Subgroups — #44552594

Order 32128.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632067	C₂	1	1	yes	maximal, characteristic
2	#30631490	C₂	1	4	yes	maximal
3	C₄×C₂₅₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#500×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₀₀₄	—	2	1	no	—
6	C₂×Dic₁₀₀₄	C₂²	1	2	yes	—
7	#22548715	—	2	1	no	—
8	C₂×C₂₅₁⋊C₁₆	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₂₅₁	—	2	1	no	—
10	#107×C₂₅₁	—	2	1	no	—
11	#107×C₂₅₁	C₂²	1	2	yes	—
12	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₅₀₂	—	2	2	no	—
14	Dic₁₀₀₄	—	2	4	no	—
15	Dic₁₀₀₄	—	4	2	no	—
16	C₄⋊Dic₂₅₁	—	4	1	no	—
17	C₄⋊Dic₂₅₁	—	4	1	no	—
18	C₂₅₁⋊C₁₆	—	2	2	no	—
19	C₂×Q₈×C₂₅₁	—	4	1	no	—
20	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₂₅₁	—	2	2	no	—
23	C₄⋊C₄×C₂₅₁	—	4	1	no	—
24	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
25	Dic₅₀₂	—	4	2	no	—
26	Dic₅₀₂	—	4	2	no	—
27	C₂×Dic₂₅₁	—	4	2	no	—
28	Q₈×C₂₅₁	—	8	2	no	—
29	C₂×C₄×C₂₅₁	—	4	2	no	—
30	C₂×C₄×C₂₅₁	D₈	1	1	yes	characteristic
31	C₂×C₄×C₂₅₁	D₈	1	1	yes	characteristic
32	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
33	C₈×C₂₅₁	SD₁₆	1	2	yes	—
34	C₈×C₂₅₁	#41	1	2	yes	—
35	C₂²×C₂₅₁	C₂×D₈	1	1	yes	characteristic
36	C₄×C₂₅₁	—	8	2	no	—
37	C₄×C₂₅₁	SD₃₂	1	2	yes	—
38	C₄×C₂₅₁	Q₃₂	1	2	yes	—
39	C₄×C₂₅₁	#124	1	1	yes	characteristic
40	C₄×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
41	Dic₂₅₁	—	8	2	no	—
42	C₂×C₂₅₁	#486	1	1	yes	characteristic
43	C₂×C₂₅₁	C₂×SD₃₂	1	1	yes	characteristic
44	C₂×C₂₅₁	C₂×Q₃₂	1	1	yes	characteristic
45	C₂₅₁	#2241	1	1	yes	characteristic, complemented, Sylow
46	#2241	—	251	1	no	maximal, Sylow
47	#500	D₂₅₁	1	1	yes	characteristic
48	#494	—	251	1	no	—
49	#358	—	251	4	no	—
50	C₄×C₁₆	—	251	1	no	—
51	C₂×Q₁₆	—	502	1	no	—
52	C₂×Q₁₆	—	251	2	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	502	1	no	—
55	C₂×C₁₆	—	251	2	no	—
56	#107	—	2	1	no	—
57	#107	C₂×D₂₅₁	1	2	yes	—
58	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	502	2	no	—
61	Q₁₆	—	1004	2	no	—
62	Q₁₆	—	502	4	no	—
63	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	1004	1	no	—
68	C₄⋊C₄	—	1004	1	no	—
69	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
70	C₁₆	—	502	2	no	—
71	Q₈	—	8	2	no	—
72	Q₈	—	1004	2	no	—
73	Q₈	—	1004	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	1004	2	no	—
76	C₂×C₄	#16860795	1	1	yes	characteristic
77	C₂×C₄	#16860795	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
79	C₈	#16860797	1	2	yes	—
80	C₈	#16860811	1	2	yes	—
81	C₂²	C₂×#16860795	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	2008	2	no	—
84	C₄	#22548659	1	2	yes	—
85	C₄	#22548661	1	2	yes	—
86	C₄	#22548704	1	1	yes	characteristic
87	C₄	C₂×#16860797	1	1	yes	characteristic
88	C₂	#30632035	1	1	yes	characteristic
89	C₂	C₂×#22548659	1	1	yes	characteristic
90	C₂	C₂×#22548661	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#16860795
commutator, cosocle	C₂×C₈×C₂₅₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	#500×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	C₂×D₈
hypercenter	#500	D₂₅₁
hypocenter	C₂₅₁	#2241
LCS	G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
45. C₂₅₁	#2241	semidirect

Maximal subgroups

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

subgroup	index
#30632067	2
#30631490	2
C₄×C₂₅₁⋊C₁₆	2
#500×C₂₅₁	2
#2241	251