dic₅₀₂	—	4	1	no	—
20	D₁₀₀₄	—	4	1	no	—
21	C₂²⋊D₂₅₁	—	8	1	no	—
22	C₄×D₂₅₁	—	8	1	no	—
23	C₂×C₄×C₂₅₁	D₈	1	1	yes	characteristic
24	C₈×C₂₅₁	—	8	1	no	—
25	C₈×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₅₁	SD₁₆	1	1	yes	characteristic
27	C₂²×C₂₅₁	SD₃₂	1	1	yes	characteristic
28	C₂×D₂₅₁	—	8	1	no	—
29	C₄×C₂₅₁	#102	1	1	yes	characteristic
30	C₄×C₂₅₁	D₁₆	1	1	yes	characteristic
31	Dic₂₅₁	—	8	1	no	—
32	C₂×C₂₅₁	—	2	1	no	—
33	C₂×C₂₅₁	#357	1	1	yes	characteristic
34	D₂₅₁	—	16	1	no	—
35	C₂₅₁	#1417	1	1	yes	characteristic, complemented, Sylow
36	#1417	—	251	1	no	maximal, Sylow
37	#508	—	251	1	no	—
38	C₂×C₃₂	—	251	1	no	—
39	#368	D₂₅₁	1	1	yes	characteristic, complemented
40	#135	—	502	1	no	—
41	Q₃₂	—	251	1	no	—
42	SD₃₂	—	502	1	no	—
43	D₁₆	—	251	1	no	—
44	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
45	#108	—	2	1	no	—
46	C₃₂	—	502	1	no	—
47	#41	—	1004	1	no	—
48	Q₁₆	—	502	1	no	—
49	SD₁₆	—	1004	1	no	—
50	D₈	—	502	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
53	C₁₆	C₄×D₂₅₁	1	1	yes	characteristic
54	C₁₆	D₁₀₀₄	1	1	yes	characteristic
55	Q₈	—	1004	1	no	—
56	D₄	—	2008	1	no	—
57	D₄	—	1004	1	no	—
58	C₂×C₄	—	2008	1	no	—
59	C₂×C₄	#16860795	1	1	yes	characteristic
60	C₈	—	8	1	no	—
61	C₈	(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
62	C₈	#16860797	1	1	yes	characteristic
63	C₂²	—	2008	1	no	—
64	C₂²	#22548658	1	1	yes	characteristic
65	C₄	—	2008	1	no	—
66	C₄	#22548635	1	1	yes	characteristic
67	C₄	#22548656	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4016	1	no	—
70	C₂	#30631487	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#22548656
commutator	C₁₆×C₂₅₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₅₁
Fitting	#368×C₂₅₁	C₂
socle	C₂×C₂₅₁	#357
prime radical	#30632119	C₂
cosocle	C₂×C₁₆×C₂₅₁	C₂²
hypercenter	#368	D₂₅₁
hypocenter	C₂₅₁	#1417
LCS	G ▷ C₁₆×C₂₅₁ ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₅₁⋊C₃₂	C₂	semidirect
3. #368×C₂₅₁	C₂	semidirect
35. C₂₅₁	#1417	semidirect
39. #368	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632119	2
C₂×C₂₅₁⋊C₃₂	2
#368×C₂₅₁	2
#1417	251