Subgroups — #35203658

Order 20288.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#23778723	C₂	1	2	yes	maximal, complemented
3	#23778721	C₂	1	2	yes	maximal, complemented
4	#23778720	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₂₆₈	—	2	1	no	—
7	D₂₅₃₆	—	2	1	no	—
8	D₂₅₃₆	C₂²	1	2	yes	—
9	#19099549	C₂²	1	2	yes	complemented
10	#41×C₃₁₇	—	2	1	no	—
11	Q₁₆×C₃₁₇	C₂²	1	1	yes	characteristic, complemented
12	SD₁₆×C₃₁₇	—	2	1	no	—
13	D₈×C₃₁₇	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
15	C₂²×D₃₁₇	—	4	1	no	—
16	D₁₂₆₈	—	2	2	no	—
17	D₁₂₆₈	—	4	1	no	—
18	Q₈×C₃₁₇	—	2	1	no	—
19	D₄×C₃₁₇	—	2	1	no	—
20	D₄×C₃₁₇	—	4	1	no	—
21	C₂×C₄×C₃₁₇	—	4	1	no	—
22	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
23	C₈×C₃₁₇	D₄	1	1	yes	characteristic
24	C₈×C₃₁₇	C₂³	1	1	yes	characteristic
25	C₂²×C₃₁₇	—	4	1	no	—
26	C₂²×C₃₁₇	D₈	1	1	yes	characteristic
27	C₂×D₃₁₇	—	4	2	no	—
28	C₂×D₃₁₇	—	8	2	no	—
29	C₄×C₃₁₇	—	4	1	no	—
30	C₄×C₃₁₇	D₈	1	1	yes	characteristic
31	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃₁₇	—	2	1	no	—
33	C₂×C₃₁₇	—	8	1	no	—
34	C₂×C₃₁₇	C₂×D₈	1	1	yes	characteristic
35	D₃₁₇	—	8	2	no	—
36	C₃₁₇	#509	1	1	yes	characteristic, complemented, Sylow
37	#509	—	317	1	no	maximal, Sylow
38	#135	D₃₁₇	1	1	yes	characteristic, complemented
39	C₂×D₈	—	317	1	no	—
40	SD₃₂	—	317	2	no	—
41	D₁₆	—	317	2	no	—
42	M₃₂	—	317	1	no	—
43	#41	—	2	1	no	—
44	C₂×D₄	—	634	1	no	—
45	Q₁₆	C₂×D₃₁₇	1	1	yes	characteristic, complemented
46	SD₁₆	—	2	1	no	—
47	D₈	—	317	2	no	—
48	D₈	—	634	1	no	—
49	D₈	C₂×D₃₁₇	1	1	yes	characteristic, complemented
50	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
51	C₁₆	—	317	2	no	—
52	C₂³	—	1268	1	no	—
53	Q₈	—	2	1	no	—
54	D₄	—	2	1	no	—
55	D₄	—	4	1	no	—
56	D₄	—	634	2	no	—
57	D₄	—	1268	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	C₂²⋊D₃₁₇	1	1	yes	characteristic
60	C₈	C₂²⋊D₃₁₇	1	1	yes	characteristic
61	C₈	C₂²×D₃₁₇	1	1	yes	characteristic
62	C₂²	—	4	1	no	—
63	C₂²	—	2536	2	no	—
64	C₂²	—	1268	2	no	—
65	C₂²	#19099559	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#19099559	1	1	yes	characteristic
68	C₄	C₂×C₂²⋊D₃₁₇	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	8	1	no	—
71	C₂	—	2536	2	no	—
72	C₂	C₂×#19099559	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19099559
commutator, cosocle	C₈×C₃₁₇	C₂³
Frattini	C₈	C₂²×D₃₁₇
Fitting	#135×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂×D₈
hypercenter	#135	D₃₁₇
hypocenter	C₃₁₇	#509
LCS	G ▷ C₈×C₃₁₇ ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₅₃₆	C₂	semidirect
2. #23778723	C₂	semidirect
3. #23778721	C₂	semidirect
4. #23778720	C₂	semidirect
5. #135×C₃₁₇	C₂	semidirect
9. #19099549	C₂²	semidirect
11. Q₁₆×C₃₁₇	C₂²	semidirect
13. D₈×C₃₁₇	C₂²	semidirect
36. C₃₁₇	#509	semidirect
38. #135	D₃₁₇	semidirect
45. Q₁₆	C₂×D₃₁₇	semidirect
49. D₈	C₂×D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₅₃₆	2
#23778723	2
#23778721	2
#23778720	2
#135×C₃₁₇	2
#509	317