Subgroups — #35203655

Order 20288.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778802	C₂	1	2	yes	maximal, complemented
2	Dic₂₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778725	C₂	1	2	yes	maximal, complemented
4	D₅₀₇₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099574	—	2	2	no	—
7	Dic₁₂₆₈	C₂²	1	2	yes	—
8	#19099563	—	2	2	no	—
9	D₂₅₃₆	C₂²	1	2	yes	—
10	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
11	C₁₆×C₃₁₇	C₂²	1	1	yes	characteristic
12	C₁₆×C₃₁₇	C₂²	1	1	yes	characteristic
13	Dic₆₃₄	—	2	2	no	—
14	D₁₂₆₈	—	2	2	no	—
15	C₂²⋊D₃₁₇	—	4	2	no	—
16	C₄×D₃₁₇	—	4	2	no	—
17	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
18	C₈×C₃₁₇	D₄	1	1	yes	characteristic
19	C₈×C₃₁₇	C₂³	1	1	yes	characteristic
20	C₂²×C₃₁₇	D₈	1	1	yes	characteristic
21	C₂×D₃₁₇	—	4	2	no	—
22	C₄×C₃₁₇	D₈	1	1	yes	characteristic
23	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
24	Dic₃₁₇	—	4	2	no	—
25	C₂×C₃₁₇	—	2	1	no	—
26	C₂×C₃₁₇	C₂×D₈	1	1	yes	characteristic
27	D₃₁₇	—	8	2	no	—
28	C₃₁₇	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	317	1	no	maximal, Sylow
30	#135	—	317	2	no	—
31	Q₃₂	—	317	1	no	—
32	SD₃₂	—	317	2	no	—
33	D₁₆	—	317	1	no	—
34	C₂×C₁₆	D₃₁₇	1	1	yes	characteristic, complemented
35	#41	—	634	2	no	—
36	Q₁₆	—	317	2	no	—
37	SD₁₆	—	634	2	no	—
38	D₈	—	317	2	no	—
39	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
40	C₁₆	C₂×D₃₁₇	1	1	yes	characteristic
41	C₁₆	C₂×D₃₁₇	1	1	yes	characteristic
42	Q₈	—	634	2	no	—
43	D₄	—	634	2	no	—
44	D₄	—	1268	2	no	—
45	C₂×C₄	—	1268	2	no	—
46	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
47	C₈	D₁₂₆₈	1	1	yes	characteristic
48	C₈	C₂²×D₃₁₇	1	1	yes	characteristic
49	C₂²	—	1268	2	no	—
50	C₂²	D₂₅₃₆	1	1	yes	characteristic
51	C₄	—	1268	2	no	—
52	C₄	D₂₅₃₆	1	1	yes	characteristic
53	C₄	C₂×D₁₂₆₈	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	2536	2	no	—
56	C₂	C₂×D₂₅₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₅₃₆
commutator, cosocle	C₈×C₃₁₇	C₂³
Frattini	C₈	C₂²×D₃₁₇
Fitting	C₂×C₁₆×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂×D₈
hypercenter	C₂×C₁₆	D₃₁₇
hypocenter	C₃₁₇	#508
LCS	G ▷ C₈×C₃₁₇ ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #23778802	C₂	semidirect
2. Dic₂₅₃₆	C₂	semidirect
3. #23778725	C₂	semidirect
4. D₅₀₇₂	C₂	semidirect
5. C₂×C₁₆×C₃₁₇	C₂	semidirect
28. C₃₁₇	#508	semidirect
34. C₂×C₁₆	D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778802	2
Dic₂₅₃₆	2
#23778725	2
D₅₀₇₂	2
C₂×C₁₆×C₃₁₇	2
#508	317