Subgroups — #35203960

Order 20288.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778774	C₂	1	2	yes	maximal, complemented
2	#23778761	C₂	1	3	yes	maximal, complemented
3	#23778752	C₂	1	6	yes	maximal, complemented
4	#23778748	C₂	1	3	yes	maximal, complemented
5	#127×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₃₁₇	C₂²	1	6	yes	—
8	C₂²×Dic₃₁₇	C₂²	1	3	yes	complemented
9	#19099553	C₂²	1	6	yes	complemented
10	C₂²⋊Dic₃₁₇	C₂²	1	3	yes	—
11	C₂²⋊Dic₃₁₇	C₂²	1	3	yes	complemented
12	#19099551	C₂²	1	6	yes	—
13	C₂×D₄×C₃₁₇	C₂²	1	3	yes	—
14	C₂×D₄×C₃₁₇	C₂²	1	3	yes	complemented
15	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²×D₃₁₇	—	4	3	no	—
17	C₂²×D₃₁₇	C₂³	1	2	yes	—
18	C₂²⋊D₃₁₇	—	4	6	no	—
19	C₂×Dic₃₁₇	—	4	3	no	—
20	C₂×Dic₃₁₇	C₂³	1	6	yes	—
21	C₂³×C₃₁₇	C₂³	1	1	yes	characteristic
22	C₂³×C₃₁₇	C₂³	1	3	yes	—
23	D₄×C₃₁₇	—	4	3	no	—
24	D₄×C₃₁₇	—	4	3	no	—
25	C₂×C₄×C₃₁₇	C₂³	1	3	yes	—
26	C₂²×C₃₁₇	—	2	3	no	—
27	C₂²×C₃₁₇	—	2	6	no	—
28	C₂²×C₃₁₇	—	2	3	no	—
29	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
30	C₂×D₃₁₇	—	2	6	no	—
31	C₂×D₃₁₇	—	4	4	no	—
32	C₄×C₃₁₇	—	2	3	no	—
33	Dic₃₁₇	—	2	6	no	—
34	C₂×C₃₁₇	—	4	1	no	—
35	C₂×C₃₁₇	—	4	3	no	—
36	C₂×C₃₁₇	H(5,𝔽₂)	1	3	yes	—
37	D₃₁₇	—	4	2	no	—
38	C₃₁₇	#560	1	1	yes	characteristic, complemented, Sylow
39	#560	—	317	1	no	maximal, Sylow
40	#127	D₃₁₇	1	1	yes	characteristic, complemented
41	#126	—	317	2	no	—
42	#123	—	317	3	no	—
43	#121	—	317	6	no	—
44	#120	—	317	3	no	—
45	C₂⁴	—	317	1	no	—
46	C₂×D₄	—	317	6	no	—
47	C₂×D₄	C₂×D₃₁₇	1	3	yes	—
48	C₂×D₄	C₂×D₃₁₇	1	3	yes	complemented
49	C₂²×C₄	—	317	3	no	—
50	C₄⋊C₄	—	317	6	no	—
51	C₂²⋊C₄	—	317	3	no	—
52	C₂²⋊C₄	—	317	3	no	—
53	C₂²⋊C₄	—	317	6	no	—
54	C₄²	C₂×D₃₁₇	1	1	yes	characteristic, complemented
55	C₂³	—	317	2	no	—
56	C₂³	—	1268	3	no	—
57	C₂³	C₂²×D₃₁₇	1	1	yes	characteristic
58	C₂³	C₂²×D₃₁₇	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	1268	6	no	—
62	C₂×C₄	—	317	6	no	—
63	C₂×C₄	—	1268	3	no	—
64	C₂×C₄	C₂²×D₃₁₇	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	634	6	no	—
69	C₂²	—	1268	4	no	—
70	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	634	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	1268	2	no	—
76	C₂	#23778825	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁₇
commutator, socle, cosocle	C₂²×C₃₁₇	C₂⁴
Fitting	#127×C₃₁₇	C₂
hypercenter	#127	D₃₁₇
hypocenter	C₃₁₇	#560
LCS	G ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #23778774	C₂	semidirect
2. #23778761	C₂	semidirect
3. #23778752	C₂	semidirect
4. #23778748	C₂	semidirect
5. #127×C₃₁₇	C₂	semidirect
8. C₂²×Dic₃₁₇	C₂²	semidirect
9. #19099553	C₂²	semidirect
11. C₂²⋊Dic₃₁₇	C₂²	semidirect
14. C₂×D₄×C₃₁₇	C₂²	semidirect
15. C₄²×C₃₁₇	C₂²	semidirect
38. C₃₁₇	#560	semidirect
40. #127	D₃₁₇	semidirect
48. C₂×D₄	C₂×D₃₁₇	semidirect
54. C₄²	C₂×D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778774	2
#23778761	2
#23778752	2
#23778748	2
#127×C₃₁₇	2
#560	317