Subgroups — #23778802

Order 10144.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19099574	C₂	1	2	yes	maximal, complemented
2	Dic₁₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#19099563	C₂	1	2	yes	maximal, complemented
4	D₂₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₆₃₄	C₂²	1	2	yes	—
7	D₁₂₆₈	C₂²	1	2	yes	—
8	C₂²⋊D₃₁₇	—	2	2	no	—
9	C₄×D₃₁₇	—	2	2	no	—
10	C₂×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
11	C₈×C₃₁₇	C₂²	1	1	yes	characteristic
12	C₈×C₃₁₇	C₂²	1	1	yes	characteristic
13	C₂²×C₃₁₇	D₄	1	1	yes	characteristic
14	C₂×D₃₁₇	—	2	2	no	—
15	C₄×C₃₁₇	D₄	1	1	yes	characteristic
16	C₄×C₃₁₇	C₂³	1	1	yes	characteristic
17	Dic₃₁₇	—	2	2	no	—
18	C₂×C₃₁₇	—	2	1	no	—
19	C₂×C₃₁₇	C₂×D₄	1	1	yes	characteristic
20	D₃₁₇	—	4	2	no	—
21	C₃₁₇	#135	1	1	yes	characteristic, complemented, Sylow
22	#135	—	317	1	no	maximal, Sylow
23	#41	—	317	2	no	—
24	Q₁₆	—	317	1	no	—
25	SD₁₆	—	317	2	no	—
26	D₈	—	317	1	no	—
27	C₂×C₈	D₃₁₇	1	1	yes	characteristic, complemented
28	Q₈	—	317	2	no	—
29	D₄	—	317	2	no	—
30	D₄	—	634	2	no	—
31	C₂×C₄	—	634	2	no	—
32	C₂×C₄	C₂×D₃₁₇	1	1	yes	characteristic
33	C₈	C₂×D₃₁₇	1	1	yes	characteristic
34	C₈	C₂×D₃₁₇	1	1	yes	characteristic
35	C₂²	—	634	2	no	—
36	C₂²	D₁₂₆₈	1	1	yes	characteristic
37	C₄	—	634	2	no	—
38	C₄	D₁₂₆₈	1	1	yes	characteristic
39	C₄	C₂²×D₃₁₇	1	1	yes	characteristic
40	C₂	—	2	1	no	—
41	C₂	—	1268	2	no	—
42	C₂	C₂×D₁₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₂₆₈
commutator, cosocle	C₄×C₃₁₇	C₂³
Frattini	C₄	C₂²×D₃₁₇
Fitting	C₂×C₈×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂×D₄
hypercenter	C₂×C₈	D₃₁₇
hypocenter	C₃₁₇	#135
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #19099574	C₂	semidirect
2. Dic₁₂₆₈	C₂	semidirect
3. #19099563	C₂	semidirect
4. D₂₅₃₆	C₂	semidirect
5. C₂×C₈×C₃₁₇	C₂	semidirect
21. C₃₁₇	#135	semidirect
27. C₂×C₈	D₃₁₇	semidirect