Subgroups — #44552149

Order 32128.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30631725	C₂	1	2	yes	maximal
2	#30631515	C₂	1	2	yes	maximal, complemented
3	#30631513	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	#22548674	—	2	2	no	—
6	#22548649	C₂²	1	4	yes	—
7	#22548647	—	2	2	no	—
8	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₂₅₁	C₄	1	4	yes	complemented
10	C₂×C₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₅₁	—	2	4	no	—
13	C₂²⋊Dic₂₅₁	—	4	2	no	—
14	C₄×Dic₂₅₁	—	4	2	no	—
15	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
16	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
18	C₂×C₈×C₂₅₁	Q₈	1	2	yes	—
19	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
20	C₁₆×C₂₅₁	C₂×C₄	1	2	yes	—
21	C₁₆×C₂₅₁	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₅₁	—	4	4	no	—
23	C₂³×C₂₅₁	D₈	1	1	yes	characteristic
24	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₂₅₁	D₈	1	1	yes	characteristic
26	C₂×C₄×C₂₅₁	Q₁₆	1	2	yes	—
27	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
28	C₈×C₂₅₁	C₄⋊C₄	1	2	yes	—
29	C₈×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
30	C₈×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₂₅₁	—	2	2	no	—
32	C₂²×C₂₅₁	#107	1	2	yes	—
33	C₂²×C₂₅₁	C₂×D₈	1	1	yes	characteristic
34	C₄×C₂₅₁	#107	1	2	yes	—
35	C₄×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₄×C₂₅₁	C₂×Q₁₆	1	1	yes	characteristic
37	Dic₂₅₁	—	8	4	no	—
38	C₂×C₂₅₁	—	2	2	no	—
39	C₂×C₂₅₁	C₂×#107	1	1	yes	characteristic
40	C₂×C₂₅₁	#508	1	2	yes	—
41	C₂₅₁	#2158	1	1	yes	characteristic, complemented, Sylow
42	#2158	—	251	1	no	maximal, Sylow
43	C₂²×C₁₆	D₂₅₁	1	1	yes	characteristic
44	#427	—	251	2	no	—
45	#367	—	251	2	no	—
46	#366	—	251	2	no	—
47	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
48	#117	—	502	2	no	—
49	C₂×C₁₆	Dic₂₅₁	1	4	yes	complemented
50	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
51	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
52	#107	—	251	4	no	—
53	#106	—	502	2	no	—
54	C₂²×C₄	D₁₀₀₄	1	1	yes	characteristic
55	C₂×C₈	C₂×Dic₂₅₁	1	2	yes	—
56	C₂×C₈	D₁₀₀₄	1	1	yes	characteristic
57	C₂×C₈	Dic₅₀₂	1	2	yes	—
58	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
59	C₄⋊C₄	—	502	4	no	—
60	C₂²⋊C₄	—	1004	2	no	—
61	C₄²	—	1004	2	no	—
62	C₁₆	C₂×Dic₂₅₁	1	2	yes	—
63	C₁₆	C₂×Dic₂₅₁	1	2	yes	—
64	C₂³	D₂₀₀₈	1	1	yes	characteristic
65	C₂×C₄	—	1004	4	no	—
66	C₂×C₄	C₄⋊Dic₂₅₁	1	2	yes	—
67	C₂×C₄	D₂₀₀₈	1	1	yes	characteristic
68	C₂×C₄	Dic₁₀₀₄	1	2	yes	—
69	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
70	C₈	C₄⋊Dic₂₅₁	1	2	yes	—
71	C₈	C₂²×Dic₂₅₁	1	1	yes	characteristic
72	C₈	C₂×Dic₅₀₂	1	1	yes	characteristic
73	C₂²	—	2	2	no	—
74	C₂²	#22548649	1	2	yes	—
75	C₂²	C₂×D₂₀₀₈	1	1	yes	characteristic
76	C₄	—	2008	4	no	—
77	C₄	#22548649	1	2	yes	—
78	C₄	C₂×C₄⋊Dic₂₅₁	1	1	yes	characteristic
79	C₄	C₂×Dic₁₀₀₄	1	1	yes	characteristic
80	C₂	—	2	2	no	—
81	C₂	C₂×#22548649	1	1	yes	characteristic
82	C₂	#30632119	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₂₀₀₈
commutator	C₈×C₂₅₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	C₂²×C₁₆×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×D₈
prime radical	C₂³×C₂₅₁	D₈
cosocle	C₂×C₈×C₂₅₁	C₂³
hypercenter	C₂²×C₁₆	D₂₅₁
hypocenter	C₂₅₁	#2158
LCS	G ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #30631515	C₂	semidirect
3. #30631513	C₂	semidirect
9. C₂×C₁₆×C₂₅₁	C₄	semidirect
41. C₂₅₁	#2158	semidirect
49. C₂×C₁₆	Dic₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30631725	2
#30631515	2
#30631513	2
C₂²×C₁₆×C₂₅₁	2
#2158	251