Subgroups — #30632129

Order 16064.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22548738	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#22548661	C₂	1	1	yes	maximal, characteristic, complemented
4	#22548660	C₂	1	1	yes	maximal, characteristic, complemented
5	#22548658	C₂	1	1	yes	maximal, characteristic, complemented
6	#22548654	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₃₂×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#16860811	—	2	1	no	—
9	Q₈×D₂₅₁	—	2	1	no	—
10	Dic₁₀₀₄	C₂²	1	1	yes	characteristic
11	#16860800	—	2	1	no	—
12	#16860797	—	2	1	no	—
13	D₂₀₀₈	C₂²	1	1	yes	characteristic
14	C₈×D₂₅₁	C₂²	1	1	yes	characteristic
15	C₂₅₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
17	Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
18	C₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
19	Dic₅₀₂	—	2	1	no	—
20	Dic₅₀₂	—	4	1	no	—
21	D₁₀₀₄	—	2	1	no	—
22	D₁₀₀₄	—	4	1	no	—
23	C₄×D₂₅₁	—	4	1	no	—
24	C₄×D₂₅₁	—	4	1	no	—
25	C₄×D₂₅₁	D₄	1	1	yes	characteristic
26	C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
27	Q₈×C₂₅₁	—	2	1	no	—
28	Q₈×C₂₅₁	—	2	1	no	—
29	C₈×C₂₅₁	C₂³	1	1	yes	characteristic
30	C₂×D₂₅₁	—	4	1	no	—
31	C₂×D₂₅₁	D₈	1	1	yes	characteristic
32	C₄×C₂₅₁	—	4	1	no	—
33	C₄×C₂₅₁	—	4	1	no	—
34	C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
35	Dic₂₅₁	—	4	1	no	—
36	Dic₂₅₁	D₈	1	1	yes	characteristic
37	C₂×C₂₅₁	C₂×D₈	1	1	yes	characteristic
38	D₂₅₁	—	2	1	no	—
39	D₂₅₁	—	8	1	no	—
40	C₂₅₁	#510	1	1	yes	characteristic, complemented, Sylow
41	#510	—	251	1	no	maximal, Sylow
42	#135	—	251	1	no	—
43	C₂×Q₁₆	—	251	1	no	—
44	Q₃₂	—	251	1	no	—
45	Q₃₂	D₂₅₁	1	1	yes	characteristic, complemented
46	SD₃₂	—	251	1	no	—
47	SD₃₂	—	251	1	no	—
48	M₃₂	—	251	1	no	—
49	#41	—	502	1	no	—
50	C₂×Q₈	—	502	1	no	—
51	Q₁₆	—	502	1	no	—
52	Q₁₆	—	251	1	no	—
53	Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
54	Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
55	SD₁₆	—	502	1	no	—
56	D₈	—	251	1	no	—
57	C₂×C₈	—	251	1	no	—
58	C₁₆	—	251	1	no	—
59	C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	2	1	no	—
62	Q₈	—	1004	1	no	—
63	Q₈	—	502	1	no	—
64	D₄	—	1004	1	no	—
65	D₄	—	502	1	no	—
66	C₂×C₄	—	1004	1	no	—
67	C₂×C₄	—	1004	1	no	—
68	C₂×C₄	—	251	1	no	—
69	C₈	—	251	1	no	—
70	C₈	C₂²×D₂₅₁	1	1	yes	characteristic
71	C₂²	—	1004	1	no	—
72	C₂²	—	251	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	1004	1	no	—
76	C₄	—	251	1	no	—
77	C₄	D₄×D₂₅₁	1	1	yes	characteristic
78	C₂	—	2008	1	no	—
79	C₂	—	502	1	no	—
80	C₂	D₈×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₂₅₁
commutator, cosocle	C₈×C₂₅₁	C₂³
Frattini	C₈	C₂²×D₂₅₁
Fitting	Q₃₂×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₂×D₈
prime radical	#22548738	C₂
hypercenter	Q₃₂	D₂₅₁
hypocenter	C₂₅₁	#510
LCS	G ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₂₅₁	C₂	semidirect
3. #22548661	C₂	semidirect
4. #22548660	C₂	semidirect
5. #22548658	C₂	semidirect
6. #22548654	C₂	semidirect
7. Q₃₂×C₂₅₁	C₂	semidirect
40. C₂₅₁	#510	semidirect
45. Q₃₂	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#22548738	2
Q₁₆×D₂₅₁	2
#22548661	2
#22548660	2
#22548658	2
#22548654	2
Q₃₂×C₂₅₁	2
#510	251