Subgroups — #44552611

Order 32128.

132 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	#30632067	C₂	1	1	yes	maximal, characteristic
3	#30631492	C₂	1	4	yes	maximal
4	#346×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	#22548707	—	2	1	no	—
9	#22548649	C₂²	1	2	yes	—
10	#22548647	—	2	1	no	—
11	C₂×C₁₆×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	—	2	2	no	—
17	C₄⋊Dic₂₅₁	—	4	2	no	—
18	Dic₂₅₁⋊C₄	—	4	2	no	—
19	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
20	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
21	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
22	C₁₆×C₂₅₁	—	2	2	no	—
23	Dic₅₀₂	—	4	2	no	—
24	Dic₅₀₂	—	4	2	no	—
25	C₂×Dic₂₅₁	—	4	2	no	—
26	C₂×Dic₂₅₁	—	4	2	no	—
27	C₂×C₄×C₂₅₁	D₈	1	2	yes	—
28	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
29	C₈×C₂₅₁	SD₁₆	1	2	yes	—
30	C₈×C₂₅₁	#41	1	2	yes	—
31	C₂²×C₂₅₁	C₂×D₈	1	1	yes	characteristic
32	C₄×C₂₅₁	—	2	2	no	—
33	C₄×C₂₅₁	#124	1	1	yes	characteristic
34	C₄×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
35	Dic₂₅₁	—	8	2	no	—
36	Dic₂₅₁	—	8	2	no	—
37	C₂×C₂₅₁	#486	1	1	yes	characteristic
38	C₂×C₂₅₁	#510	1	2	yes	—
39	C₂₅₁	#2245	1	1	yes	characteristic, complemented, Sylow
40	#2245	—	251	1	no	maximal, Sylow
41	#499	—	251	1	no	—
42	#494	—	251	1	no	—
43	#358	—	251	4	no	—
44	#346	D₂₅₁	1	1	yes	characteristic
45	C₂×Q₁₆	—	502	1	no	—
46	C₂×Q₁₆	—	251	2	no	—
47	C₄⋊Q₈	—	502	1	no	—
48	#125	—	502	1	no	—
49	C₂×C₁₆	C₂×D₂₅₁	1	2	yes	—
50	#107	—	251	2	no	—
51	#106	—	502	1	no	—
52	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
53	C₂×Q₈	—	502	2	no	—
54	Q₁₆	—	1004	2	no	—
55	Q₁₆	—	502	4	no	—
56	C₂×C₈	D₁₀₀₄	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
58	C₄⋊C₄	—	1004	2	no	—
59	C₄⋊C₄	—	1004	2	no	—
60	C₄⋊C₄	—	502	2	no	—
61	C₄²	D₁₀₀₄	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	Q₈	—	1004	2	no	—
64	Q₈	—	1004	2	no	—
65	C₂×C₄	—	1004	2	no	—
66	C₂×C₄	—	1004	2	no	—
67	C₂×C₄	D₂₀₀₈	1	2	yes	—
68	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
69	C₈	#16860799	1	2	yes	—
70	C₈	#16860810	1	2	yes	—
71	C₂²	C₂×D₂₀₀₈	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	2008	2	no	—
74	C₄	—	2008	2	no	—
75	C₄	#22548703	1	1	yes	characteristic
76	C₄	C₂×#16860799	1	1	yes	characteristic
77	C₂	#30632034	1	1	yes	characteristic
78	C₂	#30632130	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₀₀₈
commutator, cosocle	C₂×C₈×C₂₅₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	#346×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	C₂×D₈
hypercenter	#346	D₂₅₁
hypocenter	C₂₅₁	#2245
LCS	G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
39. C₂₅₁	#2245	semidirect

Maximal subgroups

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

subgroup	index
#30632089	2
#30632067	2
#30631492	2
#346×C₂₅₁	2
#2245	251