Subgroups — #35203657

Order 20288.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778799	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778725	C₂	1	1	yes	maximal, characteristic, complemented
4	#23778724	C₂	1	1	yes	maximal, characteristic, complemented
5	#23778721	C₂	1	1	yes	maximal, characteristic, complemented
6	#23778719	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₆×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#19099573	—	2	1	no	—
9	D₄×D₃₁₇	—	2	1	no	—
10	Dic₁₂₆₈	C₂²	1	1	yes	characteristic, complemented
11	#19099562	—	2	1	no	—
12	D₂₅₃₆	C₂²	1	1	yes	characteristic
13	#19099559	—	2	1	no	—
14	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
15	#19099549	C₂²	1	1	yes	characteristic, complemented
16	D₈×C₃₁₇	C₂²	1	1	yes	characteristic
17	D₈×C₃₁₇	C₂²	1	1	yes	characteristic, complemented
18	C₁₆×C₃₁₇	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₃₁₇	—	4	1	no	—
20	Dic₆₃₄	—	2	1	no	—
21	D₁₂₆₈	—	2	1	no	—
22	C₂²⋊D₃₁₇	—	4	1	no	—
23	C₂²⋊D₃₁₇	—	4	1	no	—
24	C₂×Dic₃₁₇	—	4	1	no	—
25	C₄×D₃₁₇	D₄	1	1	yes	characteristic
26	#14579261	D₄	1	1	yes	characteristic
27	D₄×C₃₁₇	—	2	1	no	—
28	D₄×C₃₁₇	—	2	1	no	—
29	C₈×C₃₁₇	C₂³	1	1	yes	characteristic
30	C₂²×C₃₁₇	—	4	1	no	—
31	C₂²×C₃₁₇	—	4	1	no	—
32	C₂×D₃₁₇	—	4	1	no	—
33	C₂×D₃₁₇	—	8	2	no	—
34	C₂×D₃₁₇	D₈	1	1	yes	characteristic
35	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
36	Dic₃₁₇	—	4	1	no	—
37	Dic₃₁₇	D₈	1	1	yes	characteristic
38	C₂×C₃₁₇	—	8	1	no	—
39	C₂×C₃₁₇	—	8	1	no	—
40	C₂×C₃₁₇	C₂×D₈	1	1	yes	characteristic
41	D₃₁₇	—	2	1	no	—
42	D₃₁₇	—	8	1	no	—
43	C₃₁₇	#509	1	1	yes	characteristic, complemented, Sylow
44	#509	—	317	1	no	maximal, Sylow
45	#135	—	317	1	no	—
46	C₂×D₈	—	317	1	no	—
47	SD₃₂	—	317	1	no	—
48	SD₃₂	—	317	1	no	—
49	D₁₆	—	317	1	no	—
50	D₁₆	D₃₁₇	1	1	yes	characteristic, complemented
51	M₃₂	—	317	1	no	—
52	#41	—	634	1	no	—
53	C₂×D₄	—	634	1	no	—
54	Q₁₆	—	317	1	no	—
55	SD₁₆	—	634	1	no	—
56	D₈	—	317	1	no	—
57	D₈	—	634	1	no	—
58	D₈	C₂×D₃₁₇	1	1	yes	characteristic
59	D₈	C₂×D₃₁₇	1	1	yes	characteristic, complemented
60	C₂×C₈	—	317	1	no	—
61	C₁₆	—	317	1	no	—
62	C₁₆	C₂×D₃₁₇	1	1	yes	characteristic, complemented
63	C₂³	—	1268	1	no	—
64	Q₈	—	634	1	no	—
65	D₄	—	2	1	no	—
66	D₄	—	2	1	no	—
67	D₄	—	634	1	no	—
68	D₄	—	1268	1	no	—
69	D₄	—	1268	1	no	—
70	C₂×C₄	—	317	1	no	—
71	C₂×C₄	—	1268	1	no	—
72	C₈	—	317	1	no	—
73	C₈	C₂²×D₃₁₇	1	1	yes	characteristic
74	C₂²	—	4	1	no	—
75	C₂²	—	4	1	no	—
76	C₂²	—	317	1	no	—
77	C₂²	—	2536	2	no	—
78	C₂²	—	1268	1	no	—
79	C₄	—	317	1	no	—
80	C₄	—	1268	1	no	—
81	C₄	D₄×D₃₁₇	1	1	yes	characteristic
82	C₂	—	8	1	no	—
83	C₂	—	8	1	no	—
84	C₂	—	634	1	no	—
85	C₂	—	2536	1	no	—
86	C₂	D₈×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₃₁₇
commutator, cosocle	C₈×C₃₁₇	C₂³
Frattini	C₈	C₂²×D₃₁₇
Fitting	D₁₆×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂×D₈
hypercenter	D₁₆	D₃₁₇
hypocenter	C₃₁₇	#509
LCS	G ▷ C₈×C₃₁₇ ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. #23778799	C₂	semidirect
2. D₈×D₃₁₇	C₂	semidirect
3. #23778725	C₂	semidirect
4. #23778724	C₂	semidirect
5. #23778721	C₂	semidirect
6. #23778719	C₂	semidirect
7. D₁₆×C₃₁₇	C₂	semidirect
10. Dic₁₂₆₈	C₂²	semidirect
15. #19099549	C₂²	semidirect
17. D₈×C₃₁₇	C₂²	semidirect
18. C₁₆×C₃₁₇	C₂²	semidirect
43. C₃₁₇	#509	semidirect
50. D₁₆	D₃₁₇	semidirect
59. D₈	C₂×D₃₁₇	semidirect
62. C₁₆	C₂×D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778799	2
D₈×D₃₁₇	2
#23778725	2
#23778724	2
#23778721	2
#23778719	2
D₁₆×C₃₁₇	2
#509	317