Subgroups — #35204028

Order 20288.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778826	C₂	1	2	yes	maximal, complemented
2	#23778808	C₂	1	6	yes	maximal, complemented
3	#23778802	C₂	1	3	yes	maximal, complemented
4	C₂×D₂₅₃₆	C₂	1	3	yes	maximal, complemented
5	#131×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099575	—	2	2	no	—
7	#19099574	C₂²	1	6	yes	complemented
8	C₂×D₁₂₆₈	C₂²	1	6	yes	complemented
9	D₄×D₃₁₇	—	2	6	no	—
10	Dic₁₂₆₈	C₂²	1	1	yes	characteristic, complemented
11	#19099563	C₂²	1	6	yes	complemented
12	D₂₅₃₆	C₂²	1	3	yes	complemented
13	D₂₅₃₆	C₂²	1	6	yes	complemented
14	#41×C₃₁₇	C₂²	1	1	yes	characteristic
15	M₁₆×C₃₁₇	C₂²	1	3	yes	complemented
16	C₂×C₈×C₃₁₇	C₂²	1	3	yes	complemented
17	C₂²×D₃₁₇	—	2	6	no	—
18	Dic₆₃₄	C₂³	1	2	yes	—
19	D₁₂₆₈	—	2	6	no	—
20	D₁₂₆₈	C₂³	1	6	yes	—
21	C₂²⋊D₃₁₇	—	2	6	no	—
22	C₄×D₃₁₇	—	2	6	no	—
23	Q₈×C₃₁₇	D₄	1	1	yes	characteristic
24	D₄×C₃₁₇	D₄	1	3	yes	—
25	C₂×C₄×C₃₁₇	C₂³	1	3	yes	—
26	C₈×C₃₁₇	C₂³	1	1	yes	characteristic
27	C₈×C₃₁₇	C₂³	1	3	yes	—
28	C₂²×C₃₁₇	C₂×D₄	1	3	yes	—
29	C₂×D₃₁₇	—	2	6	no	—
30	C₂×D₃₁₇	—	8	6	no	—
31	C₄×C₃₁₇	C₂×D₄	1	3	yes	—
32	C₄×C₃₁₇	C₂⁴	1	1	yes	characteristic
33	Dic₃₁₇	—	2	2	no	—
34	C₂×C₃₁₇	—	2	3	no	—
35	C₂×C₃₁₇	C₂²×D₄	1	1	yes	characteristic
36	D₃₁₇	—	4	6	no	—
37	C₃₁₇	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	317	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	317	2	no	—
40	AGL(1,ℤ/8)	—	317	6	no	—
41	#135	—	317	3	no	—
42	C₂×D₈	—	317	3	no	—
43	#131	D₃₁₇	1	1	yes	characteristic, complemented
44	#41	—	317	6	no	—
45	#41	—	634	2	no	—
46	#41	C₂×D₃₁₇	1	1	yes	characteristic
47	C₂×D₄	—	317	6	no	—
48	C₂×D₄	—	634	6	no	—
49	Q₁₆	—	317	1	no	—
50	SD₁₆	—	317	6	no	—
51	D₈	—	317	3	no	—
52	D₈	—	317	6	no	—
53	M₁₆	C₂×D₃₁₇	1	3	yes	complemented
54	C₂×C₈	C₂×D₃₁₇	1	3	yes	complemented
55	C₂³	—	634	6	no	—
56	Q₈	—	317	2	no	—
57	Q₈	D₁₂₆₈	1	1	yes	characteristic
58	D₄	—	317	6	no	—
59	D₄	—	634	6	no	—
60	D₄	—	634	6	no	—
61	D₄	D₁₂₆₈	1	3	yes	—
62	C₂×C₄	—	634	6	no	—
63	C₂×C₄	C₂²×D₃₁₇	1	3	yes	—
64	C₈	C₂²×D₃₁₇	1	1	yes	characteristic
65	C₈	C₂²×D₃₁₇	1	3	yes	—
66	C₂²	—	634	6	no	—
67	C₂²	—	2536	6	no	—
68	C₂²	C₂×D₁₂₆₈	1	3	yes	—
69	C₄	—	634	2	no	—
70	C₄	C₂×D₁₂₆₈	1	3	yes	—
71	C₄	C₂³×D₃₁₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	1268	6	no	—
74	C₂	C₂²×D₁₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₂₆₈
commutator, cosocle	C₄×C₃₁₇	C₂⁴
Frattini	C₄	C₂³×D₃₁₇
Fitting	#131×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂²×D₄
hypercenter	#131	D₃₁₇
hypocenter	C₃₁₇	#576
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #23778826	C₂	semidirect
2. #23778808	C₂	semidirect
3. #23778802	C₂	semidirect
4. C₂×D₂₅₃₆	C₂	semidirect
5. #131×C₃₁₇	C₂	semidirect
7. #19099574	C₂²	semidirect
8. C₂×D₁₂₆₈	C₂²	semidirect
10. Dic₁₂₆₈	C₂²	semidirect
11. #19099563	C₂²	semidirect
12. D₂₅₃₆	C₂²	semidirect
13. D₂₅₃₆	C₂²	semidirect
15. M₁₆×C₃₁₇	C₂²	semidirect
16. C₂×C₈×C₃₁₇	C₂²	semidirect
37. C₃₁₇	#576	semidirect
43. #131	D₃₁₇	semidirect
53. M₁₆	C₂×D₃₁₇	semidirect
54. C₂×C₈	C₂×D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778826	2
#23778808	2
#23778802	2
C₂×D₂₅₃₆	2
#131×C₃₁₇	2
#576	317