Subgroups — #23755741

Order 9952.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19097315	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₆₂₂	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₂₄₄	C₂	1	2	yes	maximal, complemented
4	#19097304	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₃₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₆₂₂	—	2	1	no	—
7	Dic₆₂₂	C₂²	1	2	yes	—
8	Dic₆₂₂	C₂²	1	1	yes	characteristic
9	D₁₂₄₄	C₂²	1	1	yes	characteristic
10	C₂²⋊D₃₁₁	—	2	1	no	—
11	C₂×Dic₃₁₁	—	2	1	no	—
12	C₄×D₃₁₁	—	2	1	no	—
13	C₂×C₄×C₃₁₁	C₂²	1	1	yes	characteristic
14	C₈×C₃₁₁	C₂²	1	2	yes	—
15	C₂²×C₃₁₁	D₄	1	1	yes	characteristic
16	C₂×D₃₁₁	—	2	1	no	—
17	C₄×C₃₁₁	D₄	1	1	yes	characteristic
18	C₄×C₃₁₁	C₂³	1	1	yes	characteristic
19	Dic₃₁₁	—	2	1	no	—
20	Dic₃₁₁	—	2	2	no	—
21	C₂×C₃₁₁	—	2	1	no	—
22	C₂×C₃₁₁	C₂×D₄	1	1	yes	characteristic
23	D₃₁₁	—	4	1	no	—
24	C₃₁₁	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	311	1	no	maximal, Sylow
26	#41	—	311	1	no	—
27	C₂×Q₈	—	311	1	no	—
28	Q₁₆	—	311	2	no	—
29	SD₁₆	—	311	2	no	—
30	M₁₆	D₃₁₁	1	1	yes	characteristic, complemented
31	Q₈	—	311	2	no	—
32	Q₈	—	311	1	no	—
33	Q₈	—	622	1	no	—
34	D₄	—	311	1	no	—
35	D₄	—	622	1	no	—
36	C₂×C₄	—	622	1	no	—
37	C₂×C₄	—	622	1	no	—
38	C₂×C₄	C₂×D₃₁₁	1	1	yes	characteristic
39	C₈	C₂×D₃₁₁	1	2	yes	—
40	C₂²	—	622	1	no	—
41	C₂²	D₁₂₄₄	1	1	yes	characteristic
42	C₄	—	622	1	no	—
43	C₄	—	622	2	no	—
44	C₄	D₁₂₄₄	1	1	yes	characteristic
45	C₄	C₂²×D₃₁₁	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	1244	1	no	—
48	C₂	C₂×D₁₂₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₂₄₄
commutator, cosocle	C₄×C₃₁₁	C₂³
Frattini	C₄	C₂²×D₃₁₁
Fitting	M₁₆×C₃₁₁	C₂
socle	C₂×C₃₁₁	C₂×D₄
prime radical	#19097315	C₂
hypercenter	M₁₆	D₃₁₁
hypocenter	C₃₁₁	#137
LCS	G ▷ C₄×C₃₁₁ ▷ C₂×C₃₁₁ ▷ C₃₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₆₂₂	C₂	semidirect
3. Dic₁₂₄₄	C₂	semidirect
4. #19097304	C₂	semidirect
5. M₁₆×C₃₁₁	C₂	semidirect
24. C₃₁₁	#137	semidirect
30. M₁₆	D₃₁₁	semidirect