Subgroups — #23778786

Order 10144.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19099574	C₂	1	1	yes	maximal, characteristic
2	#19099558	C₂	1	1	yes	maximal, characteristic, complemented
3	#19099557	C₂	1	2	yes	maximal, complemented
4	C₈×D₃₁₇	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₆₃₄	C₄	1	1	yes	characteristic
7	D₁₂₆₈	C₄	1	1	yes	characteristic
8	C₂²⋊D₃₁₇	C₄	1	2	yes	—
9	C₄×D₃₁₇	C₂²	1	2	yes	—
10	#14579261	C₂²	1	2	yes	—
11	C₂×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
12	C₈×C₃₁₇	C₂²	1	1	yes	characteristic
13	C₈×C₃₁₇	C₂²	1	1	yes	characteristic
14	C₂²×C₃₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×D₃₁₇	C₂×C₄	1	2	yes	—
16	C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
17	C₄×C₃₁₇	C₂³	1	1	yes	characteristic
18	Dic₃₁₇	C₂×C₄	1	2	yes	—
19	C₂×C₃₁₇	—	2	1	no	—
20	C₂×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
21	D₃₁₇	—	2	2	no	—
22	C₃₁₇	#131	1	1	yes	characteristic, complemented, Sylow
23	#131	—	317	1	no	maximal, Sylow
24	#41	—	317	1	no	—
25	M₁₆	—	317	1	no	—
26	M₁₆	—	317	2	no	—
27	C₂×C₈	—	317	2	no	—
28	C₂×C₈	D₃₁₇	1	1	yes	characteristic, complemented
29	Q₈	—	317	1	no	—
30	D₄	—	317	2	no	—
31	D₄	—	317	1	no	—
32	C₂×C₄	—	317	2	no	—
33	C₂×C₄	C₂×D₃₁₇	1	1	yes	characteristic
34	C₈	—	317	2	no	—
35	C₈	C₂×D₃₁₇	1	1	yes	characteristic
36	C₈	C₂×D₃₁₇	1	1	yes	characteristic
37	C₂²	—	317	2	no	—
38	C₂²	C₄×D₃₁₇	1	1	yes	characteristic
39	C₄	—	317	2	no	—
40	C₄	C₄×D₃₁₇	1	1	yes	characteristic
41	C₄	C₂²×D₃₁₇	1	1	yes	characteristic
42	C₂	—	2	1	no	—
43	C₂	—	634	2	no	—
44	C₂	C₂×C₄×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₃₁₇
commutator, socle	C₂×C₃₁₇	C₂²×C₄
Frattini	C₄	C₂²×D₃₁₇
Fitting	C₂×C₈×C₃₁₇	C₂
prime radical	#19099574	C₂
cosocle	C₄×C₃₁₇	C₂³
hypercenter	C₂×C₈	D₃₁₇
hypocenter	C₃₁₇	#131
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. #19099558	C₂	semidirect
3. #19099557	C₂	semidirect
4. C₈×D₃₁₇	C₂	semidirect
5. C₂×C₈×C₃₁₇	C₂	semidirect
22. C₃₁₇	#131	semidirect
28. C₂×C₈	D₃₁₇	semidirect